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The Bethe Ansatz in the Heisenberg model and the Square Ice model

A.H. van Denzen

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

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.

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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.
Abstract This chapter verifies the conjecture for the wave function, the Bethe ansatz wave function, of the anisotropic Heisenberg quantum spin chain by examining first the cases for one, two, and three spin deviations. The equations determining the quasi- momenta are the Bethe ansatz equations, now obtained from the coordinate Bethe ansatz. The Bethe ansatz equations derive from the eigenvalue equation in combination with boundary conditions, here periodic boundary conditions. These quasi-momenta also determine the energy eigenvalue. However, solving the Bethe ansatz equations to obtain a particular state requires more considerations. New variables, called rapidities, are useful. The consideration of the thermodynamic limit then allows to extract information about the ground state and low-lying excitations of the anisotropic quantum spin chain from the Bethe ansatz equations. Furthermore, complex solutions of the Bethe ansatz equations, called strings, are considered.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.
Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.