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arXiv:1510.02980 · Physical Review A

Harmonically trapped fermions in two dimensions: Ground-state energy and contact of SU(2) and SU(4) systems via a nonuniform lattice Monte Carlo method

Zhihuan Luo, Casey Berger, Joaquín E. Drut

Published Mar 2, 2016• 9 citations
arXivHTMLPDF

Abstract

We study harmonically trapped, unpolarized fermion systems with attractive interactions in two spatial dimensions with spin degeneracies ${N}_{f}^{}=2$ and 4 and $N/{N}_{f}^{}=1,\phantom{\rule{0.28em}{0ex}}3,\phantom{\rule{0.28em}{0ex}}5$, and 7 particles per flavor. We carry out our calculations using our recently proposed quantum Monte Carlo method on a nonuniform lattice. We report on the ground-state energy and contact for a range of couplings, as determined by the binding energy of the two-body system, and show explicitly how the physics of the ${N}_{f}$-body sector dominates as the coupling is increased.

DOI
10.1103/physreva.93.033604

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We study harmonically trapped, unpolarized fermion systems with attractive interactions in two spatial dimensions with spin degeneracies Nf=2{N}_{f}^{}=2Nf​=2 and 4 and N/Nf=1,3,5N/{N}_{f}^{}=1,\phantom{\rule{0.28em}{0ex}}3,\phantom{\rule{0.28em}{0ex}}5N/Nf​=1,3,5, and 7 particles per flavor. We carry out our calculations using our recently proposed quantum Monte Carlo method on a nonuniform lattice. We report on the ground-state energy and contact for a range of couplings, as determined by the binding energy of the two-body system, and show explicitly how the physics of the Nf{N}_{f}Nf​-body sector dominates as the coupling is increased.
We determine the ground-state energy and Tan's contact of attractively interacting few-fermion systems in a one-dimensional harmonic trap, for a range of couplings and particle numbers. Complementing those results, we show the corresponding density profiles. The calculations were performed with a lattice Monte Carlo approach based on a nonuniform discretization of space, defined via Gauss-Hermite quadrature points and weights. This particular coordinate basis is natural for systems in harmonic traps, and can be generalized to traps of other shapes. In all cases, it yields a position-dependent coupling and a corresponding nonuniform Hubbard-Stratonovich transformation. The resulting path integral is performed with hybrid Monte Carlo as a proof of principle for calculations at finite temperature and in higher dimensions. We present results for N=4,...,20N=4,...,20N=4,...,20 particles (although the method can be extended beyond that) to cover the range from few- to many-particle systems. This method is exact up to statistical and systematic uncertainties, which we account for---and thus also represents an ab initio calculation of this system, providing a benchmark for other methods and a prediction for ultracold-atom experiments.
We calculate the ground-state properties of unpolarized two-dimensional attractive fermions in the range from few to many particles. Using first-principles lattice Monte Carlo methods, we determine the ground-state energy, Tan's contact, momentum distribution, and single-particle correlation function. We investigate those properties for systems of $N=4,8,\ensuremath{\cdots},40$ particles and for a wide range of attractive couplings. As the attractive coupling is increased, the thermodynamic limit is reached at progressively lower NNN due to the dominance of the two-body sector. At large momenta kkk, the momentum distribution displays the expected ${k}^{\ensuremath{-}4}$ behavior, but its onset shifts from $k\ensuremath{\simeq}1.8{k}_{F}^{}$ at weak coupling towards higher kkk at strong coupling.
Using the lattice Monte Carlo method, we compute the energy and Tan's contact in the ground state as well as the first excited state of few- to many-fermion systems in a one-dimensional periodic box. We focus on unpolarized systems of N=4,6,...,12N=4,6,...,12N=4,6,...,12 particles, with a zero-range interaction, and a wide range of attractive couplings. In addition, we provide extrapolations to the infinite-volume and thermodynamic limits.
The system of interacting, trapped fermions in one dimension has been of interest in both the theoretical and experimental communities. This system is realizable experimentally using ultracold atoms in traps, where the interactions can be tuned to simulate a number of important situations in nuclear theory, condensed matter, quantum information, and QCD. Theoretically, however, this system remains a challenge to treat, and no known benchmarks exist for the ground state energy, Tan's contact, or density profiles for the few- to many-body regime. This project implements a lattice Monte Carlo (LMC) method to solve for these quantities. The method blends hybrid Monte Carlo (HMC) - a pillar of lattice quantum chromodynamics (lattice QCD) - with a non-uniform lattice defined using Gauss-Hermite quadrature points and weights. This coordinate basis is the natural one for the harmonic oscillator trapping potential, and can be generalized to traps of other shapes. Using this method, we determine the ground-state energy and Tan's contact of attractively interacting few-fermion systems in a one-dimensional harmonic trap, for a range of couplings and particle numbers. Complementing those results, we show the corresponding density profiles. We present results for N=4,...,20N = 4,...,20N=4,...,20 particles - and the method is capable of extending beyond that. The method is the first lattice calculation of its kind, and is exact up to statistical and systematic uncertainties, which we account for. Our results are therefore a benchmark for other methods and a prediction for ultracold-atom experiments.
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.