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Journal of Mathematical Physics

Quantum Heisenberg ferromagnets and stochastic exclusion processes

Lawrence E. Thomas

Published Jul 1, 1980• 14 citations
Publisher

Abstract

Let H be the isotropic spin-s quantum ferromagnetic Heisenberg Hamiltonian associated with a finite volume in Zd. In an appropriate representation, the restriction of −H to the N-spin wave sector −HN can be regarded as the generator of a Markov semigroup exp(−tHN). For s= 1/2 the corresponding process XN(t) is identified as a simple N-particle exclusion process; for higher spin it is a more complicated exclusion process involving different species of particles. A Feynman–Kac, Cameron–Martin perturbation formula for exp(−tHN) is developed in the s= 1/2 case, and the formula is used to obtain some thermodynamic correlation inequalities.

DOI
10.1063/1.524610

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Let H be the isotropic spin-s quantum ferromagnetic Heisenberg Hamiltonian associated with a finite volume in Zd. In an appropriate representation, the restriction of −H to the N-spin wave sector −HN can be regarded as the generator of a Markov semigroup exp(−tHN). For s= 1/2 the corresponding process XN(t) is identified as a simple N-particle exclusion process; for higher spin it is a more complicated exclusion process involving different species of particles. A Feynman–Kac, Cameron–Martin perturbation formula for exp(−tHN) is developed in the s= 1/2 case, and the formula is used to obtain some thermodynamic correlation inequalities.
In this paper we study a class of quantum Markov semigroups whose restriction to an abelian sub-algebra coincides, on the configurations with finite support, with the exclusion type semigroups introduced in Liggett's book14 of exchange rates [Formula: see text] not symmetric in the index site r, s. We find a sufficient condition for the existence of infinitely many faithful diagonal (or classical) invariant states for the semigroup, that satisfy a quantum detailed balance condition. This class of semigroups arises naturally in the stochastic limit of quantum interacting particles in the sense of Accardi and Kozyrev.1 We call these semigroups asymmetric exclusion quantum Markov semigroups and the associated processes asymmetric exclusion quantum processes.
Abstract In this note we relate the Hamiltonian of the integrable non-compact spin s XXZ chain to the Markov generator of a stochastic particle process. The hopping rates of the continuous-time process are identified with the ones of a q-Hahn asymmetric zero range model. The main difference with the asymmetric simple exclusion process, which can be mapped to the ordinary XXZ spin chain, is that multiple particles can occupy one and the same site. For the non-compact spin XXZ chain the associated stochastic process reduces to the multiparticle asymmetric diffusion model introduced by Sasamoto–Wadati.
We propose a dynamical matrix product ansatz describing the stochastic dynamics of two species of particles with excluded-volume interaction and the quantum mechanics of the associated quantum spin chains, respectively. The time-dependent algebra which is obtained from the action of the Markov generator of the exclusion process (or quantum Hamiltonian of the spin chain, respectively) is given in terms of a set of quadratic relations. By analysing the permutation consistency of the induced cubic relations, we obtain sufficient conditions on the hopping rates (i.e. the quantum mechanical interaction constants) which allow us to identify integrable models. From the dynamical algebra we construct the quadratic algebra of Zamolodchikov type, the associativity of which is a Yang–Baxter equation. We also obtain directly from the dynamical matrix product ansatz the Bethe ansatz equations for the spectra of these models.
Motivated by an exact mapping between anisotropic half integer spin quantum Heisenberg models and asymmetric diffusions on the lattice, we consider an anisotropic simple exclusion process with NNN particles in a rectangle of $\bbZ^2$. Every particle at row hhh tries to jump to an arbitrary empty site at row h±1h\pm 1h±1 with rate q+1q^{+ 1}q+1, where q∈(0,1)q\in (0,1)q∈(0,1) is a measure of the drift driving the particles toward the bottom of the rectangle. We prove that the spectral gap of the generator is uniformly positive in NNN and in the size of the rectangle. The proof is inspired by a recent interesting technique envisioned by E. Carlen, M. C. Carvalho and M. Loss to analyze the Kac model for the nonlinear Boltzmann equation. We then apply the result to prove precise upper and lower bounds on the energy gap for the spin-SSS, $S\in \ov2\bbN$, $\mbox{\textit{{XXZ}}}$ chain and for the 111 interface of the spin-SSS $\mbox{\textit{{XXZ}}}$\vspace{-1pt} Heisenberg model, thus generalizing previous results valid only for spin $\ov2$.
We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Using the Bethe ansatz and similarity transformations this yields new exact results. In a complementary approach we generalize previous work and present a new description of these and other processes and the related quantum chains in terms of an operator algebra with quadratic relations. The full solution of the master equation of the process is thus turned into the problem of finding representations of this algebra. We find a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. We obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal boundary fields.
A variational approach based on Bogoliubov inequality for the free energy is employed in order to treat the quantum spin-1 anisotropic ferromagnetic Heisenberg model in the presence of a crystal field. Within the Bogoliubov scheme an improved pair approximation has been used. The temperature-dependent thermodynamic functions have been obtained and provide much better results than the previous simple mean-field scheme. In one dimension, which is still nonintegrable for quantum spin-1, we get the exact results in the classical limit, or near-exact results in the quantum case, for the free energy, magnetization, and quadrupole moment, as well for the transition temperature. In two and three dimensions the corresponding global phase diagrams have been obtained as a function of the parameters of the Hamiltonian. First-order transition lines, second-order transition lines, tricritical and tetracritical points, and critical endpoints have been located through the analysis of the minimum of the Helmholtz free energy and a Landau-like expansion in the approximated free energy. Only first-order quantum transitions have been found at zero temperature. Limiting cases, such as isotropic Heisenberg, Blume-Capel, and Ising models, have been analyzed and compared to previous results obtained from other analytical approaches as well as from Monte Carlo simulations.