Statistical mechanics of kinks and central peak phenomena inφ4theory forT<~Tc

The statistical mechanics of a system of kinks condensed from a ${\phi }^4$ field theory in one space and one time dimension is studied with a functional-path-integral formulation of field theory in a coherent-state basis. The kink-kink and phonon-phonon parts of the two-point correlation function and the dynamical structure function are calculated in lowest order. All the qualitative features of the usual free-kink-gas phenomenology are recapitulated in lowest order and quantitative values of the thermodynamic observables agree better with the molecular-dynamics computer-simulation results. The free energy of the kink gas is lower than previously calculated values. Our analysis leads to a double-peak feature in the dynamical structure function as is characteristic of a relativistic boson gas and which agrees with the results of the computer-simulation studies. Our calculation of the dynamical structure function applies for temperatures below a transition temperature $T_c$ which is defined as that temperature at which the kink velocity vanishes. The numerical values that we calculate for $T_c$ are somewhat larger than those obtained from computer-simulation studies but they agree well with the results of Hartree calculations. Below $T_c$ the temperature dependence of the soft-mode frequency is $\omega =q{[1-{\left(\frac{T}{T_c}\right)}^2]}^{\frac{1}{2}}$ in reasonable agreement with observations on SrTi${\mathrm{O}}_3$. We emphasize that the path integral is dominated by the whole class of all classical configurations which extremize the classical action and satisfy classical periodic boundary conditions, both spatial and thermal. Boundary conditions are essential in our formulation and these lead to a new statistical mechanics for the condensate (kinks) and the fluctuations (phonons).