Oscar Bolina, Pierluigi Contucci, Bruno Nachtergaele
Path Integral Representation for Interface States of the Anisotropic 
Heisenberg Model
(112K, Latex with epsf)

ABSTRACT.  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 invariant 
Hamiltonian. This new representation is used to study the interface 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 spins. 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 does not fluctuate, i.e., has 
zero variance. We also show how to construct a path integral representation in 
higher dimensions and obtain a reduction formula for the partition functions in 
two dimensions in terms of the partition function of the one-dimensional model.