Investigation of a class of one-dimensional nonlinear fields

We investigate the behavior of a class of nonlinear one-dimensional fields with symmetric and periodic interaction. The prototypical equation of this class is the sine-Gordon system. These fields have classical and static solutions with localized particlelike properties. We present a general theoretical framework to discuss the stability of these solutions under quantum and thermal fluctuations. It is shown that stability requires a phase transition of the system into an ordered equilibrium state. As an example, we calculate exactly the thermodynamics of the classical system by evaluation of a functional integral, using transfer matrix techniques. The treatment of the boundary condition, or "kink" density, as a thermodynamic variable requires a nontrivial extension of the standard methods. By a remarkable analytic continuation we show that the thermodynamic pressure is given by the eigenvalue of Hill's equation in the first unstable region. In particular, we compute the thermodynamics of the sine-Gordon field, and find that the localized behavior exhibited by the classical system at zero temperature is completely destroyed at finite temperature; i.e., the solutions are unstable with respect to thermal fluctuations.