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

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

2006-08-31

arxiv.org/abs/math-ph/0608067v1

Physics

Mathematical Physics

41 pages, 4 figures

Scientific paper

10.1016/j.physa.2007.03.032

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

Affiliated with

Also associated with

No associations

Rating

Investigation of continuous-time quantum walk on root lattice $A_n$ and honeycomb lattice does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Investigation of continuous-time quantum walk on root lattice $A_n$ and honeycomb lattice, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Investigation of continuous-time quantum walk on root lattice $A_n$ and honeycomb lattice will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-397122