COMPETING ORDERS IN S-VvAVE AND P-\"IAVE SUPERCONDUCTORS
by
QI LI
A DISSERTATION
Presented to the Department of Physics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2008
11
University of Oregon Graduate School
Confirmation of Approval and Acceptance of Dissertation prepared by:
Qi Li
Title:
"Competing Orders in s-wave and p-wave Superconductors"
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the degree in the Department of Physics by:
John Toner, Chairperson, Physics
Nilendra Deshpande, Member, Physics
Stephen Gregory, Member, Physics
Dietrich Belitz, Member, Physics
Andrew Marcus, Outside Member, Chemistry
and Richard Linton, Vice President for Research and Graduate StudieslDean of the
Graduate School for the University of Oregon.
June 14, 2008
Original approval signatures are on file with the Graduate School and the University of
Oregon Libraries.
III
An Abstract of the Dissertation of
Qi Li
in the Department of Physics
for the degree of
to be taken
Doctor of Philosophy
June 2008
Title: COMPETING ORDERS IN S-WAVE AND P-WAVE
SUPERCONDUCTORS
Approved:
Dr. Dietrich Belitz, Adviser
This dissertation investigates the interplay between, and the possible coexistence
of, magnetic and superconducting order in metals. We start with studying the
electromagnetic properties of s-wave superconductors near a ferromagnetic instability.
By using a generalized Ginzburg-Landau theory and scaling arguments, we show
that competition between magnetic order and superconducting order can change the
scaling of observables. For instance, the exponent for the temperature dependence of
the critical current can deviate from the Ginzburg-Landau value of 3/2. These results
may be relevant to understanding the observed behavior of MgCNi3 .
We then study the nature of the superconductor-to-normal-metal transition in
p-wave superconductors. Although the phase transition is continuous at a mean-
field level, a more careful renormalization-group analysis in conjunction with large-
n expansion techniques strongly suggest that the transition is first order. This
conclusion is the same as for s-wave superconductors, where these techniques also
predict a first-order transition.
In p-wave superconductors, topological excitations known as skyrmions are known
to exist in addition to the more common vortices. In the third part of this dissertation,
we study the properties of skyrmion lattices in an external magnetic field. We propose
IV
experiments to distinguish vortex lattices from skyrmion lattices by means of their
melting curves and their jlSR signatures.
vCURRICULUM VITAE
NAME OF AUTHOR: Qi Li
PLACE OF BIRTH: Lanzhou, China
DATE OF BIRTH: March 27, 1976
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon
Lanzhou University, Lanzhou, China
DEGREES
Doctor of Philosophy in Physics, 2008, University of Oregon
Master of Science in Physics, 2002, Lanzhou University
Bachelor of Science in Physics, 1999, Lanzhou University
ACADEMIC INTERESTS
Critical Phenomena and Phase Transition, Field Theory
Strongly Correlated Systems
PROFESSIONAL EXPERIENCE
Research Assistant, Department of Physics, University of Oregon, Eugene,
2005-2008
Teaching Assistant, Department of Physics, University of Oregon, Eugene,
2002-2005
VI
Teaching Assistant, Lanzhou University, Lanzhou, Gansu, China, 2000-2001
PUBLICATIONS
Qi Li, D. Belitz, and T. R. Kirkpatrick "Nearly ferromagnetic superconductors,"
Phys. Rev. B 74, 134505 (2006)
Qi Li, J. Toner, and D. Belitz, "Elasticity and Melting of Skyrmion Flux
Lattices in p-Wave Superconductors," Phys. Rev. Lett 98, 187002 (2007)
Qi Li, J. Toner, and D. Belitz, "Skyrmion versus vortex flux lattices in p
-wave superconductors" arXiv:0711.4154. (submitted to Phys. Rev. B )
------------------------ - ---.
vii
ACKNOWLEDGMENTS
First of all, I would like to acknowledge the continuous support and guidance
of my adviser, Dietrich Belitz. He taught me a lot of physics and mathematics.
More importantly, as a foreign student, I learned the importance of communicating
efficiently and clearly from him. I will never forget his patient editing of my conference
talks and papers. His attitude toward physics research is so respectful that I am
always stimulated to catch up with him.
I appreciate the help of John loner, whose insightful ideas have had a great impact
on my understanding of physics. I am constantly amazed by his ability to quickly
come up with smart solutions for complicated problems. From him, I learned not
only physics but also American jokes.
I would like to thank my teachers in the physics department. This includes
Nilendra Deshpande, Paul Csonka and too many others to be listed here. Particularly,
I appreciate the discussions after class with Nilendra. These really helped me understand
the physics better.
I would also like to thank a few postdocs of Dietrich: Jorg Rollbiihler, who
discussed a lot of physics problems with me; Sumanta Tewari who gave me some
good suggestion on the research project; Ronojoy Saha' who would always open his
door to me and talk with me for hours.
And finally I would like to thank my family. I could never have made this far
without my parents' continuous encouragements. Thanks also to my wife, Zheng Xu,
my Ph.D life was never lonely.
viii
TABLE OF CONTENTS
Chapter
1. INTRODUCTION
1.1 Phase Transitions and Critical Phenomena
1.1.1 Mean Field Treatment . . . . . . . .
1.1.2 Fluctuations and the Renormalization Group
1.2 Magnetic Order .
1.2.1 Heisenberg Model and O(n) Model .
1.2.2 X-Y Model and Topological Phase Transition
1.3 Superconducting Order
1.4 Organization . . . . . . . . . . . . . . . . . . . '.' .
II. NEARLY FERROMAGNETIC SUPERCONDUCTORS
Page
1
4
4
6
10
10
12
14
19
21
2.1 Introduction . . . . . . . . . . 21
2.2 Phenomenological Arguments . 23
2.2.1 Paramagnetic Systems . 23
2.2.1.1 Thermodynamic Critical Field 24
2.2.1.2 London Penetration Depth . . 25
2.2.1.3 Critical Current . . . . . . . . 29
2.2.2 Systems at a Ferromagnetic Instability 30
2.3 Generalized Ginzburg-Landau Theory . . . . 32
2.3.1 LGW Theory for Superconducting and Magnetic
Fluctuations. . . . . . . . . . . . ... . . . . . . . 33
2.3.2 Effective Theory for Paramagnetic Superconductors. 34
2.3.3 Superconductors at Magnetic Criticality 38
2.3.3.1 Thermodynamic Critical Field . . . . . . 38
2.3.3.2 Generalized London Equation 39
2.3.3.3 Penetration Depth, and Critical Current. 42
2.3.3.4 Critical Field Hc2 43
2.4 Discussion and Conclusion 44
Chapter
III. NATURE OF PHASE TRANSITION IN P-WAVE
SUPERCONDUCTORS
3.1 Introduction .
3.2 Model .
3.3 Nature of The Phase Transition
3.3.1 Mean Field Approximation
3.3.2 Renormalized Mean-field Theory
3.3.3 E-expansion about d = 4 .
3.3.4 lin-expansion in d = 3 .
3.4 Discussion and Conclusion ...
IV. SKYRMION IN P-WAVE SUPERCONDUCTORS
4.1 Introduction . . . . . . . . . . . . . . . . . . .
4.2 Formulation of the Skyrmion Problem . . . . .
4.2.1 The Action in The London Approximation .
4.2.2 Saddle-point Solutions of the Effective Action
4.2.2.1 Meissner Solution
4.2.2.2 Vortex Solution .
4.2.2.3 Skyrmion Solution . . . . . . . . . .
4.3 Analytic Solution of the Single-skyrmion Problem
4.3.1 Zeroth Order Solution .....
4.3.2 Perturbation Theory for R » 1
4.3.3 Energy of a Single Skyrmion. .
4.3.4 Spectral Methods . . . . . . . .
4.4 Observable Consequences of the Skyrmion Energy
4.4.1 B(H) for a Skyrmion Lattice .
4.4.2 Elastic Properties of the Skyrmion Lattice
4.4.3 flSR Signature of a Skyrmion Flux Lattice
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . .
APPENDICES
ix
Page
50
50
51
53
53
55
56
60
66
68
68
71
71
74
75
76
78
80
81
82
85
87
90
91
92
98
101
A. INTEGRAL FORMULAS USED IN CHAPTER III 102
Chapter
B. MISCELLANEOUS TECHNIQUES IN CHAPTER IV
B.1 Properties of Orthogonal Unit Vectors
B.2 Solutions of the ODEs for 9 and h
B.3 Contributions to Es .
BIBLIOGRAPHY . . . . . . .
x
Page
103
103
103
105
107
Xl
LIST OF FIGURES
Figure Page
1.1 Phase diagram of water and two paths for two different kinds of phase
transitions. [1] See the text for additional information. 2
1.2 Mean field phase diagram of type I and type II superconductors. See the
text for additional information. . . . . . . . . . . . . . . . . . . . . . .. 19
2.1 Magnetic induction schematically as a function of position at a vacuum
(V) - normal metal (N) - superconductor interface (S) (a), and at a
vacuum - superconductor interface (b). 26
2.2 Schematic phase diagram showing a normal metal (NM), a ferromagnet
(FM), a superconductor (SC), and a ferromagnetic superconductor (FMSC)
in a temperature (T) - control parameter (x plane. The solid line denotes
the superconducting transition, the dashed line, the magnetic one. Along
x = 0 there is only one phase transition at the superconducting Te . See
the text for additional explanation. . . . . . . . . . . . . . . . . . . . .. 46
2.3 Same as Fig. 2.2, but with a magnetic transition for x = 0 at a temperature
Tm < Te . On the x = 0 axis it is shown that Tm splits into the bare
magnetic transition temperature T~ and the physical transition temperature
Ts to a state with spiral magnetic order, Ref.[2]. See the text for additional
explanation. ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
3.1 Mean-field phase diagram of a p-wave superconductor as described by
Eq. (3.1). See the text for additional information. 54
3.2 Vertices from Eq. (3.3). Solid lines denote the 1/J field and dashed lines
its complex conjugate. Wavy lines denote the vector potential. Dotted
lines separate the localized interaction between paired electrons. 57
3.3 Diagrams renormalize the coupling parameters . 57
3.4 Diagrams renormalize the coupling parameter u . 58
Xll
m~re fu~
3.5 Renormalization group flows for n=500. The top right fixed point is
stable in the u-v plane, so it is the critical fixed point 59
3.6 All the self energy graphs contribute to the critical exponent. The first
row is from the ungauged part and the second row is from gauged part. 61
3.7 Dressed coupling parameter and renormalized photon propagator.. . .. 63
4.1 External field (H) vs. temperature (T) phase diagram for vortex flux
lattices. Shown are the Meissner phase, the vortex lattice phase, the
vortex liquid, and the normal state. Notice that the vortex lattice is
never stable sufficiently close to Hel . . . . . . . . . . . . . . . . . . . .. 69
4.2 Order parameter configurations showing a vortex (a), and a skyrmion
(b). The local order parameters are represented by arrows on loci of
equal distance from the center of the defect. If the order parameter space
is two-dimensional, only vortices are possible, and there is a singularity
at the center of each vortex, (a). If the order-parameter space is three-
dimensional, a skyrmion can form instead, where the spin direction changes
smoothly from "down" at the center to "up" at infinity, (b). . . . . . .. 70
4.3 Configurations of the vectors f, m, and n in a Meissner phase. All three
vectors point in the same direction everywhere. . . . . . . . . . . . . .. 76
4.4 Configurations of the vectors f, m, and n for a vortex. f is constant,
whereas m and n rotate about the vortex core. Notice that the vector
shown in Fig. 4.2(a) is n. . . . . . . . . . . . . . . . . . . . . . . . . .. 77
4.5 Configurations of the vectors f, m, and n for a skyrmion. Notice that the
vector shown in Fig. 4.2(b) is l . . . .. 82
4.6 Numerical data for the energy per skyrmion per unit length (circles)
together with the best fit to a pure 1/R behavior (dashed line) from
Ref. [3], and the perturbative analytic solution given by Eq. (4.34) (solid
line). A numerical solution using spectral methods is indistinguishable
from the perturbative one. 88
4.7 Shearing of the skyrmion lattice results in a change in the distance between
skyrmion centers, and hence in their effective interaction. See the text
for additional information. 94
Figure
Xlll
Page
4.8 External field (H) vs. temperature (T) phase diagram for skyrmion flux
lattices. In contrast to the vortex case, see Fig. 4.1, there is a direct
transition from the skyrmion flux lattice to the Meissner phase. The
theory predicts the shape of the melting curve only close to Te , see Eq.
(4.61); the rest of the curve is an educated guess. . . . . . . . . . . . .. 98
1CHAPTER I
INTRODUCTION
A material can exhibit various phases under different physical conditions. An
obvious example is that water can be in the solid, liquid, or gas phase in different
temperature regions. Different phases can be characterized by different symmetries
which reflects different orders. In the solid phase of water, it has crystal symmetry
which defines discrete periodic lattice order. In the liquid or gas phase, water
has continuous rotational and translational symmetry. As we can see from this
simple example, phase transitions exist in nature. There are two kinds of phase
transitions in this example, a discontinuous phase transition, and a continuous phase
transition. Everyday experience tells us that, during the process of ice melting or
water evaporation, heat has to be transferred to let the processes continue. This heat
is called latent heat and is always associated with a discontinuous phase transition
which is also referred to as a first order phase transition. As shown in the Fig. 1.1, an
arbitrary path A which crosses the liquid-gas coexistence curve will lead to an abrupt
density change. However, at the critical point with a pressure of about 2.2 x 108 Pa
and a temperature of about 647K, there is no way to distinguish the liquid phase from
the gas phase because they are at the same density. Any path in the phase diagram,
which goes right through the critical point, for instance, path B will have no latent
heat. This kind of phase transition is usually called continuous phase transition or
second order phase transition. In this example, the density difference between liquid
and gas characterizes the phase transition and is called order parameter. Water has
some critical phenomena which are very fascinating. One of them is called critical
opalescence. Close to the phase transition point, there are regions of larger or lesser
210·
10'
10'
10'
Solid
Liquid
.... 8
A ..~
Critical point
T.=647K
p.=2.2X108Pa
Gas
700600500400300200
10' +------.---'--....,----,.....-----,---...------,
100
Temperature{K)
FIGURE 1.1: Phase diagram of water and two paths for two different kinds of phase
transitions. [1] See the text for additional information.
densities. We can define the coherence length ~ to be the mean distance between
two nearest higher density regions. If we follow the path B in the Fig. 1.1, the
liquid phase coherence length will increase from a few interatomic distance to a few
hundred nanometers when the water gets closer to the critical point. This length scale
is comparable to the visible light wavelength, so light scatting increases dramatically.
Then the system looks "milky". Finally, exact at the critical point, the coherence
length diverges and water looks homogeneous. In short, when the order parameter
changes continuously from zero to non-zero by changing a control parameter, it is
called a continuous phase transition. Otherwise, it is called discontinuous phase
transition. Here, we will present a few more examples of continuous phase transitions.
Besides order defined in real space mentioned above, order can also occur in
spin space, leading to magnetic order. Such order comes from the electrons' spin
interaction effect. In the ferromagnetic exchange interaction case, spins tend to align
parallel to each other to minimize the free energy at a sufficiently low temperature.
3This leads to a ferromagnetic ground state. This ground state breaks the spin
rotational symmetry and is reflected by the well known fact that magnetic materials
will have a non-zero magnetization below the critical Curie temperature. So the
magnetization is the natural order parameter for the ferromagnet to paramagnet
transition.
In addition to the spin-spin interaction between electrons, there is an electron-
phonon interaction which can lead to a ground state of electron pairs which have
antiparallel spins (singlet pairing) and are called Cooper pairs. These bounded
pairs account for the mechanism of superconductivity, and the pairing wave function
for Cooper pairs can be used as the order parameter for the normal metal to
superconductor phase transition.
An interesting question is whether superconducting order and magnetic order can
coexist? Naively speaking, ferromagnetic order tries to align the spins of electrons,
whereas superconducting order wants the spins of the Cooper pair antiparallel. These
two effects seem irreconcilable. However, experiments do show possible coexistence of
ferromagnetism and superconductivity[4]. Furthermore, mean field phase boundaries
have been proposed for the s-wave ferromagnetic superconductors. It was suggested
that coexisting superconducting and magnetic order might simultaneously appear via
a continuous phase transition, but nobody has seriously looked into the magnetic
critical region. This situation is proposed to be relevant to explaining a recent
experiment on the material MgCNi3. We will discuss this issue in Chaper II by a
mean field treatment and scaling analysis of the continuous phase transition.
Another candidate for the coexistence of the two competing orders is p-wave
superconductors. In this case, electrons show triplet pairing instead of conventional
singlet pairing. Triplet pairing leads to Cooper pairs with parallel spins. An example
is the ABM state in superfluid 3He. One would expect that a similar situation can
also occur for the charged superfluid case, that is, for electrons in metals.
41.1 Phase Transitions and Critical Phenomena
Continuous phase transitions and critical phenomena have been an interesting
topic for the last few decades. A continuous phase transition occurs when a symmetry
of the system is spontaneously broken at some value of a control parameter, for
instance, temperature. We will start with a well known example, the Ising model
with Z2 symmetry, then we will discuss more general magnetic systems like the
0(3) Heisenberg model and the X-V model. Finally, critical phenomena in s-wave
superconductors and p-wave superconductors will be briefly discussed.
1.1.1 Mean Field Treatment
The Hamiltonian for a spin system can be written in the following form
(1.1)
<i,j,cx>
Here, the indexes i and j denote the lattice points where spins are situated, and
a denotes the spin component. < ... > means only the nearest spin neighbors will be
summed over and JCX is the coupling strength between nearest neighbors. Here, we
will discuss classical spins only, so spins take the values of real vector variables.
For the Ising model, the spin variable can only take the values of ±1 which means
spin up or down. Correspondingly, only one component Jcx in Eq. (1.1) is nonzero.
Such a simplified model has Z2 symmetry and is often realized by the crystal-field
effects. If the coupling constant J is positive, the minimum energy configuration is
obvious: spins point in the same direction. This leads to a non-zero magnetization
M which is called the ordered phase. Landau introduced the concept of the order
parameter (O.P) to describe continuous phase transitions. A zero O.P corresponds
to the disordered phase and a non-zero one to the ordered phase. The Landau free
energy concept is based on this idea. Because the order parameter changes value
continuously from zero to non-zero at the phase transition, it must be very small
5near the transition point. Then the Landau free energy can be expanded as a scalar
function of the order parameter and must have invariance properties corresponding
to the symmetry of the disordered system. For the Ising model, the O.P is the
magnetization and the Landau free energy takes the form
1 2 4f = -tM + uM - h· M2 (1.2)
Here, h denotes an external magnetic field, and for h = 0, the Landau free energy
indeed is invariant under spin inversion. t = (T - Te ) ITe is the distance from the
phase transition point, where its value changes from positive to negative. u is another
parameter that must be positive in order to have a continuous phase transition.
In field theory language, the tM2 term is also referred to as the mass term, and
the tl term as the potential energy term which accounts for the interaction between
particles. From this Landau energy, the behavior of some physical observables can
be predicted. In the absence of an external field h, if t is positive, to minimize the
Eq.(1.2), M = 0 has to be satisfied. This is the disordered phase where spins point
in random directions. If t is negative, the optimal M to minimize f is
~tM= -4u (1.3)
which describes a magnetically ordered phase. Throughout the following discussion,
t is assumed to be negative if not explicitly defined otherwise.
From the Landau energy, some critical exponents can be extracted. For instance,
f3 describes the temperature dependence of the order parameter via M ex: (-t)f3.
From Eq.(1.3), f3 = 1/2 is predicted for Ising systems. There are several other critical
exponents, for instance, I and 5, which are defined as the temperature dependence
of the magnetic susceptibility and the field dependence of the magnetization via
X ex: Itl-')' and M(t = 0) ex: hI/ O, respectively. These two can be deduced from
6the equation of state
tA1 + 4uM3 = h (1.4)
which is obtained from minimizing f in Eq.(1.2). The magnetic susceptibility X is
defined as
(1.5)
From Eq. (1.4), we get X = 111tl, which implies 1=1. Finally, the dependence of A1
on the external field h at the phase transition point(t = 0) is,
(1.6)
So the critical exponent 5 is 3. In the real 3d Ising systems, f3 '" 0.33, I '" 1.2,5 '" 4.3.
This discrepancy between the prediction of mean field theory and the observed values
of the critical exponent will be further discussed in the following sections.
1.1.2 Fluctuations and the Renormalization Group
Mean-field theory correctly describes the qualitative features of continuous phase
transitions, but it may not be sufficient to describe the system more accurately.
For instance, thermal fluctuations can drive the system away from its mean field
configuration and may even destroy the phase transition. This can be seen from the
following arguments. To take into consideration fluctuations, the order parameter
must be a spatially dependent field. To account for how the G.P field can fluctuate
from point to point, a spatial derivative has also to be included in the free energy
density. So the fluctuating Landau energy density is
f(x) = ~tM(X)2 + UM(X)4 + ~VM(X)2 (1. 7)
It should be noted that there is a characteristic length scale in Eq.(1.7). By
comparing the dimension of the first term and the third term, a coherence length
7is defined as ~ = .JC71tT. This length characterizes the distance over which G.P
can not change too much. At the phase transition, ~ diverges, meaning a spatially
homogeneous state.
Now the question is: when are the fluctuations large enough to lead to a breakdown
of the mean field results? This question can be answered simply by comparing the
fluctuation effects to the mean field results. [5] The fluctuation is defined as oM(x) =
M(x) - M. Here M is the mean field value as stated in Eq. (1.3) and M(x) is the
fluctuating field which can vary from its mean value by an amount of oM(x). Now
the spatially averaged fluctuation is
ve-2 r ddxddx' (oM(x)oM(x')) rv T~-(d-2)Ie
lYe
(1.8)
Here, ve ex: ~d is the characteristic volume of the system. The ( ) means a
thermal average of the function in the brackets. By doing the average with the
Gaussian approximation of Eq. (1.7), we get the approximate relation in Eq. (1.8). It
can be seen the dimensionality plays an important role in the phase transition. This is
because for a continuous phase transition, the coherence length must diverge at phase
transition point. So if the spatial dimension is less than 2, fluctuations according to
Eq. (1.8) are too large to allow for a finite temperature continuous phase transition.
Thus dzc = 2 is usually called lower critical dimension. Comparing Eq. (1.8) to the
mean field result M 2 = It1/4u, we immediately get the criteria for the mean field
theory to be valid: it is a spatial dimension d > duc = 4. This means that for physical
systems in 3d, we have to take the fluctuations into account.
Before we look into the treatment of these fluctuations in detail, we first discuss
the scaling hypothesis which can successfully relate the critical exponents with a
few assumptions. [6] The static scaling hypothesis states that the free energy density
is a generalized homogeneous function of the experimental control parameters, for
instance, reduced temperature t and magnetic field H,
8(1.9)
Here, s is an arbitrary number which is called factor of scale transformation. at
and aH are exponents which can be related to the critical exponents introduced in
the previous section. For instance,
(1.10)
Though the scaling hypothesis can interpret all the critical exponents in terms of
at and aH , it can not tell anything about their values. However, it does exploit the
properties of the system under scale transformations. This is crucial for understanding
the phase transition and is the base for the renormalization group concept.
Now we can have a close look at how the free energy Eq. (1.7) changes under
scale transformations. From statistical physics, the probability for a system to have
a particular configuration is proportional to its Boltzmann factor
P[j(x)] ex e- J f(x)dx/T (1.11)
Here, the free energy is F = J j(x)dx. Now such a probability can be transformed
under a scale transformation of j(x). This can be seen by introducing a parameter
space which is spanned by all the parameters in Eq. (1.7) ,
Jk = (t,u,c) (1.12)
Firstly, we perform a coarse graining procedure of the original j (x) which
is generally referred to as a Kadanoff transformation. In Fourier space, this
transformation can simply be expressed as the following equation.
e-FT[M(k)]/T = Je-F[M(k)]/T II D M k
A>k>A/s
(1.13)
9Here, A is an ultraviolet cutoff, and s is a scale factor which is taken to be larger
than 1, with DMk the functional integral measure. This functional integral is treated
as a perturbed Gaussian integral. The integral over the momentum shell A > k > AIs
corresponds to a scale transformation for the lattice constant a ~ sa. This means
that F r is a free energy for a larger spin block system than the original system. To
recover the size of the original system, a second scale transformation has to be applied.
It is defined as,
a ~ als
M ~ M( (1.14)
Here, ( is a parameter which is called the rescaling parameter of field and can
be chosen to let some parameter remain fixed under the scale transformation, for
instance c. After those two steps of transformation, a new set of parameters can be
written as
, (t' , ')f.-l = . ,U,c
This set of new parameters is called renormalized parameters.
(1.15)
"Unimportant"
information has been eliminated (integrated out)in the renormalization process. It is
natural to ask what the result is when the above two processes have been repeated
over and over? A simple answer is that it will reach a fixed point in the parameter
space. Let R be the renormalization processes defined as above, then a point in the
parameter space evolves under the renormalization process,
(1.16)
As can be seen from Eq.(1.13) and Eq.(1.16), the operator R must be some
complicated functions whose arguments are the parameters defined in Eq.(1.15)
10
together with the scale factor s. If the scale factor s is taken to be infinitesimally
close to 1 and the renormalization process is repeated infinitely many times, it can be
shown that Eq.(1.16) reduces to a set of ordinary differential equations(ODEs). This
set of ODEs will be referred to as flow equations.
These flow equations have a natural ending point, which is a fixed point that
satisfies p* = Rp"". In another word, it is invariant under renormalization. It is not
hard to see that fixed point (u*, r*) = (0,0) will always appear. This is due to the
fact that a Gaussian action remains Gaussian under renormalization. Such a fixed
point is sometimes called a trivial fixed point. All of the critical exponents are related
to properties of the flow equations in the vicinity of a fixed point. This will be shown
explicitly in the following section and in Chapter III.
1.2 Magnetic Order
1.2.1 Heisenberg Model and D(n) Model
An extension of the Ising models for magnets is the Heisenberg modeL Since spins
have three components, the order parameter for a Heisenberg magnet is naturally a
vector in spin space denoted by M. Now the free energy must be invariant under the
group 0(3), so it reads as,
1 cf(x) = 2tMa(x)Ma(x) + uMa(x)Ma(x)Mj3(x)Mj3(x) + 28iMa(x)8iMa(x)
(1.17)
Here, ex and f3 are the spin indexes and range from 1 to n, with n = 3. i is the
spatial index and takes values from 1 to the spatial dimension d. More generally,
we will consider models when n can be any integer and d can be any positive real
number.
The mean field result of Eq.(1.17) is, M(x) = 0 for positive t. This corresponds
to the disordered state. For negative t, M(x) = j"ifn. Here, nis an arbitrarily
11
chosen unit vector in spin space. This mean-field result is almost the same as for
the Ising model we discussed in the last section, and thus will not change the critical
exponents. So within the mean field approximation, it seems that magnetic systems
have universal critical exponents that are independent of nand d. However, if we
do the renormalization process described in last section for the O(n) model, we get
different results. The flow equations for the parameters of the Eq. (1.17) are[l]
dt(l)
dl
du(l)
dl
11, (l)
2t(l) + 4kd (n + 2) 1 + t(l)
u2(l)
fU(l) - 4kd(n + 8) (1 + t(l))2' (1.18)
Here, S = 1 + dl is the factor of scale transformation through every renormalization
process and f = 4 - d is the parameter that controls the perturbation expansion
of Eq.(1.13), with kd a positive constant for any fixed spatial dimension. For
convenience, we have chosen the rescaling parameter (2 = S(d+2) to keep the parameter
c fixed. Besides the trivial Gaussian fixed point((u*, r*) = (0,0)), there is an order of
f fixed point which reads
u*
t*
4(n + 8)kd '
-(n + 2)
2(n + 8) f. (1.19)
The following arguments can be derived from the above relation. Firstly, the
negative value of t* and positive value of u* are in the expected region of mean
field result, thus make perfect sense for the continuous phase transition. Secondly,
fixed point values are very close to the Gaussian fixed point which is consistent with
the spirit of perturbation expansion. Thirdly, the remarkable feature of those fixed
points is that they depend only on the spatial dimension and the field dimension.
----------------_...
12
Whereas mean-field theory predicts that the critical behavior is independent of these
parameters, this result clearly states that fluctuations violate the mean field prediction
and cause a diverse world of magnetic orders.
It should also be noted that if c: is negative, as a result of spatial dimension is larger
than 4, from Eq.(1.19), u* is negative! This observation doesn't necessarily mean there
must be a first order phase transition. Actually, the trivial Gaussian fixed point is
stable and all the critical exponents recover the mean field values. Here, stability of a
fixed point is defined whether the flow near it will be attracted in or expelled out. In
the vicinity of a stable fixed point, there is unique direction (qualititatively about t
direction) which will guide the flow to either a positive t region (higher temperature,
disorder phase) or a negative t region (lower temperature, ordered phase), whereas
from all the directions, flows will be attracted to the fixed point.
We focus on the d < 4 case. So critical exponents can be extracted from those
non-trivial fixed points by the method introduced by [7]. We only list some of them
for future reference,
3c:f3 1/2 - 2(n + 8)'
(n + 2)c: (1.20)"'( 1+ 2(n + 8)
5 3+c: (1.21)
For d = 3(c: = 1), these predicted critical exponents fit the experimental values
much better than the mean field values. As c: goes to zero, the system behaves more
mean field like.
1.2.2 X-Y Model and Topological Phase Transition
The X-Y model is the model from the above discussion with n = 2. For a long time
it was thought to be nothing more than an anisotropic Heisenberg model. Namely,
13
the exchange coupling constant JIY. has only two non-zero components. This can be
realized by a crystal field in a plane. So the results for the O(n) model in the last
section are still valid. However, a novel topological phase in this model was discovered
in 1973[8]. Here, we will briefly introduce the concept oftopological phase transitions.
This will be revisited in Chapter III.
The 0(2) model has two independent real valued components of the O.P and is
equivalent to a U(l) field description. So we can write the 0(2) order parameter
as 'IjJ (x) ei</>(x) , where 'IjJ(x) E R is the amplitude of the complex field and ¢(x) is the
phase which takes values in the range [0, 27r]. It should be noted that ¢(x) can also be
interpreted as the angle between nearest neighbor spins. If we keep the amplitude of
the field fixed equal to the optimal mean field value and only consider the fluctuations
of the phase, we get a free energy density
f(x) = ~V¢(X)2
Minimization of this free energy requires
(1.22)
(1.23)
A trivial solution is ¢(x) = canst, which corresponds to a configuration where
spins align everywhere. This is the ground state. However, there are other
possibilities. For instance, in dimension 2, we can add a source term on the right
hand side of Eq. (1.23) to describe a non-differentiable angle ¢ at x = 0 which can
also be interpreted as a topological point defect.
(1.24)
Here 0 is the Dirac delta function in 2d. If we interpret ¢(x) as the electrostatic
potential, this equation is exactly the 2d Poisson equation which describes a Coulomb
problem with Q as the charge. Q must be an integer because ¢(x) is periodic. Due
14
to this reason, Q is usually called the topological charge number or winding number
or fiuxiod number in the superconductor context.
A solution to Eq. (1.24) can be easily obtained and the corresponding free energy
is proportional to log[R] , where R is the size of the defect. This is larger than the
ground state energy. However, the energy cost to align all the spins in the same
direction(i.e, to destroy the defect) is of order R which is much larger than the defect
energy log[R] if R is large. This leads to the conclusion that the defects are stable.
Correlation functions of such topological excitations can also be calculated which show
an algebraically decay. To distinguish this from long range order, when correlation
functions do not decay at all, such order is often referred to as quasi-Iong-range order.
In other words, at some finite temperature, topological defects will appear and cause a
weakly correlated system. This kind of phase transition is called Kosterlitz-Thouless
transition and it is a particular example of a topological phase transition.
1.3 Superconducting Order
The microscopic mechanism for conventional superconductivity was explained
by the famous BCS theory. The wave function 'ljJ(x) for the Cooper pairs can be
considered the order parameter, with 1'ljJ(x) 1 2 the density of Cooper pairs. In the
simplest case, 'ljJ(x) is a complex scalar, which means that the electrons forming pairs
have opposite spins (spin singlet case), so the total spin of a pair is zero. Besides
spin, Cooper pairs also carry charge and therefore couple to photons. This can be
described as follows. The gradient of the wave function is usually interpreted as a
velocity in quantum mechanics. For the charged Cooper pairs, it is a charge current
and naturally will couple to the photon field. So the free energy for the s-wave
15
superconductors is,
F JdX [_1 I(V-iqA(x))1/J(x)12+~I1/J(x)12+ ull1/J(x)14+~B2(X)2m 2 4 8n
-~ H(x) . B(X)]. (1.25)
4n
Here, A is the vector potential and H is the external field, with q the charge
of the Cooper pair, and B = V x A is the magnetic induction. The first term in
the above equation is usually called the supercurrent energy, the second and third
terms are called the condensation energy. The fourth term is the magnetic energy of
a superconductor in the absence of an external field, and the fifth term is a magnetic
energy in an external field. The difference between those two kinds of magnetic
energy is that the first one denotes the self property of a superconductor, namely
the U(l) gauge symmetry requires this term to be present, whereas the latter is
due to the interaction energy between the superconductor and the external field.
In the coefficient of the magnetic energy, we have included the vacuum magnetic
permeability f-lo and we choose cgs units to let it be 1. We will see in Chapter
II that this coefficient can be renormalized by coupling the magnetization to the
superconducting a.p. It is easy to check that the above free energy is invariant under
local gauge (U (1)) transformations,
A --- A+VA(x)
¢(x) --- ¢(x)+qA(x) (1.26)
Here, ¢(x) is the phase of the complex order parameter 1/J(x) = 1/J(x)ei¢(X) , and
A(x) is an arbitrary scalar field. Since this kind of order parameter has been discussed
in the context of X-Y model, it is natural to think that s-wave superconductors
16
have the same topological defects as we have seen in the last section, and that such
topological defects may playa role in the phase transition of superconductors.
The ground state of Eq. (1.25) is 'ljJo = ~ at zero external field. The
corresponding magnetic induction B(x) decays exponentially with a characteristic
penetration depth A which sets the length scale for the magnetic induction to drop
to zero
(1.27)
Hence, the superconductor expels magnetic flux. This is called the Meissner phase.
If the amplitude of the order parameter is kept fixed at 'ljJo, and only the fluctuations
of phase ¢(x) are allowed, then the saddle point equation for the magnetic induction
can be fully decoupled from the one for the amplitude of order parameter. This
approximation is usually called London approximation. The corresponding London
equation reads
(1.28)
To get the second "=" in the Eq.(1.28), we assume there is an external field which
is applied in the z direction. In this case, the system has a cylindrical symmetry, so
phase ¢(x) can be interpreted as the polar angle, where x is confined to the x - y
plane, with <Po = hc/2e the flux quantum in SCI units. This equation is similar to
Eq. (1.24). However, the physical meaning of a defect at x = 0 is a fully penetrated
magnetic field. The solution of above equation is called a single vortex which is a
zeroth-order Hankel function of imaginary argument. It decays exponentially for large
lxi, which means vortices have a short range interaction.
There is another length scale in the free energy which can be defined by comparing
the terms quadratic in 'IjJ(x) in Eq.(1.25), ~ = V1/mltll. This length scale sets the
characteristic distance for superconducting order to drop to zero.
17
Besides these length scales, there are also two important energy scales. From
experiment, it is known that too strong an external magnetic field can destroy
the superconductivity even at very low temperature. This can be explained from
Eq. (1.25) by comparing the condensation energy and the magnetic energy. If the
magnetic energy dominates, the current induced by the magnetic induction can
destroy Cooper pairs and is referred to as a depairing current. Then the system
must be in the normal state. We will see this point more clearly in Chapter II.
These two energy scales compete with each other as a function of temperature and
magnetic field. If the condensation energy is larger, the system is superconducting,
when the magnetic energy is larger, the system will be in the normal state. The
two energy scales being equal defines a field scale He(T) which is known as the
thermodynamic critical field. For field strength below He, the superconducting state
will presumably be found. However this is not always the case, as will be seen in the
next few paragraphes.
It is also helpful to compare the two length scales defined above and define a
dimensionless parameter tl, = AI~. If tl, is much less than 1, therefore the correlations
of magnetic induction can be ignored and the system is dominated by the condensation
energy, so the Meissner phase is realized. If tl, is much greater than 1, the magnetic
energy is dominant. In this case, to minimize the free energy, 'IjJ(x) can be treated in
the London approximation. This is exactly the same situation as we have discussed
for the topological phase in the X-Y model in connection with Eq. (1.24). The above
two cases are usually categorized as type I and type II superconductors. More careful
analysis shows that tl, = 1I~ separates these two cases[9] . Though fluctuations
can induce these vortices, only an external field can stabilize them. The minimum
external field required to induce a single vortex is called lower critical field Hc1 . It
can be found by comparing the free energy of a system with no flux at just below H c1
with the free energy just above Hc1 from Eq. (1.25). By comparing energies in the
two cases, in the large tl, limit, H c1 is
18
(1.29)
This result can also be made plausible as follows. Because flux lines are far apart
from each other, and each carries a fluxoid with an length scale A, the natural guess
from dimension analysis is Hel "-' <po/A2 • From this relation, it can be seen even
at the field lower than the thermodynamic critical field He' flux lines can penetrate
superconductors, so the superconductivity is not as strong as one would expect for
type II superconductors.
Careful analysis of the magnetic induction by solving the Ginzburg-Landau(G-
L) equations which minimize the free energy given by Eq.(1.25) shows that close to
Hel , the magnetic induction is B "-' 10g(H - Hcd-2.[9]This relation shows that B is
continuous at Hel . This indicates a continuous phase transition at Hel . However,
some experiments show a B-H curve that is not continuous as predicted by the G-L
argument given here. The discussion of this point is beyond the scope of this thesis
and can be found elsewhere[9]. The last relevant concept for this thesis in the context
of type II superconductors is the upper critical field Hc2 ' A field strength stronger
than Hc2 will destroy superconductivity totally. However, for a field slightly below
H c2 , the flux lines have a large density and overlap each other. Though Hc2 can be
calculated from G-L theory, a simpler estimate can be obtained as follows. Because
~ measures the core size of a single flux line which is also the minimum size of a
flux defect, if neighboring flux lines are too close together and reach the minimum
distance-~ they can afford, then the superconductivity breaks down. This immediately
leads to the conclusion Hc2 "-' <po/e. A more careful calculation from G-L theory
gives the result H c2 = <po/21re. The upper critical field is the nucleation field for the
Cooper pairs and type II superconductors will have a continuous phase transition at
this point.
HType I superconductors
--
--
",H(T)
,0
" normal
,
,
,
Superconductor , ,
,
,
T
a)
H
Type II superconductors
Normal
T
b)
19
FIGURE 1.2: Mean field phase diagram of type I and type II superconductors. See
the text for additional information.
Now we summarize the phase transitions in an external field in Fig. 1.2, For type
I superconductors, He has the physically most important meaning, and a first order
phase transition occurs at this field strength. For type II superconductors, only Hel
and He2 make physical sense at which continuous phase transitions occur.
1.4 Organization
This dissertation is organized as follows. Chapter II will focus on the coexistence of
ferromagnetic and magnetic order in the s-wave superconductors. By using the static
scaling hypothesis and critical exponents for the magnetic system, we will discuss an
unconventional behavior of the temperature dependence of the critical current. This
is relevant to understanding an experiment on the material MgCNi3 • This work has
been published as Ref. [10] with Dietrich Belitz and Theodore R. Kirkpatrick. We
independently derived all the equations almost at the same time.
Chapter III is a study of the nature of the phase transition in p-wave
20
superconductors. Renormalization group method and large n expansion techniques
will be applied to attack the problem. These techniques strongly suggest that p-wave
superconductors have a weakly first order phase transition. However, in a generalized
model of p-wave superconductors, we find a new class of fixed points which suggests
that p-wave superconductors have novel topological excitations.
Chapter IV investigates skyrmion lattices in p-wave superconductors. Firstly, we
invent an unusual perturbation method to solve the skyrmion configuration in the
London approximation for the p-wave case. Then magnetic properties, for instance,
the magnetization curve and the melting curve are predicted to distinguish skYrmions
from Abrikosov vortices. Finally, we point out that J-LSR experiments can be applied
to distinguish these two topological excitations. A short version of this work has been
published in Ref. [11] and a longer paper is available as Ref. [12]. This work is a
result of a close collaboration with Dietrich Belitz and John Toner, who contribute
about the 2/3 of the total workload.
Some technical points are given in the appendices.
21
CHAPTER II
NEARLY FERROMAGNETIC SUPERCONDUCTORS
2.1 Introduction
The work presented in this chapter has been previously published in Ref. [10] with
coauthors D. Belitz and T. R. Kirkpatrick.
The coexistence of ferromagnetism and superconductivity was predicted[2, 13, 14]
and observed, [15, 16] later it received renewed interest in the context of experimental
observations in rare earth borocarbides. [17] More recently, interest in this subject has
been revived by the observation of coexisting superconductivity and ferromagnetism
in UGe2[18, 19] and URhGe,[20] where both types of order are believed to be due to
electrons in the same band. Recent theoretical attention has centered on the structure
of the phase diagram, [21] on the existence of spontaneous flux lattices, [17, 22, 23] and
on the question of spin-triplet versus spin-singlet superconductivity.[24]
In contrast, less is known about the properties of superconductors on the
paramagnetic side of, but close to, a ferromagnetic instability. We will refer
to "paramagnetic superconductors" to describe systems in this regime, although
the superconductivity of course leads to the usual strong diamagnetic effects.
Such paramagnetic superconductors include systems below the superconducting
transition temperature, but above the temperature below which coexistence of
superconductivity and ferromagnetism occurs, as well as systems that never develop
ferromagnetism, but are close to a ferromagnetic instability in some direction in
parameter space other than temperature. An example of the latter is believed to be
the non-oxide perovskite MgCNi3 , which superconducts below a critical temperature
22
Te ~ 8 K. [25] There is no evidence for a ferromagnetic phase in this material, but it
has been suggested that a ferromagnetic ground state can be reached upon a relatively
small amount of hole doping. [26] This system may thus be close to a ferromagnetic
instability everywhere in its superconducting phase.
A recent study of MgCNi3 microfibers, with Te = 7.8 K, has revealed an anomalous
temperature dependence of the critical current density je.[27] The critical current
density vanishes at Te according to a power law j e ex IT - Te 10<, with ex = 2 between
about 1% and 10% away from the critical point, and no crossover to the usual
Ginzburg-Landau behavior, which predicts ex = 3/2. The authors of Ref. [27] have
ruled out morphological effects as an explanation, which raises the question whether
proximity to a ferromagnetic state may be responsible. Indeed, since ferromagnetic
fluctuations are expected to weaken (singlet) superconductivity, this is a plausible
suggestion for the origin of the weaker-than-expected temperature dependence of j e.
The probable proximity to ferromagnetism has led to a debate about the nature
and symmetry of the pairing in MgCNi3 . 1 This point has not been settled;
some experimental evidence points to conventional s-wave pairing; other, to a
superconducting order parameter with nodes. The nature of the pairing in the other
materials mentioned above has not been unambiguously determined either. In this
chapter we will focus on the behavior close to Te , which is qualitatively independent
of the symmetry of the order parameter and thus expected to be the same for all
nearly ferromagnetic superconductors. We use a generalized Ginzburg-Landau theory
to theoretically investigate the electrodynamic properties of a superconductor as a
ferromagnetic instability is approached. We treat the superconductivity in the usual
mean-field approximation, but the magnetic critical behavior exactly in a scaling
sense. Somewhat counter-intuitively, strong magnetic fluctuations make, in a well-
defined sense, the superconductivity more robust in certain respects. In particular, the
penetration depth becomes anomalously short. The thermodynamic critical field, on
1For a recent summary see, R. Prozorov and R. W. Gianetta, cond-matj0605612
23
the other hand, becomes weaker, as one might intuitively expect. The temperature
dependencies of the critical field He and the penetration depth A depend on the
magnetic critical exponents 5 and 'Y, respectively. For the critical current je ex: He/A,
this results in an exponent a between 1.5 (the Ginzburg-Landau result) and 2.16
in various temperature regimes. We will discuss both the existing experimental
observations, and predictions for new experiments, in the light of these results.
This chapter is organized as follows. In Sec. 2.2 we give elementary phenomenological
arguments for the dependence of the thermodynamic critical field, the penetration
depth, and the critical current density, on a constant normal-state magnetic
permeability /In. We then generalize these results to the magnetically critical case,
where one needs to distinguish between /In and the spin susceptibility /Js in a
superconduting state, and both /In and /Js become nonanalytic functions of various
control parameters. In Sec. 2.3 we derive these results from a generalized Ginzburg-
Landau theory, and in Sec. 2.4 we give a discussion of our results.
2.2 Phenomenological Arguments
2.2.1 Paramagnetic Systems
We are interested in the electromagnetic properties of superconductors with
ferromagnetic fluctuations. We denote the normal-state spin susceptibility, which
describes the response of the spin degrees of freedom to an external magnetic field
in the absence of superconductivity, by Xn, and the corresponding spin permeability
by /In = 1 + 41fXn. This is in contrast to the spin permeability /Js = 1 + 41fXs, which
includes the effects of the superconductivity on the spin response, and the magnetic
permeability /J = 1 + 41fX, which describes the response of the total magnetization,
including the diamagnetic part. It is instructive to first recall the dependence of
supercondueting properties on a constant /In ~ 1, neglecting the distinction between
/In and /Js. [28, 29] This can be done by means of elementary arguments.
24
2.2.1.1 Thermodynamic Critical Field
Consider the free energy density f of a system in a magnetic field. It obeys
1df = df (H = 0) +- H dB,47r (2.1 )
where H is the thermodynamic magnetic field, and B is the magnetic induction. For
the sake of simplicity, we ignore the vector nature of various quantities in our free
energy considerations. For fixed B, f is the appropriate thermodynamic potential
whose minimum determines the equilibrium state. However, in an experiment H is
fixed, since (c/47r) V x H = j ext is the external current density, and only the latter
is experimentally controlled. One therefore must perform a Legendre transform to a
thermodynamic potential 9 = f - BH/47r,[9, 30] which obeys
1dg = df(H =--= 0) - - B dH.
47r
(2.2)
In a paramagnetic phase, including paramagnetic superconductors, the relation
between Band H is
B = H + 47rM = (1 + 47rX)H = JLH, (2.3)
with M the magnetization, X(T, H) the magnetic susceptibility, and JL = 1+ 47rX the
magnetic permeability. Integration of Eq. (2.2) yields
1 rHg(T, H) = f(T, H = 0) - 47r io dh [1 + 47rX(T, h)] h. (2.4)
This is generally valid. In a superconducting Meissner state, B = 0, and hence
X = -1/41r (ideal diamagnetism), and f(T,H = 0) = fa +tl7,b12/2+ul7,b14/4, with fa
the free energy density of the normal state, 7,b the superconducting order parameter,
t ex (T - Tc)/Tc the dimensionless distance from the superconducting critical point,
25
and u a parameter. In a normal metal far from a magnetic instability, and ignoring
normal-state diamagnetic effects, X(T, H) ~ const. Xn, or /1n = 1 + 47rXn = const.,
and f(T, H = 0) = fa. In a normal metal close to a ferromagnetic critical point, X is
a complicated function of T and H.
Now consider a superconductor with /1n = const. According to Eq. (2.4), the
magnetic energy density gained by the system allowing magnetic flux to penetrate,
i.e., the free energy density difference between the Meissner state with B = °and the
normal state with B = /1nH, is EmlV = /1nH2/87r. By contrast, the condensation
energy density gained by the system becoming a superconductor is EeondlV = t214u.
The thermodynamic critical field, which is defined by these two energies being equal,
is thus
He = J27rlu Itl/~= H~/~, (2.5)
with H2 = J27r lu ItI the critical field for a system with /1n = 1. An increase in /1n
thus decreases the critical field, as one might expect since the externally applied field
is amplified inside the material.
2.2.1. 2 London Penetration Depth
The dependence of the London penetration depth A on /1n is intuitively less
obvious. Consider a large superconducting sample, with linear dimension L,
surrounded by vacuum and subject to a homogeneous external magnetic field H =
(0,0, H) in z-direction. Along the left edge of the sample, the magnetic induction
will be of the form B(x) = (0,0, B(x)) with B(x) = Boe-x /).. (x > 0). To determine
Bo, imagine a thin (thickness d) layer of normal conducting material around the
superconductor. Except for the superconductivity, the normal layer should have the
same properties as the superconductor, in particular, a magnetic permeability /1n'
Then we have B = H in vacuum, and B = /1nH inside the normal layer, see Fig.
2.1(a). Now let d -+ 0. Then we have (Fig. 2.1(b))
26
B B
IlnH IlnH
H H
(a) (b)
V N S V S
o d x 0 x
FIGURE 2.1: Magnetic induction schematically as a function of position at a vacuum
(V) - normal metal (N) - superconductor interface (8) (a), and at a vacuum -
superconductor interface (b).
{
H for x < 0
B(x) =
PnH e-:V / A for x:2: 0
(2.6)
Now consider the current density associated with B(x). }rom Ampere's law we
have
j(x) = 4:Vx B(x) = (O,j(x), 0),
with c the speed of light and
j(x) = 4~A B(x).
(2.7a)
(2. 7b)
This is the total current density. It has three contributions, namely, the supercurrent
density j se, the spin or magnetization current j spin = CV x M, with M the spin
contribution to the total magnetization, and the external current density j ext =
cV X H 141f. The latter vanishes in the case we are considering. In a normal metal,
the spin current is the only contribution if we ignore normal-state diamagnetic effects.
The spin or normal-state susceptibility Xn is defined as the response of M to the total
magnetic induction B minus the contribution to B of M itself,
M = Xn(B - 41fM) = (Xn/Pn) B. (2.8)
27
For the supercurrent density j sc = j - j spin this implies
jsc = -4c V x B(x) = (O,jsc(X),O),
7rPn
with
(2.9a)
(2.9b)jsc(X) = 4 C A B(x).
7rpn
Now consider one surface (area £2) of the sample. Neglecting corner effects, and for
A« £, the total magnetic flux <1> through that surface is
(2.10)
On the other hand, the total supercurrent flowing near that surface is, from Eq.
(2.9b),
Isc = jdX j sc >::::J £21
00
dxjsc(x) = _c_ ~ <1>.
o 47rpn /\
We thus can write the flux
<1> - 47rpn A 1 _ 47rpn A N
- C £ sc - c £ qv,
(2.11)
(2.12)
where N is the number of supercurrent carrying particles, q is their charge, and v is
their velocity. If m is their mass, then Ekin = Nmv2 /2 is the total kinetic energy of
the supercurrent. The flux can thus be written
. 47rpn A ~/ ~ ~
<1> = -c- £ qy 2/mv Ny Ekin . (2.13)
Now we make two observations. First, N >::::J £2 An, with n = 11,b12 the particle number
density. Second, at the critical field strength the kinetic energy of the supercurrent
must equal the condensation energy in the region where the current is flowing, which
is (see Sec. 2.2.1.1) Econd = £2 At2/u. With Ao the London penetration depth for
28
fLn = 1,
(2.14)
this allows to write the flux at the critical field
(2.15)
where the first equality follows from Eq. (2.13), and the second one from Eq. (2.10).
We thus obtain
(2.16a)
or[28, 29]
(2.16b)
The penetration depth thus decreases with increasing fLn, as does the critical field.
This is somewhat counterintuitive, as it implies that the superconductivity becomes
in some sense more robust. It also implies that a large normal-state magnetic
permeability will make the superconductor necessarily of type 1.[13] We will come
back to this observation.
Notice that the above derivation relies only on very general energetic considerations
and on Ampere's law. Also notice that it uses an identity at the critical field strength,
where the superconductivity vanishes. This is fine for He, but the penetration
depth is a property of the supercondueting state, and hence the use of fLn is not
quite appropriate for this quantity, except in the limit A ~ 00. More generally,
A depends on fLs, which in turn depends on the superconducting properties. This
makes no difference deep inside the paramagnetic supercondueting phase, and Eq.
(2.16b) is valid there. However, as we will see it makes a crucial difference close to a
ferromagnetic instability.
29
2.2.1.3 Critical Current
In order to discuss the critical current, we assume a thin-wire geometry with wire
radius R. [9, 30] The supercurrent density, which is the total current density minus
the spin current density, can be written as a generalization of Eq. (2.9a),
c
- [V x B(x) - 4r.V x M(x)]
41r
c
4r.JL(x) V x B(x), (2.17)
where we have used Eq. (2.3) and JL(x) is the local magnetic susceptibility. Now
integrate over the cross section of the wire. Assuming a homogeneous current density
within a distance A from the surface, and using Gauss's theorem on the right-hand
side, we have
2r.RAjse = -4C fd£' (V x B(x)) = 9..RH
1rJLn 2
where we have used Eq. (2.6). The critical current density je is the one that produces
the thermodynamic critical field He, which yields the familiar London theory result[9]
(2.18)
This result is plausible: Dimensionally, j e must be a magnetic field divided by
a length. The relevant length scale is the thickness of the area that supports
diamagnetic currents, which is A. The relevant field scale should be the field
that corresponds to the condensation energy, which is He. To the extent that
JLs ~ JLn = const., as we have assumed in Sec. 2.2.1.2, Eq. (2.16a) implies that je
is independent of JLn,
. ·0
Je=Je' (2.19)
As we will see below, this result changes drastically in the vicinity of a ferromagnetic
instability.
30
2.2.2 Systems at a Ferromagnetic Instability
In the vicinity of a ferromagnetic instability of the normal metal, the normal state
magnetic susceptibility Xn, and hence the permeability Pn, become large and diverge
as the phase transition is approached. At a ferromagnetic critical point, the region of
linear response shrinks to zero, and Xn and Pn become strongly field dependent. This
field dependence is characterized by the critical exponent 6,[7]
(2.20)
The value of 6 depends on the universality class the particular magnetic system
belongs to. For all realistic universality classes, 6 R;j 5, whereas in Landau or
mean-field theory, 6 = 3. [31] Substituting Eq. (2.20) into Eq. (2.5), we find for the
thermodynamic critical field
(2.21)
This result holds for a system where the distance t from the superconducting
critical point can be changed while the system remains tuned to magnetic criticality
(more precisely, to the parameter values where magnetic criticality would occur in
the absence of superconductivity). Generically, the dimensionless distance r from
magnetic criticality will change as well if t is changed, and we will discuss such more
realistic situations in Sec. 2.4.
For the penetration depth, the situation is more complicated. In contrast to
He, which compares the normal-state magnetic energy with the superconducting
condensation energy that has nothing to do with spin magnetism, A is entirely a
property of the superconducting state, and the feedback of the superconductivity on
the spin susceptibility, or the difference between Pn and Ps, cannot be neglected. As
a result of this feedback, the magnetic transition in the presence of superconductivity
does not occur at r = 0, but rather at a value r ex -~~j Ao. [2] Here ~~ is the magnetic
correlation length at zero temperature. This suggests that the spin susceptibility
31
at r = 0 will be effectively XS ex: Ao/~~ » 1 in a mean-field approximation. More
generally, one has /-Ls ~ XS ex: (Ao/~~)I, with ry another critical exponent. Using this
in Eq. (2.16b) with /-Ln replaced by /-Ls, we obtain
(2.22)
Since ry ~ 1.4 > 0 for ferromagnetic systems, [31] this implies that the penetration
depth at magnetic criticality is anomalously short. Close to the superconducting
transition, the superconductor will therefore also be of type I, in agreement with a
conclusion drawn from studying the ferromagnetic phase. [13]
For the critical current density, Eqs. (2.18), (2.21), and (2.22) predict
with
a=2<5/(<5+1)+1/2-ry/4 , (r=O)
(2.23a)
(2.23b)
With <5 ~ 5 and ry ~ 1.4 this yields a ~ 1.8, in contrast to the Ginzburg-Landau
result a = 3/2.
These results hold at r = 0, and again we have assumed that t can be varied
independently of r. Let us relax the former condition. From the above argument
for the effective value of XS at r = 0 it also follows that Eq. (2.22) is valid only for
Irl < ~~/Ao. Since ~~ is typically on the order of a few A, while Ao is typically several
hundred A or even larger even at zero temperature, and diverges as Itl-1/ 2 for t ----> 0,
this is a very small range. By contrast, Eq. (2.20) can be valid for r as large as several
percent, provided H is not too small. Not too close to Te, where He goes to zero,
Eq. (2.21) can thus be valid in a substantial r-range, while A = Ao/y'JI;; except in an
extremely small interval around r = O. In that case,
a = 2<5/(<5 + 1) + 1/2 (1 » r » ~~/Ao) (2.23c)
32
which yields 0:' ~ 2.17 if 5 ~ 5.
Finally, at larger values of r, or sufficiently close to Te that He is small enough
to invalidate Eq. (2.20), we are back to the paramagnetic case, Eq. (2.19) holds, and
thus 0:' = 3/2.
One thus faces a rather complicated situation, where the exponent 0:' can take on
values between the Ginzburg-Landau value 3/2 and a value larger than 2, Eq. (2.23c),
depending on various parameters that are not easy to control or even determine
experimentally. We will discuss this in more detail in Sec. 2.4. Before we do so, in
the following section we will give a more technical and more detailed derivation of all
of our results.
2.3 Generalized Ginzburg-Landau Theory
We now consider a coupled field theory that describes both superconducting and
spin degrees of freedom in order to derive the above results from a more microscopic
level and gain a deeper understanding of their origin. Specifically, we consider a
generalization of the usual Ginzburg-Landau equations that includes the spin degrees
of freedom. Far from magnetic criticality, the latter can be integrated out to yield
ordinary Ginzburg-Landau theory with fLn entering the magnetic energy density. At
magnetic criticality, fLn becomes field dependent, which changes the thermodynamic
critical field. In addition, the leading term in the London equation vanishes, which
leads to a generalized London equation that describes exponential decay on a length
scale shorter than Ao, in agreement with the qualitative arguments in Sec. 2.2, and
with implications for the critical current as discussed there. Unlike in the previous
general discussion, in most of this section we will treat the magnetic critical behavior
in a mean-field approximation.
33
2.3.1 LGW theory for Superconducting and Magnetic
Fluctuations
Our starting point is an action for a complex scalar field 'I/J describing the
superconducting degrees of freedom coupled to a vector potential A, and a real vector
field M describing the spin degrees offreedom. [2, 22] We reiterate that the qualitative
behavior near the superconducting Tc does not depend on the symmetry of the order
parameter, so our restriction to a scalar order parameter does not imply a loss of
generality. The action reads
s JdX [_1 I(V-iqA(x))'I/J(x)1 2+ t11 'I/J(x)1 2+ U411'I/J(x)14+-81 B 2(x)2m 2 n
+~ (\7M(X))2 + t2M 2(x) + U2 (M2(x))2 - M(x). B(x)
2 4
-~ H(x)· B(X)]. (2.24)4n
Here and in the remainder of this section we use units such that Planck's constant
and the speed of light are unity, n= c = 1. The first line is the standard Landau-
Ginzburg-Wilson (LGW) functional for singlet superconductors. The first three terms
in the second line are a standard vector-M4 theory, with M(x) the fluctuating
magnetization. M couples to the vector potential via the M . B term,2 with
B = V x A, and the last term is necessary to relate S to the appropriate Gibbs
free energy, see Eq. (2.2). Notice that 'I/J and M are coupled only indirectly via
the vector potential A. Spin-flip scattering of electrons by the magnetic moments
does give rise to a direct coupling of the form M 2 1'I/J1 2 ,[13] but these terms are not
important for our purposes.
Minimizing this action with respect to 'I/J*, A, and M yields the following saddle-
20ne might consider it more physical to write M· (B - 41fM) , which would make t2 the inverse
(normal) magnetic susceptibility. However, this just amounts to a shift of t2 by 41f.
34
point equations,
t} 1jJ(x) + u} 11jJ(xW1jJ(x) - ~ (V - iqA(x))2 'lp(X) 0, (2.25a)
m
2
-i}L ['IV (x) V1p(X) - 1jJ(x)V1jJ* (x)] - CL 11jJ(x) 1 2A(x)
2m m
1
41f V x [B(x) - H(x) - 41fM(x)] , (2.25b)
t2M(x) - aV2M(x) + 1.l 2M 2(x)M(x) = V x A(x)(2.25c)
If we drop Eq. (2.25c) and put M = °in Eq. (2.25b) (this corresponds to dropping
M from the action) we recover the usual Ginzburg-Landau equations.[9] A non-
superconducting solution of the full equations is 1jJ = 0, B = H + 41fM, and M
determined by the magnetic equation of state
(2.26)
where r = t2 - 41f. For a small constant external field H a solution of Eq. (2.26) is
M= XnH, with
Xn = l/r (2.27)
the normal-state magnetic susceptibility At this point it is the bare susceptibility, but
it is clear that by renormalizing the spin part of the action before constructing the
saddle-point solution one can make it the physical susceptibility.
2.3.2 Effective Theory for Paramagnetic Superconductors
Now consider the full Eqs. (2.25). For a small and slowly varying M(x) we have
from Eq. (2.25c)
(2.28)
35
Substituting this into Eq. (2.25b) we obtain
jse(x) = _1_ V x B(x) - ~ V x H(x),
4KMn 4K
where
(2.29a)
-- -i!L ['Ij;*(x)V'Ij;(x) - 'Ij;(x)V'Ij;*(x)]
2m
2-~ 1'Ij;(x) 1 2A(x).
m
(2.29b)
Together with Eq. (2.25a), these are the equations of motion for an effective action [29]
Seff = J [1 . 2 t 2dx -1(V-zqA(x))'Ij;(x)1 +-I'Ij;(x)12m 2
'Ll 1 1 ]
+41'Ij;(xW + 8KMn B 2 (x) - 4K H(x)· B(x) ,
(2.30)
where we have dropped the now-superfluous subscript on the Landau parameters t and
u. The same result is of course obtained by starting with Eq. (2.24) and integrating
out M in a Gaussian approximation.
The quantity j se in Eqs. (2.29) is indeed the supercurrent, as can be seen by
comparing Eq. (2.29a) with Eq. (2.9a). It does not explicitly depend on Mn, see Eq.
(2.29b), and this is important for the flux quantum to be independent of Mn. The
magnetic energy B 2 /8KMn, which does explicitly depend on Mn, does not appreciably
contribute to the free energy of a thin film or wire sample, and the standard
determination of the critical current, Ref. [9], thus leads to the usual Ginzburg-
Landau result with no correction due to Mn i- 1. This corroborates the educated
guess in Sec. 2.2.1.3.
For all other quantities, the usual analysis of Ginzburg-Landau theory now
applies. [9] One characteristic length scale is given by the square root of the ratio
36
of the coefficients of the gradient-squared term and the 't/J2 term in Eq. (2.30). This
is the superconducting coherence length e= Vl/mltl. Another one is given by the
square root of the ratio of the coefficients of the terms quadratic in A. For a constant
7/J, this is the London penetration depth
(2.31)
This is identical with Eq. (2.16b), which had been deduced on elementary
phenomenological grounds.
For the Ginzburg-Landau parameter I), = A/e we now have I), = I),o/~, with 1),0
the value of the parameter for J1n = 1. This implies that the superconductor is of
type I or type II, respectively, for 1),0 < VJ1n/2 or 1),0 > VJ1n/2. While one can show
this by an explicit analysis of the effective action, a fast way to relate the theory
for arbitrary values of J1n to the one for J1n = 1 is to rewrite the action in terms of
dimensionless quantities. [32] In conventional Ginzburg-Landau theory, this is done by
introducing
(2.32)
Here 7/Jo = V-t/u is the superconducting order parameter scale. In terms of these
quantities, the effective action reads [32]
Self (H~~A5 Jdx [I (~ V - iA(X))~(X)I'
_1~(x)12 + ~ 1~(x)14 +~ (V X A(x))2
2 J1n
-2H(x) . (V x A(x)) ,
(2.33)
A simple further rescaling procedure shows that Seff depends only on a single
--- -------- --------
37
dimensionless parameter, rather than the two parameters /),0 and /--In' Define
(2.34)
Then
Seff (Hi~2eJdx [1(V-iA(x))~(X)12
_1~(x)12 + ~ 1~(x)14 + /),6 (v x A(X))2
2 /--In
-2~ (~iI(x)). (V x A(x)).
(2.35)
This shows that the theory with an arbitrary /--In maps onto ordinary Ginzburg-Landau
theory with the replacements
(2.36)
/),0 = J /--In/2 thus indeed marks the demarcation between type I and type II
superconductors, and the critical fields can be immediately obtained from the usual
results at /--In = 1. [9] For the thermodynamic critical field He' the upper critical field
He2 , the lower critical field Hel , and the surface critical field He3 we obtain
He H~/~, (2.37a)
He2 H~2//--ln = V2/),oH~//--ln, (2.37b)
Hcl HO g(/),o/Vfi;:) _ H2 ( /~) (2.37c)el () - y'2 9 /),0 /--In·9 /),0 /),0
He3 1.695 He2 . (2.37d)
38
where the universal function 9 has the limiting behavior
{
lnx + 0.08 + oU/x)
g(x) =
1
for x» 1/V2,
for x = 1/V2.
(2.37e)
If one neglects the weak dependence of 9 on its argument, He1 is approximately
independent of Mn.
2.3.3 Superconductors at Magnetic Criticality
As one approaches a ferromagnetic instability, Mn keeps increasing and can no
longer be treated as a constant. There are two effects that become important for our
purposes. First, in a normal metal Mn becomes strongly field or induction dependent.
At r = a this dependence is nonanalytic and described by the critical exponent o.
Second, as r becomes on the order of ~~/). (see Sec. 2.2.2) in a superconducting
phase, the difference between Mn and Ms can no longer be neglected. Related to this,
the gradient squared term in Eq. (2.25c) must be taken into account. We now consider
these effects, starting with the nonanalytic field dependence in the normal state.
2.3.3.1 Thermodynamic Critical Field
At magnetic criticality in the normal state, r = 0, one has[7]
(
_) 1/0-1
Xn(r = O,H) = XO H/Ho (2.38)
Here XO is a microscopic susceptibility, and fIo is a microscopic field scale. M and,
for small values of H, B are therefore proportional to H 1/ O, or H ex: BO. For small B,
the number Mn should thus be replaced by a function of B with the following leading
B-dependence,
(2.39)
39
with Ho = (4nXo)<>/(<>-1) fIo. The magnetic energy cost of the flux expulsion that
results from the formation of a Meissner phase (which equals minus the normal-
state magnetic energy) is now obtained by using Eq. (2.39) in Eq. (2.30). It is
Em/V = H B /4n - B6+1/8n Hg- 1 = H~-l/<>Hl+1/<> /8n. The condensation energy is
still given by Eeond/V = t2 /4u, which yields
( )
<>/(6+1)
_ 2n 1 2<>/(6+1)
He -- ---;; Ha<>-1)/(6+1) It I . (2.40)
The thermodynamical critical field is thus weaker than in the paramagnetic case, and
the t-dependence is consistent with Eq. (2.21). By comparing with Eq. (2.5), we
see that with respect to the thermodynamical critical field, /-In effectively scales like
/-In rv 1/ltI 2(<>-1)/(6+1) at magnetic criticality.
Equations (2.38) through (2.40) hold also for small but nonzero values of r as long
as one is in the field scaling regime, i.e, as long as H in appropriate units is large
compared to r to an appropriate power. We will discuss this in more detail in Sec.
2.4. At this point we only mention that, since He vanishes as It I --+ 0, sufficiently
close to Te one will lose the field scaling for any nonzero value of r, and He will be
given by Eq. (2.37a).
2.3.3.2 Generalized London Equation
The ordinary London equation is obtained from Eq. (2.25b) by dropping M(x)
and treating 'IjJ(x) - 'IjJ as a constant (London approximation). With V x H(x) = 0
this leads to
-A02 B(x) = V x V x B(x). (2.41 )
Now take M into account. Using Eq. (2.25c) in Eq. (2.25b), we can eliminate Band
derive an equation for M. Once M is known, B follows from Eq. (2.25c). Within
40
the London approximation one finds
M(x) -(A~//-ln)"l x Vx M(x) + (t~rV2M(X) + (t~r A~V x Vx V 2M(x)
-uM2(x)M(x) - UA~V x V x M 2 (x)M(x). (2.42)
Here /-In = (47r + r)lr as in Sec. 2.3.2, [~ = ~~/V47r + r y'al(47r + r), and U =
ul(47r + r).
As long as /-In ~ 1, the first term on the right-hand side of Eq. (2.42) leads
to a variation of M on a length scale A = AoI~. The second term is a small
correction to the first one since ~~ «Ao. So is the third term, which is of order
(t~)2 V 2 rv (t~) 2 IA6 « 1 relative to the first one. The linearized version of
Eq. (2.42) thus reduces to the ordinary London equation, Eq. (2.41), with Ao -+ A.
However, for r = 0 the first term vanishes. This makes the second term the leading
one, and the third term, which is of order A6V2 compared to the second one, cannot
be neglected either. The linearized equation thus reads
M(x) = (t~r V 2 [1 + A~V x VxJ M(x).
With the same interface geometry as in Sec. 2.2.1.2 this takes the form
(2.43)
(2.44)
This linear quartic ODE is solved by an exponential ansatz, M(x) = Moe- px . The
real solution that falls off for x -+ 00 shows damped oscillatory behavior. From Eq.
(2.25c) we see that B(x) shows the same behavior as M(x), up to corrections of
O(t~1Ao). With the boundary condition B(x = 0) = /-lnH we finally obtain
B(x) = /-lnHe-x/V2l9n>'Q cos (XIJ2t~AO ) . (2.45)
41
This is the solution of the linearized version of Eq. (2.42) at T = O. In addition to
leaving out the terms of O(M3 ), we have also ignored the fact that the permeability,
whether Mn or Ms, does depend on B or M at magnetic criticality. In a mean-field
approximation, Mn ex: 1/B 2 at T = 0, see Eq. (2.39), which also leads to terms of
O(M3 ) in the nonlinear equation. Depending on the ratio of the external field to
Ho, these terms mayor may not be important for the initial decay of M or B near
the normal metal-to-superconductor boundary. However, once M or B has decayed
sufficiently, these terms become subleading compared to the linear ones in Eq. (2.44),
and the asymptotic behavior as B -> 0 is always given by Eq. (2.45).
In order to make contact with the discussion in Sec. 2.2.2 for small but nonzero
values of T, let us consider the linearized Eq. (2.42) while keeping the first term.
Instead of Eq. (2.44) we then have
(2.46)
This is solved by
(2.47a)
with
1 [ 2 (-0)2
(
_ ) 2 Ao/ Mn + ~m
2A2 COo ':>m
(2.47b)
Here we have chosen the solution for p2 that yields p2 -> 1/A6 for T -> 00. Equation
(2.47b) still provides two solutions for p, and the physical solution for M is determined
by the requirement that M be real.
A discussion of Eq. (2.47b) shows that p2 becomes purely real and negative at
T = Ts = -4J1ft~/Ao + 0 ( (t~) 2 / A6). This is in agreement with the results of
42
Blount and Varma, [2] who showed that spiral magnetic order coexisting with the
superconductivity occurs at this point. For Irl « ~~IAD one has p2 f::::j -i/~~Ao,
which leads to Eq. (2.45). For r » ~~IAD one finds p2 f::::j Mnl A5, which leads to
(2.48)
in agreement with Eq. (2.31).
2.3.3.3 Penetration Depth, and Critical Current
Equation (2.45) shows that the effective penetration depth at magnetic criticality
IS
(2.49)
in agreement with the conclusions of Ref. [13] drawn from studying the ferromagnetic
phase, and with Eq. (2.22) with 1= 1. The latter approximation results from the fact
that our saddle-point equations of motion describe the magnetic equation of state in
a mean-field approximation. The discussion of Eq. (2.47b) shows that this result is
valid for Irl « ~~IAD. By comparing with Eq. (2.16b) or (2.31), we see that with
respect to the penetration depth, Mn at magnetic criticality scales like Mn rv II y'jtT
in mean-field approximation, or Mn rv I/ltl i / 2 in general. The fact that II~ in
Eqs. (2.5) and (2.16b), respectively, must be interpreted differently for }-tn ----> 00 is a
consequence of the influence of the superconductivity on the spin response.
For r » ~~I AD we have, from Eq. (2.48)
(2.50)
in agreement with Eq. (2.31).
The expression for the critical current given by Eq. (2.18) is general within the
London approximation. We have now given a derivation of the behavior of the
43
thermodynamical critical field and the penetration depth given on phenomenological
grounds in Eqs. (2.21) and (2.22), respectively. The behavior of the critical current
at or near magnetic criticality is thus given by Eqs. (2.23).
2.3.3.4 Critical Field H c2
The critical exponent I is positive C'/ ~ 1.4 for typical ferromagnetic universality
classes in three dimensions[31]). The result for A, Eq. (2.22) or (2.46) in mean-field
approximation, ofthe previous subsection therefore means that Adiverges more slowly
for ItI --+ 0 than the superconducting coherence length ~ ex II \/ltT. Consequently,
superconductors at magnetic criticality (Irl « t~1Ao) are necessarily of type 1. [13]
This observation notwithstanding, the critical field Hc2 , which in a type-II
superconductor signalizes the boundary of the vortex phase, still has a physical
meaning: It is the minimum field to which the normal metal can be 'supercooled'
before it discontinuously develops a nonzero superconducting order parameter. [9] It
is thus still of interest to determine H c2 . Furthermore, the behavior will be necessarily
of type I only for Irl in an extremely narrow region. Outside of this region, Eq. (2.50)
holds, and for a sufficiently large value of /),0 = AoI~ the superconductor will still
be of type II. The determination of H c2 is done by linearizing the Ginzburg-Landau
equation, Eq. (2.25a), in 1/J. It then turns into a Schrodinger equation for a particle in
a vector potential A, with -tI/2 = -t/2 playing the role ofthe energy eigenvalue. By
means of standard arguments [9] this leads to a critical value of the magnetic induction
B = V X H given by B c2 == H~2 = -tmlq. In a paramagnetic superconductor, this
leads to
H c2 = H~2 I f.1n (f.1n = const.), (2.51)
which is the same as Eq. (2.37b). At magnetic criticality, we have, d. Eq. (2.39),
H - B J IH(J-l) Itl Jc2 - c2 0 ex . (2.52)
44
Notice that, in this context, j1,n scales as j1,n rv 1/Itl<>-1, whereas it scales as j1,n rv 111tl
if the relevant field scale is He. Since He2 vanishes much faster than He, Eq. (2.40),
the field scaling region will be restricted to larger values of It I, and He2 will be given
by Eq. (2.51) in a substantial range of t-values. Will come back to this in Sec. 2.4.
2.4 Discussion and Conclusion
To summarize, we have determined the electrodynamic properties of superconductors
close to a ferromagnetic instability, i.e., materials that, in the absence of superconductivity,
would be paramagnetic with large ferromagnetic fluctuations. This work complements
previous studies of the coexistence of superconductivity with ferromagnetic order. [2,
13] We have treated the superconductivity in mean-field (Ginzburg-Landau) approximation.
In addition, we have employed the London approximation, treating the superconducting
order parameter as a constant. The ferromagnetic critical point we have treated
explicitly in a mean-field approximation, and we have used scaling arguments
to consider the consequences of the exact magnetic critical behavior for the
superconductivity. We have found that the thermodynamical critical field He
decreases due to the ferromagnetic fluctuations, as one would expect, and depends
on the magnetic critical exponent 0, see Eqs. (2.40) and (2.21). However, the London
penetration depth also decreases, which is intuitively less obvious. At magnetic
criticality the behavior of the magnetic induction at a vacuum-to-superconductor (or
normal metal-to-superconductor) interface is still characterized by exponential decay,
but the characteristic length scale A is different from the usual London penetration
depth Ao. Within a mean-field description of the magnetic criticality it is the
geometric mean of the zero-temperature magnetic correlation length and Ao, see Eqs.
(2.45) and (2.49); more generally, it depends on the magnetic critical exponent 1, see
Eq. (2.22). However, this behavior of the penetration depth is valid only within an
extremely small region of width {?nl Ao around magnetic criticality. Outside of this
region, but still within the ferromagnetic critical region, the temperature dependence
45
of the penetration depth is the same as in Ginzburg-Landau theory, see Eq. (2.50). For
the critical current j e <X He/.A this implies a dependence on the reduced temperature
given by Itla, where the exponent 0: depends on both 0 and "'(, or on 0 only, depeding
on the value of r, see Eqs. (2.23). With exponent values appropriate for the usual
ferromagnetic universality classes, 0: ~ 1.8 extremely close to magnetic criticality,
and 0: ~ 2.15 somewhat farther away.
Let us now discuss these results in some more detail, and relate them to the
experimental observations reported in Ref. [27].
For the temperature dependencies of various observables at magnetic criticality
we have assumed that the system stays tuned to magnetic criticality while the
temperature is varied. Let us discuss to what extent this assumption is realistic.
Consider a phase diagram in a plane spanned by the temperature and some non-
thermal control parameter x, e.g., the hole doping concentration in the case of
MgCNi3 ,[26] and consider the following two qualitatively different possibilities. Figure
2.2 shows a situation where the magnetic phase separation line does not cross the line
x = O. The stoichiometric compound thus does not enter a magnetic phase upon
cooling, although the system is close to a magnetic transition for all temperatures
below the superconducting Te . This scenario is believed to apply to MgCNi3 . Figure
2.3 shows a situation where the magnetic phase separation line does cross the line
x = 0, so that the stoichiometric compound enters a phase where superconductivity
and magnetism coexist at some temperature below Te . This is the situation that
was discussed in Refs. [2] and [13] and observed in ErRh4B4 and HoM06Ss.[15, 16]
The magnetic transition is to a phase with spiral magnetic order at a temperature Ts
slightly below the temperature T~ where ferromagnetism would occur in the absence
of superconductivity. [2]
We now can see what is required to keep r constant while varying t, namely, a
situation as shown in Fig. 2.2 with the dashed line essentially parallel to the T-axis.
r is then given by the dimensionless distance between the two lines. In order for
46
Tc To f------ SC ---.--------7
................................................ .. .. ·.....NM
FMSC
FM
x
Tx = 0 IL-- -'--_~»
o
FIGURE 2.2: Schematic phase diagram showing a normal metal (NM), a ferromagnet
(FM), a superconductor (SC), and a ferromagnetic superconductor (FMSC) in
a temperature (T) - control parameter (x plane. The solid line denotes the
superconducting transition, the dashed line, the magnetic one. Along x = 0 there
is only one phase transition at the superconducting Te . See the text for additional
explanation.
the penetration depth to display the non-Ginzburg-Landau behavior described by
Eq. (2.49) or, more generally, Eq. (2.22), the two lines would have to be extremely
close, in order to keep r smaller than ~~/AD, see Eq. (2.49). This would result in a
temperature dependence of the critical current given by Eqs. (2.23a, 2.23b). While
this is possible, it is a very non-generic situation, and it would result in a very large
magnetic susceptibility of the normal metal just above the superconducting transition
temperature.
A situation that is still very non-generic, but requires somewhat less fine-tuning, is
one where the dashed line is still essentially parallel to the T-axis, but in a somewhat
larger r-range, say, with r on the order of a few percent. In this case the penetration
depth will show the usual 1/ltI1/ 2 temperature dependence, see Eq. (2.50). The
temperature dependence of the thermodynamic critical field will be more complicated
47
--------~~ Tm Tc To ... , SC..---:;,-
,
,
,
" NMFMSC \
\
\
\
I
\
FM 'I
X ,
x=o
o
T
;:.
FIGURE 2.3: Same as Fig. 2.2, but with a magnetic transition for x = 0 at a
temperature Tm < Te . On the x = 0 axis it is shown that Tm splits into the bare
magnetic transition temperature T~ and the physical transition temperature Ts to a
state with spiral magnetic order, Ref. [2]. See the text for additional explanation.
in this case. The generalization of Eq. (2.20) to nonzero values of r is
(2.53)
with I = (3(0 - 1), (3, and 0 the usual critical exponents for the magnetic transition.
In order for Eq. (2.20) to hold, the H must be large compared to r f38 in suitable
units. The latter are not determined by any universal arguments, but an analysis of
the critical equation of state for both the high-temperature ferromagnet Ni (Tm ~
630 K) [33] and the low-temperature ferromagnet CrBr3 (Tm ~ 33K)[34] shows that
in either case the relevant energy or field scale (we use units such that kB = J.-lB = 1)
is given by Tm , which is plausible. The crossover between the field scaling that leads
to Eq. (2.21) and the static scaling that leads to Eq. (2.5) thus occurs at a crossover
field
(2.54)
(30 ~ 5/3 for ferromagnetic phase transitions, and with Tm ~ 10K and r ~ 0.1, one
48
finds H x ~ 0.02T~. For MgCNi 3 in the vicinity of Te , this leads to H x ~ 0.2T.
With Hc2 at zero temperature on the order of 14 T and K, ~ 40,[35] one expects
He(T = 0) = Hc2 / -J'iK, ~ 0.25 T. Since He vanishes at Te, this means that He will be
given by Eq. (2.21) sufficiently far away from Te, but cross over to He ex: It I near Te.
Consequently, the critical current exponent a will be given by Eq. (2.23c) at some
distance from Tel and cross over to the Ginzburg-Landau result a = 3/2 as It I ----> O.
In the experiment of Ref. [27], no such crossover was observed down to It I ~ 0.01.
At least within the London approximation, our results confirm the conclusion
of Ref. [13] that superconductors near a ferromagnetic instability are necessarily of
type 1. However, we have also shown that this conclusion is inevitable only within
an extremely small region around the (bare) magnetic critical point. The fact that
MgCNi3 is observed to be of type II[35] is therefore not necessarily in contradiction
to the notion that this material is almost ferromagnetic. However, Eq. (2.52) predicts
a strong deviation from Ginzburg-Landau behavior for the upper critical field He2 •
Since He2 goes to zero rapidly as It I ----> 0, this behavior will show only at substantial
values of ItI even if r is very small. No anomalous behavior was observed for It I up
to 0.5.[35] This is reconcilable with close proximity to a magnetic instability only if r
is very small close to Te , and grows with decreasing temperature, in which case H e2
might never show the magnetic critical behavior. A signature of this situation would
be a large magnetic susceptibility in the normal state just above Te .
The conclusion from this discussion with respect to the experimental observations
in Ref. [27] is as follows. While it is possible that proximity to a ferromagnetic
instability is the cause of the observed anomalous behavior of the critical current,
such an explanation requires fine tuning of the phase diagram, and would have to
be accompanied by a very large enhancement of the spin susceptibility in the normal
phase just above Te . Explaining the lack of an anomaly in the temperature dependence
of He2 probably requires that the material is closer to the magnetic instability near
Te than at T = 0 (i.e., the dashed line in Fig. 2.2 comes closer to the T-axis with
49
increasing T). A direct measurement of the spin susceptibility in the normal phase
would be of great interest in this context.
Finally, we discuss our predictions for the case of a superconductor that does
undergo a transition to a magnetic state below Te , i.e., the situation represented
by Fig. 2.3. In the (very small) temperature interval of width 21T~ - Tsl around
T~, both the thermodynamic critical field He and the penetration depth Awill show
an anomalous temperature dependence, and the critical current exponent will be
given by Eq. (2.23b). Outside of this region, but not too close to Te, He will be
anomalous, but Awill be conventional, and the critical current exponent will be given
by Eq. (2.23c). Upon approaching Te, He will fall below the crossover field given
by Eq. (2.54), and its temperature dependence will cross over to the usual linear
Ginzburg-Landau behavior. The critical current exponent close to Te will thus be
the conventional a = 3/2. The location of this crossover depends on the critical field
scale, and will thus be material dependent. Critical current measurements in the
materials like ErRh4B4 , or HoMo6SS ' which are believed to fall into this class, would
be very interesting.
50
CHAPTER III
NATURE OF PHASE TRANSITION IN P-WAVE
SUPERCONDUCTORS
3.1 Introduction
BCS theory predicts that the phase transition from the normal state to the
superconducting state in s-wave superconductors is continuous or second order.
However, in 1974 Halperin, Lubensky, and Ma [36] showed that the coupling between
the superconducting order parameter and the electromagnetic vector potential tends
to render the transition first order. This conclusion is inevitable for extreme type-I
superconductors where fluctuations of the order parameter are negligible and the
vector potential can be integrated out exactly, and the mechanism is analogous
to the spontaneous mass generation known in particle physics as the Coleman-
Weinberg mechanism. [37] When order parameter fluctuations cannot be neglected,
and especially for type-II superconductors, the problem cannot be solved exactly.
The authors of Ref. [36] generalized the problem by considering an n/2-dimensional
complex order parameter and conducting a renormalization-group (RG) analysis in
d = 4 - c dimensions. The physical case of interest is n = 2 and d = 3. To first order
in c they found that a RG fixed point corresponding to a continuous phase transition
exists only for n > 365.9, which suggests that for physical parameter values the
transition is first order even in the type-II case. They corroborated this conclusion
by performing a large-n expansion for fixed d = 3. To first order in lin, the critical
exponent v is positive only for n > 9.72, which again strongly suggests that the
transition in the physical case n = 2 is first order. For superconductors, the size
51
of the effect is too small to be observable, whereas for the analogous smectic-A to
nematic transition in liquid crystals it was predicted to be much larger. Experiments
that showed a clear second order transition in liquid crystals later prompted a re-
examination of the theory by Dasgupta and Halperin. [38] Using Monte Carlo data and
duality arguments, these authors argued that a type-II superconductor in d = 3 should
show a second order transition after all. The discrepancy between these theoretical
results has never been clarified.
Recently there has been substantial interest in unconventional superconductivity.
In particular, Sr2Ru04 has emerged as a convincing case of p -wave superconductivity, [39,
40] and UGe2 is another candidate.[41] This raises the question whether for such
systems the fluctuation-induced first order mechanism also is applicable, or whether
the additional order parameter fluctuations allow for a second order transition in
situations that lead to a first order transition in the s-wave case. Here we investigate
this problem. By conducting an analysis for p-wave superconductors analogous to
the one of Ref. [36] we predict a first order transition as in the s-wave case, although
the restrictions are somewhat less stringent.
This chapter is organized as follows. In Sec. 3.2 we define our model and derive the
mean-field phase diagram. In Sec. 3.3 we determine the nature of the phase transition.
We do so first in a renormalized mean-field approximation that neglects fluctuations
of the superconducting order parameter. We then take such fluctuations into account,
first by means of a renormalization-group analysis in d = 4 - f dimensions, and then
by means of a lin-expansion. In Sec. 3.4 we discuss our results.
3.2 Model
Let us consider a Landau-Ginzburg-Wilson (LGW) functional appropriate for
describing spin-triplet superconducting order. The superconducting order parameter
is conveniently written as a matrix in spin space, [42] .6.0"10"2 = :E~=1 dj.l(k) ((7j.li(72)0"10"2'
Here (71,2,3 are Pauli matrices, k is a wave vector, and the dj.l are the components of a
52
complex 3-vector d(k). p-wave symmetry implies dl-"(k) = 2..:~=1 dl-"jkj , with k a unit
wave vector. The tensor field dl-"j(x) is the general order parameter for a spin-triplet p-
wave superconductor and it allows for a very rich phenomenology. For definiteness we
will constrain our discussion to a simplified order parameter describing the so-called
;3-state,[42] which has been proposed to be an appropriate description of UGed41]
It is given by a tensor product d = 'l/J (8) ¢ of a complex vector 'l/J in spin space and
a real unit vector ¢ in orbital space. The ground state is given by 'l/J = .6.0 (1, i, 0),
¢ = (0,0,1). In a weak-coupling approximation that neglects terms of higher than
bilinear order in 7/J2, q}, and \72 the action depends only on 'l/J,
s = Jdx [tl'l/J1 2 + c1D'l/J1 2 + Uo I'l/J14 + vol'l/J x'l/J*1 2
+8~'/V X A)2]. (3.1)
Here A is the vector potential, D = V - ieA is the gauge invariant gradient with
e the Cooper pair charge, and ID'l/J12 = (Di7/Ja)(D;7/J;) with summations over i and
(J implied. fJ is the normal-state magnetic permeability, and t, c, Uo, and Vo are
the parameters of the LGW functional. The fields 'l/J and A are understood to be
functions of the position x.
For later reference we now generalize the vector 'l/J from a complex 3-vector to a
complex m-vector with components 7/Ja, so that the total number of order parameter
degrees of freedom is n = 2m. In order to generalize the term with coupling constant
v we use of the following identity for 3-vectors,
(3.2)
and notice that the right-hand side is well defined for a complex m-vector. Our
53
generalized action now reads
s Jdx [t1J;a1jJ~ +C(Di1jJa)(Di1jJ~)+u1jJa1jJ~1jJf31jJ~
+v1jJa1jJa1jJ~1jJ~+ 8~,u tijk(OjAk)tilm((JzAm)] ,
(3.3)
with Ci,(3 = 1, ... m, i,j, ... = 1,2,3, and summation over repeated indices implied.
Here we have defined new coupling constants U = Ua + Va and v = -Va. In addition
to the generalization of the order parameter to an m-vector we will also consider
the system in a spatial dimension d close to d = 4. The physical case of interest is
m = d= 3.
3.3 Nature of The Phase Transition
3.3.1 Mean Field Approximation
The simplest possible approximation ignores both the fluctuations of the order
parameter field 'ljJ and the electromagnetic fluctuations described by the vector
potential A. The order parameter is then a constant, 'ljJ(x) 'ljJ, and the free energy
density f reduces to
(3.4)
In order to determine the phase diagram we parameterize the order parameter as
follows, [3]
'ljJ = 1jJa (n cos ¢ + i msin ¢) . (3.5)
Here 1jJa is real-valued amplitude, nand m are independent real unit vectors, and ¢
54
FIGURE 3.1: Mean-field phase diagram of a p-wave superconductor as described by
Eq. (3.1). See the text for additional information.
is a phase angle. The free energy density can then be written
(3.6)
We now need to distinguish between two cases.
case 1: '110 > O. The free energy is minimized by n = m, and 'I/J'6 = -t/2uo. The
condition Uo > 0 must be fulfilled for the system to be stable.
case 2: '110 < O. The free energy is minimized by n ...L m and ¢ = 11"/4, and
'I/J'6 = -t/2(uo +'110). The condition Uo +'110> 0 must be satisfied for the system to be
stable.
The first case implies 'l/J x 'l/J* = O. This is referred to as the unitary phase. In
the second case, 'l/J x 'l/J* =J- 0, which is referred to as the non-unitary phase. In either
case, mean-field theory predicts a continuous phase transition from the disordered
phase to an ordered phase at t = O. The mean-field phase diagram in the '11'0-'110 plane
is shown in Fig. 3.1.
55
3.3.2 Renormalized Mean-field Theory
A better approximation is to still treat the order parameter as a constant, 'l/J(x) -
'l/J, but to keep the electromagnetic fluctuations. The part of the action that depends
on the vector potential then takes the form
(3.7a)
where
(3. 7b)
is the inverse London penetration depth. Since A enters SA only quadratically, it can
be integrated out exactly, and the technical development is identical to the s-wave
case. [36]The result for the leading terms in powers of I'l/J 12 in d = 3 is
(3.8)
Here w ex:~ is a positive coupling constant whose presence drives the phase
transition first order.
There are several interesting aspects of this result. First, the additional term in
the mean-field free energy, with coupling constant w, is not analytic in 1'l/J1 2• This is a
result of integrating out the vector potential, which is a soft or massless fluctuation.
Second, the resulting first-order transition is an example of what is known as the
Coleman-Weinberg mechanism in particle physics, [37] or a fluctuation-induced first-
order transition in statistical mechanics. [36]
Let us discuss the validity of the renormalized mean-field theory. The length scale
given by the London penetration depth .A = k>..l needs to be compared with the second
length scale that characterizes the action, Eq. (3.1), which is the superconducting
coherence length ~ = JC7ltT. The ratio /'i, = .A/~ is the Landau-Ginzburg parameter.
For /'i, ---+ 0, order parameter fluctuations are negligible (this is the limit of an extreme
56
type-I superconductor), and the renormalized mean-field theory becomes exact. For
nonzero values of /'1, the fluctuations of the order parameter cannot be neglected, and
the question arises whether or not they change the first-order nature of the transition.
We will investigate this question next by means of two different technical approaches.
3.3.3 t-expansion about d = 4
Here, we will follow notations of reference [43, 36] and apply the momentum shell
renormalization group to the action Eq. (3.3). By this method, we can treat both
the magnetic and superconducting fluctuations at about dimension 4. t = 4 - d
is presumed to be a small and positive parameter which will justify the asymptotic
expansion of free energy functional. Eventually t can be loosely treated to be 1, so
the critical behavior of the system at spatial dimension 3 could be extrapolated.
First, the bare propagator for the complex OP field in the Fourier space can be
identified from the quadratic terms from Eq. (3.3)
(3.9)
Coulomb gauge V . A = 0 will be used through this chapter. We denote photon
propagator as,
(3.10)
where Pij(q) is the transverse projection operator Pij(q) = Oij - q~%j. The vertices
which can be read from Eq. (3.3) are listed in the Fig. 3.2. Recursion relations for
the coupling constants can be obtained from doing the momentum shell integrals for
the diagrams in Fig. 3.3 and Fig. 3.4. Now the RG flow equations for the coupling
parameters read,
:v
57
,
<e(p.+P2), ' ,P.
lr' ,
, ,
FIGURE 3.2: Vertices from Eq. (3.3). Solid lines denote the 'IjJ field and dashed lines
its complex conjugate. Wavy lines denote the vector potential. Dotted lines separate
the localized interaction between paired electrons.
J:l __
--:---i--
FIGURE 3.3: Diagrams renormalize the coupling parameters
dt (d 2) 3ce2k 4v + (n + 2)u k (3.11)- + xt+-- d+ ddl f-l t+c
du (d 4) (n + 8)u2+ 8v2+ 8uv k 48 2 2 2 4k (3.12)- + X u - ()2 d - 7r f-l C e ddl t+c
dv (d 4) _ nv2+ 12uv k (3.13)-
dl + X v (t + C)2 d
df-l-l 27rnc3e2 (3.14)-- (d - 2 + 2XA)f-l- 1 + 3(t + C)4 (3t + c)kddl
dc 127r f-lce2 (3.15)- (d - 2 + 2X - ( ) kd)cdl t+c
de (1 + XA)e (3.16)-dl
58
Here X and XA are the rescaling parameters of fields 'l/J and A respectively kd is
the surface area of the d-dimensional(d = 4 in this case) unit sphere.
When the parameter v = 0, the above flow equations recover the recursion relation
of H.L.M [36] as should be the case.
The case e = °case which corresponds to a p-wave superfluid, will not be discussed
here. We are looking for physical fixed points for e i °case.
We choose X to keep the parameter c fixed and XA = -1 to keep the charge e
fixed and assume that the parameter t has a fixed point of order c:. We find the
value of p,*-l = 27[~kde2 from the recursion relationship Eq.(3.14). By setting the
right hand side of Eqs. (3.12,3.13) equal to zero, we get coupled quadratic algebraic
2
equations which describe the fixed point values of parameters u and v. Let u = ~d X
and v = ~: y. From Eqs. (3.12)~.13) we get
36 108
x(l +-) - ((n + 8)x2+8xy + 8y2) - -2 = °
n n
36
y(l + -) - ny2 - 12xy = °
n
(3.17a)
(3.17b)
If y = 0, Eq.(3.17a) naturally recovers s-wave case discussed by H.L.M [36] who
found a real valued fixed point for n > 365.9, there exist real valued fixed points. For
--_----:...-----
. '
. '
-.---.---
---_..._-
--r;----L.---,-
FIGURE 3.4: Diagrams renormalize the coupling parameter u
59
v
u
FIGURE 3.5: Renormalization group flows for n=500. The top right fixed point is
stable in the u-v plane, so it is the critical fixed point
Y i- 0, the coupled quadratic equations can be reduced to one quadratic equation of
y only, whose discriminant is
.6. = -248832 - 6912n - 5424n2 - 408n3 + n4 (3.18)
To have .6. positive, n must be larger than nc = 420.928. For any n > nc , there
exist new fixed points besides the s-wave ones. It can be easily verified that these
newly appearing fixed points are falling in the non-unitary region in Fig. 3.1. We will
refer to them as p-wave fixed points. The stability of a fixed point can be analyzed
by linearizing the recursion relationship at that point to get the eigenfunctions and
eigenvalues. A typical flow in the u - v plane is plotted in Fig. 3.5.
Although the s-wave fixed point appears first when n > 365.9, it never controls
the phase transition. Only when n > nc , one of the p-wave fixed points takes the role
of the critical fixed point. The critical fixed point in the large n limit takes the form
E E * ne
2
u* '" -, v* '" -, p., "'-
nnE
(3.19)
60
The non-vanishing value of v* implies as least close to dimension 4, p-wave
superconductors would belong to a different universality class from their s-wave
cousins. From Fig. 3.5, the left bottom corner and top corner show a typical "run-
away" flow. This "run-away" flow goes to the u < 0 and v > 0 region which falls
off from the continuous phase transition region in the mean field graph Fig.3.1. It
is a strong signature of first order phase transition which is also consistent with the
extreme type I case discussion in section 3.3.2.
At first sight, the fact that a p-wave critical fixed point requires a larger number of
components than the s-wave case may lead to the conclusion that it is more unlikely
for p-wave superconductors to have a secound order phase transition. However, we
need to keep in mind that the f-expansion is valid only for small f, so this fixed
point theory may only apply to dimensions very close to 4. To get n c for d = 3,
one would have to take the f-expansion to higher loop order, which is a formidable
job. Another issue is the limitation of the f-expansion method which assumes
perturbatively accessible fixed points, so it may apply only to a small parameter
region. Outside of this region, other methods may have to be applied to study the
nature of the phase transition, for instance dual theory. [38J
Another way to partly answer this question was pointed out [36J in the s-wave case:
a lower bound nc1 = 9.7 for the critical value of n in the s-wave case was obtained by
a large n expansion in d = 3. We will now apply this techniques to the p-wave case.
3.3.4 lin-expansion in d = 3
We refer to the literature, [36, 44] for the general large n technique. The basic
idea is very similar to the f-expansion we have shown. In the f-expansion case,
the expansion is performed at the Gaussian fixed point. Large n limit O(n) model
is usually referred to as spherical model and is solvable. So perturbation expansion
around its saddle point is doable. Through the expansion, the most infrared divergent
61
a)
~--
d)
.................
'.
"
" "-~, ------
b)
e)
c)
FIGURE 3.6: All the self energy graphs contribute to the critical exponent. The first
row is from the ungauged part and the second row is from gauged part.
terms will be summed and we will show they contribute to the critical exponents in
terms of lin.
We make the assumption for the following perturbed parameters,
1 1
1J, '" -, v '" - , and
n n
2 1
e "'-
n
(3.20)
This is reasonable by setting f = 1 in Eq. (3.19). In this section we set c = 1
and p., = 1 for simplicity. To get critical exponents TJ and r in terms of lin, we will
calculate a two point electron correlation function perturbatively. In Fourier space
for small momentum k at critical temperature it reads
(3.21)
Eq.(3.21) is the usual definition of critical exponent TJ . Here, c denotes the cumulant
expansion of the full action Eq. (3.3). Now we show that, given Eq. (3.20), all the
diagrammatic contributions to G(k) can be controlled by an expansion in powers of
lin. Therefore, TJ can be expressed in terms of lin. Eq.(3.21) for small TJ can be
rewritten as
(3.22)
62
So the ultimate purpose is to do the perturbation for the renormalized Green's
function, and extract the logarithmic dependence of momentum terms. Following the
notation in [44], we define the renormalized mass t and self energy ~ by
t = C- l (t, k = 0)
C- 1(t, k) = to + k2 + ~(t, k)
(3.23)
(3.24)
Where to is the bare parameter offree energy Eq.(3.3). In the following calculation,
we will use (t + k2)-1 as the electron propagator, which leads to the renormalization
rule: whenever we get a quantity of self energy insertions ~(t, k), we have to subtract
the bare self energy ~(t, k = 0). For instance, applying this rule directly to Eq(3.24),
we get
C-1(t, k) = t + k2 + ~(t, k) - ~(t, k = 0) (3.25)
A less obvious example will be given when calculating the critical exponent f.
Now compare Eq.(3.2l,3.22) and Eq.(3.25), in order to get the critical exponent
'TI, we need to extract the -k2 log k term from ~(t = 0, k) - ~(t = 0, k = 0) . The self
energy graphs of order lin are listed in Fig. 3.6. We only show order lin graphs that
contribute to this critical exponent.
In the following calculations, we will apply dimensional regularization and
Feynman integral tricks which are given in the appendix.
The self energy graphs a) and c) in Fig. 3.6 have no dependence k at all, so their
contribution to 'TI is zero. Before really calculate graph b), we refer to Fig. 3.7 for the
definition of the double dashed line. The figure shows that every bubble has double
contractions of the spin indices CY, (3, which amounts to order n, so graph b) actually
is a sum of a series of graphs which are of order lin. The single bubble repeats itself
in the sum which will be defined as polarization bubble.
63
( ' (' ('ecce:::::::: = + i + i ·· I. +I I I
dressed coupling parameler v
FIGURE 3.7: Dressed coupling parameter and renormalized photon propagator.
The polarization bubble in the Fig. 3.7 is
(3.26)
(3.27)
Using the methods explained in the appendix, this integral can be easily done in
dimension 3. with the result
1 2ft 1
II(t,k) = (47f)3/2-k-arctan(2Jt/k2)
at criticality t = 0, so II(O, k) = A.
A ladder resummation of the bubble graphs gives us the double dashed line. In
Fig. 3.7, we show how the double dashed line can be constructed from v vertices.
The double dashed line constructed by the u vertices can be obtained in the same
manner. By working out the combinatorial factor before every graph, we know this
sum is actually a geometric series, for example, if we take consideration of the graph
from v vertex as a piece for self energy of graph b). The contribution from this kind
of vertex is
J ddq 4v 1~b(O, k) = (27f)d 1 + 2vmII(0, q) (q + k)2 (3.28)
As mentioned before, the first factor in the integrand IS just the geometric
64
summation of the polarization bubbles. In the small q limit, it is
2
mII(O, q) (3.29)
The parameter v drops out from the integrand which makes sense for the expected
universal property of the critical exponent.
In d = 3, the integral in Eq. 3.28 is elementary: by extracting the -k2 10g k from
~b(O, k) - ~b(O, 0), the contribution to 7) is 3~~2. The contribution from the u vertex
is straightforward to get, the result is 3n~7r2. So the overall contribution to 7) so far is
16 8 8
7)b=--+--=-
3n-7!"2 3mr2 mr2
We now check the limit v = 0, which corresponds to s-wave case. From the above
analysis, it does give the correct 3n87r2 contribution to 7) as obtained by Ma [44].
The gauge field also contributes to the critical exponent 7), via the self energy
graph d). The double wavy line is the renormalized photon propagator as shown in
Fig. 3.7. We choose Coulomb gauge as before, so the bare photon propagator is Pi~~q) •
The self energy graph of the gauged field propagator is listed in Fig. 3.7. and reads
(3.30)
After some algebra, we get
(3.31)
The photon self energy has the same structure as the gauge propagator. So the
renormalized photon propagator is
65
Now we can calculate graph d) which reads
~d(O k) = _e2J ddp (2k - p)i(2k - p)j Pij(P)
, (21r)d (k - p)2 p2+ ~e22p
Again, we extract the -k2 log k term from above integral, and obtain '1d = ;~;~.
This is a contribution from the gauge field only and has an opposite sign to the
ungauged part. This contribution to the critical exponent '1 is the dominant part, so
the overall critical exponent '1 is
-104
'1 = 3n1r2 (3.32)
which has a negative value just as in the s-wave case though its absolute value is
somewhat smaller.
Now we calculate the critical exponent "f. By a technique similar to the one we
used for '1, here we need to extract 0log(t) from the self energy graph. In this
calculation, all the external momenta k are set to zero. We just show some typical
contribution from self energy graphs to show how it works.
From graph a), we get
This integral can be easily done and the mass dependence is proportional to 0,
so "f = 2 which is the result for the spherical model. [7]
Graph c) is an example of the Feynman rule mentioned right below Eq. (3.24).
Here, ~b (t, 0) is inserted into graph a), so the expression reads
(3.33)
-------------------------------
66
Summing all the graphs of this kind constructed from all possible vertices, we get
a contribution to I for anisotropic ungauged part is
36
1= 2(1--)
n7f2
The gauge field also contributes to I in a very similar way from graph e)
128Ie = ---
n7f2
So the overall critical exponent is
100
1= 2(1--)
n7f2
(3.34)
(3.35)
For the physical value of n = 6, this is negative. This suggests a 1st order phase
transition, and the smallest number of components to yield a positive I is 10.1.
Compared to the s-wave case, I = 2(1- ,::2), which require n > 7.7 with a physical
n = 2, we conclude it is a little more likely for the p-wave superconductors to have a
second order phase transition.
The critical exponent v = 1 - 3~;2 can be easily obtained from the relation
I = v(2 - rJ) which requires n > 11.89 for a continuous phase transition to be
realized.
3.4 Discussion and Conclusion
The critical behavior of the p-wave superconductors with m complex components
order parameter has been studied both in an (-expansion and a large n expansion
technique at dimension 3 to the first order of the control parameter. In the physical
parameter region, a fluctuation induced first order phase transition is found. However,
from one-loop (-expansion, a new kind of critical fixed point is found for parameter
n > 420.9. This corresponds to a continuous phase transition into the non-unitary
67
phase. The large n result is consistent with the RG result: to have a continuous phase
transition, a number of components n larger than some nc is required. However, the
critical lower bound of nc = 11.9 in the p-wave case is closer to the physical value
n = 6 than in the s-wave case, which suggests that p-wave critical behavior is more
likely to occur. This is the result of a leading expansion only, higher order expansions
are necessary to consolidate these results.
68
CHAPTER IV
SKYRMION IN P-WAVE SUPERCONDUCTORS
4.1 Introduction
The shorter version ofthis chapter has been published in Ref. [11] with John Toner
and Dietrich Belitz.
One of the most fascinating phenomena exhibited by conventional, s-wave, type-
II superconductors is the appearance of an Abrikosov flux lattice of vortices in the
presence of an external magnetic field H in a range Hel < IHI < He2 between a
lower critical field Hcl and an upper critical field He2 . [9] It has been known for quite
some time both theoretically[45, 46, 47, 48] and experimentally [49, 50] that these flux
lattices can melt. The melting curve separates an Abrikosov vortex lattice phase from
a vortex liquid phase, and the vortex lattice is found to melt in the vicinity of both
Hel and He2 , as shown in Fig. 4.1. The melting occurs because the elastic constants
of the flux lattice (Le., the shear, bulk, and tilt moduli) vanish exponentially near
these field values. As a result, in clean superconductors, root-mean-square positional
thermal fluctuations J(ju(x)12) grow exponentially as these fields are approached.
According to the Lindemann criterion, when these fluctuations become comparable
to the lattice constant a, the translational order of the flux lattice is destroyed; Le.,
the lattice melts.
Vortices are topological defects in the texture of the superconducting order
parameter, and in s-wave superconductors, where the order parameter is a complex
scalar, only one type of defect is possible. In p-wave superconductors, the more
complicated structure of the order parameter allows for an additional type of
69
H
,
, Normal State,
,
,
,
, H
c2,
,
Hmel~ \
\
\Vortex \
liquid \
\H \c1
\
T Tc
FIGURE 4.1: External field (H) vs. temperature (T) phase diagram for vortex flux
lattices. Shown are the Meissner phase, the vortex lattice phase, the vortex liquid,
and the normal state. Notice that the vortex lattice is never stable sufficiently close
to He!'
topological defect known as a skyrmion. In contrast to vortices, skyrmions do not
involve a singularity at the core of the defect; rather, the order parameter field is
smooth everywhere, as illustrated in Fig. 4.2.
Skyrmions were first introduced in a nuclear physics context by Skyrme,[51] and
slight variations of this concept.!
were later shown or proposed to be important in superfluid 3He,[52, 53] in the
Blue Phases of liquid crystals, [54] in Quantum Hall systems,[55, 56] in itinerant
ferromagnets, [57] and in p-wave superconductors. [3] In the latter case, skyrmions
carry a quantized magnetic flux, as do vortices, although the lowest energy skyrmion
contains two flux quanta, while the lowest energy vortex contains just one. For
strongly type-II superconductors, skyrmions have a lower free energy than vortices,
and a vortex lattice should thus be the state that occurs naturally.[3]
1Defects of this general type are known under various names in different contexts, and the action
for the defects studied here differs from the one considered by Skyrme. We follow a recent trend to
refer to all defects of this type as skyrmions.
(a) (b)
70
FIGURE 4.2: Order parameter configurations showing a vortex (a), and a skyrmion
(b). The local order parameters are represented by arrows on loci of equal distance
from the center of the defect. If the order parameter space is two-dimensional, only
vortices are possible, and there is a singularity at the center of each vortex, (a). If
the order-parameter space is three-dimensional, a skyrmion can form instead, where
the spin direction changes smoothly from "down" at the center to "up" at infinity,
(b).
Recent evidence of p-wave superconductivity in Sr2Ru04[39, 40F provides a
motivation for further exploring the properties of skyrmion flux lattices in such
systems. It was shown numerically by Knigavko et al.[3] that the interaction between
skyrmions falls off only as 1/R with distance R, as opposed to the exponentially
decaying interaction between vortices. As result, skyrmion lattices have a very
different dependence of the magnetic induction on the external magnetic field near Hcl
than do vortex lattices. In this chapter we confirm and expand on these results. We
show analytically that the skyrmion-skyrmion interaction, in addition to a leading
1/R-dependence, has a correction proportional to In R/R2 that explains a small
discrepancy between the numerical results in Ref. [3] and a strict 1/R fit, and we
calculate the interaction energy up to 0(1/R2). We further show that the melting
curve of a skyrmion lattice is qualitatively different from that of a vortex lattice.
Namely, skyrmion lattices melt nowhere in the vicinity of Hel , so there is a direct
transition from the Meissner phase to the skyrmion lattice, see Fig. 4.8 below. Finally,
we predict and discuss the magnetic induction distribution n(B) of a skyrmion lattice
state as observed in a muon spin resonance (f,lSR) experiment. For a vortex lattice,
2The precise nature of the order parameter in Sr2Ru04 is still being debated, see Ref. [58}.
71
the exponential decay of the magnetic induction B at large distances from a vortex
core implies nCB) ex: In B / B. For a skyrmion lattice, we find that B decays only
algebraically, which leads to nCB) ex: B-3/ 2• Some of these results have been reported
before in Ref. [11].
The chapter is organized as follows. In Sec. 4.2 we review the formulation in
Ref. [3] of the skyrmion problem. In particular, we start from the Ginzburg-Landau
(GL) model for p-wave superconductors and consider the free energy in a London
approximation. We parameterize the skyrmion solution of the saddle-point equations,
and express the energy in terms of the solution of the saddle-point equations. In
Sec. 4.3 we analytically solve these saddle-point equations perturbatively for large
skyrmion radius R, and we calculate the energy of a single skyrmion as a power series
in 1/R to order 1/R2• In Sec. 4.4 we determine the elastic properties of the skyrmion
lattice, and we predict the magnetic induction distribution nCB) as observed in a j.lSR
experiment.
4.2 Formulation of the Skyrmion Problem
In this section we review the formulation of the skyrmion problem presented in
Ref. [3], who derived an effective action that allows for skyrmions as saddle-point
solutions. The resulting ordinary differential equations (ODEs) describing skyrmions
[3] are the starting point for our analytic treatment.
4.2.1 The Action in The London Approximation
We start from a Landau-Ginzburg-Wilson (LGW) functional appropriate for
describing spin-triplet superconducting order,
s = Jdx £(1/J(x) ,A(x)), (4.1a)
72
with an action density
£ ('ljJ , A)
(4.1b)
Here 'ljJ (x) is a 3-component complex order parameter field,
A(x) is the electromagnetic vector potential, and D = V -iqA denotes the gauge
invariant gradient operator. m and q are the mass and the charge, respectively, of a
Cooper pair, and we use units such that n= c ~= 1. t, 'U, and v are the parameters of
the LGW theory.
Let us look for saddle-point solutions to this action. In a large part of parameter
space, namely, for v < 0 and 'U > -v, the stable saddle-point solution has the form
'ljJ(x) 'ljJ = fa (1, i, O)/V2, where the amplitude fa is determined by minimization
of the free energy. [42] This is known as the jJ-phase, and it is considered the most
likely case to be realized in any of the candidates for p-wave superconductivity.3
Fluctuations about this saddle point are conveniently parameterized by writing the
order parameter field as
'ljJ(x) = ~f(X) (n(x) + im(x)) , (4.2)
where it(x) and m(x) are unit real orthogonal vectors in order-parameter space and
f(x) is the modulus of order parameter. With this parameterization, the action
density can be written
t f2 + ('U + V)j4
+2~ [(V1)2 + f2[}(aJ)2 + (it· aim - qAi)2]]
1
+-(V X A)2,
87f
3See the discussion in Ref. [3], and references therein.
(4.3)
73
where i = it x m, summation over repeated indices is implied, and we have made use
of the identities listed in Appendix B.1.
There are two length scales associated with the action density, Eq. (4.3). The
coherence length ~ is determined by comparing the F term with the (V1)2 term,
~ = 1/J2mjt/. (4.4a)
It is the length scale over which the amplitude of the order parameter will typically
vary. The London penetration depth A is determined by comparing the A 2 term with
the (V x A)2 term,
(4.4b)
The ratio of these two length scales, K, A/~, is the Ginzburg-Landau parameter.
Now we write f(x) = fa + bf(x), with fa = J-t/2(u + v). Deep inside the
supercondueting phase, where -t > 0 is large, the amplitude fluctuations bf are
massive, and to study low-energy excitations one can integrate out f in a tree
approximation. This approximation becomes exact in the limit of large K, and is
known in this context as the London approximation. We introduce dimensionless
quantities by measuring distances in units of A and the action in units of <P6/321f3 A,
and we introduce a dimensionless vector potential a = 21fAA/<Po, with <Po = 21f/q the
magnetic flux quantum. Ignoring constant contributions to the action we can then
write the action density in London approximation as follows,[3]
(4.5)
with b = V x a. The above derivation makes it clear that this effective action is a
generalization of the 0(3) nonlinear sigma model (represented by the first term on
74
the right-hand side of Eq. (4.5)) that one obtains for a real 3-vector order parameter
by integrating out the amplitude fluctuations in tree approximation. [31]
It should be noticed the parametrization of p-wave order parameter Eq. (4.3) is
valid only in the parameter region discussed above. The stable saddle point solution
has the property 'l/J x 'l/J* # 0 which corresponds to the non-unitary phase. However,
there is another possibility which is referred as unitary phase and has been discussed
in Chapter III. Namely, it = m. In this case, the parameterization of order parameter
can be written as,
'l/J(x) = j(x)it(x) x ei<f;(x), (4.6)
This equation is nothing more than a straight forward analogy from a scalar
order parameter of s-wave to a vector one. It can be expected if some p-wave
superconductors happen to be described by Eq. (4.6), all the s-wave discussion can
be directly applied to such case. A discussion of this analogy can be found in the
reference [3], here we will focus on the non-unitary phase only.
4.2.2 Saddle-point Solutions of the Effective Action
We now are looking for saddle-point solutions to the effective field theory, Eq.
(4.5). Considering i and it independent variables, and minimizing with respect to i
subject to the constraints j2 = it2 = 1 and i· it = 0 yields
with
J= Vx b
(4.7a)
(4.7b)
the supercurrent. The variation with respect to a is straightforward and yields a
75
generalized London equation,
(4.7c)
It is convenient to take the curl of Eq. (4.7c) and use Eq. (B.3) to express the right-
hand side of the resulting equation in terms of i. We then obtain the saddle-point
equations as a set of coupled partial differential equations (PDEs) in terms of band
i only:
(4.8a)
(4.8b)
Notice that the right-hand side of Eq. (4.8a) is valid in this form only at points where
i(x) is differentiable, see Eq. (B.3). Field configurations that obey these PDEs have
an energy
(4.9)
where we have added a uniform external magnetic field h measured in units of
<T>o/21fA2 . Notice that the energy depends on it and m, whereas Eqs. (4.8) depend
only on i, and that different choices of it and m can lead to the same i. Therefore,
a field configuration satisfying Eqs. (4.8) is only necessary for making the energy
stationary, but not sufficient.
4.2.2.1 Meissner Solution
A very simple order parameter configuration consists of constant it(x) and m(x)
everywhere, see Fig. 4.3.
This leads to an i(x) i that is constant everywhere. Equation (4.8b) is then
/
1/
X- m~
-~
~---
~m "\
" 1
76
FIGURE 4.3: Configurations of the vectors £, m, and n in a Meissner phase. All
three vectors point in the same direction everywhere.
trivially satisfied. The right-hand side of Eq. (4.8a) vanishes, and hence the PDE for
b reduces to the usual London equation with a solution b(x) 0 in the bulk. This
solution describes a Meissner phase with energy EM = O.
4.2.2.2 Vortex Solution
Now consider a field configuration where n(x) and m(x) are confined to a plane
(say, the x-v plane), but rotate about an arbitrarily chosen point of origin:
n(x)
m(x)
(cos ¢, sin¢, 0),
(- sin ¢, cos ¢, 0) , (4.10)
where ¢ denotes the azimuthal angle in the x-v plane with respect to the x-axis.
This field configuration, known as a vortex and shown in Fig. 4.4, corresponds to a
constant i everywhere except at the origin, where there is a singularity. Therefore,
the right-hand side of Eq. (4.8a) is not applicable, and we return to Eq. (4.7c), which
77
FIGURE 4.4: Configurations of the vectors I, m, and ii for a vortex. I is constant,
whereas mand ii rotate about the vortex core. Notice that the vector shown in Fig.
4.2(a) is ii.
takes the form
For any closed path C in the x-y plane that surrounds the origin one has
:Ie d£· V¢(x) = 27f,
Lds· (\7 x V¢(x)) = 27f,
(4.11)
(4.12a)
(4.12b)
where A is the surface whose boundary is C. 4 This quantization condition shows that,
instead of Eq. (4.8a), we have
b(x) - V 2b(x) = 27fzb(x) b(y). (4.13)
4More generally, ¢ is an element of the circle or one-sphere 8 1, and hence fc de· V ¢(x) = 21fn
with n an integer. n is a topological invariant that characterizes the singularity (known as a vortex),
and the number of flux quanta that penetrate the vortex is N = Inl. The vortex with n = 1 has the
lowest energy within this family of solutions (apart from the trivial "non-vortex" with n = 0).
78
This is solved by a b that is equal to the boundary condition value everywhere along
the z-axis and that falls off exponentially away from the z-axis. This solution is
known as a vortex, and the amount of magnetic flux contained in one vortex is one
flux quantum <Po.
lt is the precise analog of, and, indeed, essentially identical to, the familiar vortex
in conventional s-wave superconductors.
The energy of a vortex given by Eq. (4.13), as calculated from Eq. (4.9), is
logarithmically infinite. This is due to the point-like nature of the vortex core where
the amplitude of the order parameter goes discontinuously to zero. In reality, the
amplitude cannot vary on length scales shorter then the coherence length ~, which
provides an ultraviolet cutoff. The energy is then proportional to in t\,. [9] In an
external magnetic field this energy cost is offset by the magnetic energy gain due to
letting some flux penetrate the sample. For t\, larger than a critical value t\,c== 1/V2,
and for external fields larger than the lower critical field Hel , a hexagonal lattice
of vortices has a lower energy than the Meissner phase. This state is known as
an Abrikosov flux lattice, and is precisely the same as that in conventional s-wave
superconductors. [9]
4.2.2.3 Skyrmion Solution
Due to the three-component nature of the order parameter, more complicated
solutions of the saddle-point equations can be constructed for which the vector i is
not fixed. Let B be the angle between i and the z-axis, and consider a cylindrically
symmetric field configuration parameterized as
i ezcosB(r)+ersinB(r),
it (ez sin B(r) - er cos B(r)) sin lp + e<p cos lp
m (e z sin B(r) - er cos B(r)) cos lp - e<p sin lp.
(4.14)
79
For this to minimize the energy, [ at large distances from the origin must be constant
because of the first term in the energy, Eq. (4.9), and for a skyrmion centered in a
cylinder of radius R we take [ to point in the +z-direction for r = R, B(r = R) = O.
The quantization condition analogous to Eq. (4.12b) for the vortex is
Jdx dy tij i· (aJ x a)) = 811" (4.15)
To be consistent with this, [ must point in the -z-direction at the origin, B(r = 0) = 11".
Equation (4.14) parameterizes the order parameter in terms of a function B(r ).
In addition, the energy depends on the vector potential which we take to be purely
azimuthal, in accordance with our cylindrically symmetric ansatz,
a(x) = a(r) e'P' (4.16)
With this parameterization, we obtain from Eq. (4.9) the energy per unit length,
along the cylinder axis, of a cylindrically symmetric skyrmion in a region of radius R,
E/Eo 11R drr [(B1(r))2 + :2 sin2B(r)]
+l R drr [~ (1 + cosB(r)) + a(r)r
+f drr [a~) + a'(rr (4.17)
where Eo = (<I)o/411"A)2. This expression was first obtained in Ref. [3]. The three
terms correspond to the three terms in the London action, Eq. (4.5). They represent
the energy of the nonlinear sigma model, the kinetic energy of the supercurrent, and
the magnetic energy, respectively. Minimization of E with respect to B(r) and a(r)
80
yields Euler-Lagrange equations
(4.18a)
1 1 1
a"(r) + - a'(r) - 2 a(r) = a(r) + - [1 + cose(r)].
r r r
(4.18b)
This set of coupled, nonlinear ODEs must be solved subject to the boundary
conditions e(r = 0) = 1r and e(r = R) = 0, as explained above. The solution is
known as a skyrmion, and each skyrmion contains two flux quanta. 5 Since Eqs. (4.8)
are necessary for making the energy stationary, the solution of Eqs. (4.18), inserted
in Eqs. (4.14,4.16), is guaranteed to be a solution of Eqs. (4.8) as well.
The energy of a single skyrmion is finite even in London approximation, see Sec.
4.3 below. For large values of the Ginzburg-Landau parameter f'\, a skyrmion therefore
has a lower energy than a vortex, and the value of the lower critical field Hel , at which
the Meissner phase becomes unstable, is correspondingly lower for skyrmions than for
vortices. This is the basis for the expectation that, in strongly type-II (i.e., large-f'\,)
p-wave superconductors, a skyrmion flux lattice will be realized rather than a vortex
flux lattice.
4.3 Analytic Solution of the Single-skyrmion Problem
We now need to solve the coupled ODEs (4.18). Due to their nonlinear nature,
this is a difficult task, and in Ref. [3] it was done numerically. It turns out, however,
that one can construct a perturbative analytical solution in the limit of large skyrmion
radius, R» .\ with AIR as a small parameter. This provides information about the
5More generally, Jdxdy Cij i· (ad x ojl) = 81rQ, with Q an integer. Q is a topological invariant
that characterizes the defect (known as a skyrmion), and the number of flux quanta that penetrate
the skyrmion is N = 2Q. [3] The skyrmion with Q = 1 ha..s the lowest energy within this family of
solutions.
81
superconducting state near Hcl , where the system is always in that limit. We will
construct the perturbative solution, and calculate the energy, to second order in the
small parameter. Our general strategy is as follows. We use Eq. (4.18b) to iteratively
express a in terms of 8 and its derivatives. Substitution in Eq. (4.18a) then yields a
closed ODE for 8(r) that has to be solved.
4.3.1 Zeroth Order Solution
Let us first consider R = 00. For r -------+ 00, the left-hand side of Eq. (4.18b) falls
off as 1/r2 , and hence the vector potential, to zeroth order for large r, is given by
1
aoo(r) = -- [1 + cos 8(r)] .
r
(4.19)
Note that we use the exact 8(r) in this expression, not the zeroth order approximation
to it. Since we can only compute 8(r) perturbatively, this expression for the
zeroth order vector potential will itself have to be expanded perturbatively later.
Substitution in Eq. (4.18a) yields
r2 8"(r) + r8'(r) = ~ sin(28(r)).
The solution obeying the appropriate boundary condition is[3]
800 (r) = f (r / f) ,
with
f(x) = 2arctan(1/x).
(4.20)
(4.21a)
(4.21b)
The length scale f is arbitrary at this point and will be determined later from the
82
requirement 8(r = R < 00) = O. For R » 1 it will turn out that f 0:: .fR, The
skyrmion solution is schematically shown in Fig. 4.5.
I n
FIGURE 4.5: Configurations of the vectors f, m, and n for a skyrmion. Notice that
the vector shown in Fig. 4.2(b) is i.
4.3.2 Perturbation Theory for R» 1
We now determine the corrections to the zeroth order solution. Let us write
8(r) = 8oo (r) + o8(r) and a(r) = aoo(r) + oa(r) and require loa(r)1 « laoo(r)I and
lo8(r)\ « 1.6 An inspection of the ODEs (4.18) shows that for, the corrections can
be expanded in a series in powers of 11f,
o8(r)
oa(r)
1 1
f2 9(r1f) + f4 h(r1f) + 0 (11f6) ,
1 1
f3 a(rlf) + f5 f3(rlf) + 0(1/f7 ).
(4.22a)
(4.22b)
The functions a and f3 can be determined by substituting Eq. (4.22b) in Eq. (4.18b)
and equating coefficients of powers of 1/.e. The resulting equations for a and f3 are
6We emphasize that we do not require 10B(r) I « IBoo (r) I, as this requirement is neither necessary
nor desirable. Rather, we expand, for instance, sin(Boo + oB) = sin Boo cos oB + cos Boo sinoB =
cos Boo oB + O(oB2 ), which is valid for alloB« 1, not just for those that satisfy 10B(r)l« IBoo(r)l.
83
linear algebraic equations, not ODE's, because terms involving derivatives of a and (3
only enter at higher order in lie, as one can verify by direct calculation. Hence, the
solutions for a and (3 can be read off at once, and are:
where
a(x)
(3(x)
16x
(1 +X2)3'
2 (3x4 - 6.1:2 - 1) (x) _ 2 (3x 2 - 1) ,(x)
x 2 (1 + X 2 )3 9 x(l + X 2 )2 9
2
+ 2 gff (x) + A (x) ,l+x
(4.23a)
(4.23b)
A(x) 1 1aff(x) +- a'(x) - 2 a(x)
x x
384x(x2 - 1)
(1+x2)5 . (4.23c)
Similarly, by comparing coefficients in Eq. (4.18a) we find ODEs for the functions 9
and h,
gff(X) + ~ g'(x) - ~ cos(2j(x)) g(x) -~ sin(f(x)) a(x), (4.24a)
x x x
hff(x) + ~ h'(x) - ~ cos(2j(x)) h(x) -~ sin(f(x)) (3(x)
x x x
1 2
-2 sin(2j(x)) l(x) - - cos(f(x)) a(x) g(x), (4.24b)
x x
with j(x) from Eq. (4.24b).
The ODE (4.24a) for 9 can be solved by standard methods, see Appendix B.2.
The physical solution is the one that vanishes for x ----+ 0; it is proportional to x for
84
x » 1. We find
(x) = _i x[x2 (4 + X 2 ) + 2(1 + X 2 ) In(1 + x 2 )]9 3 (1 + x 2 )2 ,
(4.25a)
the large-x asymptotic behavior of which is
4 16 lnx 8 lnx
g(x» 1) = --x- --- - - +0(-).3 3 x 3x x 2 (4.25b)
This determines both the function {3(x) , Eq. (4.23b), and the inhomogeneity of the
ODE (4.24b) for h(x). The latter can again be solved in terms oftabulated functions,
see Appendix B.2, but we will need only the two leading terms for x -----+ 00. The
physical solution is again the one that vanishes for x -----+ 0, and its large-x asymptotic
behavior is
32 536h(x» 1) = -- x lnx + - x + O(1/x).9 135 (4.26)
Finally, we need to fix the length scale.e. It is determined by the requirement
O(r = R) = O. We find
(4.27a)
where
(4.27b)
85
and
c
d
4/3,
536/135,
(4.27c)
(4.27d)
are the absolute values of the coefficients of the terms proportional to x in the
large-x expansions of g(x) and h(x), respectively. We see that, for R » 1, e is
indeed proportional to VR , as we had anticipated above. That is, the characteristic
skyrmion length scale e is the geometric mean of the London penetration depth .\
(recall that we measure all lengths in units of .\) and the skyrmion size R. We now
can also check our requirement 15B « 1: from Eq. (4. 22a) we see that for r « e,
I5B(r) ex 1/R, while for r » e, 15B(r) is bounded by a term proportional to 1/R1/ 2 .
For R large compared to the penetration depth the condition is thus fulfilled for all
r. Similarly, l5a is found to be small compared to aoo for all r.
4.3.3 Energy of a Single Skyrmion
By using our perturbative solution in Eq. (4.17), we are now in a position to
calculate the energy of a single skyrmion to 0(1/R2 ). It is convenient to first expand
the energy in powers of 1/e2 , and then determine the R-dependence by using Eqs.
(4.27) .
Let us first consider the supercurrent energy E e , i.e., the second term in Eq. (4.17).
It can be written
(4.28)
86
Using Eqs. (4.23a) we find
I 32 1 ( I 6Ec Eo = 5 £4 + 0 1 £ ). (4.29)
Now consider the magnetic energy Em, which is the third term in Eq. (4.17). It
can be written
with
b(r) = ~ aoo(r) + a~(r) + ~ rSa(r) + rSa'(r) ,
r r
(4.30)
(4.31a)
the magnetic induction in our reduced units. Notice that in calculating aoo(r), e(r)
in Eq. (4.19) needs to be expanded to first order in rS e, as noted earlier. The two
leading contributions to b2 are then
where x = r I£. Performing the integral yields
8 1 112 1 6
E lEo = - - - - - + 0(1/£ ).
m 3 £2 135 £4
(4.31b)
(4.32)
Finally, we need to calculate the energy E s coming from the gradient terms in the
first term in Eq. (4.17). The expansion of the two terms in the integrand yields seven
integrals that contribute to the desired order, they are listed in Appendix B.3. The
87
result is
(4.33)
Adding the three contributions, and using Eqs. (4.27), we find our final result for
the energy of a skyrmion of radius R » 1,
E/Eo
(4.34)
Knigavko et al. [3] solved the Eqs. (4.18) numerically, and thereby numerically
determined the energy, which they fit to a 1/R-dependence. Their results are shown
in Fig. 4.6 together with the analytical result given in Eq. (4.34). The perturbative
solution up to O(ln R/R 2) was first given in Ref. [11]. We have also solved the
equations numerically, using spectral methods to convert the boundary value problem
to a set of algebraic equations for the unknown coefficients in an expansion in
Chebyshev polynomials[59]. For the R-range shown, and on the scale of the figure,
the result is indistinguishable from the perturbative one.
4.3.4 Spectral Methods
In this section, we will briefly review some of the most important concepts in the
spectral method and apply the method to solve Eqs. (4.18).
Spectral methods have been utilized to solve both the differential and integral
88
50 100 150 200 250 300
RIA.
FIGURE 4.6: Numerical data for the energy per skyrmion per unit length (circles)
together with the best fit to a pure 1/R behavior (dashed line) from Ref. [3], and the
perturbative analytic solution given by Eq. (4.34) (solid line). A numerical solution
using spectral methods is indistinguishable from the perturbative one.
equations. The basic idea is to assume that the unknown variables in the equations
interested can be expanded by N+1 basis functions rPn (x).
N
u(x) ~ UN(X) L anrPn(x)
n=O
(4.35)
A candidate for the basis functions could be the 8m functions, the Hermit
polynomials, or the Chebyshev functions, etc. In our case, since we know the
perturbation solutions are smooth rational functions without any periodic nodes,
we will choose the Chebyshev series. The Chebyshev functions are defined as,
Tn(cos B) - cos(n B) (4.36)
So Tn(x) is a polynomial functions in x. It can be seen easily that the Chebyshev
polynomial are defined in the range [-1, 1]. This means that in order to use the
89
Chebyshev basis, the boundary value problems have to be transformed to this range
by using a linear variable transformation.
The Chebyshev basis has many nice properties, for instance, it has a larger
convergence basin than the Taylor power series with faster convergence. These
properties can guarantee that for not very large N, the approximated sum of series
in Eq. (4.35) will acquire a high enough accuracy.
In general, a differential equation can be written as
Lu(x) = f(x) (4.37)
Here, L is a differential operator which has a linear or nonlinear property. If the
variable u(x) is replaced by the approximation UN(X) with fixed order N, in general,
Eq. (4.37) will not be satisfied and let it be denoted as a residual function R(x),
R(x) - LUN(X) - f(x) (4.38)
We want to minimize R(x) by determining N+1 unknown coefficients an. So a
differential equation is transformed to a set of algebraic equations. A N+1 conditions
have to be artificially input to let these algebraic equation for an have a closed form.
One way is to choose at most N+1 collocation (interpolation) points in the range
[-1,1]. It should be noted that boundary conditions will also reduce the number of
chosen collocation points.
In our case, the appropriate boundary condition is a(O) = 0, a(R) = llrrR,
e(O) = 7r, and e(R) = O. These boundary conditions naturally reduce N + 1 to
N - 1 interpolation points. The simplest choice of interpolation points is an evenly
distributed N - 1 points between [-1,1]. Though this choice leads to a qualitatively
accurate result, it is not the best choice. This is because that the true solution
theta(r) as we have already known is sharp at the left bound and decays extremely
slowly at the right bound. So it is natural to think a better choice could be larger
90
point density at the left bound and diluter point density at the right end. It turns
out a Chebyshev grid choice is a better choice,
2i - 1
Xi - cos[2(N + 1) 7l'], i = 1,2, .... , N - 1 (4.39)
After the collocation points are chosen, we are facing to solve 2N +2 coupled non-
linear algebraic equations (NLAEs). An reasonable guess of the roots of these NLAEs
is taken to be the coefficients of the Chebyshev transformation of the zeroth order
solutions, Eq.(4.21b) and Eq.(4.19). Then sophisticated computational packages, for
instance, command FindRoot in Mathematica, can be applied to find the roots of
those NLAEs. It turns out the computed Chebyshev series solution fit extremely well
to our analytic solution for a reasonably large value of R. For small R, our analytic
solution breaks down and this numerical solution will provide a complementary
method to give the correct solution.
4.4 Observable Consequences of the Skyrmion Energy
Our calculation of the skyrmion energy in Sec. 4.3 has been for a cylindrically
symmetric skyrmion. The result shows that each skyrmion will try to maximize its
radius in order to minimize the energy, which leads to a repulsive interaction between
skyrmions whose potential is proportional to 1/R. Skyrmions are thus expected
to form a lattice structure, as do vortices, and they will thus not be cylindrically
symmetric, since the lattice is not. One expects a hexagonal lattice, as in the case
of the vortex lattice, and our treatment involves the same approximation as in the
numerical work of Ref. [3]; namely, approximating the hexagonal unit cell by a circle
of the same area. We expect this approximation to recover the correct scaling of the
energy, and to reproduce the coefficients of that scaling to the same accuracy as radius
of the circle of the same area reproduces the distance from the center of a hexagon
to the nearest point on its edge; i.e., V2V3/7l' - 1 ~ 0.05. We will now proceed to
91
calculate observable consequences of the dependence of the energy on the radius of
the unit cell. These include the relation B(H) between the magnetic induction Band
the external magnetic field H, the elastic properties of the skyrmion lattice and the
resulting phase diagram in the H-T-plane, and the ,uSR signature of the skyrmion
lattice.
4.4.1 B(H) for a Skyrmion Lattice
We start by calculating the dependence of the equilibrium lattice constant Ron
an external magnetic field H. This is done by minimizing the energy per unit volume,
which is the energy per unit length per skyrmion, Eq. (4.34), divided by the area per
skyrmion, 1fR2 , plus a reduction in the energy of - 2CPoH / 41f due to the external field.
The latter is obtained from the last term in Eq. (4.9) by noting that the magnetic
flux Jdxdy (2 . b) = 2CPo for each skyrmion in the lattice. This negative external
field contribution must also be divided by 1fR2 to give the energy per unit volume.
Returning to ordinary units, we thus find a Gibbs free energy per unit volume
( ) =~ [_~ 4V6A ° (A2 In(R/A))]9 R 41f2 R2 + 3R3 + R4 '
(4.40)
where K = CP6/21f A2 , and
(4.41)
with Bel K /ZCPo. For H < Hel , we have .6. > 0, and the free energy is minimized
by R = 00; i.e., the skyrmion density is zero. This is the Meissner phase. For H > Hel
the free energy is minimized by
R = Ro = ZV6A/.6., (4.42)
92
and there is a nonzero skyrmion density. We see that H e1 is indeed the lower critical
field. Note that the equilibrium flux lattice constant Ro diverges as 1/tl, whereas
in the case of a vortex lattice it diverges only logarithmically as In(1/tl).[9] For the
averaged magnetic induction B = 2<I>o/nR6 this implies
(4.43)
For H -----j. He1 from above, B(H) in the case of a skyrmion lattice thus vanishes
with zero slope, whereas in the case of a vortex lattice it vanishes with an infinite
slope. [9] This result, with a slightly different prefactor, was first obtained from the
aforementioned numerical determination of E(R) in Ref. [3]. Note that the only
material parameter that appears in this expression for B is H e1 .
4.4.2 Elastic Properties of the Skyrmion Lattice
Now we turn to the elastic properties of skyrmion lattice. Let the equilibrium
position of the i th skyrmion line be described by a two-dimensional lattice vector
~ = (Xi, Yi), and the actual position by
(4.44)
where u = (ux , Uy) is the two-dimensional displacement vector, and we use z as the
parameter of the skyrmion line. The strain tensor ua {3 is defined as
(4.45)
For a hexagonal lattice of lines parallel to the z-axis, the elastic Hamiltonian is[60]
~Jdx [2p (ua {3(x)ua {3(x)) + AL (uaa (x))2
+Ktiltlozu(x) 1 2] . (4.46)
93
Here summation over repeated indices is implied. J-l, AL' and K ti1t are the shear, bulk,
and tilt moduli, respectively, of the lattice, and we now need to determined these
elastic constants.
The combination J-l + AL can be obtained by considering the energy change of the
system upon a dilation of the lattice. Let R change from Ro to Ro(1 + t), with a
dilation factor t « 1. Such a dilation corresponds to a displacement field u(x) = t X..l,
where X..l is the projection of x perpendicular to the z-axis. [60] The strain tensor is
thus ua f3 = t ba f3. Inserting this in the elastic Hamiltonian, Eq. (4.46), yields the
energy per unit volume for the dilation,
(4.47a)
This should be compared with the energy as given by Eq. (4.40),
(4.47b)
Comparing Eqs. (4.47a) and (4.47b) yields
(4.48)
To obtain J-l (or Ad separately, we should consider shear deformations, which
change the shape, but not the area, of the unit cell. Since we have already
approximated the hexagonal unit cell by a circle, this is difficult to do, and we
resort to the following heuristic method, which will give the correct scaling of J-l
with ~ (but not the correct prefactors). To this end we observe that our result for
the Gibbs free energy, Eq. (4.40), is of the same form we would have obtained if
the skyrmions interacted via a pair potential U(r) that for distances r ;S Ro is of
order KA/r, and for larger distances falls off sufficiently rapidly that only nearest-
94
neighbor interactions need to be considered. Treating the skyrmion lattice as if such
an "equivalent potential" were the origin of the skyrmion energy allows us to calculate
the shear modulus as follows:
If the lattice is subjected to a uniform x - y shear - i.e., a displacement field
u(x) = 2fyX - for which uxy = uyx = f, and all other components of ua {3 = 0, the
elastic energy, Eq. (4.46) predicts an elastic energy per unit volume of
(4.49)
Such a shear skews each fundamental triangle of the skyrmion lattice by displacing
the top (or bottom, for the downward-pointing triangles) to the right (or the left, for
downward-pointing triangles) by an amount of order fRo, where Ro is the skyrmion
lattice spacing found earlier, Eq. (4.42) (see Fig. 4.7). This shortens the length of
FIGURE 4.7: Shearing of the skyrmion lattice results in a change in the distance
between skyrmion centers, and hence in their effective interaction. See the text for
additional information.
one bond of the triangle by an amount of order fRo, and increases the opposite
bond's length by the same amount. Hence, the linear in f change in the "equivalent
potentials" of these two bonds cancels, and the total change (6.E jtriangle) in the
energy per unit length of fundamental triangle, per triangle, is given by:
(4.50)
95
where the 0(1) factor includes both geometrical factors (e.g., sines and cosines),
and counting factors (e.g., to avoid multiple counting of each triangle). If we take
U(r) = K Air as suggested above, we have
U" (Ro) = ~; x 0(1).
Inserting this into Eq. (4.50) gives
.!:::..E = K A(2 X 0 (1).
tnangle Ro
(4.51)
(4.52)
This is the change in energy per unit cell. To get the energy per unit volume, we
must divide by the unit cell area, which is nR6. Doing so gives
Comparing this with Eq. (4.49) then determines f-l:
KA
f-l = R3 x 0(1).
o
Using Eq. (4.42) for Ro then leads to our final result for f-l:
K!:::..3
f-l =~ x 0(1).
(4.53)
(4.54)
(4.55a)
From Eq. (4.48) we see that the bulk modulus or Lame coefficient is given given by
the same expression,
(4.55b)
We now turn to the tilt modulus K tilt . This can be obtained by considering
a uniform tilt of the axes of the skyrmions away from the z-axis, i.e., away from
96
the direction of the external magnetic field H, by an angle {) «1. For small {),
{) = 18u/8zl. Therefore, the tilt energy in Eq. (4.46) is identical with the change
of the B . H term in Eq. (4.9). This contribution to the energy is, per unit length
and in ordinary units, given by -1>0 H cos e/21f, and its change due to tilting is
1>0 H(l- cos {))/21f R:j 1>0 H {)2 /41f = 1>0 H 18z uI 2/41f. Dividing this result by the unit
cell area 1fR6, using Eq. (4.42) for Ro, and identifying the result with the tilt term in
the elastic Hamiltonian, Eq. (4.46), yields J{tilt in the vicinity of Hel ,
1 2 2
J{tilt = - H el ,6. .121f (4.56)
We now are in a position to calculate the mean-square positional fluctuations
(lu(x)1 2). Taking the Fourier transform of Eq. (4.46), and using the equipartition
theorem, yields
2 kBT '"' 1(lu(x)1 h = 11 L...J 2 +J{ 2
qEBZ f-L ql. tilt qz
for the transverse fluctuations, and
(4.57a)
(4.57b)
for the longitudinal ones. Here ql. and qz are the projections of the wave vector
q orthogonal to and along the z-direction, respectively. The Brillouin zone BZ of
the skyrmion lattice is a hexagon (which we have approximated by a circle) of edge
length 0(1)/Ro in the plane perpendicular to the z-axis, and extends infinitely in the
z-direction.
Since f-L and AL are the same apart from a prefactor of 0(1) which we have not
determined, see Eqs. (4.55), the same is true for the transverse and longitudinal
contributions to the fluctuations, and it suffices to consider the former. Performing
97
the integral over qz yields
(4.58)
The remammg integral over the perpendicular part of the Brillouin zone is
proportional to liRa, and using Eqs. (4.55) and (4.42) we obtain
(4.59)
Using Eq. (4.42) again we see that, near Hel , (lu(x)1 2) ex R~/2« R6. That is, in this
regime the positional fluctuations are small compared to the lattice constant, which
tells us that the lattice will be stable against melting. To elaborate on this, let us
consider the Lindemann criterion for melting, which states that the lattice will melt
when the ratio f L = (lu(x)12)IR6 exceeds a critical value f e = 0(1). In our case,
(4.60)
As H ----+ Hel , ,6. ----+ 0, and the Lindemann ratio vanishes. Hence, the skyrmion lattice
does not melt at any temperature for H close to Hel .
We finally determine the shape of the melting curve Hm(T) near the supercondueting
transition temperature Te . Since, in mean field theory, Hel ex (Te - T), and
A ex II JTe - T,[9] we find from Eq. (4.60) by putting f L = const. = 0(1),
The resulting phase diagram is shown schematically in Fig. 4.8. Comparing
with Fig. 4.1 we see the qualitative difference between the vortex and skyrmion flux
98
lattices: whereas the vortex lattice always melts near H el , the skyrmion lattice melts
nowhere near Hel . This is a direct consequence of the long-ranged interaction between
skyrmions, as opposed to the screened Coulomb interaction between vortices.
Skyrmion
Lattice
H
"-
"
"\. Normal State
\.
\
\
\ H 2\ c
\
\
\
\
\
H \me~ \
\
\
\
T T
G
FIGURE 4.8: External field (H) vs. temperature (T) phase diagram for skyrmion
flux lattices. In contrast to the vortex case, see Fig. 4.1, there is a direct transition
from the skyrmion flux lattke to the Meissner phase. The theory predicts the shape of
the melting curve only close to Te , see Eq. (4.61); the rest of the curve is an educated
guess.
4.4.3 ,uSR Signature of a Skyrmion Flux Lattice
Muon spin rotation (,uSR) is a powerful tool which has been extensively applied
to study the vortex state in type-II superconductors. [61, 62] A crucial quantity in
this type of experiment is the ,uSR line shape n(B), which is the probability density
that a muon experiences a local magnetic induction B and precesses at the Larmor
frequency that corresponds to B. It is defined as
n(B) =(5(B(x) - B)), (4.62)
99
where B(x) is the magnitude of the local magnetic induction, and (... ) denotes the
spatial average over a flux lattice unit cell.
To predict the {lSR line shape for a skyrmion flux lattice near Hel it is sufficient,
for large Ro, to use only the lowest solution for the magnetic induction obtained in
Sec. 4.3.1. Inserting Eqs. (4.21) into Eq. (4.19), we find for the magnetic induction
in reduced units
(4.63)
Restoring physical units then gives
(4.64)
where we've dropped the minus sign since only the magnitude of B can be detected
in {lSR measurements.
From Eq. (4.62) we then find, for H near Hell where our theory is valid,
3
() 1 (Hc1~)2 1 ( .)n B = 24V2 ~ H
c1
skyrmlOns. (4.65)
Of course, n(B) is only non-zero for those values of B that actually occur inside
the unit cell ofthe skyrmion lattice. From Eq. (4.64), we see that the maximum value
of B will occur at the center of the unit cell (r = 0), which gives
I I Hc1A.2 Hc1~IB max = B(r = 0)1 =~ = -8-' (4.66a)
The minimum value of B occurs at the edge of the unit cell (i.e., r = R), where Eq.
(4.64) gives
(4.66b)
100
In the second equalities in Eqs. (4.66) we have used Eqs. (4.27) and (4.42) to express
f in terms of R and R in terms of ~, respectively.
To summarize: the prediction of our cylindrical approximation for n(B) is that the
simple power law Eq. (4.65) holds for B min < B < Bmax . For B < B min or B > Bmax ,
n(B) = O.
Since the above results were derived in the cylindrical approximation, we expect
the numerical coefficients in Eqs. (4.66) to be off by the approximately 5% mentioned
in the opening paragraph of Sec. 4.4 throughout most of the range Bmin < B < B max .
When B gets close to B min , however, we expect more radical departures from the
cylindrical approximation. This is because contours of constant B near the edge of
the hexagonal unit cell will, for B within 5% or so of B min or so, start intersecting
the unit cell boundary, leading to van Hove-like singularities in n(B). Such subtleties
cannot be captured within the cylindrical approximation. Note, however, that they
only occur over a very small range of B; for the remainder of the large window
B min < B < B max (which spans three decades even for ~ as big as 0.2), Eq. (4.65)
holds, up to the aforementioned 5% numerical error in its overall coefficient.
To compare this result with the corresponding one for a vortex flux lattice, we
recall that in that case B (r) is given by a modified Bessel function which for distances
r » A takes the form
1B(r) ex -- e- r / A•
yITJ5
For small B, we then find from Eq. (4.62)
(4.67)
() In(l/B)n B ex B (vortices) . (4.68)
We see that the p,SR line shape is qualitatively different in the two cases, due to
the long-range nature of B(r) in the skyrmion case versus the exponential decay in
the vortex case.
101
4.5 Conclusion
In summary, we have considered properties of a flux lattice formed by the
topological excitations commonly referred to as skyrmions, rather than by ordinary
vortices. For strongly type-II materials in the ,B-phase, skyrmions are more stable than
vortices. [3] We have presented an analytical calculation of the energy of a cylindrically
symmetric skyrmion of radius R up to 0(1/R2 ) in an expansion in powers of 1/R. This
provides excellent agreement with numerical solutions of the skyrmion equations. The
interaction between skyrmions is long-ranged, falling off only as the inverse distance,
in contrast to the exponentially decaying interaction between vortices. As a result,
the elastic properties of a skyrmion flux lattice are very different from those of a vortex
flux lattice, which leads to qualitatively different melting curves for the two systems.
The phase diagram thus provides a smoking gun for the presence of skyrmions. In
addition, the ,uSR line width for skyrmions is qualitatively different from the vortex
case.
We finally mention two limitations of our discussion. First, we have restricted
ourselves to a discussion of a particular p-wave ground state, namely, the non-
unitary state sometimes referred to as the ,B-phase. This state breaks time-reversal
symmetry and the recently reported absence of experimental evidence for the latter
in Sr2Ru04[58] suggests to also consider other possible p-wave states and their
topological excitations, in analogy to the rich phenomenology in Helium 3. [53] Second,
in a real crystalline material, crystal-field effects will invalidate our isotropic model
at very long distances, and cause the skyrmion interaction to fall off exponentially.
This is the same effect that makes, for instance, the isotropic Heisenberg model
of ferromagnetism inapplicable at very long distances and gives the ferromagnetic
magnons a small mass. It should be emphasized that this is usually an extremely
weak effect that is also material dependent. Once p-wave superconductivity has been
firmly established in a particular material, this point needs to be revisited in order
to determine the energy scales on which the above analysis is valid.
102
APPENDIX A
INTEGRAL FORMULAS USED IN CHAPTER III
Feynman integral tricks are useful to transform the anisotropic denominator into
isotropic one.
1 f(a + b) t xa - 1(1 - X)b-l
AaBb = f(a)f(b) Jo [Ax + B(1 - x)]a+b
Dimensional regularization integral formula which is extensively used in this paper.
J ddp 1 _ 1 r(n - d/2) _1 n-d/2(27f)d(p2+~2)n- (47f)d/2 r(n) (~2)
J ddp p2 _ 1 d r(n - d/2 - 1) (_1 )n-d/2-1(27f)d (p2 + ~2)n - (47f)d/22 r(n) ~2
Let R = a + bx + cx2 .
Jdx -1 . 2cx + brn = r-;. arcsm( F/5.. )yR y-C -~
where ~ = 4ac - b2 . The above integral is true if c < 0 and ~ < 0 .
(A.l)
103
APPENDIX B
MISCELLANEOUS TECHNIQUES IN CHAPTER IV
B.l Properties of Orthogonal Unit Vectors
Let it and m be orthogonal real unit vectors, and i = it x m. Then the
normalization condition ni'Pti = mi'fhi = 1 and the orthogonality condition nimi = 0
imply
(B.la)
(B.lb)
With these relations it is straightforward to show that
(B.2)
Finally, in regions where i(x) is differentiable the Mermin-Ho relation[63] holds,
(B.3)
B.2 Solutions of the ODEs for 9 and h
The functions 9 and h in Sec. 4.3.2 both satisfy an ODE of the form (see Eqs.
104
(4.24))
II() 1 I() (x4 - 6x2 + 1) () ()F x + - F x - 2( 2)2 F X = q X ,X X 1 + X (B.4)
with an inhomogeneity q given by the right-hand side of Eq. (4.24a) or (4.24b),
respectively. It is easy to check that the corresponding homogeneous equation,
obtained from Eq. (B.4) by putting q(x) 0, is solved by
(B.5)
(This is the solution that vanishes as X ~ O. The second solution diverges in this
limit.) Now write F(x) = Fh(x) G(x), and let y(x) = QI(x). Then y is found to obey
the elementary first-order ODE
with
The solution is
yl(X) + p(x)y(x) = q(X)/Fh(X), (B.6a)
(B.6b)
(B. 7)y(x) = e-!dxp [C1 +Jdxqe!dXP] ,
with C1 an integration constant. A second integration yields G(x), and hence F(x) in
terms of two integration constants. The latter can be determined by requiring that for
small X the solution coincides with the asymptotic solution that vanishes as X ~ O. By
using a power-law ansatz for 9 and h in Eqs. (4.24) we find g(x ~ 0) = -8x3 +O(x4 ,
and h(x ~ 0) = 256x3 + O(x4 ), which suffices to fix the integration constants. For
105
g(x) we find the expression given in Eq. (4.25a). For h(x) we obtain
h(x) = (270X(11+ X2)4) (592 + 2x2(8( -1,119 + 90x2+ 286x4 + 240x6 + 30x8) +
10,320(1 + X2)3) + 2, 296(-1 + x2)(1 + X2)4 + 4x2(1 + X2)3 + 1, 704 In x
+321n(1 + x2) ( -30 + 142x2+ 276x4 + 171x6 + 52x8 - 15x10 - 15(3 + x2)
(x + x3)21n (1 + x2)) - 1, 920x2(1 + x2)3Li2(_X2)) , (B.8)
with Li the polylogarithm function. The asymptotic behavior for large x is given by
Eq. (4.26).
B.3 Contributions to Es
By expanding the integrand of the first term in Eq. (4.17), we can express the
energy Es to 0(1/R 2 ) in terms of seven integrals,
7
Es/Eo = LIi + 0(1/£6), (B.9)
i=1
with
l R /1! x (B.10a)II 4 0 dx (1 + x2)2 ,
12 2 l R /1! 1 ((x2 - 1) ')£2 0 dx 1 + x2 x2 + 1 9 (x) - x9 (x) ,
(B.10b)
13 1 l R /1! [ I 2 (x4 - 6x2+ 1) 2 ]2£4 0 dx x (g (x)) + x(l + X2)2 9 (x) ,
(B.lOc)
2 rR/i! 1 ((X2 -1) ')
14 £4 Jo dx 1 + x2 x2 + 1 h(x) - xh (x) ,
1 fR/i!
£6 J0 dx (Xg' (X) hi (X)
(x4 -6x2 +1) )
+ x(l + X2)2 g(X)h(x) ,
4 rR/i! (x2 - 1) 3
- 3£6 Jo dx (1 + X2)2 9 (x),
1 rR/i! (x2 - I? 4
- 6£8 Jo dx x(l + X2)2 9 (X).
Evaluating the integrals to 0(1/£4) yields Eq. (4.33).
106
(B.10d)
(B.10e)
(B.10f)
(B.10g)
107
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