Projective-truncation-approximation study of the one-dimensional $\phi^4$ lattice model

TL;DR

This paper introduces a projective truncation approximation method for classical models, applied to the 1D $^4$ lattice, providing detailed phonon and correlation insights with exact temperature exponents.

Contribution

It develops the PTA method within the Green's function formalism and applies it to the 1D $^4$ lattice, achieving results comparable to advanced variational approaches.

Findings

Exact phonon dispersion and correlation functions obtained.

Power-law temperature dependence with exact exponents identified.

Results extend beyond quadratic variational approximation.

Abstract

In this paper, we first develop the projective truncation approximation (PTA) in the Green's function equation of motion (EOM) formalism for classical statistical models. To implement PTA for a given Hamiltonian, we choose a set of basis variables and projectively truncate the hierarchical EOM. We apply PTA to the one-dimensional $\phi^4$ lattice model. Phonon dispersion and static correlation functions are studied in detail. Using one- and two-dimensional bases, we obtain results identical to and beyond the quadratic variational approximation, respectively. In particular, we analyze the power-law temperature dependence of the static averages in the low- and high-temperature limits, and we give exact exponents.