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arXiv:1505.02131 · Physical Review A

Few-fermion systems in one dimension: Ground- and excited-state energies and contacts

Lukas Rammelmüller, William J. Porter, Andrew C. Loheac, Joaquín E. Drut

Published Jul 27, 2015• 28 citations
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Abstract

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 $N=4,6,...,12$ 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.

DOI
10.1103/physreva.92.013631

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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.
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,...,12 particles, with a zero-range interaction, and a wide range of attractive couplings. In addition, we determine the critical coupling for bound-state formation and provide extrapolations to the infinite volume and thermodynamic limits. For the latter, we find agreement with previous calculations based on the Bethe ansatz.
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.
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 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.
Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box.We use lattice Monte Carlo methods to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles.Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps).In non-interacting systems, the characteristic length scale of the oscillations is set by (2k F ) -1 , where k F is the Fermi momentum, while repulsive interactions tend to generate Wignercrystal oscillations of period (4k F ) -1 .Based on the non-interacting result, we find a remarkably simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix.While the total momentum is not a conserved quantity in the presence of hard walls, the magnitude of the momentum does provide a good quantum number.We are therefore able to provide a detailed characterization of the (quasi-)momentum distribution, which displays rather robust discontinuity at the Fermi surface.In addition, we determine the spatially varying on-site density-density correlation, which in turn yields Tan's contact density and, upon integration, Tan's contact.As is well known, the latter fully determines the short-range correlations and plays a crucial role in a multitude of equilibrium and non-equilibrium sum rules.
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.