Moment-method in a hard-sphere problem on a lattice, and physical applications. I. Uniform chain

Abstract

An exact representation is obtained for the propagator of two ’’hard spheres’’ on a one-dimensional (uniform) lattice where only nearest-neighbor hops are considered. The problem is solved within a random-walk formulation, with a general expression for the nth moment of the spectral density being given. These formal results are then applied to the discussion of several physical applications in a unified manner. In particular, we show that within the framework of the Hubbard Hamiltonian one may obtain, from the appropriate generalized spectral density, the optical absorption spectrum of a strongly correlated one-dimensional band of electrons. This theory, as applied, for example, to magnetic insulators or to the TCNQ (tetra-cyanoquinodimethane) salts, predicts a logarithmic divergence of the optical spectrum, for which a novel interpretation is given here in terms of an equivalent ’’surface’’ problem, shown to be isomorphic to the one studied. This is the problem of a two-dimensional rectangular crystal which has been cleaved along a main-diagonal line ’’surface.’’ From this, we also indicate the effects of chain dimerization (or Peierls transition) in a simple case of the spectral density. A simple counter example is also given which shows the nonuniqueness of the moment-method reconstruction for functions of unbounded variation. The partition function is also obtained analytically, and from this follows the energy of the two holes one-dimensional spinless fermion band. Finally, a physical argument is presented which supports the spinless fermion prediction of infinite mobility (or d.c. conductivity) for the hard spheres.