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Next nearest neighbour interactions in the Heisenberg model

J. Moens

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

The Bethe ansatz is used to analytically solve the Heisenberg model with nearest neighbour interactions for spin-1/2 particles in an one dimensional chain. Our goal is to find out if the Bethe ansatz can be extended to nearest and next nearest neighbour interactions and to gain insight into the system with exact diagonalization. We find that this cannot be done. Because of certain assumptions made in the Bethe ansatz, the method fails for next nearest neighbour interactions.

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The Bethe ansatz is used to analytically solve the Heisenberg model with nearest neighbour interactions for spin-1/2 particles in an one dimensional chain. Our goal is to find out if the Bethe ansatz can be extended to nearest and next nearest neighbour interactions and to gain insight into the system with exact diagonalization. We find that this cannot be done. Because of certain assumptions made in the Bethe ansatz, the method fails for next nearest neighbour interactions.
Two physical models are discussed. We start out with the Heisenberg spin chain, which consists of a number of atoms in a circular con?guration, each with spin up or down. We set up a Hamiltonian of this system considering only nearest-neighbour interactions, and determine the energy of the spin chain. The mathematical description of the state in which these atoms will be, i.e. the wave functions of these atoms, satis?es a certain template, which is called the Bethe Ansatz. Using this Ansatz, we can solve the eigenvalue equation of the Hamiltonian and thus calculate the energy of the system. What follows is a discussion on a model for water ice, of which we want to determine the entropy at absolute zero. This leads to a set of equations that is very similar to that encountered in the Heisenberg model, and it turns out that the Bethe Ansatz is useful in this model as well: it enables us to explicitly calculate the entropy of an infinitely big system.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.