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arXiv:cond-mat/0501080 · Physical Review Letters

Phase Separation and an Upper Bound for a Generalized Superfluid Gap for Cold Fermi Fluids in the Unitary Regime

Thomas D. Cohen

Published Sep 14, 2005• 33 citations
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Abstract

An upper bound is derived for $\ensuremath{\Delta}$ for a cold dilute fluid of equal amounts of two species of fermion in the unitary limit ${k}_{f}a\ensuremath{\rightarrow}\ensuremath{\infty}$ (where ${k}_{f}$ is the Fermi momentum, $a$ is the scattering length, and $\ensuremath{\Delta}$ is a pairing energy: the difference in energy per particle between adding to the system a macroscopic number (but infinitesimal fraction) of particles of one species compared to adding equal numbers of both. The bound is $\ensuremath{\delta}\ensuremath{\le}\frac{5}{3}[2(2\ensuremath{\xi}{)}^{2/5}\ensuremath{-}(2\ensuremath{\xi})]$ where $\ensuremath{\xi}=ϵ/{ϵ}_{\mathrm{FG}}$, $\ensuremath{\delta}=2\ensuremath{\Delta}/{ϵ}_{\mathrm{FG}}$; $ϵ$ is the energy per particle and ${ϵ}_{\mathrm{FG}}$ is the energy per particle of a noninteracting Fermi gas. If the bound is saturated, then systems with unequal densities of the two species will separate spatially into a superfluid phase with equal numbers of the two species and a normal phase with the excess. If the bound is not saturated, then $\ensuremath{\Delta}$ is the usual superfluid gap. If the superfluid gap exceeds the maximum allowed by the inequality, phase separation occurs.

DOI
10.1103/physrevlett.95.120403

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An upper bound is derived for \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ for a cold dilute fluid of equal amounts of two species of fermion in the unitary limit kfa\ensuremath→\ensuremath∞{k}_{f}a\ensuremath{\rightarrow}\ensuremath{\infty}kf​a\ensuremath→\ensuremath∞ (where kf{k}_{f}kf​ is the Fermi momentum, aaa is the scattering length, and \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is a pairing energy: the difference in energy per particle between adding to the system a macroscopic number (but infinitesimal fraction) of particles of one species compared to adding equal numbers of both. The bound is \ensuremathδ\ensuremath≤53[2(2\ensuremathξ)2/5\ensuremath−(2\ensuremathξ)]\ensuremath{\delta}\ensuremath{\le}\frac{5}{3}[2(2\ensuremath{\xi}{)}^{2/5}\ensuremath{-}(2\ensuremath{\xi})]\ensuremathδ\ensuremath≤35​[2(2\ensuremathξ)2/5\ensuremath−(2\ensuremathξ)] where \ensuremathξ=ϵ/ϵFG\ensuremath{\xi}=ϵ/{ϵ}_{\mathrm{FG}}\ensuremathξ=ϵ/ϵFG​, \ensuremathδ=2\ensuremathΔ/ϵFG\ensuremath{\delta}=2\ensuremath{\Delta}/{ϵ}_{\mathrm{FG}}\ensuremathδ=2\ensuremathΔ/ϵFG​; ϵϵϵ is the energy per particle and ϵFG{ϵ}_{\mathrm{FG}}ϵFG​ is the energy per particle of a noninteracting Fermi gas. If the bound is saturated, then systems with unequal densities of the two species will separate spatially into a superfluid phase with equal numbers of the two species and a normal phase with the excess. If the bound is not saturated, then \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is the usual superfluid gap. If the superfluid gap exceeds the maximum allowed by the inequality, phase separation occurs.
Superfluidity in the cold atomic two-species Fermi gas system in the unitary limit of infinite scattering length remains incompletely understood. In particular, a pseudogap phase has been proposed to exist above the superfluid critical temperature. Here we apply the auxiliary-field quantum Monte Carlo method to perform ab initio calculations of the temperature dependence of three quantities---the energy-staggering pairing gap, the condensate fraction, and the heat capacity---in a trapped finite-size cold atom system in the unitary limit of infinite scattering length. As the calculations of the energy-staggering pairing gap require the use of the canonical ensemble, we employ an efficient algorithm for the stabilization of particle-number projection that is essential for reaching convergence in the size of the model space. We observe clear signatures of the superfluid phase transition in all three quantities, including a signature of the recently measured $\ensuremath{\lambda}$ peak in the heat capacity. We also find that the energy-staggering gap does not lead the condensate fraction as the temperature decreases.
We analyze strongly interacting Fermi gases in the unitary regime by considering the generalization to an arbitrary number NNN of spin-1∕21∕21∕2 fermion flavors with Sp(2N)\mathrm{Sp}(2N)Sp(2N) symmetry. For $N\ensuremath{\rightarrow}\ensuremath{\infty}$ this problem is exactly solved by the Bardeen-Cooper-Schrieffer--Bose-Einstein condensate mean-field theory, with corrections small in the parameter 1∕N1∕N1∕N. The large-NNN expansion provides a systematic way to determine corrections to mean-field predictions, allowing the calculation of a variety of thermodynamic quantities at (and in proximity to) unitarity, including the energy, the pairing gap, and the upper-critical polarization (in the case of a polarized gas) for the normal to superfluid instability. For the physical case of N=1N=1N=1, among other quantities, we predict in the unitarity regime, the energy of the gas to be $\ensuremath{\xi}=0.28$ times that for the noninteracting gas and the pairing gap to be 0.52 times the Fermi energy.
Ground-state energies and superfluid gaps are calculated for degenerate Fermi systems interacting via long attractive scattering lengths such as cold atomic gases, neutron, and nuclear matter. In the intermediate region of densities, where the interparticle spacing $(\ensuremath{\sim}{1/k}_{F})$ is longer than the range of the interaction but shorter than the scattering length, the superfluid gaps and the energy per particle are found to be proportional to the Fermi energy and thus differ from the dilute and high-density limits. The attractive potential increase linearly with the spin-isospin or hyperspin statistical factor such that, e.g., symmetric nuclear matter undergoes spinodal decomposition and collapses whereas neutron matter and Fermionic atomic gases with two hyperspin states are mechanically stable in the intermediate density region. The regions of spinodal instabilities in the resulting phase diagram are reduced and do not prevent a superfluid transition.
In this work we investigate the general properties and the ground state of an asymmetrical dilute gas of cold fermionic atoms, formed by two particle species having different densities. We have shown in a recent paper, that a mixed phase composed of normal and superfluid components is the energetically favored ground state of such a cold fermionic system. Here we extend the analysis and verify that in fact, the mixed phase is the preferred ground state of an asymmetrical superfluid in various situations. We predict that the mixed phase can serve as a way of detecting superfluidity and estimating the magnitude of the gap parameter in asymmetrical fermionic systems.
The phase diagrams of low density Fermi-Fermi mixtures with equal or unequal masses and equal or unequal populations are described at zero and finite temperatures in the strong attraction limit. In this limit, the Fermi-Fermi mixture can be described by a weakly interacting Bose-Fermi mixture, where the bosons correspond to Feshbach molecules and the fermions correspond to excess atoms. First, we discuss the three and four fermion scattering processes, and use the exact boson-fermion and boson-boson scattering lengths to generate the phase diagrams in terms of the underlying fermion-fermion scattering length. In three dimensions, in addition to the normal and uniform superfluid phases, we find two stable nonuniform states corresponding to (i) phase separation between pure unpaired (excess) and pure paired fermions (molecular bosons); and (ii) phase separation between pure excess fermions and a mixture of excess fermions and molecular bosons. Lastly, we also discuss the effects of the trapping potential in the density profiles of condensed and noncondensed molecular bosons, and excess fermions at zero and finite temperatures, and discuss possible implications of our findings to experiments involving mixtures of ultracold fermions.