Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Crossover from BCS to Bose superconductivity: A functional integral approach

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

Published Apr 1, 1993• 0 citations
OpenAlex

Abstract

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.

Related papers

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

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.

Physical Review Letters•Nov 8, 1993•861 citations

Crossover between BCS and performed-Boson theories with increasing interactions

Mohit Randeria

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.

•Aug 1, 1993•0 citations

BCS to Bose crossover: Broken-symmetry state

Jan R. Engelbrecht, Mohit Randeria, C.A.R. Sáde Melo

A functional integral formulation, used previously to calculate ${\mathrm{T}}_{\mathrm{c}}$ and describe normal state properties of the BCS-Bose crossover, is extended to T${\mathrm{T}}_{\mathrm{c}}$. The saddle point approximation is shown to be qualitatively correct for T\ensuremath{\ll}${\mathrm{T}}_{\mathrm{c}}$ for all couplings, in contrast to the situation above ${\mathrm{T}}_{\mathrm{c}}$. 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.

Physical review. B, Condensed matter•Jun 1, 1997•400 citations

BCS-to-BEC Quantum Phase Transition in High-Tc Superconductors and Fermionic Atomic Gases: A Functional Integral Approach

Sergio S. Botelho

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.

•Sep 12, 2005•2 citations

Dynamics of Fluctuactions in the BCS-BEC Crossover in Superfluid Fermi Gases

H Vivas C

The BCS-BEC crossover phenomenon in a 3D attractive Fermi gas is revisited by using finite temperature Green functions formalism derived from the quantum path integral technique and Mean Field Approximation (MFA). Collective modes for weak and strong coupling particle-particle interaction g are obtained after recovering the dynamical coefficients in the nonlinear Time Dependent Ginzburg Landau (TDGL) equation, and numerically simulated under those particular states for the filling energy g ~ 0 and order parameter Δ( x , τ ) ~ Δ 0 . Explicit formulas for the first order average fluctuations ⟨η 2 q,ω ⟩ and the scaling law for the thermal energy associated to the non-linear collective modes are also reported.

Journal of Physics Conference Series•Dec 8, 2014•0 citations

Strong-coupling limit in the evolution from BCS superconductivity to Bose-Einstein condensation

P. Pieri, Giancarlo Calvanese Strinati

We consider the problem of the crossover from BCS superconductivity to Bose-Einstein condensation in three dimensions for a system of fermions with a mutual attractive interaction, for which we adopt the simplifying assumption of a suitably regularized point-contact interaction. We examine in a critical way the fermionic (self-consistent) T-matrix approximation, which has been widely utilized in the literature to describe this crossover above the superconducting critical temperature, and show that it fails to yield the correct behavior of the system in the strong-coupling limit, where composite bosons form as tightly bound fermion pairs. We then set up the correct approximation for a ``dilute'' system of composite bosons and show that a class of diagrams has to be considered in the place of the fermionic T-matrix approximation for the self-energy. This class of diagrams correctly describes both the weak- and strong-coupling limits, and consequently results in an improved interpolation scheme for the intermediate (crossover) region. In this context, we provide also a systematic mapping between the corresponding diagrammatic theories for the composite bosons and the constituent fermions. As a preliminary result to demonstrate the numerical effect of our class of diagrams on physical quantities, we calculate the value of the scattering length for composite bosons in the strong-coupling limit and show that it is considerably modified with respect to the result obtained within the self-consistent fermionic T-matrix approximation.

arXiv:cond-mat/9811166•Physical review. B, Condensed matter•Jun 1, 2000•185 citations
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.
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.
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.
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 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.
The BCS-BEC crossover phenomenon in a 3D attractive Fermi gas is revisited by using finite temperature Green functions formalism derived from the quantum path integral technique and Mean Field Approximation (MFA). Collective modes for weak and strong coupling particle-particle interaction g are obtained after recovering the dynamical coefficients in the nonlinear Time Dependent Ginzburg Landau (TDGL) equation, and numerically simulated under those particular states for the filling energy g ~ 0 and order parameter Δ( x , τ ) ~ Δ 0 . Explicit formulas for the first order average fluctuations ⟨η 2 q,ω ⟩ and the scaling law for the thermal energy associated to the non-linear collective modes are also reported.
We consider the problem of the crossover from BCS superconductivity to Bose-Einstein condensation in three dimensions for a system of fermions with a mutual attractive interaction, for which we adopt the simplifying assumption of a suitably regularized point-contact interaction. We examine in a critical way the fermionic (self-consistent) T-matrix approximation, which has been widely utilized in the literature to describe this crossover above the superconducting critical temperature, and show that it fails to yield the correct behavior of the system in the strong-coupling limit, where composite bosons form as tightly bound fermion pairs. We then set up the correct approximation for a ``dilute'' system of composite bosons and show that a class of diagrams has to be considered in the place of the fermionic T-matrix approximation for the self-energy. This class of diagrams correctly describes both the weak- and strong-coupling limits, and consequently results in an improved interpolation scheme for the intermediate (crossover) region. In this context, we provide also a systematic mapping between the corresponding diagrammatic theories for the composite bosons and the constituent fermions. As a preliminary result to demonstrate the numerical effect of our class of diagrams on physical quantities, we calculate the value of the scattering length for composite bosons in the strong-coupling limit and show that it is considerably modified with respect to the result obtained within the self-consistent fermionic T-matrix approximation.