Investigation of continuous-time quantum walk on root lattice $A_n$ and honeycomb lattice

TL;DR

This paper analyzes the behavior of continuous-time quantum walks on root and honeycomb lattices using spectral methods, association schemes, and orthogonal polynomials, revealing their algebraic and supersymmetric structures.

Contribution

It introduces a spectral distribution approach to CTQW on complex lattices, constructs associated algebraic schemes, and uncovers supersymmetric properties of honeycomb lattices.

Findings

Spectral distribution method effectively describes CTQW on large lattices.

Construction of $n$-variable orthogonal polynomials related to generalized Gegenbauer polynomials.

Identification of supersymmetric structure in finite honeycomb lattices.

Abstract

The continuous-time quantum walk (CTQW) on root lattice $A_n$ (known as hexagonal lattice for $n=2$) and honeycomb one is investigated by using spectral distribution method. To this aim, some association schemes are constructed from abelian group $Z^{\otimes n}_m$ and two copies of finite hexagonal lattices, such that their underlying graphs tend to root lattice $A_n$ and honeycomb one, as the size of the underlying graphs grows to infinity. The CTQW on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on Terwilliger algebra, where we get the required results for root lattice $A_n$ and honeycomb one, from large enough underlying graphs. Moreover, by using the stationary phase method, the long time behavior of CTQW on infinite graphs is approximated with finite ones. Also it is shown that the Bose-Mesner algebras of our constructed association schemes (called $n$-variable $P$-polynomial) can be generated by $n$ commuting generators, where raising, flat and lowering operators (as elements of Terwilliger algebra) are associated with each generator. A system of $n$-variable orthogonal polynomials which are special cases of \textit{generalized} Gegenbauer polynomials is constructed, where the probability amplitudes are given by integrals over these polynomials or their linear combinations. Finally the suppersymmetric structure of finite honeycomb lattices is revealed. Keywords: underlying graphs of association schemes, continuous-time quantum walk, orthogonal polynomials, spectral distribution. PACs Index: 03.65.Ud