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Physical review. B, Condensed matter

Exact solution of a lattice band problem related to an exactly soluble many-body problem: The missing-states problem

Bill Sutherland

Published Oct 1, 1988• 9 citations
Publisher

Abstract

We introduce and solve a lattice version of an easily soluble one-dimensional continuum quantum system with a periodic potential. The relationship of the one problem to the other is identical to the relationship of the one-dimensional quantum many-body lattice gas recently introduced by Shastry [Phys. Rev. Lett. 60, 639 (1988)] and by Haldane [Phys. Rev. Lett. 60, 635 (1988)] to the one-dimensional quantum many-body continuum gas earlier solved by Sutherland. Thus it is hoped that by understanding the "missing" states in this simple case, we might better understand the missing states in the more complicated many-body case, which in many instances comprise all the states. The results are not encouraging.

DOI
10.1103/physrevb.38.6689

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We introduce and solve a lattice version of an easily soluble one-dimensional continuum quantum system with a periodic potential. The relationship of the one problem to the other is identical to the relationship of the one-dimensional quantum many-body lattice gas recently introduced by Shastry [Phys. Rev. Lett. 60, 639 (1988)] and by Haldane [Phys. Rev. Lett. 60, 635 (1988)] to the one-dimensional quantum many-body continuum gas earlier solved by Sutherland. Thus it is hoped that by understanding the "missing" states in this simple case, we might better understand the missing states in the more complicated many-body case, which in many instances comprise all the states. The results are not encouraging.
An exactly soluble one-dimensional three-body problem is presented, in which the interaction between the particles consists of local two-body potentials between each two particles. Infinitely high step functions are chosen for the form of the three potential functions. This interaction allows only three-body bound states and no continuum states. We have considered three different choices of the mass ratios of the three particles and we give formulas in closed form for the energies and for the wavefunctions of all states.
We propose a model for the quantum harmonic oscillator on a discrete lattice which can be written in a supersymmetric form, in contrast with the more direct discretization of the harmonic oscillator. Its ground state is easily found to be annihilated by the annihilation operator defined here, and its excitation spectrum is obtained numerically. We then define an operator whose continuum limit corresponds to an angular momentum in terms of the creation–annihilation operators of our model. Coherent states with the correct continuum limit are also constructed. The versatility of the model is then used to calculate, in a simple way, the generalized position-dependent scattering length for a particle colliding with a single static impurity in a periodic potential and the exact ground state of an interacting many-body problem in a one-dimensional ring.
We consider a general class of discrete unitary dynamical models on the lattice. We show that generically such models give rise to a wavefunction satisfying a Schroedinger equation in the continuum limit, in any number of dimensions. There is a simple mathematical relationship between the mass of the Schroedinger particle and the eigenvalues of a unitary matrix describing the local evolution of the model. Second quantized versions of these unitary models can be defined, describing in the continuum limit the evolution of a nonrelativistic quantum many-body theory. An arbitrary potential is easily incorporated into these systems. The models we describe fall in the class of quantum lattice gas automata, and can be implemented on a quantum computer with a speedup exponential in the number of particles in the system. This gives an efficient algorithm for simulating general nonrelativistic interacting quantum many-body systems on a quantum computer.
We study the quantum lattice gas model in one dimension introduced by Lesanovsky [Lesanovsky, Phys. Rev. Lett. 108, 105301 (2012)], who showed that the exact ground state and a couple of excited states can be obtained analytically. The Hamiltonian of the model depends solely on the parameter zzz, the meaning of which is a fugacity in the corresponding classical lattice gas model. For small zzz (0<z<10<z<10<z<1), we prove that there is an energy gap between the ground state and the excited states by applying Knabe's method [Knabe, J. Stat. Phys. 52, 627 (1988)].