Time-Dependent Correlations in a Solvable Ferromagnetic Model

Abstract

The exact equation of state, two-time correlation function, and linear response function, are calculated in the limit of infinite $N$ for a classical $N$-spin system with a ferromagnetic phase transition, in a particular nonuniform magnetic field. The correlation function can be analytically continued in temperature (or magneticfield strength) from the nonferromagnetic to the ferromagnetic region of the $T\ensuremath{-}H$ plane; the result of such continuation is not, however, the correlation function for the ferromagnetic region, but a function which grows exponentially in time. The frequency-dependent linear response function has a pole at zero frequency throughout the ferromagnetic region due to a broken symmetry; the corresponding function in the nonferromagnetic region developes a pole at zero frequency as the ferromagnetic region is approached, but when the function is continued in temperature (or field strength) into the ferromagnetic region, the pole detaches itself from the origin and moves up into the complex frequency plane, signifying an exponential growth in time of the linear response. The purpose of the model is to demonstrate that this kind of behavior does not contradict any general structural properties of equilibrium thermodynamic correlation or response functions. The possible general significance of such behavior for a theory of metastable states is discussed.