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arXiv:2009.06195 · Progress of Theoretical and Experimental Physics

Perfect state transfer in two dimensions and the bivariate dual-Hahn polynomials

Hiroshi Miki, Satoshi Tsujimoto, Luc Vinet

Published Apr 14, 2022• 2 citations
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Abstract

Abstract A new solvable two-dimensional spin lattice model defined on a regular grid of triangular shape is proposed. The hopping amplitudes between sites are related to recurrence coefficients of certain bivariate dual-Hahn polynomials. For a specific choice of the parameters, perfect state transfer and fractional revival are shown to take place.

DOI
10.1093/ptep/ptac069

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Abstract A new solvable two-dimensional spin lattice model defined on a regular grid of triangular shape is proposed. The hopping amplitudes between sites are related to recurrence coefficients of certain bivariate dual-Hahn polynomials. For a specific choice of the parameters, perfect state transfer and fractional revival are shown to take place.
Quantum-state transfer in a triangular domain of a two-dimensional, equally spaced, spin lattice with nonhomogeneous nearest-neighbor couplings is analyzed. An exact solution of the one-excitation dynamics is provided in terms of two-variable Krawtchouk orthogonal polynomials that have been recently defined. The probability amplitude for an excitation to transit from one site to another is given. For some values of the parameters, perfect transfer is shown to take place from the apex of the lattice to the boundary hypotenuse.
Abstract We introduce an analytical XX spin chain with asymmetrical transport properties. It has an even number N + 1 of sites labeled by . It does not exhibit perfect state transfer (PST) from end-to-end but rather from the first site to the next to last one. In fact, PST of one-excitation states takes place between the even sites: , ; while states localized at a single odd site undergo fractional revival (FR) over odd sites only. Perfect return is witnessed at double the PST/FR time. The couplings and local magnetic fields are related to the recurrence coefficients of the dual −1 Hahn polynomials.
It is shown that the hopping of a single excitation on certain triangular spin lattices with non-uniform couplings and local magnetic fields can be described as the projections of quantum walks on graphs of the ordered Hamming scheme of depth 2. For some values of the parameters the models exhibit perfect state transfer between two summits of the lattice. Fractional revival is also observed in some instances. The bivariate Krawtchouk polynomials of the Tratnik type that form the eigenvalue matrices of the ordered Hamming scheme of depth 2 give the overlaps between the energy eigenstates and the occupational basis vectors.