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The anisotropic XY model on the inhomogeneous periodic chain

J. P. de Lima·L. L. Goncalves·T. F. A. Alves
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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 $1/n\sum_{m=1}^{n}μ_{m}\left< S_{j,m}^{z}\right>$ ($j$ denotes the cell, $m$ the site within the cell, $μ_{m}$ 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, $χ_{q}^{zz}(ω),$ are also obtained at arbitrary temperatures. Explicit results are presented in the limit T=0, where the critical behaviour occurs, for the static susceptibility $χ_{q}^{zz}(0)$ as a function of the transverse field $h$, and for the frequency dependency of dynamic susceptibility $χ_{q}^{zz}(ω)$.

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J. P. de Lima, L. L. Goncalves, T. F. A. Alves. 2007-04-09. The anisotropic XY model on the inhomogeneous periodic chain. https://doi.org/10.1103/physrevb.75.214406

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