Path integral representation for interface states of the anisotropic Heisenberg model

We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation i nvariant Hamiltonian. This new representation is used to study the interfac e ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spin s. As an application of this bound, we prove that the total third component of the spin in a large interval of even length centered on the interface d oes not fluctuate, i.e. has zero variance. We also show how to construct a path integral representation in higher dimensions and obtain a reduction fo rmula for the partition functions in two dimensions in terms of the partiti on function of the one-dimensional model.