Cambridge University Press eBooks

Transfer matrix method

Gleb Eduardovich Arutyunov

Published Apr 27, 2026• 0 citations

Abstract

The generalized Bethe hypothesis, though conceptually powerful, becomes increasingly unwieldy when approached through direct state-by-state analysis, particularly for systems with multiple excitations. While one- and two-particle states allow straightforward generalizations, a universal proof valid for arbitrary permutation modules remains a central challenge. This chapter introduces the transfer matrix method as a systematic and elegant framework to address this issue. Based on the spin chain realization of permutation modules, the method facilitates the analysis of a broad class of integrable systems and serves as a powerful computational tool. A particularly appealing feature is the diagonal form of the transfer matrix on the common eigenspace of scattering operators. For the case gl 2 , the inhomogeneous spin chain transfer matrix is diagonalized using Lieb’s method, thereby confirming the generalized Bethe hypothesis in this setting. In addition, representation theory is applied to classify spin chain states according to their transformation properties under the global symmetry algebra, highlighting the rich algebraic structure underlying the solvability of these models.

DOI: 10.1017/9781009664837.005

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