 99291 Oscar Bolina, Pierluigi Contucci, Bruno Nachtergaele
 Path Integral Representation for Interface States of the Anisotropic
Heisenberg Model
(112K, Latex with epsf)
Aug 2, 99

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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
twodimensional 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 onedimensional model.
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