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Bethe Ansatz and Integrability

Tadeusz Lulek

Published Nov 26, 2017• 0 citations
Publisher
DOI
10.1142/9789813234345_0003

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Presented is a method, which we call Bethe ansatz cluster expansion (BACE) method, to investigate statistical mechanics of quantum integrable systems. By use of only the Bethe ansatz equation, n -particle partition functions are exactly calculated. From the expression for the partition function, the n -particle cluster integrals are derived. The results completely agree with those of the thermal Bethe ansatz (TBA) by Yang and Yang. Thus, the BACE method directly proves the validity of the TBA. Partition theory and graphical representation are used to make the presentation simple and transparent.
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.