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arXiv:0911.1688 · Physical Review B

Two-Cooper-pair problem and the Pauli exclusion principle

W. V. Pogosov, Monique Combescot, Michel Crouzeix

Published May 13, 2010• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.81.174514

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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.
Chapter 10 deals with the exact solution to the BCS problem, as obtained by analytically solving Richardson-Gaudin equations. The chapter first provides a simple derivation of these equations and shows how to solve them for arbitrary N. The explicit N dependence of the N–Cooper pair energy obtained from this resolution provides invaluable insights into the role of the Pauli exclusion principle in paired fermion systems because the reduced BCS potential does not directly act between Cooper pairs. Revelation of this subtle point makes it possible to understand (i) why Cooper pairs can strongly overlap without breaking, whereas, at large density, an exciton system dissociates into an electron-hole plasma, and (ii) why the gap energy is far larger than the average pair energy. This understanding is further supported by the Cooper pair many-body parameter, which has the same form as the ones for Wannier or Frenkel excitons do.
This Letter provides the solution to a yet unsolved basic problem of Solid State Physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 year old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these NNN pairs via NNN parameters coupled through NNN non-linear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the NNN-pair energy depends on NNN as N(N−1)N(N-1)N(N−1) only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.
Typically visualized from an independent particle viewpoint, the Pauli principle's role in collective motion is analyzed leading to a reimagination of the microscopic dynamics underlying superfluidity/superconductivity and a reinterpretation of several interrelated phenomena: Cooper pairs, the Fermi sea, and Pauli blocking. The current approach, symmetry-invariant perturbation theory is a first principles method with no adjustable parameters. An adiabatic evolution is employed to transfer the well-known Pauli restrictions for identical, independent particles with two spin values to restrictions on the collective modes of an ensemble of ``spin up'' ``spin down'' particles. The collective modes, analytic N-body normal modes, are obtained from a group theoretic exact solution of the first-order equations. Cooper pairing is reinterpreted not as the pairing of two fermions with total zero momentum, but as the convergence of the momentum of the entire ensemble to two values, +k and -k, as the particles in the normal mode move back and forth with a single frequency and phase. The Fermi sea and Pauli blocking, commonly described using independent fermions that occupy lower states to create a ``sea'' in energy space and block occupation is redescribed as a collective energy phenomena of the entire ensemble. Superfluidity, which has always been viewed as a collective phenomena as Cooper pairs are assumed to condense into a macroscopic occupation of a single lowest state, is now reimagined without two-body pairing in real space, but as a macroscopic occupation of a low-energy phonon normal mode resulting in the convergence of the momentum to two equal and opposite values. The expected properties of superfluidity including the rigidity of the wave function, interactions between fermions in different pairs, convergence of the momentum and the gap in the excitation spectrum are discussed.
Relation between the binding energy of a pair of electrons in Cooper's configuration and the magnitude of the attractive interaction between them is investigated theoretically for a two-dimensional electron gas when the electron pair has (a) zero and (b) finite center of mass momentum (CMM) first (i) in the background of a passive Fermi sea and (ii) with inclusion of the effect of Pauli exclusion principle in the presence of an active Fermi sea, at zero temperature. In the Cooper's approach (Cooper 1956 Phys. Rev. 104 1189) two pairing electrons are considered above the Fermi sea. Due to Pauli exclusion principle, these two electrons are forbidden to go into the states whose energies are less than the Fermi energy. Consequently, the role of the electrons which form the Fermi sea is passive in formation of electron pair (Fujita and Godoy 2001 Theory of High Temperature Superconductivity (Kluwer Academic Publishers)). However, in the case of an active Fermi sea, implying possible presence of multiple pairs of electrons coming out of the Fermi sea, the two electrons will have less available region in the momentum space for pair formation due to the Pauli blocking effect. In both the situations, threshold values of attractive interaction are found to exist to bind the electrons having finite CMM even with respect to the Fermi energy while with arbitrarily small value of attractive interaction they can be bound only in the case of zero CMM and that too in the presence of a passive Fermi sea. However, the presence of an active Fermi sea strongly opposes the pair formation having lower CMM. Furthermore, raising the range of the attractive interaction below and above the Fermi surface makes the pair formation more difficult due to enhanced Pauli blocking in this case. In the case of a passive Fermi sea however, with the increase in the range of the attractive interaction it becomes easier for the electron pairs to be in both Cooper pair state (CP) and true bound state (TB). These results are totally distinct from those obtained by Randeria et al (1989 Phys. Rev. Lett. 62 981, 1990 Phys. Rev. B 41 327) that a fermionic bound state in true vacuum is a necessary and sufficient condition for the Cooper pairing instability in the presence of a passive Fermi sea, independent of attractive interaction strength implying that for a potential which is attractive everywhere in two dimension, a two-body bound state exists for an arbitrarily weak attraction. Our calculations show, in the mean field approximation, that co-existence of TB state and CP state with various CMMs can occur only beyond a critical attractive interaction strength. This could lead to the formation of incoherent pairs even under strong coupling situation, which may hint onto the pseudo gap formation leading to some of the exotic features in the anomalous normal state seen in some of the exotic superconductors.
We start from a Hamiltonian describing non-interacting fermions and add bosons to the model, with a Jaynes-Cummings-like interaction between the bosons and fermions. Because of the specific form of the interaction the model can be solved exactly. In the ground state, part of the electrons form bound pairs with opposite momentum and spin. The model also shows a gap in the kinetic energy of the fermions, but not in the spectrum of the full Hamiltonian. This gap is not of a mean-field nature, but is due to the Pauli exclusion principle.
This Letter provides the solution to a yet unsolved basic problem of solid state physics: the ground state energy of an arbitrary number of Cooper pairs interacting via the Bardeen-Cooper-Schrieffer potential. We here break a 50 yr old math problem by analytically solving Richardson-Gaudin equations which give the exact energy of these N pairs via N parameters coupled through N nonlinear equations. Our result fully supports the standard BCS result obtained for a pair number equal to half the number of states feeling the potential. More importantly, it shows that the interaction part of the N-pair energy depends on N as N(N-1) only from N=1 to the dense regime, a result which evidences that Cooper pairs interact via Pauli blocking only.