Thermal Broadening of Phonon Spectral Function in Classical Lattice Models: Projective Truncation Approximation

Thermal broadening of the quasi-particle peak in the spectral function is an important physical feature in many statistical systems, but it is difficult to calculate. To tackle this problem, we propose the $H$-expanded basis within the projective truncation approximation (PTA) of the Green's function equation of motion. A zeros-removing technique is introduced to stabilize the iterative solution of the PTA equations. Benchmarking calculations on the classical one-variable anharmonic oscillator model and the one-dimensional $\phi^4$ lattice model show that the thermal broadened quasi-particle peak in the spectral function can be produced on a semi-quantitative level. Using this method, we discuss the low- and high- temperature power-law behaviors of the spectral width $\Gamma_k(T)$ of the one-dimensional $\phi^4$ model, finding it in contradiction with the assumption of effective phonon theory. A short-chain limit of this model is also discovered. Issues of extending the $H$-expanded basis to quantum systems and of the applicability of the Debye formula for thermal conductivity are discussed.