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Journal of Mathematical Physics

Quantum Many-Body Problem in One Dimension: Thermodynamics

Bill Sutherland

Published Feb 1, 1971• 402 citations
Publisher

Abstract

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.

DOI
10.1063/1.1665585

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Exact results on quantum many-body systems in one dimension

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In the course of the completion of this thesis we have addressed several problems concerning quantum many-body systems and one-dimensional lattice models. Before getting into details, let me briefly outline some of them together with our main contributions, and sketch out some central concepts we shall develop later. To begin with, we have studied the entanglement properties of the ground state of several one-dimensional systems (see Refs. [P1]–[P4]), and investigated several exact spectral properties of lattice models in one dimension, obtaining an analytic expression for the partition function of certain models with long-range interactions (see [P4] and [P5] for a detailed description of the models involved). We have completely characterized the spectrum of the so-called long range t-J model for the first time and, in particular, rigorously proved a conjecture proposed by Saiga and Kuramoto in the nineties for the spin 1/2 case...

•Jun 20, 2019•0 citations
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 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.
In the course of the completion of this thesis we have addressed several problems concerning quantum many-body systems and one-dimensional lattice models. Before getting into details, let me briefly outline some of them together with our main contributions, and sketch out some central concepts we shall develop later. To begin with, we have studied the entanglement properties of the ground state of several one-dimensional systems (see Refs. [P1]–[P4]), and investigated several exact spectral properties of lattice models in one dimension, obtaining an analytic expression for the partition function of certain models with long-range interactions (see [P4] and [P5] for a detailed description of the models involved). We have completely characterized the spectrum of the so-called long range t-J model for the first time and, in particular, rigorously proved a conjecture proposed by Saiga and Kuramoto in the nineties for the spin 1/2 case...