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Physical Review Letters

Exact Solution for the Ground State of a One-Dimensional Quantum Lattice Gas with Coulomb-Like Interaction

Bill Sutherland

Published Mar 16, 1998• 2 citations
Publisher

Abstract

We present exact wave functions for the ground state of a lattice gas of spinless one-dimensional fermions, at filling factors of $1/M$. The one-body hopping operator decreases exponentially. The two-body interaction is Coulomb-like: A neutralizing background is required for the system to be extensive. There is a parameter in the problem, which can be taken to be the strength of the charge. All correlation functions are evaluated, and the system is shown to have long-ranged crystalline order. For filling factor $1/2$, the momentum distribution and pair correlation function are explicitly evaluated. There is no Fermi surface. The crystalline order parameter is evaluated.

DOI
10.1103/physrevlett.80.2445

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We present exact wave functions for the ground state of a lattice gas of spinless one-dimensional fermions, at filling factors of 1/M1/M1/M. The one-body hopping operator decreases exponentially. The two-body interaction is Coulomb-like: A neutralizing background is required for the system to be extensive. There is a parameter in the problem, which can be taken to be the strength of the charge. All correlation functions are evaluated, and the system is shown to have long-ranged crystalline order. For filling factor 1/21/21/2, the momentum distribution and pair correlation function are explicitly evaluated. There is no Fermi surface. The crystalline order parameter is evaluated.
For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.
We study the correlations in a family of ground-state wave functions for one-dimensional quantum many-body systems. For a particular ordering of the M particles, these wave functions are given as the absolute value of the free-fermion ground-state wave function, raised to an arbitrary power \ensuremath{\lambda}. For other permutations of the particles, we use the particle statistics---boson or fermion---to construct the wave function. This prescription works equally well on the continuum line of length L or on the discrete lattice of size N. The theory of random matrices singles out the ``special'' values of \ensuremath{\lambda}=1/2, 1, and 2, and provides techniques for calculating all correlations. Results obtained in this work include analytic expressions for the lattice momentum distributions, accurate numerical results for all momentum distributions and their critical exponents, and explicit results for the lattice pair-correlation functions. We emphasize that this is only for the stated special values of \ensuremath{\lambda}. Finally, these wave functions are evaluated as candidates for variational wave functions.
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.
We consider a two-dimensional gas of spinless fermions with Coulomb and short-range interactions on a square lattice at T=0T=0T=0. Using the exact-diagonalization technique we study finite clusters up to 16 particles at filling factors $\ensuremath{\nu}=1/2$ and 1/6. By increasing the hopping amplitude we obtain the low-energy spectrum of the system in a wide range from the classical Wigner crystal to an almost free gas of fermions. Particular effort is made to study the mechanism of the structural and insulator-metal transitions. We show that both transitions are determined by the energy band of the defect with the lowest energy in the Wigner crystal.
We study the ground state properties of a quasi-one-dimensional electron gas interacting via an effective potential with a harmonic transversal confinement and long-range Coulomb tail. The exact correlation energy has been calculated for a wide range of electron densities by using the lattice regularized diffusion Monte Carlo method, which is a recent development of the standard projection Monte Carlo technique. In this case it is particularly useful as it allows one to sample the exact ground state of the system, even in the low-density regime when the exchange between electrons is extremely small. For different values of the width parameter bbb $(0.1\phantom{\rule{0.3em}{0ex}}{a}_{0}^{*}\ensuremath{\leqslant}b\ensuremath{\leqslant}4\phantom{\rule{0.3em}{0ex}}{a}_{0}^{*})$, we give a simple parametrization of the correlation energy, which provides an accurate local density energy functional for quasi-one-dimensional systems. Moreover, we show that static correlations are in qualitative agreement with those obtained for the Luttinger liquid model with long-range interactions.
We present an exactly solvable Hamiltonian which consists of nearest neighbor hopping and long-range interaction. The ground state and the thermodynamic quantities are analytically obtained in any dimensions. In a repulsive case, the Luttinger's theorem breaks down and the hypothesis of the adiabatic continuation no longer holds. At half filling, we show the existence of a metal-insulator transition as a function of the strength of the interaction. The zero energy excitation disappears at the critical value of the interaction when we increase the interaction. In this case, the system is an insulator. The charge compressibility behaves as κ≈|1-ρ| 1-2/ d near half filling in d dimensions where ρ is the density of electrons. At other fillings, there is always a zero energy excitation and the system is metallic. When the interaction is attractive, the ground state is composed of pairs of electrons with opposite spins. The low energy excitations, however, is not a particle-hole type of a non-interacting case but a pairwise excitation across the Fermi surface.