Anisotropic XY model on the inhomogeneous periodic chain

Abstract

The static and dynamic properties of the anisotropic $XY$ model $(s=1∕2)$ on the inhomogeneous periodic chain, composed of $N$ cells with $n$ different exchange interactions and magnetic moments, in a transverse field $h$, are determined exactly at arbitrary temperatures. The properties are obtained by introducing the Jordan-Wigner fermionization and by reducing the problem to a diagonalization of a finite matrix of $n$th order. The quantum transitions are determined exactly by analyzing, as a function of the field, the induced magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}{\ensuremath{\mu}}_{m}⟨{S}_{j,m}^{z}⟩$ ($j$ denotes the cell, $m$ the site within the cell, ${\ensuremath{\mu}}_{m}$ the magnetic moment at site $m$ within the cell) and the spontaneous magnetization $1∕n{\ensuremath{\sum}}_{m=1}^{n}⟨{S}_{j,{m}^{\ensuremath{'}}}^{x}⟩$ which is obtained from the correlations $⟨{S}_{j,m}^{x}{S}_{j+r,m}^{x}⟩$ for large spin separations. These results, which are obtained for infinite chains, correspond to an extension of the ones obtained by Tong and Zhong [Physica B 304, 91 (2001)]. The dynamic correlations, $⟨{S}_{j,m}^{z}(t){S}_{{j}^{\ensuremath{'}},{m}^{\ensuremath{'}}}^{z}(0)⟩$, and the dynamic susceptibility, ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$, are also obtained at arbitrary temperatures. Explicit results are presented in the limit $T=0$, where the critical behavior occurs, for the static susceptibility ${\ensuremath{\chi}}_{q}^{zz}(0)$ as a function of the transverse field $h$, and for the frequency dependency of dynamic susceptibility ${\ensuremath{\chi}}_{q}^{zz}(\ensuremath{\omega})$.