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

The correlation functions ${\rho }_{\mathrm{xx}}(R,t)$ and ${\rho }_{\mathrm{yy}}(R,t)$ are calculated for the one-dimensional Ising model with a transverse field $h$ at $T=0\mathrm{}$. This model corresponds to the $X Y$ model with $\gamma =1\mathrm{}$, and is equivalent to the two-dimensional anisotropic ($J_1\to \infty$, $J_2\to 0$) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of $h$, ${\rho }_{\mathrm{yy}}\mathrm{}(R,t)=-\left(\frac{1}{h^2}\right)\left(\frac{{\partial }^2}{\partial t^2}\right){\rho }_{\mathrm{xx}}\mathrm{}(R,t)\mathrm{}$. At the critical field $h=h_c=1\mathrm{}$, ${\rho }_{\mathrm{xx}}\mathrm{}(R,t)\mathrm{}\sim {(R^2-t^2)}^{\frac{-1\mathrm{}}{8}}$. For $h<\mathrm{}1,h>\mathrm{}1\mathrm{}$ 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 $T=0\mathrm{}$.