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arXiv:math-ph/9908004 · Reviews in Mathematical Physics

PATH INTEGRAL REPRESENTATION FOR INTERFACE STATES OF THE ANISOTROPIC HEISENBERG MODEL

Oscar Bolina, Pierluigi Contucci, Bruno Nachtergaele

Published Oct 1, 2000• 17 citations

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.

DOI: 10.1142/s0129055x00000496

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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.

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We show that the ground states of the three-dimensional XXZ Heisenberg ferromagnet\n with a 111 interface have excitations localized in a subvolume of linear size R with\n energies bounded by O(1/R^2). As part of the proof we show the equivalence of ensembles for\n the 111 interface states in the following sense: In the thermodynamic limit the states with\n fixed magnetization yield the same expectation values for gauge invariant local observables\n as a suitable grand canonical state with fluctuating magnetization. Here, gauge invariant\n means commuting with the total third component of the spin, which is a conserved quantity\n of the Hamiltonian. As a corollary of equivalence of ensembles we also prove the\n convergence of the thermodynamic limit of sequences of canonical states (i.e., with fixed\n magnetization).

The authors develop a quantum Monte Carlo procedure for the xxz spin chain. The quantum model in one dimension is mapped by a path integral method based on the Trotter formula into the infinite anisotropy limit of a classical two-dimensional spin system. The expectation values of the energy density, susceptibility and static s z -s z structure function are computed in the classical model for finite but increasing anisotropy. The computation method is a Monte Carlo simulation of the classical system. The results are systematically extrapolated to the infinite anisotropy limit.

The correlation functions of the

XXZ

Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers

The

XXZ

Heisenberg chain is considered for two specific limits of the anisotropy parameter:

\Delta \to 0

and

\Delta \to -\infty

. The corresponding wave functions are expressed in terms of symmetric Schur functions. Certain expectation values and thermal correlation functions of the ferromagnetic string operators are calculated over the basis of

N

-particle Bethe states. The thermal correlator of the ferromagnetic string is expressed through the generating function of the lattice paths of random walks of vicious walkers. A relationship between the expectation values obtained and the generating functions of strict plane partitions in a box is discussed. An asymptotic estimate of the thermal correlator of the ferromagnetic string is obtained in the zero temperature limit. It is shown that its amplitude is related to the number of plane partitions.

The SU(2) coherent-state path-integral representation is used to derive a fairly explicit expression for the partition function

{\mathit{Z}}_{0}

[\ensuremath{\psi}\ensuremath{\rightarrow}] of noninteracting quantum spins in the presence of an arbitrary time-dependent external field \ensuremath{\psi}\ensuremath{\rightarrow}. By performing then the standard Gaussian average of

{\mathit{Z}}_{0}

[\ensuremath{\psi}\ensuremath{\rightarrow}] over \ensuremath{\psi}\ensuremath{\rightarrow} the partition function of the ferromagnetically coupled Heisenberg model is represented in a way which allows one to simplify the derivation of results such as obtained by Leibler and Orland [Ann. Phys. (N.Y.) 132, 277 (1981)] and to generalize them to arbitrary spin quantum number.