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arXiv:cond-mat/0306726

Generalized Cooper pairing and Bose-Einstein condensation

Valdir C. Aguilera-Navarro, Mauricio Fortes, MANUEL DE LLANO

Published Jun 30, 2003• 0 citations
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Abstract

A Bethe-Salpeter treatment of Cooper pairs (CPs) based on an ideal Fermi gas (IFG) "sea"' yields the familiar negative-energy, two-particle bound-state if two-hole CPs are ignored, but is meaningless otherwise as it gives purely-imaginary energies. However, when based on the BCS ground state, legitimate two-particle "moving" CPs emerge but as positive-energy, finite-lifetime resonances for nonzero center-of-mass momentum, with a \textit{linear}dispersion leading term. Bose-Einstein condensation of such pairs may thus occur in exactly two dimensions as it cannot with quadratic dispersion.

DOI
10.48550/arxiv.cond-mat/0306726

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A Bethe-Salpeter treatment of Cooper pairs (CPs) based on an ideal Fermi gas (IFG) "sea"' yields the familiar negative-energy, two-particle bound-state if two-hole CPs are ignored, but is meaningless otherwise as it gives purely-imaginary energies. However, when based on the BCS ground state, legitimate two-particle "moving" CPs emerge but as positive-energy, finite-lifetime resonances for nonzero center-of-mass momentum, with a \textit{linear}dispersion leading term. Bose-Einstein condensation of such pairs may thus occur in exactly two dimensions as it cannot with quadratic dispersion.
We review Cooper pairing starting from its simplest, original 1956 version of two electrons interacting above the Fermi sea of an ideal Fermi gas (IFG). The two-electron interaction assumed extensively (if not exclusively), is the attractive two-parameter Cooper, and then BCS, model interactions. Hole Cooper pairs (CPs) and electron-hole CPs are then included along with the initial electron-CPs in terms of the single-fermion Green functions implied by the Bethe-Salpeter (BS) integral equation in the ladder approximation. A purely-imaginary CP energy "instability" is recovered that is well-documented in the literature at least since the late 1950's. A novel interpretation of this instability is that an unperturbed Hamiltonian different from the IFG one first used by Cooper suffices to obtain meaningful CPs. Instead of the IFG sea, a BCS-correlated Fermi "sea" used in the BS equation interpreted as the associated unperturbed Hamiltonian leads to real CP energies (with small imaginary terms implying damping). We survey how this has been achieved in 1D, 2D and 3D, and give a more detailed treatment in 2D. A vital distinction is that the original and generalized CPs are true bosons in contrast with BCS pairs that are not ordinary bosons but rather "hard-core bosons" as they do not obey strict Bose commutation rules. Another important common element of the original or generalized CPs (particularly in 2D where ordinary Bose-Einstein condensation (BEC) does not occur) is their linear dispersion relation in leading order in the total (or, center-of-mass) momentum power-series expansion of the CP energy. This theory encompasses, in principle, all empirically known superconductors including quasi-2D superconductors such as cuprates and the ET organic compounds, as well as quasi-1D ones such as the organometallic Bechgaard salts and nanotubes.
Cooper pairing in two dimensions is analyzed with a set of renormalized equations to determine its binding energy for any fermion number density and all coupling assuming a generic pairwise residual interfermion interaction. Also considered are Cooper pairs (CP's) with nonzero center-of-mass momentum (CMM) and their binding energy is expanded analytically in powers of the CMM up to quadratic terms. A Fermi-sea-dependent linear term in the CMM dominates the pair excitation energy in weak coupling (also called the BCS regime) while the more familiar quadratic term prevails in strong coupling (the Bose regime). The crossover, though strictly unrelated to BCS theory per se, is studied numerically as it is expected to play a central role in a model of superconductivity as a Bose-Einstein condensation of CPs where the transition temperature vanishes for all dimensionality d< 2d<~2d< 2 for quadratic dispersion, but is nonzero for all d> 1d>~1d> 1 for linear dispersion.
Unification of the Bardeen, Cooper and Schrieffer (BCS) and the Bose‐Einstein condensation (BEC) theories is surveyed in terms of a generalized BEC (GBEC) finite‐temperature statistical formalism. A vital distinction is that Cooper pairs (CPs) are true bosons that may suffer a BEC since they obey BE statistics, in contrast with BCS pairs that are “hard‐core bosons” at best. A second crucial ingredient is the explicit presence of hole‐pairs (2h) alongside the usual electron‐pairs (2e). A third critical element (particularly in 2D where ordinary BEC does not occur) is the linear dispersion relation of CPs in leading order in the center‐of‐mass momentum (CMM) power‐series expansion of the CP energy. The GBEC theory reduces in limiting cases to all five continuum (as opposed to “spin”) statistical theories of superconductivity, from BCS on one extreme to the BEC theory on the other, as well as to the BCS‐Bose “crossover” picture and the 1989 Friedberg‐Lee BEC theory. It accounts for 2e‐ and 2h‐CPs in arbitrary proportions while BCS theory can be deduced from the GBEC theory but allows only equal (50%‐50%) BE condensed‐mixtures of both kinds of CPs. As it yields the precise BCS gap equation for all temperatures as well as the precise BCS zero‐temperature condensation energy for all couplings, it suggests that the BCS condensate is a BE condensate of a ternary mixture of kinematically independent unpaired electrons coexisting with equally proportioned weakly‐bound zero‐CMM 2e‐ and 2h‐CPs. Without abandoning the electron‐phonon mechanism in moderately weak coupling, and fortuituously insensitive to the BF interactions, the GBEC theory suffices to reproduce the unusually high values of Tc (in units of the Fermi temperature TF) of 0.01–0.05 empirically found in the so‐called “exotic” superconductors of the Uemura plot, including cuprates, in contrast to the low values of Tc/TF⩽10−3 roughly reproduced by BCS theory for conventional (mostly elemental) superconductors.
Although BCS pairs of fermions are known to obey neither Bose–Einstein (BE) commutation relations nor BE statistics, we show how Cooper pairs (CPs), whether the simple original ones or the CPs recently generalized in a many-body Bethe–Salpeter approach, being clearly distinct from BCS pairs at least obey BE statistics. Hence, contrary to widespread popular belief, CPs can undergo BE condensation to account for superconductivity if charged, as well as for neutral-atom fermion superfluidity where CPs, but uncharged, are also expected to form.