Effective potential and finite-temperature renormalization of the ${\phi }^4$ chain

The quantum thermodynamics of the nonintegrable ${\phi }^4$ one-dimensional chain is studied by means of a classical effective potential, which includes in a fully quantum way the linear modes of the field. In contrast to the case of the sine-Gordon chain, integrable in the continuum limit, exact results are not available, so that approximate quantum calculations appear to be useful. This effective potential is determined by a variational approach developed in previous papers, which is based on the path-integral formulation of statistical mechanics. The temperature renormalization is studied, in the limit of low temperature, by means of a self-consistent saddle-point method, both for the vacuum and the one-kink sectors, and the results of the semiclassical approximation are recovered. Important results are obtained by a new low-coupling expansion for the effective potential. Its range of validity in temperature is much wider than the range of previous high-temperature expansions. The results for the nonlinear contributions to internal energy and specific heat, obtained by means of original transfer-matrix computations, are finally presented and discussed.