The isotropic XY model on the inhomogeneous periodic chain

Abstract

The static and dynamic properties of the isotropic XY-model $(s=1/2)$ on the inhomogeneous periodic chain, composed of \emph{N} segments with \emph{n} different exchange interactions and magnetic moments, in a transverse field \emph{h} are obtained exactly at arbitrary temperatures. The properties are determined by introducing the generalized Jordan-Wigner transformation and by reducing the problem to a diagonalization of a finite matrix of \emph{n-th} order. The diagonalization procedure is discussed in detail and the critical behaviour induced by the transverse field, at T=0, is presented. The quantum transitions are determined by analyzing the behaviour of the induced magnetization, defined as $(1/n){\sum }_{m=1}^n{\mu }_m<S_{j,m}^z>$ where ${\mu }_m$ is the magnetic moment at site \emph{m} within the segment \emph{j}, as a function of the field, and the critical fields determined exactly. The dynamic correlations, $<S_{j,m}^z(t)S_{j^{{}^{\prime }},m^{{}^{\prime }}}^z(0)>$, and the dynamic susceptibility ${\chi }_q^{zz}(\omega )$ are also obtained at arbitrary temperatures. Explicit results are also presented in the limit T=0, where the critical behaviour occurs, for the static susceptibility ${\chi }_q^{zz}(0)$ as a function of the transverse field \emph{h}, and for the frequency dependency of dynamic susceptibility ${\chi }_q^{zz}(\omega )$. Also in this limit, the transverse time-correlation $<S_{j,m}^x(t)S_{j^{{}^{\prime }},m^{{}^{\prime }}}^x(0)>$, the dynamic and isothermal susceptibilities, ${\chi }_q^{xx}(\omega )$ and ${\chi }_T^{xx}$, are obtained for the transverse field greater or equal than the saturation field.

38 pages, 11 figures