Variational approach to renormalized phonon in momentum-nonconserving nonlinear lattices

A previously proposed variational approach for momentum-conserving systems [J. Liu et.al., Phys. Rev. E 91, 042910 (2015)] is extended to systematically investigate general momentum-nonconserving nonlinear lattices. Two intrinsic identities characterizing optimal reference systems are revealed, which enables us to derive explicit expressions for optimal variational parameters. The resulting optimal harmonic reference systems provide information for the band gap as well as the dispersion of renormalized phonons in nonlinear lattices. As a demonstration, we consider the one-dimensional \phi^?4 lattice. By combining the transfer integral operator method, we show that the phonon band gap endows a simple power-law temperature dependence in the weak stochasticity regime where predicted dispersion is reliable by comparing with numerical results. In addition, an exact relation between ensemble averages of the \phi^?4 lattice in the whole temperature range is found, regardless of the existence of the strong stochasticity threshold.