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Physical Review Letters

Crossover from BCS to Bose superconductivity: Transition temperature and time-dependent Ginzburg-Landau theory

C. A. R. Sá de Melo, Mohit Randeria, Jan R. Engelbrecht

Published Nov 8, 1993• 861 citations
Publisher

Abstract

We use a functional integral formulation to study the finite temperature crossover from cooperative Cooper pairing to independent bound state formation and condensation. We show the inadequacy of mean field results for normal state properties obtained at the saddle point level as the coupling increases. The importance of quantum (temporal) fluctuations is pointed out and an interpolation scheme for ${\mathit{T}}_{\mathit{c}}$ is derived from this point of view. The time-dependent Ginzburg-Landau (TDGL) equation near ${\mathit{T}}_{\mathit{c}}$ is shown to describe a damped mode in the BCS limit, and a propagating one in the Bose limit. A singular point is identified at intermediate coupling where a simple TDGL description fails.

DOI
10.1103/physrevlett.71.3202

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We use a functional integral formulation to study the finite temperature crossover from cooperative Cooper pairing to independent bound state formation and condensation. We show the inadequacy of mean field results for normal state properties obtained at the saddle point level as the coupling increases. The importance of quantum (temporal) fluctuations is pointed out and an interpolation scheme for Tc{\mathit{T}}_{\mathit{c}}Tc​ is derived from this point of view. The time-dependent Ginzburg-Landau (TDGL) equation near Tc{\mathit{T}}_{\mathit{c}}Tc​ is shown to describe a damped mode in the BCS limit, and a propagating one in the Bose limit. A singular point is identified at intermediate coupling where a simple TDGL description fails.
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.
A functional integral formulation, used previously to calculate Tc{\mathrm{T}}_{\mathrm{c}}Tc​ and describe normal state properties of the BCS-Bose crossover, is extended to TTc{\mathrm{T}}_{\mathrm{c}}Tc​. The saddle point approximation is shown to be qualitatively correct for T\ensuremath{\ll}Tc{\mathrm{T}}_{\mathrm{c}}Tc​ for all couplings, in contrast to the situation above Tc{\mathrm{T}}_{\mathrm{c}}Tc​. Several features of the crossover are described. The difference between the T=0 ``pair size'' and the (prefactor of the T dependent) Ginzburg-Landau coherence length is pointed out: the two quantities are the same only in the BCS limit. The evolution of the collective modes from the BCS to the Bose regime is discussed together with the mixing of the amplitude and phase in the absence of a particle-hole symmetry.
The author reviews recent work on Fermi systems with attractive interactions of arbitrary strength. First, he describes the results of a functional integral formulation to study the crossover from cooperative Cooper pairing to independent bound state formation and condensation. 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. Next, results from a quantum Monte Carlo simulation of the 2D attractive Hubbard model are described. The intermediate coupling normal state is found to deviate markedly from a canonical Fermi liquid: {open_quotes}spin-gap{close_quotes} behavior, with 1/T{sub 1}T {approximately} {chi}(T), is found in a degenerate Fermi system.
This chapter derives the time-dependent Ginzburg–Landau equation from first principles, by relying on the same formal approach that has led to the time-dependent Gross–Pitaevskii equation of Chapter 28. Specifically, the time-dependent Ginzburg–Landau equation holds close to the critical temperature of the initial equilibrium preparation and in the (extreme) BCS limit of the BCS–BEC crossover, when the Cooper pairs are largely overlapping with each other. Care has to be exerted when dealing with the analytic properties in the wave-vector and frequency space of the normal and anomalous particle–particle bubbles.
Abstract Starting from a model of free Fermions in two dimensions with an arbitrary strong effective interaction, we derive a Ginzburg‐Landau theory describing the crossover from BCS‐superconductivity to Bose‐condensation. We find a smooth crossover from the standard BCS‐limit to a Gross‐Pitaevski type equation for the order parameter in a Bose superfluid. The mean field transition temperature exhibits a maximum at a coupling strength, where the behaviour crosses over from BCS to Bose like with corresponding values of 2 Δ0/Tc ≈ 5 which are characteristic for high Tc superconductors.
By means of the functional-integral formulation we discuss the occurrence of superconductivity resulting from Bose-Einstein condensation within a model describing preformed pairs which coexist in thermal equilibrium with unpaired fermions. At saddle-point level we study the thermodynamic properties of the model as functions of the boson excitation energy $\ensuremath{\nu},$ discussing the results in connection with the phenomenology of the pseudogap state of the superconducting cuprates. For temperatures above the Bose-Einstein condensation temperature Tc{T}_{c}Tc​ we also analyze the effects of Gaussian fluctuations on the critical temperature and on the boson spectrum. Finally, we compare the bosonic effective action which controls thermal fluctuations of the mixture to the time-dependent Ginzburg-Landau functional of an interacting fermionic system at intermediate coupling.