We have computed the sequence of memory functions ${K}_{n}$ (t) and their Laplace transforms, for the one-dimensional Ising model in a transverse field at T=\ensuremath{\infty}. The form of these functions shows explicitly why the Mori continued-fraction approach, with traditional truncation schemes, fails for this model.We have computed the sequence of memory functions ${K}_{n}$(t) and their Laplace transforms, for the one-dimensional Ising model in a transverse field at T=\ensuremath{\infty}. The form of these functions shows explicitly why the Mori continued-fraction approach, with traditional truncation schemes, fails for this model.

Physical review. B, Condensed matterSep 1, 19895 citations

Space-time—dependent spin correlation of the one-dimensional Ising model with a transverse field. Application to higher dimensions

Janine Lajzerowicz, Pierre M. Pfeuty

The correlation functions ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$ and ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field $h$ at $T=0$ . This model corresponds to the $X Y$ model with $\ensuremath{\gamma}=1$, and is equivalent to the two-dimensional anisotropic (${J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${J}_{2}\ensuremath{\rightarrow}0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of $h$ , ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)=\ensuremath{-}(\frac{1}{{h}^{2}})(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}}){\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$. At the critical field $h={h}_{c}=1$ , ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)\ensuremath{\sim}{({R}^{2}\ensuremath{-}{t}^{2})}^{\frac{\ensuremath{-}1}{8}}$. For $h<1,h>1$ the results already obtained for the $\mathrm{XY}$ model are recovered. We give some consequences from this equivalence in higher dimension, concerning the behavior of the correlation function at the critical field at $T=0$ .The correlation functions ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$ and ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field $h$ at $T=0$. This model corresponds to the $X Y$ model with $\ensuremath{\gamma}=1$, and is equivalent to the two-dimensional anisotropic (${J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${J}_{2}\ensuremath{\rightarrow}0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of $h$, ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)=\ensuremath{-}(\frac{1}{{h}^{2}})(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}}){\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$. At the critical field $h={h}_{c}=1$, ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)\ensuremath{\sim}{({R}^{2}\ensuremath{-}{t}^{2})}^{\frac{\ensuremath{-}1}{8}}$. For $h<1,h>1$ the results already obtained for the $\mathrm{XY}$ model are recovered. We give some consequences from this equivalence in higher dimension, concerning the behavior of the correlation function at the critical field at $T=0$.

Physical review. B, Solid stateJun 1, 197535 citations

Ising model in a transverse random field

L. A. Serkov, V. O. Shvalev

Physics of the Solid StateNov 1, 19940 citations

We have computed the sequence of memory functions

{K}_{n}

(t) and their Laplace transforms, for the one-dimensional Ising model in a transverse field at T=\ensuremath{\infty}. The form of these functions shows explicitly why the Mori continued-fraction approach, with traditional truncation schemes, fails for this model.

The correlation functions ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$ and ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field

h

T=0

. This model corresponds to the

X Y

model with $\ensuremath{\gamma}=1$, and is equivalent to the two-dimensional anisotropic (${J}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${J}_{2}\ensuremath{\rightarrow}0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of

h

, ${\ensuremath{\rho}}_{\mathrm{yy}}(R,t)=\ensuremath{-}(\frac{1}{{h}^{2}})(\frac{{\ensuremath{\partial}}^{2}}{\ensuremath{\partial}{t}^{2}}){\ensuremath{\rho}}_{\mathrm{xx}}(R,t)$. At the critical field

h={h}_{c}=1

, ${\ensuremath{\rho}}_{\mathrm{xx}}(R,t)\ensuremath{\sim}{({R}^{2}\ensuremath{-}{t}^{2})}^{\frac{\ensuremath{-}1}{8}}$. For

h<1,h>1

the results already obtained for the

\mathrm{XY}

model are recovered. We give some consequences from this equivalence in higher dimension, concerning the behavior of the correlation function at the critical field at

T=0