Edwards Localization

TL;DR

This work studies localization phenomena in the Edwards model of a particle interacting with a bath of stochastic scattering centers, using Monte Carlo averaging over center configurations and both canonical and affine quantization. It demonstrates static and dynamic ground-state localization in a 1D setting with delta-function impurities, showing that localization strengthens with coupling $g$ and with disorder, while affine quantization yields localization that is independent of $g$. The analysis connects to a polaron-like picture with a Bose-Einstein condensed background and outlines a path-integral perspective for dynamics. Overall, the results advance understanding of localization in quantum stochastic systems and suggest practical methods for studying disorder-averaged states and their evolution.

Abstract

We study the localization problem in quantum stochastic mechanics. We start from the Edwards model for a particle in a bath of scattering centers and prove static localization of the ground state wavefunction of the particle in a one dimensional square well coupled to Dirac delta like scattering centers in arbitrary but fixed positions. We see how the localization increases for increasing coupling $g$. Then we choose the scattering centers positions as pseudo random numbers with a uniform probability distribution and observe an increase in the localization of the average of the ground state over the many positions realizations. We discuss how this averaging procedure is consistent with a picture of a particle in a Bose-Einstein condensate of of non interacting boson scattering centers interacting with the particle with Dirac delta functions pair potential. We then study the dynamics of the ground state wave function. We conclude with a discussion of the affine quantization version of the Lax model which reduces to a system of contiguous square wells with walls in arbitrary positions independently of the coupling constant $g$.

Figures (2)

• Figure 1: We show the ground state from Eq. (\ref{['eq:esmc']}) for three choices of increasing $M$ and $N=3$. Low $g=1$ in the left panel and high $g=100$ in the right panel. From Eq. (\ref{['eq:evmc']}) the resulting ground state energy is as follows: for $g=1\longrightarrow$ ($\widetilde{E}_0=741.956$ for $M=1$, $\widetilde{E}_0=148.428$ for $M=10$, and $\widetilde{E}_0=613.504$ for $M=100$) and for $g=100\longrightarrow$ ($\widetilde{E}_0=3027.62$ for $M=1$, $\widetilde{E}_0=2343.16$ for $M=10$, and $\widetilde{E}_0=3698.51$ for $M=100$).
• Figure 2: We show the time evolution of $|\widetilde{\Psi}_0(x,t)|^2$ from Eq. (\ref{['eq:dyn2']}) for $N=3, g=100, M=100$ at $t=0,1,3,30$.