The BCS–BEC crossover: From ultra-cold Fermi gases to nuclear systems

Abstract

This report adresses topics and questions of common interest in the fields of ultra-cold gases and nuclear physics in the context of the BCS-BEC crossover. By this crossover, the phenomena of Bardeen-Cooper-Schrieffer (BCS) superfluidity and Bose-Einstein condensation (BEC), which share the same kind of spontaneous symmetry breaking, are smoothly connected through the progressive reduction of the size of the fermion pairs involved as the fundamental entities in both phenomena. This size ranges, from large values when Cooper pairs are strongly overlapping in the BCS limit of a weak inter-particle attraction, to small values when composite bosons are non-overlapping in the BEC limit of a strong inter-particle attraction, across the intermediate unitarity limit where the size of the pairs is comparable with the average inter-particle distance.The BCS-BEC crossover has recently been realized experimentally, and essentially in all of its aspects, with ultracold Fermi gases. This realization, in turn, has raised the interest of the nuclear physics community in the crossover problem, since it represents an unprecedented tool to test fundamental and unanswered questions of nuclear manybody theory. Here, we focus on the several aspects of the BCS-BEC crossover, which are of broad joint interest to both ultra-cold Fermi gases and nuclear matter, and which will likely help to solve in the future some open problems in nuclear physics (concerning, for instance, neutron stars). Similarities and differences occurring in ultra-cold Fermi gases and nuclear matter will then be emphasized, not only about the relative phenomenologies but also about the theoretical approaches to be used in the two contexts. Common to both contexts is the fact that at zero temperature the BCS-BEC crossover can be described at the mean-field level with reasonable accuracy. At finite temperature, on the other hand, inclusion of pairing fluctuations beyond mean field represents an essential ingredient of the theory, especially in the normal phase where they account for precursor pairing effects.After an introduction to present the key concepts of the BCS-BEC crossover, this report discusses the mean-field treatment of the superfluid phase, both for homogeneous and inhomogeneous systems, as well as for symmetric (spinor isospin-balanced) and asymmetric (spin-or isospin-imbalanced) matter. Pairing fluctuations in the normal phase are then considered, with their manifestations in thermodynamic and dynamic quantities. The last two Sections provide a more specialized discussion of the BCS-BEC crossover in ultra-cold Fermi gases and nuclear matter, respectively. The separate discussion in the two contexts aims at cross communicating to both communities topics and aspects which, albeit arising in one of the two fields, share a strong common interest.

BCS-BEC Crossover and Pairing Fluctuations in a Two Band Superfluid/Superconductor: A T Matrix Approach

“…Clearly, the NSR and T matrix approaches are essentially equivalent when the self-energy corrections are small, while when this does not occur, quantitative differences are expected in their results for thermodynamic quantities, such as, e.g., the chemical potential µ and critical temperature T c . For the BCS-BEC crossover in a single band, these differences turn out to be rather moderate across the whole BCS-BEC crossover, even when the self-energy is not small [21]. We will show below that the same occurs in the two band system considered in this work.…”

“…As another direction toward the unconventional properties of superconductors, the crossover from the weakly coupled BCS state to the Bose-Einstein condensate (BEC) of a tightly bound molecule with increasing the two body attractive interaction has extensively been discussed [15,16]. In fact, this crossover has been realized in ultracold Fermi atomic gases [17,18], and currently, various properties and fluctuation phenomena of this system have been investigated theoretically and experimentally [19][20][21]. Interestingly, recent experiments of FeSe multi-band superconductors also indicated that this electron system is in the BCS-BEC crossover regime [22][23][24].…”

Bardeen-Cooper-Schrieffer–type pairing in a spin- 12 Bose gas with spin-orbit coupling

“…Following the standard prescription for the BCS-BEC crossover problem [3], we substitute the bare interaction strength g between the ↑ and ↓ bosons with the associated s-wave scattering length a s in vacuum through the relation 1/g = −mV /(4πa s )+ k 1/(2 k ), where V is the volume. In addition, we define a length scale k 0 through an analogy with the number equation N 0 = k 3 0 V /(3π 2 ) of a free Fermi gas at T = 0, along with the corresponding energy scale 0 = k 2 0 /(2m).…”

“…This intrinsic stability is what lies behind the long-sought realization of the so-called BCS-BEC crossover, when a two-component Fermi gas is magnetically swept across a Feshbach resonance [1]. Having witnessed more than a decade of tremendous successes since their creation, the ultra-cold Fermi gases has become a thriving field in modern quantum physics as it keeps enriching its toolbox with a widerange of applications for the strongly-correlated phenomena in a much broader context [2,3].…”

“…Then, assuming that the instability towards a BCS-like paired state is favored against the competing states, e.g., collapse, fragmentation, phase separation, etc. [4-11, 20, 21], one may treat the inter-component attraction through a close analogy with the theory of paired fermions [2,3]. For this purpose, we apply the functional path-integral approach, and derive a mean-field theory that is based on the saddle-point approximation for pairing, and then a Ginzburg-Landau theory by including the Gaussian fluctuations on top.…”