Multiparticle States on a Lattice

Low-energy excitation spectrum (energy versus momentum) of the Heisenberg spin-ladder for J/J' = ½. Beside the two-particle continuum (grey shaded) and the elementary triplet excitation (dotted line), there are three massive quasiparticles: a singlet bound state (solid line), a triplet bound state (dashed line), and a quintet antibound state(dash-dotted line).

The properties of multiparticle states can tell us a great deal about the dynamics of a lattice model in theoretical physics. We have developed a new method for calculating these properties which should be useful in several different areas.

Quantum lattice models are used in many areas of theoretical physics. For instance: the Hubbard model is used for electrons hopping about in a crystalline material; the Heisenberg model is used for atomic spins in a magnetic material; and lattice gauge theories are used to simulate Quantum Chromodynamics, the theory of quarks and gluons. In each case the properties of multiparticle states can reveal a great deal about the dynamics of the system: the nature of the elementary excitations, and the forces between them. (A "particle" here might refer to an electron in the Hubbard model, a glueball in lattice gauge theory, or a "magnon" or spin-wave in the Heisenberg model).

For many years, we have been using high-order perturbation series expansions to study these systems. In 1996, Gelfand discovered a new and efficient "linked cluster" expansion method for single-particle states. We have now been able to generalize this to multiparticle excitations, and particularly two-particle bound states. We can estimate their energy spectra and dispersion relations, and hence deduce a great deal about the forces acting between the elementary excitations. We hope to find many uses for this technique.

Chris Hamer and Zheng Weihong