Physics Theses and Dissertations

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https://hdl.handle.net/1794/2048

This collection contains some of the theses and dissertations produced by students in the University of Oregon Physics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.

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Browsing Physics Theses and Dissertations by Author "Belitz, Dietrich"

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    Effective Soft-Mode Theory of Strongly Interacting Fermions in Dirac Semimetals
    (University of Oregon, 2019-01-11) de Coster, George; Belitz, Dietrich
    We present an effective field theory for interacting electrons in clean semimetals (both three dimensional Dirac semimetals and graphene) in terms of their soft or massless bosonic degrees of freedom. We show, by means of a Ward identity, that the intrinsic semimetal ground state breaks the Sp(4M) symmetry of the theory. In Fermi liquids this enables one to identify the massive, non-Goldstone modes of the theory and integrate them out. Due to the vanishing density of states in semimetals, unlike in Fermi liquids, both Goldstone and non-Goldstone modes are equally soft, and so all two-particle correlations need to be kept. The resulting theory is not perturbative with respect to the electron-electron interaction; rather, it is controlled by means of a systematic loop expansion and allows one to determine the exact asymptotic form of observables in the limits of small frequencies and/or wave vectors. Equivalently, it provides a mechanism of determining the long time-tail and long wavelength behavior of observables and excitations. As a representative application, we use the theory to compute the zero-bias anomaly for the density of states for both short and long-range interactions in two and three dimensions. We find that the leading nonanalyticity in semimetals with a long-ranged interaction appears at the same order in frequency as the one in Fermi liquids, since the effects of the vanishing density of states at the Fermi level are offset by the breakdown of screening. Consequently, we are able to provide a logical scheme to determine the leading non-analytical behavior of observables in semimetals using knowledge of the corresponding non-analyticities in a Fermi liquid.
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    On a Spectral Method for Calculating the Electrical Resistivity of a Low Temperature Metal from the Linearized Boltzmann Equation
    (University of Oregon, 2022-05-10) Amarel, James; Belitz, Dietrich
    While it is well known that transport equations may be derived diagrammatically, both this approach and that of Boltzmann inevitably encounter an integral equation that both is difficult to solve and, for the most part, has yielded only to uncontrolled approximations. Even though the popular approximations, which are typically either variational in nature or involve dropping memory effects, can be expected to capture the temperature scaling of the kinetic coefficients, it is desirable to exactly obtain the prefactor by way of a mathematically justifiable approximation. For the purpose of so precisely resolving the distribution function that governs the elementary excitations of a metal perturbed by an externally applied static electric field, a spectral method was developed that makes use of the temperature as a control parameter to facilitate an asymptotic inversion of the collision operator; the technique leverages a singularity that is inherent to the Boltzmann equation in the low temperature limit, i.e. when the dissipating Boson bath is all but frozen out. This present dissertation is mainly concerned with the anomalous transport behavior that is commonly observed in quantum magnets; throughout a wide range of their phase diagram, materials such as the metallic ferromagnet ZrZn$_2$ display a power law behavior of the electrical resistivity $\rho \propto T^s$ with $s < 2$. As is thoroughly established, this non-Fermi-liquid like exponent $s$ does not arise solely due to the scattering of conduction electrons by phonons, magnons, or screened Coulomb fluctuations, for each of these soft excitations leads to $s > 2$ at temperatures $T \approx 10K$ (where ZrZn$_2$ exhibits $1.5 < s < 1.7$). After preliminarily investigating the electron-phonon system by way of rigorous reasoning, I will argue that the observed scaling of the residual resistivity $\rho \propto T^{3/2}$ in metallic ferromagnets can be attributed to interference between two scattering mechanisms: ferromagnons and static impurities. This dissertation includes previously published co-authoredmaterial.
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    Singular Symmetric Hyperbolic Systems and Cosmological Solutions to the Einstein Equations
    (University of Oregon, 2014-06-17) Ames, Ellery; Belitz, Dietrich
    Characterizing the long-time behavior of solutions to the Einstein field equations remains an active area of research today. In certain types of coordinates the Einstein equations form a coupled system of quasilinear wave equations. The investigation of the nature and properties of solutions to these equations lies in the field of geometric analysis. We make several contributions to the study of solution dynamics near singularities. While singularities are known to occur quite generally in solutions to the Einstein equations, the singular behavior of solutions is not well-understood. A valuable tool in this program has been to prove the existence of families of solutions which are so-called asymptotically velocity term dominated (AVTD). It turns out that a method, known as the Fuchsian method, is well-suited to proving the existence of families of such solutions. We formulate and prove a Fuchsian-type theorem for a class of quasilinear hyperbolic partial differential equations and show that the Einstein equations can be formulated as such a Fuchsian system in certain gauges, notably wave gauges. This formulation of Einstein equations provides a convenient general framework with which to study solutions within particular symmetry classes. The theorem mentioned above is applied to the class of solutions with two spatial symmetries -- both in the polarized and in the Gowdy cases -- in order to prove the existence of families of AVTD solutions. In the polarized case we find families of solutions in the smooth and Sobolev regularity classes in the areal gauge. In the Gowdy case we find a family of wave gauges, which contain the areal gauge, such that there exists a family of smooth AVTD solutions in each gauge.