Quantum Walks of SU(2)_k Anyons on a Ladder

TL;DR

This paper investigates the dynamics of SU(2)_k anyons using a quantum walk model on a ladder, revealing how braiding interactions influence spreading velocities and probability distributions, with implications for understanding non-Abelian anyon behavior.

Contribution

It introduces a simplified quantum walk model for anyons that accounts for braiding effects and fusion degrees of freedom, enabling longer simulations and analysis of spreading behaviors.

Findings

Abelian anyons exhibit ballistic spreading similar to trivial particles.

Non-Abelian anyons show linearly increasing spreading velocity over time.

The probability distribution for the k=2 case matches a classical unbiased random walk.

Abstract

We study the effects of braiding interactions on single anyon dynamics using a quantum walk model on a quasi-1-dimensional ladder filled with stationary anyons. The model includes loss of information of the coin and nonlocal fusion degrees of freedom on every second time step, such that the entanglement between the position states and the exponentially growing auxiliary degrees of freedom is lost. The computational complexity of numerical calculations reduces drastically from the fully coherent anyonic quantum walk model, allowing for relatively long simulations for anyons which are spin-1/2 irreps of SU(2)_k Chern-Simons theory. We find that for Abelian anyons, the walk retains the ballistic spreading velocity just like particles with trivial braiding statistics. For non-Abelian anyons, the numerical results indicate that the spreading velocity is linearly dependent on the number of time steps. By approximating the Kraus generators of the time evolution map by circulant matrices, it is shown that the spatial probability distribution for the k=2 walk, corresponding to Ising model anyons, is equal to the classical unbiased random walk distribution.