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International Journal of Modern Physics B

QUANTUM SUPERSPIN CHAINS

Timothy H. Baker, Peter Jarvis

Published Nov 1, 1994• 1 citation
Publisher

Abstract

Supersymmetric formulations of spin-½ quantum chains, associated with integrable models, are considered. One class of super-extensions, based on low-dimensional classical and exceptional superalgebras containing sl(2), is illustrated with the case of osp(1/2). A more radical generalization, in which the algebra of Pauli matrices is identified with the algebra of supersymmetric quantum mechanics, is also presented, and some of its algebraic properties discussed.

DOI
10.1142/s0217979294001536

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Supersymmetric formulations of spin-½ quantum chains, associated with integrable models, are considered. One class of super-extensions, based on low-dimensional classical and exceptional superalgebras containing sl(2), is illustrated with the case of osp(1/2). A more radical generalization, in which the algebra of Pauli matrices is identified with the algebra of supersymmetric quantum mechanics, is also presented, and some of its algebraic properties discussed.
An algebraic view of supersymmetric quantum mechanics is taken, emphasising the couplings between bosonic and fermionic modes in the supercharges. A class of model Hamiltonians is introduced wherein the fermionic (bosonic) operators are canonical and the bosonic (fermionic) ones satisfy a Lie algebra (superalgebra) whose representation theory permits the complete solution of the model in principle. The kinematical symmetry of such models is also described. The examples of one and two bosonic models, with 5U(2) and 5U(3) dynamical algebras respectively, are analysed in detail.
For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0-th time of the hierarchy. This implies a remarkable relation between the quantum supersymmetric spin chains and classical many-body integrable systems of particles of the Ruijsenaars-Schneider type. As an outcome, we obtain a system of algebraic equations for the spectrum of the spin chain Hamiltonians.