Path Integral Representation for Interface States of the Anisotropic Heisenberg Model

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.

23 pages, 9 figures

References in corpus (1)

• Finite-volume excitations of the 111 interface in the quantum XXZ model

Cited by in corpus (10)

• Thermodynamic Limit for the Mallows Model on $S_n$
• Some properties for the low-lying spectrum of the ferromagnetic, quantum XXZ spin system
• Asymmetric diffusion and the energy gap above the 111 ground state of the quantum XXZ model
• The spectral gap for the ferromagnetic spin-J XXZ chain
• The large-spin asymptotics of the ferromagnetic XXZ chain
• Expansions for Droplet States in the Ferromagnetic XXZ Heisenberg Chain
• Isolated Eigenvalues of the Ferromagnetic Spin-J XXZ Chain with Kink Boundary Conditions
• Path integral representations for the spin-pinned quantum XXZ chain
• Ferromagnetic Domain Wall Ground States in One-Dimensional Deformed Flat-Band Hubbard Model
• Density matrix for the kink ground state of the ferromagnetic XXZ chain