Path Integral Representation for Interface States of the Anisotropic Heisenberg Model

TL;DR

This paper introduces a geometric path integral representation for the ground states of the anisotropic Heisenberg model, enabling analysis of interface states and their statistical properties.

Contribution

It develops a novel path integral model for the XXZ ferromagnetic chain, linking quantum ground states to classical statistical mechanics and extending to higher dimensions.

Findings

Probability of down spins decays exponentially with distance from the interface.

Total spin component in a large interval centered on the interface has zero variance.

Provides a reduction formula for partition functions in two dimensions.

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.