Universal behavior of repulsive two-dimensional fermions in the vicinity of the quantum freezing point

We show by a meta-analysis of the available Quantum Monte-Carlo (QMC) results that two-dimensional fermions with repulsive interactions exhibit universal behavior in the strongly-correlated regime, and that their freezing transition can be described using a quantum generalization of the classical Hansen-Verlet freezing criterion. We calculate the liquid-state energy and the freezing point of the 2D dipolar Fermi gas (2DDFG) using a variational method by taking ground state wave functions of 2D electron gas (2DEG) as trial states. A comparison with the recent fixed-node diffusion Monte-Carlo analysis of the 2DDFG shows that our simple variational technique captures more than 95% of the correlation energy, and predicts the freezing transition within the uncertainty bounds of QMC. Finally, we utilize the ground state wave functions of 2DDFG as trial states and provide a variational account of the effects of finite 2D confinement width. Our results indicate significant beyond mean-field effects. We calculate the frequency of collective monopole oscillations of the quasi-2D dipolar gas as an experimental demonstration of correlation effects. An intriguing behavior of fermions with strong repulsive interactions is the spontaneous breaking of the translation symmetry in the ground-state and the formation of the so-called Wigner crystal (WC) phase. While originally proposed for the electron gas [1], a large body of evidence from Quantum Monte-Carlo (QMC) simulations and firstprinciple considerations have shown that the WC transition is indeed a universal aspect of repulsively interacting particles, independent of their quantum statistics, number of spatial dimensions, interaction law or spin degeneracy. Some of the extensively studied models that exhibit the WC transition are the electron gas in 2D [2–4] and 3D [5], 2D Coulomb bosons [6], 2D Yukawa bosons [7], 2D dipolar bosons [8] and fermions [9], and 2D hard-core bosons [10] and fermions [11]. The conventional explanation of WC transition at zero temperature is based on the competition between quantum fluctuations (kinetic energy) and the inter-particle repulsion, favoring delocalized and localized states, respectively. The symmetry broken state is energetically favorable when the ratio of the interaction over kinetic energy becomes sufficiently large. The ordered state is a triangular crystal in 2D which has the largest packing ratio. While general arguments from the Landau-Ginzburg theory suggest that the WC transition is a direct firstorder transition, for interaction laws falling slower than 1/r [12], the first-order transition may be replaced by a series of second-order transitions through intermediate “microemulsion” phases such as stripes and bubbles [13]. Such transitions, however, generally take place only over a very narrow window of densities and remain yet to be observed in QMC simulations due to finite size limitations [9, 14]. In this Letter, we investigate the features of the strongly-correlated liquid phase of fermions in the vicinity of the WC transition and show that models with significantly different interaction laws exhibit universal features. For concreteness, we restrict our analysis to singlecomponent fermions in 2D. Throughout this letter, we use the terminology “universal” to refer to properties that dep-1 ar X iv :1 21 2. 14 93 v2 [ co nd -m at .q ua nt -g as ] 1 3 Ju n 20 13