The anisotropic XY model on the inhomogeneous periodic chain

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 ($j$ 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 $\chi_{q}^{zz}(\omega)$.Comment: 33 pages, 13 figures, 01 table. Revised version (minor corrections) accepted for publiction in Phys. Rev.

Spin dynamics in finite cyclic XY model

Evolution of the z-component of a single spin in the finite cyclic XY spin 1/2 chain is studied. Initially one selected spin is polarized while other spins are completely unpolarized and uncorrelated. Polarization of the selected spin as a function of time is proportional to the autocorrelation function at infinite temperature. Initialization of the selected spin gives rise to two wave packets moving in opposite directions and winding over the circle. We express the correlation function as a series in winding number and derive tractable approximations for each term. This allows to give qualitative explanation and quantitative description to various finite-size effects such as partial revivals and transition from regular to erratic behavior.Comment: v2: substantially extended; v3: references added, accepted to Phys. Rev.