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

The correlation functions ρₓₓ(R,t) and ρyy(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 γ=1, and is equivalent to the two-dimensional anisotropic (J₁→∞, J₂→0) Ising model. The additional dimension in the classical model is the imaginary time of the quantum model. For all values of h, ρyy(R,t)=-(1h²)(∂²∂t²)ρₓₓ(R,t). At the critical field h=hc=1, ρₓₓ(R,t)~(R²-t²)^-18. For $h<1,h>1$ the results already obtained for the 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 $T=0$.