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Stochastic Processes and Random MatricesLecture Notes of the Les Houches Summer School: Volume 104, July 2015$
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Grégory Schehr, Alexander Altland, Yan V. Fyodorov, Neil O'Connell, and Leticia F. Cugliandolo

Print publication date: 2017

Print ISBN-13: 9780198797319

Published to Oxford Scholarship Online: January 2018

DOI: 10.1093/oso/9780198797319.001.0001

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Quantum spin chains and classical integrable systems

Quantum spin chains and classical integrable systems

(p.578) 13 Quantum spin chains and classical integrable systems
Stochastic Processes and Random Matrices

Anton Zabrodin

Oxford University Press

This chapter is a review of the recently established quantum-classical correspondence for integrable systems based on the construction of the master T-operator. For integrable inhomogeneous quantum spin chains with gl(N)-invariant R-matrices in finite-dimensional representations, the master T-operator is a sort of generating function for the family of commuting quantum transfer matrices depending on an infinite number of parameters. Any eigenvalue of the master T-operator is the tau-function of the classical modified KP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0th time of the hierarchy. This implies a remarkable relation between the quantum spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, a system of algebraic equations can be obtained for the spectrum of the spin chain Hamiltonians.

Keywords:   quantum-classical correspondence, tau-functions, KP hierarchy, integrable systems, Ruijsenaars-Schneider systems

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