Spin lattices, state transfer, and bivariate Krawtchouk polynomials

Genest, Vincent X.; Miki, Hiroshi; Vinet, Luc; Zhedanov, Alexei

doi:10.1139/cjp-2014-0568

Cited by 3 publications

(4 citation statements)

References 21 publications

(36 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is to be expected that the burgeoning theory of multivariate orthogonal polynomials would be instrumental in the construction of higher dimensional simplexes with interesting transport properties. Work in this direction has been initiated [18,37,42]. PST in graphs has also been the object of much attention.…”

Section: Resultsmentioning

confidence: 99%

Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results

Bossé¹,

Vinet²

2017

SIGMA

Self Cite

View full text Add to dashboard Cite

Abstract. The analytic specifications of photonic lattices with fractional revival (FR) and perfect state transfer (PST) are reviewed. The approach to their design which is based on orthogonal polynomials is highlighted. A compendium of analytic models with PST is offered. New results on their FR properties are included. The nearest-neighbour approximation is adopted in most of the review; one analytic example with next-to-nearest neighbour interactions is also presented.

show abstract

Section: Resultsmentioning

confidence: 99%

Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results

Bossé¹,

Vinet²

2017

SIGMA

Self Cite

View full text Add to dashboard Cite

show abstract

“…Furthermore, in some special case, another perfect state transfer from the apex to another plane is simultaneously realized. Such situation does not occur in [5][6][7]. We trust this communication will allow further exploration of phenomena inherent to quantum walks in higher dimensions.…”

Section: Discussionmentioning

confidence: 73%

“…However, in higher dimensions, it is difficult to observe perfect state transfer and even very few spin lattices are known to be solvable. Recently, 2-dimensional solvable spin lattices have been proposed from bivariate Krawtchouk polynomials and extended notion of perfect state transfer is observed in these models [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

3-dimensional solvable XX spin lattice Hamiltonian derived from 3-variable Krawtchouk polynomials

Miki

Miura

2016

JSIAM Letters

Self Cite

View full text Add to dashboard Cite

A new solvable 3-dimensional spin lattice Hamiltonian with inhomogeneous couplings, which can be diagonalized by 3-variable Krawtchouk polynomials, is proposed. The model is defined on the lattice of rectangular pyramid and describes near-neighbor interactions instead of nearest-neighbor ones. Using the properties of 3-variable Krawtchouk polynomials, quantum state transfer in the model is analyzed. In particular, for some value of the parameters, perfect state transfer is shown to occur from the apex to the diagonal plane.

show abstract

“…[4]). In particular, orthogonal polynomials of the Askey scheme [5] are eigenfunctions of Hamiltonians characterized by a certain tridiagonal interaction structure describing 'triangular' two-dimensional spin lattices associated with the XY spin chain with nearest neighbor non-homogeneous couplings and non-zero magnetic field (see [6] and references therein). According to the interaction considered, the parameters entering in the definition of the q−hypergeometric eigenfunctions are restricted.…”

Section: Introductionmentioning

confidence: 99%

A bispectral q-hypergeometric basis for a class of quantum integrable models

Baseilhac

Martín

2018

Journal of Mathematical Physics

View full text Add to dashboard Cite

Abstract. For the class of quantum integrable models generated from the q−Onsager algebra, a basis of bispectral multivariable q−orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with N variables and N + 3 parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed. In a second part, finite dimensional modules are constructed and studied for a certain class of parameters and if the N variables belong to a discrete support. In this case, the bispectral property finds a natural interpretation within the framework of tridiagonal pairs. In a third part, eigenfunctions of the q−Dolan-Grady hierarchy are considered in the polynomial basis. In particular, invariant subspaces are identified for certain conditions generalizing Nepomechie's relations. In a fourth part, the analysis is extended to the special case q = 1. This framework provides a q−hypergeometric formulation of quantum integrable models such as the open XXZ spin chain with generic integrable boundary conditions (q = 1).MSC: 81R50; 81R10; 81U15; 39A70; 33D50; 39A13.

show abstract

Spin lattices, state transfer, and bivariate Krawtchouk polynomials

Cited by 3 publications

References 21 publications

Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results

Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results

3-dimensional solvable XX spin lattice Hamiltonian derived from 3-variable Krawtchouk polynomials

A bispectral q-hypergeometric basis for a class of quantum integrable models

Contact Info

Product

Resources

About