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

arXiv:0806.3492

Cooper pairing reexamined

Mauricio Fortes, MANUEL DE LLANO, M. A. Solís

Published Jun 21, 2008• 0 citations
arXivHTMLPDF

Abstract

When both two-electron \textit{and} two-hole Cooper-pairing are treated on an equal footing in the ladder approximation to the Bethe-Salpeter (BS) equation, the zero-total-momentum Cooper-pair energy is found to have two \textit{real} solutions $\mathcal{E}_{0}^{BS}=\pm 2\hbar ω_{{D}%}/\sqrt{{e}^{2/λ}+{1}}$ which coincide with the zero-temperature BCS energy gap $Δ=\hbar ω_{D}/\sinh (1/λ) $ in the weak coupling limit. Here, $\hbar ω_{D}$ is the Debye energy and $λ\geq 0$ the BCS model interaction coupling parameter. The interpretation of the BCS energy gap as the binding energy of a Cooper-pair is often claimed in the literature but, to our knowledge, never substantiated even in weak-coupling as we find here. In addition, we confirm the two purely-\textit{imaginary} solutions assumed since at least the late 1950s as the \textit{only} solutions, namely, $\mathcal{E}_{0}^{BS}=\pm i2\hbar ω_{D}/\sqrt{{e}^{2/λ}{-1}}.$

DOI
10.48550/arxiv.0806.3492

Related papers

GENERALIZED COOPER PAIRING IN SUPERCONDUCTORS

MANUEL DE LLANO, James F. Annett

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.

International Journal of Modern Physics B•Aug 20, 2007•19 citations

Generalized Cooper pairing and Bose-Einstein condensation

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

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.

arXiv:cond-mat/0306726•Jun 30, 2003•0 citations

BEC, BCS and BCS-Bose Crossover Theories in Superconductors and Superfluids

MANUEL DE LLANO, Francisco J. Sevilla, M. A. Solís +1 more

For the Cooper/BCS model interaction in superconductors (SCs) it is shown: a) how BCS-Bose crossover picture transition temperatures Tc, defined self-consistently by both the gap and fermion-number equations, require unphysically large couplings to differ significantly from the Tc of ordinary BCS theory defined without the number equation since here the chemical potential is assumed equal to the Fermi energy; how although ignoring either hole- or electron-Cooper-pairs in the recent "complete boson-fermion model": b) one obtains the precise BCS gap equation for all temperatures T, but c) only half the zero-temperature BCS condensation energy emerges. Results (b) and (c) are also expected to hold for neutral-fermion superfluids (SFs)--such as liquid $^3$He, neutron matter and trapped ultra-cold fermion atomic gases--where the pair-forming two-fermion interaction of course differs from the Cooper/BCS one for SCs.

arXiv:cond-mat/0412418•Dec 16, 2004•0 citations

Cooper Pairing Revisited

Valdir C. Aguilera-Navarro, Mauricio Fortes, MANUEL DE LLANO +1 more

We recall the fundamental fact that Cooper pairs defined without ignoring two- hole pairs along with two-particle ones leads to a purely imaginary pair energy when the problem is based on the ideal Fermi gas sea. However, bound finite- lifetime pairs are recovered when it is based on the BCS ground-state Fermi sea. Their excitation energy is gapped at zero total momenta and then rises linearly in either 2D or 3D, making them similar to 3D plasmons which however rise quadratically.

arXiv:cond-mat/0312001•Nov 28, 2003•0 citations

Two-Cooper-pair problem and the Pauli exclusion principle

W. V. Pogosov, Monique Combescot, Michel Crouzeix

While the one-Cooper-pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for $N=2$ already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to $N$ pairs but no precise description of this continuous change has been given so far. Using Richardson's procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs---the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.

arXiv:0911.1688•Physical Review B•May 13, 2010•22 citations

Role of superconducting energy gap in extended BCS-Bose crossover theory

Israel Chávez, L. A. García, MANUEL DE LLANO +1 more

The generalized Bose–Einstein condensation (GBEC) theory of superconductivity (SC) is briefly surveyed. It hinges on three distinct new ingredients: (i) Treatment of Cooper pairs (CPs) as actual bosons since they obey Bose statistics, in contrast to BCS pairs which do not obey Bose commutation relations; (ii) inclusion of two-hole Cooper pairs (2hCPs) on an equal footing with two-electron Cooper pairs (2eCPs), thus making this a complete boson–fermion (BF) model; and (iii) inclusion in the resulting ternary ideal BF gas with particular BF vertex interactions that drive boson formation/disintegration processes. GBEC subsumes as special cases both BCS (having its 50–50 symmetry of both kinds of CPs) and ordinary BEC theories (having no 2hCPs), as well as the now familiar BCS-Bose crossover theory. We extended the crossover theory with the explicit inclusion of 2hCPs and construct a phase diagram of [Formula: see text] versus [Formula: see text], where [Formula: see text] and [Formula: see text] are the critical and Fermi temperatures, [Formula: see text] is the total number density and [Formula: see text] that of unbound electrons at [Formula: see text]. Also, with this extended crossover one can construct the energy gap [Formula: see text] versus [Formula: see text] for some elemental SCs by solving at least two equations numerically: a gap-like and a number equation. In 50–50 symmetry, the energy gap curve agrees quite well with experimental data. But ignoring 2hCPs altogether leads to the gap curve falling substantially below that with 50–50 symmetry which already fits the data quite well, showing that 2hCPs are indispensable to describe SCs.

International Journal of Modern Physics B•Aug 21, 2017•3 citations
When both two-electron \textit{and} two-hole Cooper-pairing are treated on an equal footing in the ladder approximation to the Bethe-Salpeter (BS) equation, the zero-total-momentum Cooper-pair energy is found to have two \textit{real} solutions \mathcal{E}_{0}^{BS}=\pm 2\hbar ω_{{D}%}/\sqrt{{e}^{2/λ}+{1}} which coincide with the zero-temperature BCS energy gap Δ=ℏωD/sinh⁡(1/λ)Δ=\hbar ω_{D}/\sinh (1/λ) Δ=ℏωD​/sinh(1/λ) in the weak coupling limit. Here, ℏωD\hbar ω_{D}ℏωD​ is the Debye energy and λ≥0λ\geq 0λ≥0 the BCS model interaction coupling parameter. The interpretation of the BCS energy gap as the binding energy of a Cooper-pair is often claimed in the literature but, to our knowledge, never substantiated even in weak-coupling as we find here. In addition, we confirm the two purely-\textit{imaginary} solutions assumed since at least the late 1950s as the \textit{only} solutions, namely, E0BS=±i2ℏωD/e2/λ−1.\mathcal{E}_{0}^{BS}=\pm i2\hbar ω_{D}/\sqrt{{e}^{2/λ}{-1}}.E0BS​=±i2ℏωD​/e2/λ−1​.
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.
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.
For the Cooper/BCS model interaction in superconductors (SCs) it is shown: a) how BCS-Bose crossover picture transition temperatures Tc, defined self-consistently by both the gap and fermion-number equations, require unphysically large couplings to differ significantly from the Tc of ordinary BCS theory defined without the number equation since here the chemical potential is assumed equal to the Fermi energy; how although ignoring either hole- or electron-Cooper-pairs in the recent "complete boson-fermion model": b) one obtains the precise BCS gap equation for all temperatures T, but c) only half the zero-temperature BCS condensation energy emerges. Results (b) and (c) are also expected to hold for neutral-fermion superfluids (SFs)--such as liquid 3^33He, neutron matter and trapped ultra-cold fermion atomic gases--where the pair-forming two-fermion interaction of course differs from the Cooper/BCS one for SCs.
While the one-Cooper-pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for N=2N=2N=2 already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to NNN pairs but no precise description of this continuous change has been given so far. Using Richardson's procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs---the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.
The generalized Bose–Einstein condensation (GBEC) theory of superconductivity (SC) is briefly surveyed. It hinges on three distinct new ingredients: (i) Treatment of Cooper pairs (CPs) as actual bosons since they obey Bose statistics, in contrast to BCS pairs which do not obey Bose commutation relations; (ii) inclusion of two-hole Cooper pairs (2hCPs) on an equal footing with two-electron Cooper pairs (2eCPs), thus making this a complete boson–fermion (BF) model; and (iii) inclusion in the resulting ternary ideal BF gas with particular BF vertex interactions that drive boson formation/disintegration processes. GBEC subsumes as special cases both BCS (having its 50–50 symmetry of both kinds of CPs) and ordinary BEC theories (having no 2hCPs), as well as the now familiar BCS-Bose crossover theory. We extended the crossover theory with the explicit inclusion of 2hCPs and construct a phase diagram of [Formula: see text] versus [Formula: see text], where [Formula: see text] and [Formula: see text] are the critical and Fermi temperatures, [Formula: see text] is the total number density and [Formula: see text] that of unbound electrons at [Formula: see text]. Also, with this extended crossover one can construct the energy gap [Formula: see text] versus [Formula: see text] for some elemental SCs by solving at least two equations numerically: a gap-like and a number equation. In 50–50 symmetry, the energy gap curve agrees quite well with experimental data. But ignoring 2hCPs altogether leads to the gap curve falling substantially below that with 50–50 symmetry which already fits the data quite well, showing that 2hCPs are indispensable to describe SCs.