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

Ground-state properties of the BCS–Bose Einstein crossover in a dx2−y2 -wave superconductor

Byron C. den Hertog

Published Jul 1, 1999• 32 citations
Publisher

Abstract

We examine the ground-state properties of the crossover between BCS superconductivity and Bose Einstein condensation within a model that exhibits ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing symmetry. We compare results for zero temperature with known features of s-wave systems and show that bosonic degrees of freedom are likely to emerge only in the dilute limit. We relate the change in properties of the ground state as a function of density to the effect of the exclusion principle and show that the one-particle distribution function undergoes a significant change when bosonic behavior appears.

DOI
10.1103/physrevb.60.559

Related papers

Groundstate properties of the BCS-Bose Einstein crossover in a $d_{x^{2}-y^{2}}$ wave superconductor

Byron C. den Hertog

We examine the groundstate properties of the crossover between BCS superconductivity and Bose Einstein condensation within a model which exhibits $d_{x^{2} - y^{2}}$ pairing symmetry. We compare results for zero temperature with known features of s-wave systems and show that bosonic degrees of freedom are likely to emerge only in the dilute limit. We relate the change in properties of the groundstate as a function of density to the effect of the exclusion principle and show that the one particle distribution function undergoes a significant change when bosonic behavior appears.

arXiv:cond-mat/9808051•Aug 5, 1998•0 citations

BCS to Bose-Einstein crossover phase diagram at zero temperature for a dx2−y2 order parameter superconductor: Dependence on the tight-binding structure

M. B. Soares, F. Kokubun, José Jesús Rodríguez-Núñez +1 more

We study the effect of the next-nearest-neighbor (NNN) hopping matrix element ${t}^{\ensuremath{'}}$ on the zero-temperature $(T=0\mathrm{K})$ crossover phase diagram, namely, from the BCS limit to the Bose-Einstein regime, for a model which presents a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor order parameter symmetry. This theoretical generalization is required by angle-resolved photoemission spectroscopy experiments. Our NNN hopping is characterized by the parameter $\ensuremath{\alpha}\ensuremath{\equiv}{t}^{\ensuremath{'}}/t\ensuremath{\ne}0,$ where t is the nearest neighbor hopping matrix element. For \ensuremath{\alpha}\ensuremath{\geqslant}0, there is a crossover of BCS binding energy type as described [B. C. den Hertog, Phys. Rev. B 60, 559 (1999)] by keeping the density fixed. However, for \ensuremath{\alpha}0, the system is always in the Bose-Einstein regime for $n~{n}_{c}\ensuremath{\approx}0.12.$ We conclude that the presence of the NNN hopping matrix element drastically influences the crossover diagram for a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor.

Physical review. B, Condensed matter•Apr 16, 2002•11 citations

Pair excitations, collective modes, and gauge invariance in the BCS–Bose-Einstein crossover scenario

Ioan Kosztin, Qijin Chen, Ying-Jer Kao +1 more

In this paper we study the BCS Bose-Einstein condensation (BEC) crossover scenario within the superconducting state, using a T-matrix approach which yields the ground state proposed by Leggett. Here we extend this ground state analysis to finite temperatures T and interpret the resulting physics. We find two types of bosoniclike excitations of the system: long lived, incoherent pair excitations and collective modes of the superconducting order parameter, which have different dynamics. Using a gauge invariant formalism, this paper addresses their contrasting behavior as a function of T and superconducting coupling constant g. At a more physical level, our paper emphasizes how, at finite T, BCS-BEC approaches introduce an important parameter ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2}={\ensuremath{\Delta}}^{2}\ensuremath{-}{\ensuremath{\Delta}}_{\mathrm{sc}}^{2}$ into the description of superconductivity. This parameter is governed by the pair excitations and is associated with particle-hole asymmetry effects that are significant for sufficiently large g. In the fermionic regime, ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2}$ represents the difference between the square of the excitation gap ${\ensuremath{\Delta}}^{2}$ and that of the superconducting order parameter ${\ensuremath{\Delta}}_{\mathrm{sc}}^{2}.$ The parameter ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2},$ which is necessarily zero in the BCS (mean field) limit increases monotonically with the strength of the attractive interaction g. It follows that there is a significant physical distinction between this BCS-BEC crossover approach (in which g is the essential variable which determines ${\ensuremath{\Delta}}_{\mathrm{pg}})$ and the widely discussed phase fluctuation scenario in which the plasma frequency is the tuning parameter. Finally, we emphasize that in the strong coupling limit, there are important differences between the composite bosons that arise in crossover theories and the usual bosons of the (interacting) Bose liquid. Because of constraints imposed on the fermionic excitation gap and chemical potential, in crossover theories, the fermionic degrees of freedom can never be fully removed from consideration.

arXiv:cond-mat/9906180•Physical review. B, Condensed matter•May 1, 2000•74 citations

Comment on “BCS to Bose-Einstein crossover phase diagram at zero temperature for a dx2−y2 order parameter superconductor: Dependence on the tight-binding structure”

A. Perali, P. Pieri, Giancarlo Calvanese Strinati

The work by Soares et al. [Phys. Rev. B 65, 174506 (2002)] investigates the BCS to Bose-Einstein crossover for d-wave pairing in the two-dimensional attractive Hubbard model. Contrary to their claims, we found that a nonpairing region does not exist in the density vs coupling phase diagram. The gap parameter at $T=0,$ as obtained by solving analytically as well as numerically the BCS equations, is in fact finite for any nonzero density and coupling, even in the weak-coupling regime.

arXiv:cond-mat/0211132•Physical review. B, Condensed matter•Aug 28, 2003•16 citations

Properties of strongly paired fermionic condensates

Jesper Levinsen, Victor Gurarie

We study a gas of fermions undergoing a wide resonance $s$-wave BCS--Bose-Einstein-condensate (BEC) crossover, in the BEC regime at zero temperature. We calculate the chemical potential and the speed of sound of this Bose-Einstein-condensed gas, as well as the condensate depletion, in the low-density approximation. We discuss how higher-order terms in the low-density expansion can be constructed. We demonstrate that the standard BCS-BEC gap equation is invalid in the BEC regime and is inconsistent with the results obtained here. The low-density approximation we employ breaks down in the intermediate BCS-BEC crossover region. Hence our theory is unable to predict how the chemical potential and the speed of sound evolve once the interactions are tuned towards the BCS regime. As a part of our theory, we derive the well-known result for the bosonic scattering length diagrammatically and check that there are no bound states of two bosons.

arXiv:cond-mat/0510672•Physical Review A•May 10, 2006•62 citations

On the theory of temperature BCS–BEC crossover in two-dimensional square-lattice superconductors

В. М. Локтев, Volodymyr Turkowski

Physical properties of two-dimensional (2D) superconductors is an active area of long time researches in which many, even principle, questions are still open. One of them is a possibility of a smooth crossover of the system from the standard Bardeen–Cooper–Schrieffer (BCS) regime of strongly overlapping Cooper pairs to the regime of Bose–Einstein condensation (BEC) of almost non-overlapping local pairs. The adequate to the real process description of this transition which can happen either by changing the doping or temperature would lead to a better understanding of the superconducting properties of materials. Though the crossover is not observed in one of the most important layered systems—namely, cuprate superconductors—we demonstrate that in systems with stronger electron-electron attraction it is possible. In particular, in the d-wave channel at sufficiently high doping the crossover can take place with growth of temperature two times, i.e., as BEC→BCS, and followed by reverse BCS→BEC condensation. The physical mechanisms of such transitions and possibility to detect them experimentally in real systems are discussed.

Low Temperature Physics•Aug 1, 2025•2 citations
We examine the ground-state properties of the crossover between BCS superconductivity and Bose Einstein condensation within a model that exhibits dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ pairing symmetry. We compare results for zero temperature with known features of s-wave systems and show that bosonic degrees of freedom are likely to emerge only in the dilute limit. We relate the change in properties of the ground state as a function of density to the effect of the exclusion principle and show that the one-particle distribution function undergoes a significant change when bosonic behavior appears.
Groundstate properties of the BCS-Bose Einstein crossover in a dx2−y2d_{x^{2}-y^{2}}dx2−y2​ wave superconductor
We examine the groundstate properties of the crossover between BCS superconductivity and Bose Einstein condensation within a model which exhibits dx2−y2d_{x^{2} - y^{2}}dx2−y2​ pairing symmetry. We compare results for zero temperature with known features of s-wave systems and show that bosonic degrees of freedom are likely to emerge only in the dilute limit. We relate the change in properties of the groundstate as a function of density to the effect of the exclusion principle and show that the one particle distribution function undergoes a significant change when bosonic behavior appears.
We study the effect of the next-nearest-neighbor (NNN) hopping matrix element ${t}^{\ensuremath{'}}$ on the zero-temperature (T=0K)(T=0\mathrm{K})(T=0K) crossover phase diagram, namely, from the BCS limit to the Bose-Einstein regime, for a model which presents a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor order parameter symmetry. This theoretical generalization is required by angle-resolved photoemission spectroscopy experiments. Our NNN hopping is characterized by the parameter $\ensuremath{\alpha}\ensuremath{\equiv}{t}^{\ensuremath{'}}/t\ensuremath{\ne}0,$ where t is the nearest neighbor hopping matrix element. For \ensuremath{\alpha}\ensuremath{\geqslant}0, there is a crossover of BCS binding energy type as described [B. C. den Hertog, Phys. Rev. B 60, 559 (1999)] by keeping the density fixed. However, for \ensuremath{\alpha}0, the system is always in the Bose-Einstein regime for $n~{n}_{c}\ensuremath{\approx}0.12.$ We conclude that the presence of the NNN hopping matrix element drastically influences the crossover diagram for a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductor.
In this paper we study the BCS Bose-Einstein condensation (BEC) crossover scenario within the superconducting state, using a T-matrix approach which yields the ground state proposed by Leggett. Here we extend this ground state analysis to finite temperatures T and interpret the resulting physics. We find two types of bosoniclike excitations of the system: long lived, incoherent pair excitations and collective modes of the superconducting order parameter, which have different dynamics. Using a gauge invariant formalism, this paper addresses their contrasting behavior as a function of T and superconducting coupling constant g. At a more physical level, our paper emphasizes how, at finite T, BCS-BEC approaches introduce an important parameter ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2}={\ensuremath{\Delta}}^{2}\ensuremath{-}{\ensuremath{\Delta}}_{\mathrm{sc}}^{2}$ into the description of superconductivity. This parameter is governed by the pair excitations and is associated with particle-hole asymmetry effects that are significant for sufficiently large g. In the fermionic regime, ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2}$ represents the difference between the square of the excitation gap ${\ensuremath{\Delta}}^{2}$ and that of the superconducting order parameter ${\ensuremath{\Delta}}_{\mathrm{sc}}^{2}.$ The parameter ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2},$ which is necessarily zero in the BCS (mean field) limit increases monotonically with the strength of the attractive interaction g. It follows that there is a significant physical distinction between this BCS-BEC crossover approach (in which g is the essential variable which determines ${\ensuremath{\Delta}}_{\mathrm{pg}})$ and the widely discussed phase fluctuation scenario in which the plasma frequency is the tuning parameter. Finally, we emphasize that in the strong coupling limit, there are important differences between the composite bosons that arise in crossover theories and the usual bosons of the (interacting) Bose liquid. Because of constraints imposed on the fermionic excitation gap and chemical potential, in crossover theories, the fermionic degrees of freedom can never be fully removed from consideration.
The work by Soares et al. [Phys. Rev. B 65, 174506 (2002)] investigates the BCS to Bose-Einstein crossover for d-wave pairing in the two-dimensional attractive Hubbard model. Contrary to their claims, we found that a nonpairing region does not exist in the density vs coupling phase diagram. The gap parameter at T=0,T=0,T=0, as obtained by solving analytically as well as numerically the BCS equations, is in fact finite for any nonzero density and coupling, even in the weak-coupling regime.
We study a gas of fermions undergoing a wide resonance sss-wave BCS--Bose-Einstein-condensate (BEC) crossover, in the BEC regime at zero temperature. We calculate the chemical potential and the speed of sound of this Bose-Einstein-condensed gas, as well as the condensate depletion, in the low-density approximation. We discuss how higher-order terms in the low-density expansion can be constructed. We demonstrate that the standard BCS-BEC gap equation is invalid in the BEC regime and is inconsistent with the results obtained here. The low-density approximation we employ breaks down in the intermediate BCS-BEC crossover region. Hence our theory is unable to predict how the chemical potential and the speed of sound evolve once the interactions are tuned towards the BCS regime. As a part of our theory, we derive the well-known result for the bosonic scattering length diagrammatically and check that there are no bound states of two bosons.
Physical properties of two-dimensional (2D) superconductors is an active area of long time researches in which many, even principle, questions are still open. One of them is a possibility of a smooth crossover of the system from the standard Bardeen–Cooper–Schrieffer (BCS) regime of strongly overlapping Cooper pairs to the regime of Bose–Einstein condensation (BEC) of almost non-overlapping local pairs. The adequate to the real process description of this transition which can happen either by changing the doping or temperature would lead to a better understanding of the superconducting properties of materials. Though the crossover is not observed in one of the most important layered systems—namely, cuprate superconductors—we demonstrate that in systems with stronger electron-electron attraction it is possible. In particular, in the d-wave channel at sufficiently high doping the crossover can take place with growth of temperature two times, i.e., as BEC→BCS, and followed by reverse BCS→BEC condensation. The physical mechanisms of such transitions and possibility to detect them experimentally in real systems are discussed.