Scaling Phenomena in a Unitary Model of Directed Propagating Waves with Applications to One-Dimensional Electrons in a Time-Varying Potential

TL;DR

This paper investigates a 2D lattice model of directed waves, exploring its scaling behavior, interference effects under magnetic flux, and propagation in random media, with applications to quantum particles and electromagnetic waves.

Contribution

It introduces a unitary model for directed wave propagation, analyzes interference effects due to magnetic flux, and studies scaling properties in random media, revealing novel behaviors distinct from existing models.

Findings

Amplitude becomes more collimated with larger flux quantum ratios.

Scaling of wave width with distance differs from known models.

Probability moments decay as t^{-n/2} at the origin.

Abstract

We study a 2D lattice model of forward-directed waves in which the integrated intensity for classical waves (or probability for quantum mechanical particles) is conserved. The model describes the time evolution of 1D quantum particle in a time-varying potential and also applies to propagation of electromagnetic waves in two dimensions within the parabolic approximation. We present a closed form solution for propagation in a uniform system. Motivated by recent studies of non-unitary directed models for localized 2D electrons tunneling in a magnetic field, we then address related theoretical questions of how the interference pattern between constrained-forward paths in this unitary model is affected by the addition of phases corresponding to such a magnetic field. The behavior is found to depend sensitively on the value of $\Phi/\Phi_{0}$, where $\Phi$ is flux per plaquette and $\Phi_{0}$ is the unit of flux quantum. For $\Phi/\Phi_{0} = p/q$ we find the amplitude to be more collimated the larger is the value of $q$. We next consider propagation in a random forward scattering media. In particular the scaling properties associated with the transverse width $x$ of the wave, as function of its distance $t$ from point source, are addressed. We find the moments of $x$ to scale with $t$ in a very different way from what is known for either off-lattice unitary or on-lattice non-unitary systems. The scaling of the moments of the probability $[P^{n}(x,t)]$ (or intensity) at a point $(x=0,t)$ is found to be consistent with a simple behavior $[P^{n}(0,t)] \sim t^{-\frac{n}{2}}$. Implications to the behavior of one-dimensional lattice quantum particles in a dynamically fluctuating