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

Physical review. B, Condensed matter

Thermodynamics of superconductivity beyond the Gorkov decoupling

Shun-ichiro Koh

Published Apr 1, 1994• 11 citations
Publisher

Abstract

Superconductivity beyond the Gorkov decoupling is examined with an emphasis on the thermodynamic properties. In the Gorkov decoupling, the interaction term is factorized into the pairing correlation. Therefore, in contrast to the strong pair-pair correlation in coordinate space, the correlation between the Cooper pairs in momentum space is neglected. In this paper, in addition to the conventional pairing interaction, we consider a more general type of attractive interaction that cannot be factorized into the product of the pairing correlation. Accordingly, it becomes necessary to include the neglected correlation. We assume a variational functional where the general type of interaction is taken into account, and determine the state self-consistently at finite temperature. The following results are obtained: (1) The state obtained is more stable at low temperature, compared with the BCS state with the same parameters. (2) As the temperature is increased, our state approaches the BCS state. (3) At the same temperature as in the BCS theory, our system undergoes a phase transition from the superconducting phase to the normal phase.

DOI
10.1103/physrevb.49.8983

Related papers

Thermodynamics of a superconductor with strongly bound Cooper pairs

M. Keller, Walter Metzner, Ulrich Schollwöck

We study the thermodynamics of a superconductor with strongly bound Cooper pairs by solving the self-consistent T-matrix equations for the attractive Hubbard model within a local approximation. We determine the transition temperature as a function of the interaction strength at quarter filling, and compute the temperature dependence of the superconducting order parameter, the specific heat, the spin susceptibility, and the density of local pairs in the normal and superconducting phase. Pronounced deviations from BCS behavior are found already for moderate interactions, where superconductivity sets in at a critical temperature above the coherence temperature for Fermi liquid behavior.

Physical review. B, Condensed matter•Aug 1, 1999•39 citations

Cooper pairing with finite angular momentum via a central attraction: From the BCS to the Bose limits

Jorge Quintanilla, Balazs L. Gyorffy, James F. Annett +1 more

In the context of a simple model featuring an explicit, central interaction potential, and using a standard functional-integral technique, we study superconductivity with angular momentum quantum number $l=2$ as an emergent property of the many-body system. Our interaction potential is attractive at a finite distance ${r}_{0},$ and the breaking of the rotational symmetry is the result of an interplay between ${r}_{0}$ and the interparticle distance ${r}_{s}.$ This interplay is generic to interactions of this type and is responsible for the existence of d-wave pairing for a range of densities. However, we find that $l=2$ pairing takes place only in the BCS limit. In contrast, as the Bose-Einstein (BE) limit is approached the internal energy of the ``preformed pairs'' becomes the dominant contribution and there is a quantum phase transition in which the s-wave symmetry is restored. We also find that the limiting value of the critical temperature is ${k}_{B}{T}_{c}\ensuremath{\rightarrow}3.315{\ensuremath{\Elzxh}}^{2}{/2m}^{*}[n/2(2l+1){]}^{2/3},$ which coincides with the usual result only for $l=0;$ for $l>0,$ it differs in the degeneracy factor $1/(2l+1),$ which lowers ${T}_{c}.$ Our results thus place constraints on exotic pairing in the BE limit, while at the same time indicating a particularly interesting route to pairing with $l>0$ in a BCS superconductor.

arXiv:cond-mat/0106250•Physical review. B, Condensed matter•Dec 31, 2002•10 citations

Evolution from BCS superconductivity to Bose-Einstein condensation: Current correlation function in the broken-symmetry phase

Natascia Andrenacci, P. Pieri, Giancarlo Calvanese Strinati

We consider the current correlation function for a three-dimensional system of fermions embedded in a homogeneous background and mutually interacting via an attractive short-range potential, below the (superconducting) critical temperature. Diagrammatic contributions in the broken-symmetry phase are identified, which yield for the (wave-vector and frequency-dependent) current correlation function the fermionic BCS approximation in the weak-coupling limit and the bosonic Bogoliubov approximation in the strong-coupling limit (whereby composite bosons form as bound-fermion pairs). The temperature dependence of the superfluid density (from the BCS exponential behavior at weak coupling to a power-law behavior at strong coupling) and the form of the Pippard-like kernel at zero temperature are explicitly obtained from weak to strong coupling. Quite generally, it is shown that the Pippard-like kernel is the sum of a local (London-like) term and of a nonlocal component, the local term being dominant in the strong-coupling limit and the nonlocal component in the BCS (weak-coupling) limit. It is also shown that the range of the nonlocal component is determined by the coherence length measuring the spatial correlations of the amplitude of the order parameter, namely, the correlations among different Cooper pairs (or composite bosons), rather than between the fermionic partners of a given pair. In addition, a prescription is provided for mapping the fermionic onto the bosonic diagrammatic theories in the broken-symmetry phase, thus complementing what has been already done in the normal phase.

arXiv:cond-mat/0305324•Physical review. B, Condensed matter•Oct 6, 2003•51 citations

Effects of electron correlation on superconductivity in the Hatsugai–Kohmoto model*

Huai-Shuang Zhu, Qiang Han

Understanding how electrons form pairs in the presence of strong electron correlations demands going beyond the BCS paradigm. We study a correlated superconducting model where the correlation effects are accounted for by a U term local in momentum space. The electron correlation is treated exactly while the electron pairing is treated approximately using the mean-field theory. The self-consistent equation for the pair potential is derived and solved. Somewhat contrary to expectation, a weak attractive U comparable to the pair potential can destroy the superconductivity, whereas for weak to intermediate repulsive U , the pair potential can be enhanced. The fidelity of the mean-field ground state is calculated to describe the strength of the elelectron correlation. We show that the pair potential is not equal to the single-electron superconducting gap for the strongly correlated superconductors, in contrast to the uncorrelated BCS limit.

Chinese Physics B•Mar 5, 2021•8 citations

Superconducting state with a finite-momentum pairing mechanism in zero external magnetic field

Florian Loder, A. P. Kampf, Thilo Kopp

In the BCS theory of superconductivity, one assumes that all Cooper pairs have the same center-of-mass momentum. This is indeed enforced by self-consistency if the pairing interaction is momentum independent. Here, we show that for an attractive nearest-neighbor interaction, this is different. In this case, stable solutions with pairs with momenta $\mathbf{q}$ and $\ensuremath{-}\mathbf{q}$ coexist and, for a sufficiently strong interaction, one of these states becomes the ground state of the superconductor. The possibility for a finite-momentum pairing state emerges only for nodal superconductors and is accompanied by a charge order with wave vector $2\mathbf{q}$. For a weak pairing interaction, the ground state is a $d$-wave superconductor.

arXiv:0907.4345•Physical Review B•Jan 28, 2010•57 citations

BCS superconductivity in quantum critical metals

Jian-Huang She, Jan Zaanen

We present a simple phenomenological scaling theory for the pairing instability of a quantum critical metal. It can be viewed as a minimal generalization of the classical Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity (SC) for normal Fermi-liquid metals. We assume that attractive interactions are induced in the fermion system by an external ``bosonic glue'' that is strongly retarded. Resting on the small Migdal parameter, all the required information from the fermion system needed to address the superconductivity enters through the pairing susceptibility. Asserting that the normal state is a strongly interacting quantum critical state of fermions, the form of this susceptibility is governed by conformal invariance and one only has the scaling dimension of the pair operator as free parameter. Within this scaling framework, conventional BCS theory appears as the ``marginal'' case but it is now easily generalized to the (ir)relevant scaling regimes. In the relevant regime an algebraic singularity takes over from the BCS logarithm with the obvious effect that the pairing instability becomes stronger. However, it is more surprising that this effect is strongest for small couplings and small Migdal parameters, highlighting an unanticipated important role of retardation. Using exact forms for the finite-temperature pair susceptibility from $1+1\text{D}$ conformal field theory as models, we study the transition temperatures, finding that the gap to transition temperature ratios is generically large compared to the BCS case, showing, however, an opposite trend as a function of the coupling strength compared to the conventional Migdal-Eliashberg theory. We show that our scaling theory naturally produces the superconducting ``domes'' surrounding the quantum critical points (QCPs), even when the coupling to the glue itself is not changing at all. We argue that hidden relations will exist between the location of the crossover lines to the Fermi liquids away from the quantum critical points and the detailed form of the dome when the glue strength is independent of the zero-temperature control parameter. Finally, we discuss the behavior of the orbital-limited upper critical magnetic field as a function of the zero-temperature coupling constant. Compared to the variation in the transition temperature, the critical field might show a much stronger variation pending the value of the dynamical critical exponent.

Physical Review B•Nov 30, 2009•48 citations
Superconductivity beyond the Gorkov decoupling is examined with an emphasis on the thermodynamic properties. In the Gorkov decoupling, the interaction term is factorized into the pairing correlation. Therefore, in contrast to the strong pair-pair correlation in coordinate space, the correlation between the Cooper pairs in momentum space is neglected. In this paper, in addition to the conventional pairing interaction, we consider a more general type of attractive interaction that cannot be factorized into the product of the pairing correlation. Accordingly, it becomes necessary to include the neglected correlation. We assume a variational functional where the general type of interaction is taken into account, and determine the state self-consistently at finite temperature. The following results are obtained: (1) The state obtained is more stable at low temperature, compared with the BCS state with the same parameters. (2) As the temperature is increased, our state approaches the BCS state. (3) At the same temperature as in the BCS theory, our system undergoes a phase transition from the superconducting phase to the normal phase.
We study the thermodynamics of a superconductor with strongly bound Cooper pairs by solving the self-consistent T-matrix equations for the attractive Hubbard model within a local approximation. We determine the transition temperature as a function of the interaction strength at quarter filling, and compute the temperature dependence of the superconducting order parameter, the specific heat, the spin susceptibility, and the density of local pairs in the normal and superconducting phase. Pronounced deviations from BCS behavior are found already for moderate interactions, where superconductivity sets in at a critical temperature above the coherence temperature for Fermi liquid behavior.
In the context of a simple model featuring an explicit, central interaction potential, and using a standard functional-integral technique, we study superconductivity with angular momentum quantum number l=2l=2l=2 as an emergent property of the many-body system. Our interaction potential is attractive at a finite distance r0,{r}_{0},r0​, and the breaking of the rotational symmetry is the result of an interplay between r0{r}_{0}r0​ and the interparticle distance rs.{r}_{s}.rs​. This interplay is generic to interactions of this type and is responsible for the existence of d-wave pairing for a range of densities. However, we find that l=2l=2l=2 pairing takes place only in the BCS limit. In contrast, as the Bose-Einstein (BE) limit is approached the internal energy of the ``preformed pairs'' becomes the dominant contribution and there is a quantum phase transition in which the s-wave symmetry is restored. We also find that the limiting value of the critical temperature is ${k}_{B}{T}_{c}\ensuremath{\rightarrow}3.315{\ensuremath{\Elzxh}}^{2}{/2m}^{*}[n/2(2l+1){]}^{2/3},$ which coincides with the usual result only for l=0;l=0;l=0; for l>0,l>0,l>0, it differs in the degeneracy factor 1/(2l+1),1/(2l+1),1/(2l+1), which lowers Tc.{T}_{c}.Tc​. Our results thus place constraints on exotic pairing in the BE limit, while at the same time indicating a particularly interesting route to pairing with l>0l>0l>0 in a BCS superconductor.
We consider the current correlation function for a three-dimensional system of fermions embedded in a homogeneous background and mutually interacting via an attractive short-range potential, below the (superconducting) critical temperature. Diagrammatic contributions in the broken-symmetry phase are identified, which yield for the (wave-vector and frequency-dependent) current correlation function the fermionic BCS approximation in the weak-coupling limit and the bosonic Bogoliubov approximation in the strong-coupling limit (whereby composite bosons form as bound-fermion pairs). The temperature dependence of the superfluid density (from the BCS exponential behavior at weak coupling to a power-law behavior at strong coupling) and the form of the Pippard-like kernel at zero temperature are explicitly obtained from weak to strong coupling. Quite generally, it is shown that the Pippard-like kernel is the sum of a local (London-like) term and of a nonlocal component, the local term being dominant in the strong-coupling limit and the nonlocal component in the BCS (weak-coupling) limit. It is also shown that the range of the nonlocal component is determined by the coherence length measuring the spatial correlations of the amplitude of the order parameter, namely, the correlations among different Cooper pairs (or composite bosons), rather than between the fermionic partners of a given pair. In addition, a prescription is provided for mapping the fermionic onto the bosonic diagrammatic theories in the broken-symmetry phase, thus complementing what has been already done in the normal phase.
Understanding how electrons form pairs in the presence of strong electron correlations demands going beyond the BCS paradigm. We study a correlated superconducting model where the correlation effects are accounted for by a U term local in momentum space. The electron correlation is treated exactly while the electron pairing is treated approximately using the mean-field theory. The self-consistent equation for the pair potential is derived and solved. Somewhat contrary to expectation, a weak attractive U comparable to the pair potential can destroy the superconductivity, whereas for weak to intermediate repulsive U , the pair potential can be enhanced. The fidelity of the mean-field ground state is calculated to describe the strength of the elelectron correlation. We show that the pair potential is not equal to the single-electron superconducting gap for the strongly correlated superconductors, in contrast to the uncorrelated BCS limit.
In the BCS theory of superconductivity, one assumes that all Cooper pairs have the same center-of-mass momentum. This is indeed enforced by self-consistency if the pairing interaction is momentum independent. Here, we show that for an attractive nearest-neighbor interaction, this is different. In this case, stable solutions with pairs with momenta q\mathbf{q}q and $\ensuremath{-}\mathbf{q}$ coexist and, for a sufficiently strong interaction, one of these states becomes the ground state of the superconductor. The possibility for a finite-momentum pairing state emerges only for nodal superconductors and is accompanied by a charge order with wave vector 2q2\mathbf{q}2q. For a weak pairing interaction, the ground state is a ddd-wave superconductor.
We present a simple phenomenological scaling theory for the pairing instability of a quantum critical metal. It can be viewed as a minimal generalization of the classical Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity (SC) for normal Fermi-liquid metals. We assume that attractive interactions are induced in the fermion system by an external ``bosonic glue'' that is strongly retarded. Resting on the small Migdal parameter, all the required information from the fermion system needed to address the superconductivity enters through the pairing susceptibility. Asserting that the normal state is a strongly interacting quantum critical state of fermions, the form of this susceptibility is governed by conformal invariance and one only has the scaling dimension of the pair operator as free parameter. Within this scaling framework, conventional BCS theory appears as the ``marginal'' case but it is now easily generalized to the (ir)relevant scaling regimes. In the relevant regime an algebraic singularity takes over from the BCS logarithm with the obvious effect that the pairing instability becomes stronger. However, it is more surprising that this effect is strongest for small couplings and small Migdal parameters, highlighting an unanticipated important role of retardation. Using exact forms for the finite-temperature pair susceptibility from 1+1D1+1\text{D}1+1D conformal field theory as models, we study the transition temperatures, finding that the gap to transition temperature ratios is generically large compared to the BCS case, showing, however, an opposite trend as a function of the coupling strength compared to the conventional Migdal-Eliashberg theory. We show that our scaling theory naturally produces the superconducting ``domes'' surrounding the quantum critical points (QCPs), even when the coupling to the glue itself is not changing at all. We argue that hidden relations will exist between the location of the crossover lines to the Fermi liquids away from the quantum critical points and the detailed form of the dome when the glue strength is independent of the zero-temperature control parameter. Finally, we discuss the behavior of the orbital-limited upper critical magnetic field as a function of the zero-temperature coupling constant. Compared to the variation in the transition temperature, the critical field might show a much stronger variation pending the value of the dynamical critical exponent.