Quantization of \beta-Fermi-Pasta-Ulam Lattice with Nearest and Next-nearest Neighbour Interactions

TL;DR

This paper quantizes the eta-FPU lattice with nearest and next-nearest neighbor interactions, revealing multi-phonon bound states that correspond to quantum breathers, and analyzes how extended interactions influence the spectrum and correlations.

Contribution

It introduces a numerical quantization approach for the eta-FPU model with extended interactions and identifies quantum breather states, extending previous classical and mean field analyses.

Findings

Excellent agreement between numerical and analytical spectra

Existence of multi-phonon bound states with particle-like properties

Next-nearest neighbor interactions alter the eigenvalue spectrum and correlations

Abstract

We quantize the \beta-Fermi-Pasta-Ulam (FPU) model with nearest and next-nearest neighbour interactions using a number conserving approximation and a numerical exact diagonalization method. Our numerical mean field bi-phonon spectrum shows excellent agreement with the analytic mean field results of Ivi\'{c} and Tsironis ((2006) Physica D 216 200), except for the wave vector at the midpoint of the Brillouin zone. We then relax the mean field approximation and calculate the eigenvalue spectrum of the full Hamiltonian. We show the existence of multi-phonon bound states and analyze the properties of these states as the system parameters vary. From the calculation of the spatial correlation function we then show that these multi-phonon bound states are particle like states with finite spatial correlation. Accordingly we identify these multi-phonon bound states as the quantum equivalent of the breather solutions of the corresponding classical FPU model. The four-phonon spectrum of the system is then obtained and its properties are studied. We then generalize the study to an extended range interaction and consider the quantization of the \beta-FPU model with next-nearest-neighbour interactions. We analyze the effect of the next-nearest-neighbour interactions on the eigenvalue spectrum and the correlation functions of the system.