Frequency-dependent Couplings Research Articles

A quantum field theory of resonant modes in 3-d nonlinear lattices at finite temperatures

We establish the Heisenberg equation for phonon operators in 3-d nonlinear lattices at finite temperatures. Applying the one loop approximation to the nonlinear potential in the Heisenberg equation, we obtain a dynamical equation with a self-consistent potential created by a spatially localized mode. Under the assumption that the mode is well-localized at a lattice site, we obtain an eigenvalue equation with the frequency dependent couplings corresponding to that for the force constant defects in linear lattices. However, the frequency dependent couplings allow only frequencies near the middle of the phonon bands, while the constant couplings allow low frequencies.