The anisotropic XY model on the inhomogeneous periodic chain

TL;DR

This paper provides an exact analysis of the static and dynamic properties of the anisotropic XY model on an inhomogeneous periodic chain, revealing quantum phase transitions and dynamic correlations at arbitrary temperatures.

Contribution

It extends previous work by exactly solving the inhomogeneous XY model with multiple exchange interactions and magnetic moments, including dynamic correlations and susceptibilities.

Findings

Exact determination of quantum transitions via magnetization analysis.

Explicit results for static susceptibility at T=0.

Frequency-dependent dynamic susceptibility behavior.

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 $nth$ order. The quantum transitions are determined exactly by analyzing, as a function of the field, the induced magnetization $1/n\sum_{m=1}^{n}\mu_{m}\left< S_{j,m}^{z}\right>$ ($j$ denotes the cell, $m$ the site within the cell, $\mu_{m}$ the magnetic moment at site $m$ within the cell) and the spontaneous magnetization $1/n\sum_{m=1}^{n}\left< S_{j,m}^{x},\right>$ which is obtained from the correlations $\left< S_{j,m}^{x}S_{j+r,m}^{x}\right>$ for large spin separations. These results, which are obtained for infinite chains, correspond to an extension of the ones obtained by Tong and Zhong(\textit{Physica B} \textbf{304,}91 (2001)). The dynamic correlations, $\left< S_{j,m}^{z}(t)S_{j^{\prime},m^{\prime}}^{z}(0)\right>$, and the dynamic susceptibility, $\chi_{q}^{zz}(\omega),$ are also obtained at arbitrary temperatures. Explicit results are 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 $h$, and for the frequency dependency of dynamic susceptibility $\chi_{q}^{zz}(\omega)$.