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arXiv:1111.3793 · International Symposium on Lattice Field Theory (LATTICE)

Extended study for unitary fermions on lattice using cumulant expansion technique

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

Published Jul 5, 2012• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.22323/1.139.0203

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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.
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=3N=3N=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 0.30.3%0.3 uncertainty; as additional benchmarks we reproduce precision calculations of N=3,...,6N=3,...,6N=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 NNN considered.
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 present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy as well as the correlation functions in the ground state are exactly determined as series expansions in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applications of the analytic character of the present approach is discussed.
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.
We apply the general principles of effective field theories to the construction of effective interactions suitable for few- and many-body calculations in a no-core shell model framework. We calculate the spectrum of systems with three and four two-component fermions in a harmonic trap. In the unitary limit, we find that three-particle results are within 10% of known semianalytical values even in small model spaces. The method is very general, and can be readily extended to other regimes, more particles, different species (e.g., protons and neutrons in nuclear physics), or more-component fermions (as well as bosons). As an illustration, we present calculations of the lowest-energy three-fermion states away from the unitary limit and find a possible inversion of parity in the ground state in the limit of trap size large compared to the scattering length. Furthermore, we investigate the lowest positive-parity states for four fermions, although we are limited by the dimensions we can currently handle in this case.
Abstract A systematic nonperturbative scheme for calculating the ground state energy is adapted for studying systems of strongly correlated electrons on a lattice. It includes a method for calculating the cumulants of the Hamiltonian and a method, using the t-expansion, for constructing successive approximations to the ground state energy by these cumulants. The scheme is applied to spinless fermion and Hubbard models, and a method is proposed to overcome the problems found in previous attempts to use this scheme to study the Hubbard model.