Lifshitz Point in the Phase Diagram of Resonantly Interacting ^6Li - ^{40}K Mixtures
📝 Abstract
We consider a strongly interacting ${}^{6} $Li- ${}^{40} $K mixture, which is imbalanced both in the masses and the densities of the two fermionic species. At present, it is the experimentalist’s favorite for reaching the superfluid regime. We construct an effective thermodynamic potential that leads to excellent agreement with Monte Carlo results for the normal state. We use it to determine the universal phase diagram of the mixture in the unitarity limit, where we find, in contrast to the mass-balanced case, the presence of a Lifshitz point. This point is characterized by the effective mass of the Cooper pairs becoming negative, which signals an instability towards a supersolid phase.
💡 Analysis
We consider a strongly interacting ${}^{6} $Li- ${}^{40} $K mixture, which is imbalanced both in the masses and the densities of the two fermionic species. At present, it is the experimentalist’s favorite for reaching the superfluid regime. We construct an effective thermodynamic potential that leads to excellent agreement with Monte Carlo results for the normal state. We use it to determine the universal phase diagram of the mixture in the unitarity limit, where we find, in contrast to the mass-balanced case, the presence of a Lifshitz point. This point is characterized by the effective mass of the Cooper pairs becoming negative, which signals an instability towards a supersolid phase.
📄 Content
Introduction. -Ultracold atomic Fermi mixtures are presently at the center of attention of both experimental and theoretical physicists. Due to the amazing control over this system many fundamental discoveries have been made, while more are likely to follow soon. These discoveries started with reaching the so-called BEC regime where the balanced two-component Fermi mixture turns superfluid due to the Bose-Einstein condensation (BEC) of molecules [1,2]. By using a Feshbach resonance to vary the interaction strength between atoms in different spin states, the BEC-BCS crossover between a Bose-Einstein condensate of molecules and a BCS superfluid of Cooper pairs could be directly observed [3]. Since pairing is optimal for an equal amount of atoms in each spin state and is absent for the noninteracting fully polarized system, a phase transition occurs as a function of spin imbalance [4,5]. The phase diagram of the polarized mixture was for strong interactions found to be governed by a tricritical point that resulted in the observation of phase separation [5,6]. The presence of gapless Sarma superfluidity is also likely to be present in this system [7,8], but has not been unambiguously identified yet.
Most recently, experiments have indicated that the physical consequences of yet another parameter can be explored, namely that of a mass imbalance between the two fermionic components. A very promising mixture in this respect consists of 6 Li and 40 K, which has a mass ratio of 6.7. Several accessible Feshbach resonances are identified in the mixture [9], while both species have also been simultaneously cooled into the degenerate regime [10]. So far, experimental interest has focused on the BEC side of the Feshbach resonance, where molecules are formed. Although the mass imbalance itself does not lead to fundamental new physics here, this situation changes when the heteronuclear molecules are optically pumped to their ground state [11]. Then the molecules acquire a large electric dipole moment, which gives rise to anisotropic long-ranged interactions. In an optical lattice, this can lead to supersolid phases [12].
In this Letter we focus on a different regime, namely the so-called unitarity limit, where the s-wave scattering length of the interspecies interaction diverges. Here, the size of the Cooper pairs is comparable to the average interparticle distance and the pairing is a many-body effect. The mass imbalance has a profound effect on the pairing now, because it strongly alters the Fermi spheres. We show that for the sufficiently large mass ratio of the 6 Li-40 K mixture, the phase diagram not only encompasses all the exciting physics known from the mass-balanced case, but is even much richer. Similar to the solely spinimbalanced case is the presence of phase separation [13], which can occur due to the mismatch of the Fermi surfaces. Also similar is that gapless Sarma superfluidity is unstable at zero temperature [13], while there is a predicted crossover to the Sarma phase at nonzero temperatures [7]. However, the most exciting difference that we find is the presence of a Lifshitz point in the phase diagram.
At a Lifshitz point the transition to the superfluid phase undergoes a dramatic change of character. Rather than preferring a homogeneous order parameter, the system now becomes an inhomogeneous superfluid. This exotic possibility was early investigated for the weakly interacting mass-balanced case by Larkin and Ovchinnikov (LO), who considered a superfluid with a single standingwave order parameter [16]. This is energetically more favorable than the plane-wave case studied by Fulde and Ferrell (FF) [17]. Since the LO phase results in periodic modulations of the particle densities, it is a supersolid [18]. The FF and LO phases have intrigued the physics community for many decades, but so far remained elusive in experiments with atomic Fermi mixtures. Typically, Lifshitz points are predicted at weak interactions where the critical temperatures are very low. However, in this Letter we show that the very special phase diagram of the 6 Li-40 K mixture contains both a Lifshitz and a tricritical point in the unitarity limit, as shown in Fig. 1. This is in sharp contrast to the mass-balanced case, where at unitarity a large body of theory only finds a tricritical point, in agreement with experiments [6].
In first instance, all these expectations follow from mean-field theory, which is useful for a qualitative de-
, where m is twice the reduced mass and n is the total particle density. The result of mean-field theory is shown in panel a). For a majority of light 6 Li atoms there is a tricritical point (TCP), at which the normal state (N), the homogeneous superfluid state (SF), and the forbidden region (FR) meet each other. For a majority of heavy 40 K atoms there is a Lifshitz point (LP), where there is an instability towards supersolidity (SS). The size of the supersolid stability region is not calculated wi
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