Variational approach to the thermodynamics of a quantum sine-Gordon field with out-of-plane fluctuations

The quantum corrections to the partition function of nonlinear systems with a nonlocal kinetic energy are calculated by extending a variational approach based on the path integral and developed in a previous paper, which allows one to take into account in the quantum scheme of the quadratic part of the Hamiltonian. This extension can be useful to study the role of the out-of-plane fluctuations which cause deviations from the sine-Gordon model for some magnetic chains.