Abstract
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.
Highlights
Supersymmetric extensions of quantum integrable spin chains were introduced in [1, 2].Such models, called graded magnets in [2], are based on solutions of the graded Yang-Baxter equation in the same way as ordinary integrable spin chains are built from quantum R-matrices satisfying the Yang-Baxter RRR = RRR relation
For integrable inhomogeneous supersymmetric spin chains 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
It is a polynomial in the spectral parameter which is identified with the 0-th time of the hierarchy
Summary
Supersymmetric extensions of quantum integrable spin chains were introduced in [1, 2]. This means that any eigenvalue T (x, t) of T(x, t) is a tau-function of the mKP hierarchy In this way, the commutative algebras of susy-XXX spin chain Hamiltonians, for all possible gradings, appear to be embedded into the infinite integrable hierarchy of non-linear differential-difference equations, the mKP hierarchy [11,12,13]. The commutative algebras of susy-XXX spin chain Hamiltonians, for all possible gradings, appear to be embedded into the infinite integrable hierarchy of non-linear differential-difference equations, the mKP hierarchy [11,12,13] This is a further development of the earlier studies [14,15,16,17] clarifying the role of classical integrable hierarchies in quantum integrable models.
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