Quantum percolation and transition point of a directed discrete-time quantum walk

TL;DR

This paper investigates quantum percolation using a directed discrete-time quantum walk model on various 2D lattices, determining the critical connectivity for percolation and supporting findings with analytical and numerical methods, relevant for quantum transport and information.

Contribution

It introduces a numerical and analytical study of quantum percolation thresholds for a two-state particle on different 2D lattices using a quantum walk model, highlighting the transition point behavior.

Findings

Percolation transition point tends towards unity with increasing lattice size.

Analytical expression for percolation probability on square lattice matches numerical results.

Quantum walk model reveals critical connectivity for quantum percolation in 2D lattices.

Abstract

Quantum percolation describes the problem of a quantum particle moving through a disordered system. While certain similarities to classical percolation exist, the quantum case has additional complexity due to the possibility of Anderson localisation. Here, we consider a directed discrete-time quantum walk as a model to study quantum percolation of a two-state particle on a two-dimensional lattice. Using numerical analysis we determine the fraction of connected edges required (transition point) in the lattice for the two-state particle to percolate with finite (non-zero) probability for three fundamental lattice geometries, finite square lattice, honeycomb lattice, and nanotube structure and show that it tends towards unity for increasing lattice sizes. To support the numerical results we also use a continuum approximation to analytically derive the expression for the percolation probability for the case of the square lattice and show that it agrees with the numerically obtained results for the discrete case. Beyond the fundamental interest to understand the dynamics of a two-state particle on a lattice (network) with disconnected vertices, our study has the potential to shed light on the transport dynamics in various quantum condensed matter systems and the construction of quantum information processing and communication protocols.