A τ-function on the inhomogeneous lattice is introduced by a deformation of that of the Toda lattice hierarchy. An associated discrete-time integrable system is constructed in Sato and Lax formalisms. It is shown that a one-dimensional reduction of the system gives a Bäcklund transformation of the classical isotropic Heisenberg spin chain, which is equivalent to the R II -chain proposed by Spiridonov and Zhedanov.

Journal of Physics A Mathematical and TheoreticalOct 20, 20084 citations

arXiv:cond-mat/0501525 · Journal of Magnetism and Magnetic Materials

The isotropic model on the inhomogeneous periodic chain

J.P. de Lima, T. F. A. Alves, L.L. Gonçalves

Published Mar 29, 2005• 7 citations

arXiv HTML PDF

DOI: 10.1016/j.jmmm.2005.03.001

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The isotropic

XY

model

(s=1/2)

in a transverse field, with uniform long-range interactions among the transverse components of the spins, on the inhomogeneous periodic chain, is studied. The model, composed of

N

segments with

n

different exchange interactions and magnetic moments, is exactly solved by introducing the integral Gaussian transformation and the generalized Jordan--Wigner transformation, which reduce the problem to the diagonalization of a finite matrix of

n\text{th}

order. The quantum transitions induced by the transverse field are determined by analyzing the induced magnetization of the cell and the equation of state. The phase diagrams for the quantum transitions, in the space generated by the transverse field and the interaction parameters, are presented. As expected, the model presents multiple, first- and second-order quantum transitions induced by the transverse field, and it corresponds to an extension of the models recently considered by the authors. Detailed results are also presented, at

T=0

, for the induced magnetization and isothermal susceptibility ${\ensuremath{\chi}}_{T}^{zz}$ as a function of the transverse field.