Equation of State and Phases of Polarized Unitary Fermi Gas

Dilute fermion gases such as those of 6 Li and 40 K are quantum mechanical systems with controllable short range and strong interactions. They offer an ideal test bed for our knowledge of the quantum many-body physics. The properties of these Fermi gases can be probed experimentally [1]. The interaction is described by the dimensionless parameter a s k F where a s is the swave scattering length and k F is the Fermi momentum. In the weakly interacting (so-called BCS) regime where 1/a s k F << 0 and up to the strongly interacting regime with 1/a s k F ≈ 0 (also called unitary regime), one or more non-trivial phases have been suggested [2,3,4,5] for the Fermi gases with spin imbalance. These polarized atomic gases hold resemblance to the case of magnetized superconductivity [6]. Here, instead of the external magnetic field, we assume unequal chemical potentials. The constraints given on the chemical potentials of the different fermion species suggest the existence of one or more intermediate polarized phases [7]. However, these phases are hard to study theoretically since the mean field approaches are not quantitatively accurate, and the numerical techniques such as the Fixed Node Diffusion Monte Carlo (FN-DMC) method requires the knowledge of the physically motivated guiding functions in the first quantized form.

The system we consider in this article is that of the idealized Fermi gas consisting of two(↑,↓-spin) species with equal masses. We assume control on each one of the chemical potentials (µ ↑ ,µ ↓ ) and the physically measurable quantities are the densities n ↑ and n ↓ (0 ≤ n ↓ ≤ n ↑ ). In the spin symmetric phase, the existence of the gap is manifest in the fact that the densities are not sensitive to a small difference of the chemical potentials δµ ≡ (µ ↑ -µ ↓ )/2. The superfluid phase imposes the constraint δµ ≤ ∆ [8]. Here ∆ is the usual superfluid pairing gap of the symmetric system. This condition gives the lower bound on the critical y ≡ µ ↓ /µ ↑ defined as Y 1 [7]. We use a capitalized notation Y x to indicate the upper or lower bounds while the lower case notation y x corresponds to the actual critical value at the phase transition. At a specific value of y 1 ≥ Y 1 (or correspondingly at a critical δµ), the fully paired superfluid (SF ) undergoes phase transition into the partially polarized phase (P P , see Fig 3) and the densities become unequal (n ↓ < n ↑ ). Recently, two possibilities were considered. In the first, the SF phase could transition into the polarized normal phase going through a phase separated mixture of the superfluid and the partially polarized normal fluid [7,9]. This transition that is assumed to be of the first order, is characteristic of the weakly interacting regime and also of the unitary limit. Another possibility is that the SF phase undergoes the second order phase transition and becomes a homogeneous polarized superfluid that accommodates the excess of one species. This was suggested for small polarizations in the unitary limit by Carlson et al. [10]. This phase is alternatively called gapless or polarized superfluid phase (SF p ). The recent work by Pilati et al. [11] considers both possibilities and suggests that the gapless homogeneous phase may occur in the 1/a s k F > 0 regime for moderate polarization. In the limit of the complete polarization, the system is in the normal fully polarized(N F P ) phase with N ↑ spin up particles and µ ↑ = h2 (6π 2 n ↑ ) 2/3 2m > 0. This system is also insensitive to changes of δµ as long as µ ↓ « 0. When µ ↓ ≥ the energy difference between N ↑ + 1 ↓ and N ↑ systems, the system phase transitions into the partially polarized(P P ) phase. This defines an upper bound for y known as Y 0 [7]. Several authors have shown [7,9,12,13,14] that a simple variational solution of non-interacting N F P + interacting impurity gives an upper bound Y 0 reasonably close to the actual threshold value y 0 .

In this letter, we construct the equation of state of the unitary Fermi gas connecting the limits x ≡ n ↓ n ↑ = 0 and x = 1. Then we estimate the actual y 1 (≥ Y 1 ) and y 0 (≈ Y 0 ) as a direct application of the knowledge of the equation of state. We also verify the consistency with the previously reported values of Y 0 and Y 1 . We also present the equation of state in terms of the grand canonical potential (pressure) and the density profiles of the trapped gas. In order to do so, we implement the canonical ensemble auxiliary field Monte Carlo (AFMC) formalism at zero temperature. AFMC is usually formulated in the second quantized form. In principle, it does not depend on the particular choice of the basis. It can be applied to the finite temperature [16,17,18] as well as zero temperature [17,20] Fermi systems. With the cost of introducing a set of additional integration variables, the time propagator can be expressed in the basis set of one particle orbitals. Then, the multi-dimensional integrations over the additional auxiliary variabl

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