Long-Time Asymptotics for the Autocorrelation Function of the Transverse Ising Chain at the Critical Magnetic Field

Abstract

We consider the particular case of the spin $- \frac{1}{2}XY$ model in a magnetic field with Hamiltonian

$$H = - \frac{1}{2}\sum\limits_{{\ell \in \mathbb{Z}}} {(\sigma _{\ell }^{x}\sigma _{{\ell + 1}}^{x} + } \sigma _{\ell }^{z})$$

(1.1)

where $\sigma _{\ell }^{x},\sigma _{\ell }^{z}$, are the standard Pauli matrices at the ${{\ell }^{{th}}}$ site of a one-dimensional lattice. As is well known, the Hamiltonain H can clearly be identified with the transverse Ising model at the critical transverse magnetic field ([LSM]). We will study the long-time behavior of the autocorrelation function X(t) of the first spin component

$$\begin{array}{*{20}{c}} {X(t) = \left\langle {\sigma _{0}^{x}(t)\sigma _{0}^{x}} \right\rangle T} \\ { = \frac{{Tr({{e}^{{ - \beta H}}}({{e}^{{ - iHt}}}\sigma _{0}^{x}{{e}^{{iHt}}})\sigma _{0}^{x})}}{{Tr({{e}^{{ - \beta H}}})}}} \\ \end{array}$$

(1.2)

where ß = 1/T is the inverse temperature.