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

Oscar Bolina, Pierluigi Contucci, Bruno Nachtergaele, Shannon L. Starr

Published Jan 1, 1999• 11 citations
OpenAlex

Abstract

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).

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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).
We introduce a quantum Monte Carlo method at finite temperature for interacting fermionic models in the canonical ensemble, where the conservation of the particle number is enforced. Although general thermodynamic arguments ensure the equivalence of the canonical and the grand-canonical ensembles in the thermodynamic limit, their approach to the infinite-volume limit is distinctively different. Observables computed in the canonical ensemble generically display a finite-size correction proportional to the inverse volume, whereas in the grand-canonical ensemble the approach is exponential in the ratio of the linear size over the correlation length. We verify these predictions by quantum Monte Carlo simulations of the Hubbard model in one and two dimensions in the grand-canonical and the canonical ensemble. We prove an exact formula for the finite-size part of the free energy density, energy density and other observables in the canonical ensemble and relate this correction to a susceptibility computed in the corresponding grand-canonical ensemble. This result is confirmed by an exact computation of the one-dimensional classical Ising model in the canonical ensemble, which for classical models corresponds to the so-called fixed-magnetization ensemble. Our method is useful for simulating finite systems which are not coupled to a particle bath, such as in nuclear or cold atom physics.
The statistical thermodynamics of the quantum spin- s XXZ Heisenberg model in the Weiss (mean-field) limit rests on the outer multiplicities of the irreducible representations in the Clebsch–Gordan decomposition of the n -fold spin- s tensor product, which determine the energy-level degeneracies and hence the partition function. Using known closed forms for these multiplicities in terms of generalized binomial coefficients of order 2 s + 1 , we write the exact finite-size partition function of the effective collective Hamiltonian and evaluate its thermodynamic limit by a large-deviation (Laplace) analysis validated against direct summations for systems of up to n = 10 4 spins. The free energy then follows from a two-parameter variational principle in the total spin and the magnetization per site. The ferromagnetic ordering is driven by the rotationally invariant part of the Hamiltonian, so that the Curie temperature T c = 1 3 J s ( s + 1 ) is independent of the anisotropy Δ, and the transition is of second order for every spin s , with classical mean-field exponents. The anisotropy reshapes the phase diagram: for Δ < 1 the field-driven first-order line of the isotropic ( Δ = 1 ) model opens into a dome of easy-plane ferromagnetic order, bounded by a continuous transition that restores the residual U(1) symmetry, and no first-order transitions or metastability occur throughout − 1 ≤ Δ < 1 ; the isotropic Curie-Weiss ferromagnet is recovered as Δ → 1. In two appendices we detail the asymptotics of the outer multiplicities down to the Gaussian-fluctuation prefactor and discuss the q -number and Jacobi-theta structures underlying the model, including the partial-theta-type structure that obstructs a standard theta-function reduction of the residual total-spin sum.
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.
In this thesis we present three results about the ferromagnetic quantum XXZ model: 1) Existence of a spectral gap above all infinite-volume ground states in one dimension for any choice of spin S>1/2 (for S=1/2 this was already known); 2) Characterization of one-dimensional "droplet" states as ground states of the spin-1/2 model with up spin boundary fields; 3) An upper bound for the spectral gap above the spin-1/2 interface ground states in dimension two and higher. The upper bound vanishes as 1/R^2, where R is the linear size of the system. This thesis contains research performed in collboration with other authors: O. Bolina, P. Contucci, T. Koma, B. Nachtergaele.
We show that the well-known translation invariant ground states and the recently discovered kink and antikink ground states are the complete set of pure infinite-volume ground states (in the sense of local stability) of the spin-5 ferromagnetic XXZ chains with Hamiltonian H = -EJS^S^ 4-Si 2) S% + ASi^S^l for all A > 1, and all 5 € |N.For the isotropic model (A = 1) we show that all ground states are translation invariant.For the proof of these statements we propose a strategy for demonstrating completeness of the list of the pure infinite-volume ground states of a quantum many-body system, of which the present results for the XXX and XXZ chains can be seen as an example.The result for A > 1 can also be proved by an easy extension to general S of the method used in [T.Matsui, Lett.Math.Phys.37 (1996) 397] for the spin-1/2 ferromagnetic XXZ chain with A > 1.However, our proof is e-print archive:
This paper is a short review of recent results on interface states in the Falicov-Kimball model and the ferromagnetic XXZ Heisenberg model. More specifically, we discuss the following topics: 1) The existence of interfaces in quantum lattice models that can be considered as perturbations of classical models. 2) The rigidity of the 111 interface in the three-dimensional Falicov-Kimball model at sufficiently low temperatures. 3) The low-lying excitations and the scaling of the gap in the 111 interface ground state in the ferromagnetic XXZ Heisenberg model in three dimensions. 4) The existence of droplet states in the XXZ chain and their properties.