Solution of the Bloch Equation for Many-Particle Lattice Systems and the Heisenberg Magnets in Terms of the Path Summation

Abstract

The solution of the Bloch equation for many-particle systems on a finite lattice is given in terms of the “path summation”. The boundary conditions we consider are “zero on the boundary” and “the periodic boundary condition”. The obtained expression under the periodic boundary condition is appropriate for the Monte Carlo calculation of the density matrix and the ground state energy of Bose lattice systems and the Heisenberg magnet with a ferromagnetic interaction. For the antiferromagnetic Heisenberg magnet, the present formalism is useful only for the lattices in which each of the closed paths has an even number of edges.