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

The continuous-time quantum walk (CTQW) on root lattice AnAn (known as hexagonal lattice for n=2n=2) and honeycomb one is investigated by using spectral distribution method. To this aim, some association schemes are constructed from abelian group Zm⊗n and two copies of finite hexagonal lattices, such that their underlying graphs tend to root lattice AnAn 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 AnAn and honeycomb one, from large enough underlying graphs. Moreover, by using the stationary phase method, the long time behavior of CTQW on infinite hexagonal lattice is approximated with finite ones for finite distances from the origin while for large distances, the scaling behavior of the probability distribution is deduced. Also, it is shown that, the probability amplitudes at finite times possess the point group symmetry in the sense that two lattice points have the same probability amplitudes if and only if they belong to the same orbit (stratum of the graph) of the point group of the lattice. Apart from physical results, 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 generalized Gegenbauer polynomials is constructed, where the probability amplitudes are given by integrals over these polynomials or their linear combinations. Finally the supersymmetric structure of finite honeycomb lattices is revealed.