The dynamics of BCS (Bardeen-Cooper-Schrieffer) superconductors is fairly well understood due to the availability of a mean-field solution for the pairing Hamiltonian, a solution which gives the quantum state of a superconductor as a state of almost-free fermions interacting only with a condensate. As a result, transition probabilities may be computed, and expressed in terms of matrix elements of electron creation and annihilation operators between approximate eigenstates. These matrix elements are also called ``coherence factors.'' Mean-field theory is however not sufficient to describe all eigenstates of a superconductor, a deficiency which is hardly important in (or very close to) equilibrium, but one that becomes relevant in certain out-of-equilibrium situations. We report here on a computation of matrix elements (coherence factors) for the pairing Hamiltonian between any ``two-arc'' eigenstates in the thermodynamic limit.
The dynamics of BCS (Bardeen-Cooper-Schrieffer) superconductors is fairly well understood due to the availability of a mean-field solution for the pairing Hamiltonian, a solution which gives the quantum state of a superconductor as a state of almost-free fermions interacting only with a condensate. As a result, transition probabilities may be computed, and expressed in terms of matrix elements of electron creation and annihilation operators between approximate eigenstates. These matrix elements are also called coherence factors.'' Mean-field theory is however not sufficient to describe all eigenstates of a superconductor, a deficiency which is hardly important in (or very close to) equilibrium, but one that becomes relevant in certain out-of-equilibrium situations. We report here on a computation of matrix elements (coherence factors) for the pairing Hamiltonian between any two-arc'' eigenstates in the thermodynamic limit.
We present a derivation of a previously announced result for matrix elements between exact eigenstates of the pairing Hamiltonian. Our results, which generalize the well-known Bardeen–Cooper–Schrieffer (BCS) (Bardeen et al 1957 Phys. Rev. 108 1175; 1957 Phys. Rev. 106 162) expressions for what are known as ‘coherence factors’, are derived based on the Slavnov (1989 Theor. Math. Phys. 79 502) formula for overlaps between Bethe-ansatz states, thus making use of the known connection between the exact diagonalization of the BCS Hamiltonian, due to Richardson (1963 Phys. Lett. 3 277; 1964 Nucl. Phys. A 52 221), and the algebraic Bethe ansatz. The resulting formula has a compact form after a suitable parameterization of the energy plane. Although we apply our method here to the pairing Hamiltonian, it may be adjusted to study what is termed the ‘Sutherland limit’ (Sutherland 1995 Phys. Rev. Lett. 74 816) for exactly solvable models, namely where a macroscopic number of rapidities form a large string.
BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).
We review efforts to unify both the Bardeen, Cooper and Schrieffer (BCS) and Bose-Einstein condensation (BEC) pictures of superconductivity. We have finally achieved this in terms of a "\textit{complete} boson-fermion (BF) model" (CBFM) that reduces in special cases to all the main continuum (as opposed to "spin") statistical theories of superconductivity. Our BF model is "complete" in the sense that not only two-electron (2e) but also two-hole (2h) Cooper pairs (CPs) are allowed in arbitrary proportions. In contrast, BCS-Bogoliubov theory--which can also be considered as the theory of a mixture of kinematically independent electrons, 2e- and 2h-CPs--allows only equal, 50%-50%, mixtures of the two kinds of CPs. This is obvious from the perfect symmetry about μ, the electron chemical potential, of the well-known Bogoliubov v2(ε) and u2(ε) coefficients, where ε is the electron energy. The CBFM is then applied to see: a) whether the BCS model interaction for the electron-phonon dynamical mechanism is sufficient to predict the unusually high values of Tc (in units of the Fermi temperature) of ≃0.01−0.1 exhibited by the so-called ``exotic'' superconductors \cite{Brandow} in both 2D and 3D--relative to the low values of ≲10−3 more or less correctly predicted by BCS theory for conventional, elemental superconductors; and b) whether it can at least suggest, if not explain, why "hole superconductors" have higher Tc's.
Abstract The superfluid density ns(T) n s ( T ) of a superconductor is calculated based on the generalized Bose–Einstein condensation (GBEC) theory that addresses a fully-interacting ternary boson-fermion gas mixture of free electrons as fermions, plus two-electron Cooper pairs (2eCPs) and also, explicitly, two-hole Cooper pairs (2hCPs), both as bosons. Here we consider two special cases (i) 100%–0% (i.e., with no condensed 2hCPs) and (ii) 0%–100% (i.e., with no condensed 2eCPs). Subsumed in GBEC are the Bardeen–Cooper–Schrieffer (BCS) and Bose–Einstein condensation (BEC) theories along with the BCS-BEC crossover theory extended with 2hCPs. We find that in the weak-coupling regime ns(0) n s ( 0 ) agrees with data from the Uemura et al. (2004) graph for several elemental SCs by taking in 3D with a quadratic energy-dispersion relation while in 2D with a linear relation are much too far below the data. In the strong-coupling regime the linear behavior of critical temperature Tc T c vs ns(0) n s ( 0 ) obtained here is just as Božović et al. (2016) found. However, in 2D with a linear relation accounting for 0%–100%, ns(T)/ns(0) n s ( T ) / n s ( 0 ) compares well with some high- Tc T c -cuprate SC data between the two coupling regimes.
Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.
The problem of the evolution from BCS theory with cooperative Cooper pairing to the formation and condensation of composite bosons has attracted considerable attention for the past several decades. It has gained renewed impetus in the mid-eighties with the discovery of the high-Tc superconductors, which have a coherence length comparable to the interparticle spacing. More recently, this subject has spurred a great deal of research activity in connection with experiments involving dilute atomic gases of fermionic atoms. The initial objective of this work will be to use functional integral techniques to analyze the low-temperature BCS-to-BEC evolution of d-wave superconductors within the saddle point (mean field) approximation for a continuum model. Then, the same mathematical formalism will be applied to the problem of the BCS-to-BEC evolution of fully spin-polarized p-wave Fermi gases in two dimensions. We find that a quantum phase transition occurs for both systems as they are driven from the BCS-like regime of weakly interacting fermionic pairs to the opposite BEC-like regime of strongly interacting bosonic molecules. This is in contrast to the smooth crossover predicted and observed in systems that exhibit s-wave pairing symmetry. We calculate several spectroscopic and thermodynamic properties that signal the occurrence of this phase transition, and suggest some possible experimental realizations. Finally, fluctuations about the saddle point solution are included in the calculations, and the effects of such correction are analyzed in the low (T~0) and high (T~Tc) temperature limits. We conclude that, at high temperatures, the bosonic degrees of freedom that arise from two-particle bound states become essential to describe the strong coupling limit,\nas the saddle point approximation alone becomes unreliable.