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arXiv:0803.2293 · Physical Review A

Superfluid-to-insulator phase transition in a unitary Fermi gas

Nir Barnea

Published Nov 24, 2008• 8 citations
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Abstract

We study the properties of a dilute low-energy two-component Fermi gas. To this end, we approximate the Fermi gas by the Hubbard lattice Hamiltonian and solve it using the dynamical mean-field approximation. At unitarity, this approximation yields the value $\ensuremath{\xi}\ensuremath{\approx}0.44$ for the ratio between the ground-state energy per particle of the interacting and free systems $(E∕N=\ensuremath{\xi}{E}_{FG})$, in remarkable agreement with full quantum Monte Carlo simulations. Investigating the evolution of the system with temperature, we have found that below the critical temperature ${T}_{c}$, the system is a superfluid and the energy gap is decreasing monotonously. For temperatures above ${T}_{c}$, the system is an insulator and the corresponding energy gap is monotonously increasing.

DOI
10.1103/physreva.78.053629

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•Jan 18, 2018•1 citation
We study the properties of a dilute low-energy two-component Fermi gas. To this end, we approximate the Fermi gas by the Hubbard lattice Hamiltonian and solve it using the dynamical mean-field approximation. At unitarity, this approximation yields the value \ensuremathξ\ensuremath≈0.44\ensuremath{\xi}\ensuremath{\approx}0.44\ensuremathξ\ensuremath≈0.44 for the ratio between the ground-state energy per particle of the interacting and free systems (E∕N=\ensuremathξEFG)(E∕N=\ensuremath{\xi}{E}_{FG})(E∕N=\ensuremathξEFG​), in remarkable agreement with full quantum Monte Carlo simulations. Investigating the evolution of the system with temperature, we have found that below the critical temperature Tc{T}_{c}Tc​, the system is a superfluid and the energy gap is decreasing monotonously. For temperatures above Tc{T}_{c}Tc​, the system is an insulator and the corresponding energy gap is monotonously increasing.
We investigate the approach to the universal regime of the dilute unitary Fermi gas as the density is reduced to zero in a lattice model. To this end we study the chemical potential, superfluid order parameter and internal energy of the attractive Hubbard model in three different lattices with densities of states (DOSs) which share the same low-energy behavior of fermions in three-dimensional free space: a cubic lattice, a ``Bethe lattice'' with a semicircular DOS, and a ``lattice gas'' with parabolic dispersion and a sharp energy cutoff that ensures the normalization of the DOS. The model is solved using dynamical mean-field theory, that treats directly the thermodynamic limit and arbitrarily low densities, eliminating finite-size effects. At densities on the order of one fermion per site the lattice and its specific form dominate the results. The evolution to the low-density limit is smooth and it does not allow to define an unambiguous low-density regime. Such finite-density effects are significantly reduced using the lattice gas, and they are maximal for the three-dimensional cubic lattice. Even though dynamical mean-field theory is bound to reduce to the more standard static mean field in the limit of zero density due to the local nature of the self-energy and of the vertex functions, it compares well with accurate Monte Carlo simulations down to the lowest densities accessible to the latter.
We study the system of a dilute gas of fermions in three dimensions, with attractive interactions tuned to the unitarity point, using the nonperturbative restricted path-integral Monte Carlo method. The pairing and superfluid properties of this system are calculated at finite temperature. The total energy at very low temperature from our results agrees closely with that of previous ground-state quantum Monte Carlo calculations. We identify the temperature ${T}^{*}\ensuremath{\approx}0.70{ϵ}_{F}$, below which pairing correlations develop, and estimate the critical temperature for the superfluid transition ${T}_{c}\ensuremath{\approx}0.245{ϵ}_{F}$ from a finite size scaling analysis of the superfluid density.
The unitary Fermi gas is a many-body system of two-component fermions with zero-range interactions tuned to infinite scattering length. Despite much activity and interest in unitary Fermi gas and its universal properties, there have been great difficulties in performing accurate calculations of the superfluid condensate fraction and pairing wave function. In this paper, we present auxiliary-field lattice Monte Carlo simulations using a lattice interaction which accelerates the approach to the continuum limit, thereby allowing for robust calculations of these difficult observables. As a benchmark test, we compute the ground-state energy of 33 spin-up and 33 spin-down particles. As a fraction of the free Fermi gas energy EFG{E}_{\text{FG}}EFG​, we find E0/EFG=0.369(2),0.372(2),{E}_{0}/{E}_{\text{FG}}=0.369(2),0.372(2),E0​/EFG​=0.369(2),0.372(2), using two different definitions of the finite-system energy ratio, in agreement with the latest theoretical and experimental results. We then determine the condensate fraction by measuring off-diagonal long-range order in the two-body density matrix. We find that the fraction of condensed pairs is $\ensuremath{\alpha}=0.43(2)$. We also extract the pairing wave function and find the pair correlation length to be ${\ensuremath{\zeta}}_{p}{k}_{F}=1.8(3)\ensuremath{\hbar}$, where kF{k}_{F}kF​ is the Fermi momentum. Provided that the simulations can be performed without severe sign oscillations, the methods we present here can be applied to superfluid neutron matter as well as more exotic PPP-wave and DDD-wave superfluids.
The equation of state of the partially polarized two component Fermi gas at zero temperature in the unitary limit is computed by the ab initio auxiliary field Monte Carlo method. From this, we obtain the critical ratio of the chemical potentials ${\ensuremath{\mu}}_{\ensuremath{\downarrow}}∕{\ensuremath{\mu}}_{\ensuremath{\uparrow}}$ at the phase transitions. The value of ${\ensuremath{\mu}}_{\ensuremath{\downarrow}}∕{\ensuremath{\mu}}_{\ensuremath{\uparrow}}$ at the transition between the fully paired superfluid and the partially polarized phases is 0.11, while the critical value at the phase transition between the partially polarized phase and the fully polarized normal fluid is $\ensuremath{-}0.59$. We also determine the radial boundaries of the phase transitions of the Fermi gas in the harmonic trap as a function of the total polarization. We find that beyond the critical polarization 0.65, the fully paired superfluid core disappears in the trapped Fermi gas.
In the two-component Fermi gas with contact interactions a pseudogap regime, in which pairing correlations are present without superfluidity, can exist at temperatures between the superfluid critical temperature TcT_cTc​ and a temperature T∗>TcT^* > T_cT∗>Tc​. However, the existence of a pseudogap in the unitary limit of infinite scattering length is debated. To help address this issue, we have used finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods to study the thermodynamics of the spin-balanced homogeneous unitary Fermi gas on a lattice. We present results for the thermal energy, heat capacity, condensate fraction, a model-independent pairing gap, and spin susceptibility, and compare them to experimental data when available. Our model space consists of the complete first Brillouin zone of the lattice, and our calculations are performed in the canonical ensemble of fixed particle number. We find that the energy-staggering pairing gap vanishes above TcT_cTc​ and that T∗T^*T∗ at unitarity, as determined from the spin susceptibility, is lower than previously reported in AFMC simulations.