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arXiv:2005.08326 · Theoretical and Mathematical Physics

Wronskian-type formula for inhomogeneous $$TQ$$ equations

Rafael I. Nepomechie

Published Sep 1, 2020• 4 citations
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Abstract

It is known that the transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain with nondiagonal boundary magnetic fields satisfy a $$TQ$$ equation with an inhomogeneous term. We derive a discrete Wronskian-type formula relating a solution of this inhomogeneous $$TQ$$ equation to the corresponding solution of a dual inhomogeneous $$TQ$$ equation.

DOI
10.1134/s004057792009007x

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Wronskian-type formula for inhomogeneous TQTQTQ equations
It is known that the transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain with nondiagonal boundary magnetic fields satisfy a TQTQTQ equation with an inhomogeneous term. We derive a discrete Wronskian-type formula relating a solution of this inhomogeneous TQTQTQ equation to the corresponding solution of a dual inhomogeneous TQTQTQ equation.
We consider the open spin- s XXZ quantum spin chain with nondiagonal boundary terms. By exploiting certain functional relations at roots of unity, we derive a generalized form of T – Q relation involving more than one independent Q ( u ), which we use to propose the Bethe-ansatz-type expressions for the eigenvalues of the transfer matrix. At most two of the boundary parameters are set to be arbitrary and the bulk anisotropy parameter has values η = iπ/2,iπ/4,.... We also provide numerical evidence for the completeness of the Bethe-ansatz-type solutions derived, using the s = 1 case as an example.
We construct Q-operators for the open spin-12 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang–Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.
In this paper, we studied the exact solution of the C2(1) invariant quantum spin chain with off-diagonal open boundary condition. We obtain a solution of the reflection equation where the all matrix element of reflection matrix are nonzeros. By using the technique of fusion, we construct the fused transfer matrix and find the closed recursive relations among the transfer matrices. Based on the algebraic analysis, we obtain the eigenvalue of the system and express it as the inhomogeneous T−Q relation.