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arXiv:1909.05155 · Nuclear Physics B

Lax matrices for a 1-parameter subfamily of van Diejen–Toda chains

B.G. Pusztai

Published Nov 26, 2019• 0 citations
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Abstract

In this paper, we construct Lax matrices for certain relativistic open Toda chains endowed with a one-sided 1-parameter boundary interaction. Built upon the Lax representation of the dynamics, an algebraic solution algorithm is also exhibited. To our best knowledge, this particular 1-parameter subfamily of van Diejen–Toda chains has not been analyzed in earlier literature.

DOI
10.1016/j.nuclphysb.2019.114866

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In this paper, we construct Lax matrices for certain relativistic open Toda chains endowed with a one-sided 1-parameter boundary interaction. Built upon the Lax representation of the dynamics, an algebraic solution algorithm is also exhibited. To our best knowledge, this particular 1-parameter subfamily of van Diejen–Toda chains has not been analyzed in earlier literature.
Abstract We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1 + 1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin’s formula for the double-row transfer matrix is used to derive Hamilton’s equations of motion for both the Lax matrix and the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size for previously known Hamiltonians of type BC N and D N corresponding to constant and dynamical boundary matrices respectively.
Investigated is the relativistic periodic Toda chain, to each site of which the ultra-local Weyl algebra is associated. Weyl's qqq we are considering here is restricted to be inside the unit circle. Quantum Lax operators of the model are intertwined by six vertex RRR-matrix. Both independent Baxter's QQQ-operators are constructed explicitly as seria over local Weyl generators. The operator-valued wronskian of QQQ-s is also calculated.
We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1 + 1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin's formula for the double-row transfer matrix is used to derive Hamilton's equations of motion for both the Lax matrix and the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size for previously known Hamiltonians of type BC N and D N corresponding to constant and dynamical boundary matrices respectively.
The Lax pair for the one-dimensional Hubbard spin chain with open boundary conditions (BC) is explicitly constructed and two different kinds of boundary K-matrices compatible with the integrability of the model are obtained. Our construction provides an alternative and direct demonstration for the quantum integrability of the model and it is found that the model is related to a class of commuting transfer matrices of an equivalent coupled asymmetric six-vertex model with open BC. Our results show that the chemical potential term is indeed nontrivial for the underlying algebraic structure of the model.