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arXiv:1506.07334 · Physical Review A

High-temperature expansion for interacting fermions

Mingyuan Sun, X. Leyronas

Published Nov 13, 2015• 26 citations
arXivHTMLPDF

Abstract

We present a general method for the high-temperature expansion of the self-energy of interacting particles. Although the method is valid for fermions and bosons, we illustrate it for spin-one-half fermions interacting via a zero range potential, in the Bose-Einstein-condensate--Bardeen-Cooper-Schrieffer (BEC-BCS) crossover. The small parameter of the expansion is the fugacity $z$. Our results include terms of order $z$ and ${z}^{2}$, which take into account, respectively, two- and three-body correlations. We give results for the high-temperature expansion of Tan's contact at order ${z}^{3}$ in the whole BEC-BCS crossover. We apply our method to calculate the spectral function at the unitary limit. We find structures that are different from those discussed in previous approaches, which included only two-body correlations. This shows that including three-body correlations can play an important role in the structures of the spectral function.

DOI
10.1103/physreva.92.053611

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We present a general method for the high-temperature expansion of the self-energy of interacting particles. Although the method is valid for fermions and bosons, we illustrate it for spin-one-half fermions interacting via a zero range potential, in the Bose-Einstein-condensate--Bardeen-Cooper-Schrieffer (BEC-BCS) crossover. The small parameter of the expansion is the fugacity zzz. Our results include terms of order zzz and z2{z}^{2}z2, which take into account, respectively, two- and three-body correlations. We give results for the high-temperature expansion of Tan's contact at order z3{z}^{3}z3 in the whole BEC-BCS crossover. We apply our method to calculate the spectral function at the unitary limit. We find structures that are different from those discussed in previous approaches, which included only two-body correlations. This shows that including three-body correlations can play an important role in the structures of the spectral function.
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Strongly interacting Fermi gas systems play an important role in many areas of physics. Their systematic analytical study has been a difficult theoretical endeavour as it is outside the reach of ordinary perturbation theory. Much progress has been made with the development of the large-N expansion, where a fermion is generalized to have N (or 2N) components and the Lagrangian’s global symmetry is extended to the unitary group U(N) (or the compact symplectic group USp(2N)). The chosen symmetry group dictates the system’s properties as U(N) enhances the fermion-hole interaction channel and suppresses the fermion-fermion channel, whereas USp(2N) treats both channels equally by allowing two independent interaction terms in the Lagrangian density, the density-density interaction which enhances the fermion-hole channel and the pairing interaction which enhances the fermion-fermion channel. Literature has been either only focusing on the pairing interaction term or including both but restricting their respective coupling constants to be the same. The most general renormalizable quantum field theory (QFT) compatible with a global USp(2N)-symmetry contains both interaction terms with independent, dimensionless coupling constants. By studying such a system at zero temperature, we want to understand how the two interaction terms influence each other by independently tuning the coupling constants from the attractive to the repulsive regime. To this end, we calculate the ground state energy density to leading order in the large-N expansion in four different cases which are classified by the coupling constant regime and map out the phase diagram. We find that the density-density interaction renders the theory unstable if its coupling constant g₁ is tuned above 2π and that the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensate (BEC) crossover extends to 𝜇̃ /(ϵB) = −g₁/(2π) for g₁ < 2π, where ϵB is the Cooper pair binding energy in vacuum and 𝜇̃ the renormalized chemical potential. Furthermore, we find an exotic phase at the phase boundary (g₁= 2π, 𝜇̃ = −ϵB ) resulting from a flat direction in the leading-order effective potential, where the degenerate ground states span the entire BCS–BEC crossover.
We investigate the 1/N1/N1/N expansion proposed recently as a strategy to include quantum fluctuation effects in the nonrelativistic, attractive Fermi gas at and near unitarity. We extend the previous results by calculating the next-to-leading order corrections to the critical temperature along the whole crossover from Bardeen-Cooper-Schrieffer (BCS) superconductivity to Bose-Einstein condensation. We demonstrate explicitly that the extrapolation from the mean-field approximation, based on the 1/N1/N1/N expansion, provides a useful approximation scheme only on the BCS side of the crossover. We then apply the technique to the study of strongly interacting relativistic many-fermion systems. Having in mind the application to color superconductivity in cold dense quark matter, we develop, within a simple model, a formalism suitable to compare the effects of order parameter fluctuations in phases with different pairing patterns. Our main conclusion is that the relative correction to the critical temperature is to a good accuracy proportional to the mean-field ratio of the critical temperature and the chemical potential. As a consequence, it is significant even rather deep in the BCS regime, where phenomenologically interesting values of the quark--quark coupling are expected. Possible impact on the phase diagram of color-superconducting quark matter is discussed.
We use the functional renormalization group approach with partial bosonization in the particle-particle channel to study the effect of order parameter fluctuations on the BCS-Bose-Einstein condensate (BEC) crossover of superfluid fermions in three dimensions. Our approach is based on a new truncation of the vertex expansion where the renormalization group flow of bosonic two-point functions is closed by means of Dyson-Schwinger equations and the superfluid order parameter is related to the single-particle gap via a Ward identity. We explicitly calculate the chemical potential, the single-particle gap, and the superfluid order parameter at the unitary point and compare our results with experiments and previous calculations.