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arXiv:0712.3593 · Physical Review A

Off-diagonal ground-state properties of a one-dimensional gas of Fermi hard rods

Ferran Mazzanti, G. E. Astrakharchik, Jordi Boronat, J. Casulleras

Published Apr 29, 2008• 11 citations
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Abstract

A variational Monte Carlo calculation of the one-body density matrix and momentum distribution of a system of Fermi hard rods (HRs) is presented and compared with the same quantities for its bosonic counterpart. The calculation is exact within statistical errors since we sample the exact ground-state wave function, whose analytical expression is known. The numerical results are in good agreement with known asymptotic expansions valid for Luttinger liquids. We find that the difference between the absolute value of the bosonic and fermionic density matrices becomes marginally small as the density increases. In this same regime, the corresponding momentum distributions merge into a common profile that is independent of the statistics. Nonanalytical contributions to the one-body density matrix are also discussed and found to be less relevant with increasing density.

DOI
10.1103/physreva.77.043632

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A variational Monte Carlo calculation of the one-body density matrix and momentum distribution of a system of Fermi hard rods (HRs) is presented and compared with the same quantities for its bosonic counterpart. The calculation is exact within statistical errors since we sample the exact ground-state wave function, whose analytical expression is known. The numerical results are in good agreement with known asymptotic expansions valid for Luttinger liquids. We find that the difference between the absolute value of the bosonic and fermionic density matrices becomes marginally small as the density increases. In this same regime, the corresponding momentum distributions merge into a common profile that is independent of the statistics. Nonanalytical contributions to the one-body density matrix are also discussed and found to be less relevant with increasing density.
A quantum Monte Carlo simulation of a system of bosonic hard rods in one dimension is presented and discussed. The calculation is exact since the analytical form of the wave function is known and is in excellent agreement with predictions obtained from asymptotic expansions valid at large distances. The analysis of the static structure factor and the pair distribution function indicates that a solidlike and a gaslike phases exist at high and low densities, respectively. The one-body density matrix decays following a power law at large distances and produces a divergence in the low density momentum distribution at k=0 which can be identified as a quasicondensate.
We investigate the ground state of the one-dimensional fermionic system enclosed in a hard-wall trap with attractive contact ppp-wave interactions. Based on the Bethe ansatz method, the explicit wave function is derived by numerically solving the Bethe ansatz equations for the full physical regimes $(\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{c}_{F}\ensuremath{\le}0)$. With the exact wave function some quantities which are important in many-body physics are obtained, including the one-body density matrix and the momentum distribution of the ground state for finite system. It is shown that the shell structure of the density profiles disappears with the increase of the interaction and in the fermionic Tonks-Girardeau limit the density distribution shows the same behavior as that of an ideal Bose gas. However, the one-body density matrix and the momentum distribution exhibit completely different structures compared with their bosonic counterparts.
The ground-state properties of a single-component one-dimensional Coulomb gas are investigated. We use Bose-Fermi mapping for the ground-state wave function which permits solution of the Fermi sign problem in the following respects: (i) the nodal surface is known, permitting exact calculations; and (ii) evaluation of determinants is avoided, reducing the numerical complexity to that of a bosonic system and, thus, allowing simulation of a large number of fermions. Due to the mapping, the energy and local properties in one-dimensional Coulomb systems are exactly the same for Bose-Einstein and Fermi-Dirac statistics. The exact ground-state energy is calculated in homogeneous and trapped geometries using the diffusion Monte Carlo method. We show that in the low-density Wigner crystal limit an elementary low-lying excitation is a plasmon, which is to be contrasted with the high-density ideal Fermi gas/Tonks-Girardeau limit, where low-lying excitations are phonons. Exact density profiles are compared to the ones calculated within the local density approximation, which predicts a change from a semicircular to an inverted parabolic shape of the density profile as the value of the charge 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.
Abstract 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 ( 2 k F ) − 1 , where k F is the Fermi momentum, while repulsive interactions tend to generate Wigner-crystal oscillations of period ( 4 k 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.
Using the exact N-particle ground state wave function for a one-dimensional gas of hard-core bosons in a harmonic trap we develop an algorithm to compute the reduced single-particle density matrix and corresponding momentum distribution. Accurate numerical results are presented for up to N=8N=8N=8 particles, and the momentum distributions are compared to a recent analytic approximation.