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Physics and Chemistry of Liquids

Gap equation for anyon superfluids

G. G. N. Angilella, Norman Henry March, Fabio Siringo, R. Pucci

Published May 6, 2006• 1 citation
Publisher

Abstract

The free energy for superconducting electrons in the Bardeen–Cooper–Schrieffer (BCS) framework is first generalized to apply to superfluid anyons. By appropriate minimization of this free energy, the gap equation for superfluid anyons is then derived. As an application, the energy gap Δ (0), in reduced energy units k B Tc , with Tc the transition temperature, is obtained as a function of the anyon statistics parameter.

DOI
10.1080/00319100600740983

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The free energy for superconducting electrons in the Bardeen–Cooper–Schrieffer (BCS) framework is first generalized to apply to superfluid anyons. By appropriate minimization of this free energy, the gap equation for superfluid anyons is then derived. As an application, the energy gap Δ (0), in reduced energy units k B Tc , with Tc the transition temperature, is obtained as a function of the anyon statistics parameter.
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.
A nonperturbative argument is used to derive a one-electron Hamiltonian whose eigenfunctions are in one-to-one correspondence with electron or hole quasiparticles of an N-electron system. Nonlocal exchange is included exactly, and self-energy terms due to electronic correlation are included at the level of approximation of a given N-electron wave function. When parametrized by quasiparticle occupation numbers, this one-electron Hamiltonian is the functional derivative of an exact N-electron energy functional, as in the Landau theory of interacting fermion systems. The present nonperturbative derivation extends the Landau theory to ordered or localized systems for which many-body perturbation theory is not directly applicable. Analysis of the Bardeen-Cooper-Schrieffer (BCS) reduced Hamiltonian for superconductors leads to a temperature-dependent energy gap and a transition temperature in good agreement with the BCS theory, in a formalism that conserves the number of electrons.
Abstract The superfluid density ns(T)n_{s}(T)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)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 TcT_{c}Tc​ T c vs ns(0)n_{s}(0)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)ns​(T)/ns​(0) n s ( T ) / n s ( 0 ) compares well with some high- TcT_{c}Tc​ T c -cuprate SC data between the two coupling regimes.
We show how critical temperatures Tc from the BCS-Bose crossover theory (defined self-consistently by both the gap and fermion-number equations) requires unphysically large couplings for the Cooper/BCS model interaction to differ significantly from the Tc of ordinary BCS theory (where the number equation is replaced by the assumption that the chemical potential equals the Fermi energy). Furthermore, within a recent generalization of Bose-Einstein condensation (GBEC) that includes electron and also, without loss of generality, hole Cooper pairs (CPs), both species interacting with unpaired electrons and holes, it is shown that although explicitly ignoring either hole- or electron CPs yields the precise BCS gap equation for all temperatures T, only half of the T = 0 BCS rigorous-upper-bound condensation energy is obtained.
We develop a theory of superconductivity (or superfluidity) based on condensed fermion quartets focusing on the dilute spin-12\frac{1}{2}21​ systems at zero temperature. In the spirit of the Bardeen--Cooper--Schrieffer ansatz, a variational wavefunction is constructed such that, within the so-called ``dilute quartet approximation", it is the ground state of an effective quartic Hamiltonian. For a given two-body interaction in favor of quartetting, the gap parameter is suitably defined and the gap equation is also derived. As to the excited states, an intuitive physical picture based on a sixteen-dimensional ``occupation space" is depicted and the associated eigen-energies are obtained. This theory is applied to compute the superfluid fraction, which is found to be the same as in conventional superconductors, despite the interacting nature of the quartet problem.