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Particle-Hole Fluctuations in Superconductors

Mason Protter

Published Jan 1, 2019• 0 citations
Publisher

Abstract

In this thesis we investigate how corrections originating from particle-hole interactions may be introduced into the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity using functional integral methods. The attractive Fermionic contact interaction is simultaneously decoupled in the Cooper and exchange channels, with the exchange-channel decoupling field treated as a small fluctuation about its zero mean-field value. Upon integrating out the exchange-channel fluctuation field, the system is described by a new effective action for the Cooper-channel decoupling field, which models the superconducting condensate. The saddle-point condition of this effective action generates the standard mean-field BCS equations with correction terms containing particle-hole polarization bubbles. The corrections obtained in this thesis are similar in spirit, albeit different in form, to those originally found in 1961 by Gor’kov and Melik-Barkhudarov (GMB). An advantage of the technique employed here, as opposed to the GMB approach, is the methodical manner in which fluctuations can be incorporated into superconductors.

DOI
10.7939/r3-6tf8-av79

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In this thesis we investigate how corrections originating from particle-hole interactions may be introduced into the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity using functional integral methods. The attractive Fermionic contact interaction is simultaneously decoupled in the Cooper and exchange channels, with the exchange-channel decoupling field treated as a small fluctuation about its zero mean-field value. Upon integrating out the exchange-channel fluctuation field, the system is described by a new effective action for the Cooper-channel decoupling field, which models the superconducting condensate. The saddle-point condition of this effective action generates the standard mean-field BCS equations with correction terms containing particle-hole polarization bubbles. The corrections obtained in this thesis are similar in spirit, albeit different in form, to those originally found in 1961 by Gor’kov and Melik-Barkhudarov (GMB). An advantage of the technique employed here, as opposed to the GMB approach, is the methodical manner in which fluctuations can be incorporated into superconductors.
The problem of the evolution from BCS theory with cooperative Cooper pairing to the formation and condensation of composite bosons has attracted considerable attention for the past several decades. It has gained renewed impetus in the mid-eighties with the discovery of the high-Tc superconductors, which have a coherence length comparable to the interparticle spacing. More recently, this subject has spurred a great deal of research activity in connection with experiments involving dilute atomic gases of fermionic atoms. The initial objective of this work will be to use functional integral techniques to analyze the low-temperature BCS-to-BEC evolution of d-wave superconductors within the saddle point (mean field) approximation for a continuum model. Then, the same mathematical formalism will be applied to the problem of the BCS-to-BEC evolution of fully spin-polarized p-wave Fermi gases in two dimensions. We find that a quantum phase transition occurs for both systems as they are driven from the BCS-like regime of weakly interacting fermionic pairs to the opposite BEC-like regime of strongly interacting bosonic molecules. This is in contrast to the smooth crossover predicted and observed in systems that exhibit s-wave pairing symmetry. We calculate several spectroscopic and thermodynamic properties that signal the occurrence of this phase transition, and suggest some possible experimental realizations. Finally, fluctuations about the saddle point solution are included in the calculations, and the effects of such correction are analyzed in the low (T~0) and high (T~Tc) temperature limits. We conclude that, at high temperatures, the bosonic degrees of freedom that arise from two-particle bound states become essential to describe the strong coupling limit,\nas the saddle point approximation alone becomes unreliable.
We improve the Bardeen-Cooper-Schrieffer wave function with a fixed particle number so as to incorporate many-body correlations beyond the mean-field treatment. It is shown that the correlations lower the ground-state energy far more than Cooper-pair condensation in the weak-coupling region. Moreover, they naturally bring a superposition over the number of condensed particles. Thus, Cooper-pair condensation is special among the various bound-state formations of quantum mechanics in that number fluctuations are necessarily present in the condensate through the dynamical exchange of particles with the non-condensate reservoir. On the basis of this result, we propose ΔNcon⋅Δϕ≳1\varDelta N_{\rm con}\cdot \varDelta ϕ\gtrsim 1ΔNcon​⋅Δϕ≳1 as the uncertainty relation relevant to the number-phase fluctuations in superconductors and superfluids, where the number of condensed particles NconN_{\rm con}Ncon​ is used instead of the total particle number NNN. The formula implies that a macroscopic phase ϕϕϕ can be established even in number-fixed superconductors and superfluids since ΔNcon≫1\varDelta N_{\rm con}\gg 1ΔNcon​≫1.
This Thesis analyzes the role of a mean-field theory and of the fluctuations in two different fermionic systems: the BCS-BEC crossover and high-Tc superconducting cuprates. In the first part of the Thesis we introduce the mean-field theory and the Gaussian-level fluctuations for the BCS-BEC crossover, i.e. the continuous evolution observed in ultracold neutral fermionic systems, from a weakly-coupled regime where fermions form Cooper pairs, to a strongly-coupled regime where the fermions at first form bosonic molecular dimers, subsequently undergoing Bose-Einstein condensation for low enough temperatures. We then analyze some problems related to the crossover: the condensate fraction for a spin-unbalanced Fermi gas, the Beliaev decay for collective excitations and the case of a two-dimensional Fermi gas, for which we calculate the first sound velocity and the Berezinskii-Kosterlitz-Thouless critical temperature, in very good agreement with recent experimental data. Finally we analyze a regularization procedure for the equation of state in the deep-BEC limit, allowing one to analytically derive the correct ratio between the fermionic and bosonic scattering lengths. In the second part of the Thesis we introduce a gauge approach to superconductivity in cuprates in which the hole is decomposed as the product of a spinful neutral particle, the spinon, and a spinless charged particle, the holon. The statistics of each particle is modified by binding it to a gauge field providing a statistical flux, this approach being analogous to Chern-Simons bosonization. We thus obtain a model characterized by three characteristic temperatures corresponding, respectively, to the appearance of a finite density of incoherent holon pairs, of a finite density of incoherent spinon pairs and, finally, to the phase coherence leading to superconductivity. Specifically in the present Thesis within this model we study the superfluid density, demonstrating that the spinon and holon contributions sum according to a Ioffe-Larking-like rule, analogous to that they found for resistivity, showing excellent agreement with experimental data in the moderate underdoping up to optimal doping region. We also demonstrated that the formalism reproduces the quasi-universality observed in experimental data at different dopings and allows for an analytical derivation of a relation that approximates the Uemura relation for moderate dopings.
The fluctuations properties of the BCS Hamiltonian which are induced in finite systems are explored. It is shown that for very large particle numbers, the static-path approximation is an excellent approximation, while for small particle numbers, the random-phase-approximation corrections should be taken into account, especially for particle numbers small enough to cause the normal- to superconducting-phase transition. A correlated mean field containing quantal fluctuations is introduced and compared with the Hartree-Bogoliubov mean field.
The theory of superconductivity is formulated starting from the configuration-space form of the Bardeen-Cooper-Schrieffer (BCS) wave function. Self-consistent equations for treating nonu- niform systems are obtained. These equations have the advantage over the Bogoliubov-de Gennes equations that an effective condensate wave function and quasiparticle eigenstates can separately be obtained. The intrinsic orbital angular momentum density of the condensate is naturally iden- tified in this formalism due to a manifest two-particle bound-state nature in the one-particle density matrix.
We use a functional integral formulation to study the crossover from cooperative Cooper pairing to the formation and condensation of tightly bound pairs in a 3D continuum model of fermions with attractive interactions. The inadequacy of a saddle point approximation with increasing coupling is pointed out, and the importance of temporal (quantum) fluctuations for normal state properties at intermediate and strong coupling is emphasized. In addition to recovering the Nozieres-Schmitt-Pink interpolation scheme for T{sub c}, and the Leggett variational results for T = 0, we also present results for evolution of the time-dependent Ginzburg-Landau equation and collective mode spectrum as a function of the coupling.