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arXiv:1106.5725 · Physical Review A

Lattice Monte Carlo calculations for unitary fermions in a harmonic trap

Michael G. Endres, David B. Kaplan, Jong-Wan Lee, Amy Nicholson

Published Oct 31, 2011• 55 citations
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Abstract

We present a lattice Monte Carlo approach developed for studying large numbers of strongly interacting nonrelativistic fermions and apply it to a dilute gas of unitary fermions confined to a harmonic trap. In place of importance sampling, our approach makes use of high statistics, an improved action, and recently proposed statistical techniques. We show how improvement of the lattice action can remove discretization and finite volume errors systematically. For $N=3$ unitary fermions in a box, our errors in the energy scale as the inverse lattice volume, and we reproduce a previous high-precision benchmark calculation to within our $0.3%$ uncertainty; as additional benchmarks we reproduce precision calculations of $N=3,...,6$ unitary fermions in a harmonic trap to within our $\ensuremath{\sim}1%$ uncertainty. We then use this action to determine the ground-state energies of up to 70 unpolarized fermions trapped in a harmonic potential on a lattice as large as ${64}^{3}\ifmmode\times\else\texttimes\fi{}72$. In contrast to variational calculations, we find evidence for persistent deviations from the thermodynamic limit for the range of $N$ considered.

DOI
10.1103/physreva.84.043644

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We perform lattice Monte Carlo simulations for up to 66 unitary fermions in a finite box using a highly improved lattice action for nonrelativistic spin-12\frac{1}{2}21​ fermions. We obtain a value of $0.{366}_{\ensuremath{-}0.011}^{+0.016}$ for the Bertsch parameter, defined as the energy of the unitary Fermi gas measured in units of the free gas energy in the thermodynamic limit. In addition, for up to four unitary fermions, we compute the spectrum of the lattice theory by exact diagonalization of the transfer matrix projected onto irreducible representations of the octahedral group for small to moderate size lattices, providing an independent check of our few-body simulation results. We compare our exact numerical and simulation results for the spectrum to benchmark studies of other research groups as well as perform an extended analysis of our lattice action improvement scheme, including an analysis of the errors associated with higher partial waves and finite temporal discretization.
A recently developed lattice method for large numbers of strongly interacting nonrelativistic fermions exhibits a heavy tail in the distributions of correlators for large Euclidean time τ and large number of fermions N, which only allows the measurement of ground state energies for a limited number of fermions using standard techniques.In such cases, it is suggested that measuring the log of the correlator is more efficient, and a cumulant expansion of this quantity can be exactly related to the correlation function.The cumulant expansion technique allows us to determine the ground state energies of up to 66 unpolarized unitary fermions on lattices as large as 72 × 14 3 , and up to 70 unpolarized unitary fermions trapped in a harmonic potential on lattices as large as 72 × 64 3 .We have also improved our lattice action with a Galilean invariant form for the four-fermion interaction, which results in predictive volume scaling of the lowest energy of three fermions in a periodic box and in good agreement of our results for N ≤ 6 trapped unitary fermions with those from other benchmark calculations.
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I perform lattice Monte Carlo studies of universal four-component fermion systems in one spatial dimension. Continuum few-body observables (i.e., ground-state energies and integrated contact densities) are determined for both unpolarized and polarized systems of up to eight fermions confined to a harmonic trap. Estimates of the continuum energies for four and five trapped fermions show agreement with exact analytic calculations to within approximately one percent statistical uncertainties. Continuum many-body observables are determined for unpolarized systems of up to 88 fermions confined to a finite box, and 56 fermions confined to a harmonic trap. Results are reported for universal quantities such as the Bertsch parameter, defined as the energy of the untrapped many-body system in units of the corresponding free-gas energy, and its subleading correction at large but finite scattering length. Two independent estimates of these quantities are obtained from thermodynamic limit extrapolations of continuum extrapolated observables. A third estimate of the Bertsch parameter is obtained by combining estimates of the untrapped and trapped integrated contact densities with additional theoretical input from a calculation based on Thomas-Fermi theory. All estimates of the Bertsch parameter and its subleading correction are found to be consistent to within approximately one percent statistical uncertainties. Finally, the continuum restoration of virial theorems is verified for both few- and many-body systems confined to a trap.
We study a one-dimensional two-component Fermi gas in a harmonic trapping potential using finite-temperature lattice quantum Monte Carlo methods. We are able to compute observables in the canonical ensemble via an efficient projective approach. Results for density profiles, correlations, as well as energy-related observables are presented for systems with up to 80 particles and various temperatures. Our simulations reproduce known numerical results and compare well against available experimental data close to the ground state, while at higher temperature they are benchmarked against the exact solution of the two-particle system. This provides an indication that a standard lattice discretization is sufficient to capture the physics of the trapped system. In the special case of a spin-imbalanced gas, we find no sign problem in the studied parameter ranges, allowing access without the need of specialized methods. This includes simulations close to the ground state and at large population imbalance, where we present results for density correlations, indicating pairing at finite total momentum.