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Physical Review A

Exact Results for a Quantum Many-Body Problem in One Dimension. II

Bill Sutherland

Published Mar 1, 1972• 706 citations
Publisher

Abstract

We continue our previous investigation of the properties of a system of either fermions or bosons interacting in one dimension by a two-body potential $V(\mathcal{r})=\frac{g}{{\mathcal{r}}^{2}}$ with periodic boundary conditions. A method is introduced which provides a physically intuitive description of the excited states. The energies are given explicitly for finite and infinite systems, including all corrections due to quasiparticle interaction. The method is shown to apply to other one-dimensional systems as well. In particular, the results are applied to a classical one-dimensional $N$-body model of Dyson---the Coulomb gas in a Brownian medium.

DOI
10.1103/physreva.5.1372

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We continue our previous investigation of the properties of a system of either fermions or bosons interacting in one dimension by a two-body potential V(r)=gr2V(\mathcal{r})=\frac{g}{{\mathcal{r}}^{2}}V(r)=r2g​ with periodic boundary conditions. A method is introduced which provides a physically intuitive description of the excited states. The energies are given explicitly for finite and infinite systems, including all corrections due to quasiparticle interaction. The method is shown to apply to other one-dimensional systems as well. In particular, the results are applied to a classical one-dimensional NNN-body model of Dyson---the Coulomb gas in a Brownian medium.
We investigate exactly a system of either fermions or bosons interacting in one dimension by a two-body potential V(r)=gr2V(r)=\frac{g}{{r}^{2}}V(r)=r2g​ with periodic boundary conditions. In addition to rederiving known results for correlation functions and thermodynamics in the thermodynamic limit, we present expressions for the one-particle density matrix at zero temperature and particular (nontrivial) values of the coupling constant ggg, as a determinant of order $N\ifmmode\times\else\texttimes\fi{}N$. These concise expressions allow a discussion of the momentum distribution in the thermodynamic limit. In particular, for a case of repulsive bosons, the determinant is evaluated explicitly, exhibiting a weak (logarithmic) singularity at zero momentum, and vanishing outside of a "Fermi" surface.
We continue our investigation of a system of either fermions or bosons interacting in one dimension by a 2‐body potential V(r) = g/r2. We first present an approximation for the eigenstates of a general 1‐dimensional quantum many‐body system. We then apply this approximation to the g/r2 potential, allowing complete determination of the thermodynamic properties. Finally, comparing the results with those properties known exactly, we conjecture that the approximation is, in fact, exact for the g/r2 potential.
We continue our investigation of a system of either fermions or bosons interacting in one dimension by a 2-body potential V(r) = g/r2. We first present an approximation for the eigenstates of a general 1-dimensional quantum many-body system. We then apply this approximation to the g/r2 potential, allowing complete determination of the thermodynamic properties. Finally, comparing the results with those properties known exactly, we conjecture that the approximation is, in fact, exact for the g/r2 potential.
We present a semiclassical study of the spectrum of a few-body system consisting of two short-range interacting bosonic particles in one dimension, a particular case of a general class of integrable many-body systems where the energy spectrum is given by the solution of algebraic transcendental equations. By an exact mapping between δ-potentials and boundary conditions on the few-body wave functions, we are able to extend previous semiclassical results for single-particle systems with mixed boundary conditions to the two-body problem. The semiclassical approach allows us to derive explicit analytical results for the smooth part of the two-body density of states that are in excellent agreement with numerical calculations. It further enables us to include the effect of bound states in the attractive case. Remarkably, for the particular case of two particles in one dimension, the discrete energy levels obtained through a requantization condition of the smooth density of states are essentially in perfect agreement with the exact ones.
We extend to the N-body case previous techniques for solving three-body problems with repulsive interparticle potentials and periodic boundary conditions on each particle. For clarity, we begin with one-dimensional problems, although the techniques are not peculiar to them and can be generalized to three dimensions. We decompose the wavefunction ψ into N(N − 1)/2 parts, according to Ψ= ∑ i<j=1Nψij;from the Schrödinger equation we find a basic equation for a typical ψij. As a test of this equation, we apply it for N = 4 and the case of δ-function interactions, and solve it numerically to find close agreement between the energy per particle of a four-body system and a system of the same density in which N → ∞. The numerical results are consistent with the analytic one that Eg(∞), which is the ground-state energy per particle of the system with an infinite number of particles, is related to Eg(N), which is that energy for the N-body system of the same density, by Eg(N) = Eg(∞)[1 − (1/N)] for weak repulsion and by Eg(N) = Eg(∞)[1 − (1/N2)] for very strong repulsion. We begin a similar comparison for the three-dimensional hard-sphere case by working out the problem of two hard spheres with periodic boundary conditions over a length L2 such that the density is the same as for an N-body problem in volume V=LN3:2/L23=N/LN3. We find that we get the analog of the one-dimensional result, for which case Eg(2) = Eg(∞)/2 for N = 2. That is, we find Eg(2) = πa2h/mv, which is just one-half of the many-body theory result. We also calculate the two-body correlation function and find good agreement with the many-body one, as calculated by Lee, Huang, and Yang.