Finite-temperature correlations for the Ising chain in a transverse field

Two sets of nonlinear differential equations are derived and discussed for the time-dependent correlations between x-components of spins (S = 12) in an Ising chain in the presence of a transverse magnetic field. The equations are independent of temperature which enters only through the initial conditions for the correlations. The equations are valid for the (general) inhomogeneous case in which the exchange coupling as well as the magnetic field depend on the sites in the chain. In the derivation use is made of a general formulation of the thermodynamic Wick theorem. For the homogeneous case a nonlinear differential-difference equation is derived, generalizing the Painlevé III equation found previously at zero temperature in the scaling limit. The finite-temperature field theory limit is discussed also.