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Quantum Theory of Disordered Superconductors - A Replica Approach

Yat Fan Lau

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Motivated by the experimental observations of the anomalous metal states in many superconducting thin films systems, this thesis studies the interplay of disorder and interactions in superconductivity. We first demonstrate that the Fermi glass ground state is unstable in first-order perturbation theory under weak, short-ranged interaction, provided all states are localized. This finding elucidates why methods like the Hartree-Fock approximation are adequate to explain local moment formation in repulsive interactions. We then revisit the problem of dirty superconductors using a replica-symmetric BCS (RS-BCS) theory. We apply RS-BCS theory to disordered superconducting grains within the localization length, assuming the macroscopic system consists of weakly coupled grains. Besides disordered potential, We also consider regions with opposite signs of interaction exist in the grains, with net attractive interaction between electrons. Using RS-BCS mean field theory, we compute the phase diagram, tunneling density of states, and superfluid density across disorder strengths. Our results qualitatively agree with prior numerical studies for purely attractive interactions, and suggest a possible Cooper-pair-glass state when interactions of opposite signs nearly cancel.

DOI
10.14711/thesis-hdl152593

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Motivated by the experimental observations of the anomalous metal states in many superconducting thin films systems, this thesis studies the interplay of disorder and interactions in superconductivity. We first demonstrate that the Fermi glass ground state is unstable in first-order perturbation theory under weak, short-ranged interaction, provided all states are localized. This finding elucidates why methods like the Hartree-Fock approximation are adequate to explain local moment formation in repulsive interactions. We then revisit the problem of dirty superconductors using a replica-symmetric BCS (RS-BCS) theory. We apply RS-BCS theory to disordered superconducting grains within the localization length, assuming the macroscopic system consists of weakly coupled grains. Besides disordered potential, We also consider regions with opposite signs of interaction exist in the grains, with net attractive interaction between electrons. Using RS-BCS mean field theory, we compute the phase diagram, tunneling density of states, and superfluid density across disorder strengths. Our results qualitatively agree with prior numerical studies for purely attractive interactions, and suggest a possible Cooper-pair-glass state when interactions of opposite signs nearly cancel.
In this paper, we revisit the problem of dirty superconductors using a replica-symmetric BCS (RS-BCS) theory. We apply the RS-BCS theory to dirty superconductor grains of size LdL^dLd, where LLL is the localization length and ddd is the dimension of the system, assuming that the macroscopic system is composed of weakly coupled grains. Besides disordered potential, We also consider the case where regions with opposite signs of interaction exist in the grains, with net attractive interaction between electrons. Within the RS-BCS mean field theory, the system phase diagram, single-particle tunneling density of states and the superfluid density are computed within the RS-BCS theory for different strengths of disorder, We show that our result agrees qualitatively with previous numerical studies in the case with pure attractive interaction, and that a Cooper-pair-glass state may exist when regions with opposite sign of interaction exists and the net interaction between electrons is close to zero. The plausible relevance of our result to the anomalous metal state is discussed.
We study the normal (nonsuperconducting) phase of the attractive Hubbard model within the dynamical mean field theory (DMFT) using the numerical renormalization group (NRG) as an impurity solver. A wide range of attractive potentials U is considered, from the weak-coupling limit, where superconducting instability is well described by the BCS approximation, to the strong-coupling region, where the superconducting transition is described by Bose condensation of compact Cooper pairs, which are formed at temperatures much exceeding the superconducting transition temperature. We calculate the density of states, the spectral density, and the optical conductivity in the normal phase for this wide range of U, including the disorder effects. We also present the results on superconducting instability of the normal state dependence on the attraction strength U and the degree of disorder. The disorder influence on the critical temperature T c is rather weak, suggesting in fact the validity of Anderson’s theorem, with the account of the general widening of the conduction band due to disorder.
We study the normal (non-superconducting) phase of attractive Hubbard model within dynamical mean field theory (DMFT) using numerical renormalization group (NRG) as impurity solver. Wide range of attractive potentials UUU is considered, from the weak-coupling limit, where superconducting instability is well described by BCS approximation, up to the strong-coupling region, where superconducting transition is described by Bose-condensation of compact Cooper pairs, which are formed at temperatures much exceeding superconducting transition temperature. We calculate density of states, spectral density and optical conductivity in the normal phase for this wide range of UUU, including the disorder effects. Also we present the results on superconducting instability of the normal state dependence on the attraction strength UUU and the degree of disorder. Disorder influence on the critical temperature TcT_cTc​ is rather weak, suggesting in fact the validity of Anderson theorem, with the account of the general widening of the conduction band due to disorder.
We study the normal (non-superconducting) phase of attractive Hubbard model within dynamical mean field theory (DMFT) using numerical renormalization group (NRG) as impurity solver. Wide range of attractive potentials UUU is considered, from the weak-coupling limit, where superconducting instability is well described by BCS approximation, up to the strong-coupling region, where superconducting transition is described by Bose-condensation of compact Cooper pairs, which are formed at temperatures much exceeding superconducting transition temperature. We calculate density of states, spectral density and optical conductivity in the normal phase for this wide range of UUU, including the disorder effects. Also we present the results on superconducting instability of the normal state dependence on the attraction strength UUU and the degree of disorder. Disorder influence on the critical temperature TcT_cTc​ is rather weak, suggesting in fact the validity of Anderson theorem, with the account of the general widening of the conduction band due to disorder.
We study the interplay of disorder localization and strong local interactions within the Anderson-Hubbard model. Taking into account local Mott-Hubbard physics and static screening of the disorder potential, the system is mapped onto an effective single-particle Anderson model, which is studied within the self-consistent theory of electron localization. For fermions, we find rich nonmonotonic behavior of the localization length ξ, particularly in two-dimensional systems, including an interaction-induced exponential enhancement of ξ for small and intermediate disorders and a strong reduction of ξ due to hopping suppression by strong interactions. In three dimensions, we identify for half filling a Mott-Hubbard-assisted Anderson localized phase existing between the metallic and the Mott-Hubbardgapped phases. For small U there is re-entrant behavior from the Anderson localized phase to the metallic phase. For bosons, the unrestricted particle occupation number per lattice site yields a monotonic enhancement of ξ as a function of decreasing interaction, which we assume to persist until the superfluid Bose- Einstein condensate phase is entered. Besides, we study cold atomic gases expanding, by a diffusion process, in a weak random potential. We show that the density-density correlation function of the expanding gas is strongly affected by disorder and we estimate the typical size of a speckle spot, i.e., a region of enhanced or depleted density. Both a Fermi gas and a Bose-Einstein condensate (in a mean-field approach) are considered.
Our general interest is in self-consistent-field (scf) theories of disordered fermions. They generate physically relevant subensembles (``scf ensembles'') within a given Altland-Zirnbauer class. We are motivated to investigate such ensembles (i) by the possibility to discover new fixed points due to (long-range) interactions; (ii) by analytical scf theories that rely on partial self-consistency approximations awaiting a numerical validation; and (iii) by the overall importance of scf theories for the understanding of complex interaction-mediated phenomena in terms of effective single-particle pictures. In this paper we present an efficient, parallelized implementation solving scf problems with spatially local fields by applying a kernel-polynomial approach. Our first application is the Boguliubov-deGennes theory of the attractive-UUU Hubbard model in the presence of on-site disorder; the sc fields are the particle density n(r)n(\mathbf{r})n(r) and the gap function $\mathrm{\ensuremath{\Delta}}(\mathbf{r})$. For this case, we reach system sizes unprecedented in earlier work. They allow us to study phenomena emerging at scales substantially larger than the lattice constant, such as the interplay of multifractality and interactions or the formation of superconducting islands. For example, we observe that the coherence length exhibits a nonmonotonic behavior with increasing disorder strength already at moderate UUU. With respect to methodology our results are important because we establish that partial self-consistency (``energy-only'') schemes as typically employed in analytical approaches tend to miss qualitative physics such as island formation.