We consider imbalanced Fermi gases with strong attractive interactions, for which Cooper-pair formation plays an important role. The two-component mix-tures consist either of identical fermionic atoms in two different hyperfine states, or of two different atomic species both occupying only a single hyperfine state. In both cases, the number of atoms for each component is allowed to be differ-ent, which leads to a spin imbalance, or spin polarization. Two different atomic species also lead to a mass imbalance. Imbalanced Fermi gases are relevant to condensed-matter physics, nuclear physics and astroparticle physics. They have been studied intensively in recent years, following their experimental realization in ultracold atomic Fermi gases. The experimental control in such a system allows for a systematic study of the equation of state and the phase diagram as a function of temperature, spin polarization and interaction strength. In this re-view, we discuss the progress in understanding strongly-interacting imbalanced Fermi gases, where a main goal is to describe the results of the highly controlled experiments. We start by discussing Feshbach resonances, after which we treat the imbalanced Fermi gas in mean-field theory to give an introduction to the relevant physics. We encounter several unusual superfluid phases, including phase-separation, gapless Sarma superfluidity, and supersolidity. To obtain a more quantitative description of the experiments, we review also more sophisti-cated techniques, such as diagrammatic methods and the renormalization-group theory. We end the review by discussing two theoretical approaches to treat the inhomogeneous imbalanced Fermi gas, namely the Landau-Ginzburg theory and the Bogoliubov-de Gennes approach. Key words:
We investigate temperature effects in a Fermi gas with imbalanced spin populations. From the general expression of the thermal gap equation we find, in weak coupling limit , an analytical expression for the transition temperature T c as a function of various possibilities of chemical potential and mass asymmetries between the two particle species. For a range of asymmetry between certain specific values, this equation always has two solutions for T c , which has been interpreted as a reentrant phenomena or a pairing induced by a temperature effect. We show that the lower T c is never related to a stable solution. The same results are obtained in the strong coupling limit . The thermodynamical potential is carefully analyzed to avoid consideration of the unstable solutions. We also obtain the tricritical points for the chemical potential and mass imbalanced cases, and beyond these points we properly minimize the thermodynamic potential to find the stable and metastable first-order transition lines.
We present results on the effect of short-range, attractive interactions on the properties of balanced tw-dimensional Fermi gases in the nonsuperfluid (normal) phase. Our approach combines the renormalization group (RG) with perturbation theory, yielding observables such as the equation of state and compressibility. We find good agreement with recent experiments that measured the equation of state in trapped gases in the balanced regime, showing that these results are consistent with logarithmic corrections in the equation of state.
We theoretically study a possible pairing mechanism for a two-dimensional population imbalanced Fermi gas with short-range repulsive interactions, which can be realized on the upper branch of a Feshbach resonance. We use a well-controlled renormalization-group approach, which allows an unbiased study of the instabilities of imbalanced Fermi liquid without assumption of a broken symmetry and gives a numerical calculation of the transition temperature from microscopic parameters. Our results show a leading superfluid instability in the
p-wave channel for the majority species. The corresponding mechanism is that there are effective attractive interactions for the majority species, induced by the particle-hole susceptibility of the minority species, where the mismatch of the Fermi surfaces of the two species plays an important role. We also propose an experimental protocol for detecting the
p-wave superfluidity and discuss the corresponding experimental signatures.
We study the analytical structure of the effective action for spin- and mass-imbalanced Fermi mixtures at the onset of the superfluid state. Of our particular focus is the possibility of suppressing the tricritical temperature to zero, so that the transition remains continuous down to T = 0 and the phase diagram hosts a quantum critical point. At mean-field level we analytically identify such a possibility in a regime of parameters in dimensionality d = 3. In contrast, in d = 2 we demonstrate that the occurrence of a quantum critical point is (at the mean-field level) excluded. We show that the Landau expansion of the effective potential remains well-defined in the limit [Formula: see text] except for a subset of model parameters which includes the standard BCS limit. We calculate the mean-field asymptotic shape of the transition line. Employing the functional renormalization group framework we go beyond the mean field theory and demonstrate the stability of the quantum critical point in d = 3 with respect to fluctuations.
The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A
d=1 warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in
d=2and3, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function
F, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a
N1 expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in
d=2 using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.