Spin lattices, state transfer and bivariate Krawtchouk polynomials

TL;DR

This paper explores quantum state transfer in specific 2D spin lattices using bivariate Krawtchouk polynomials, providing exact solutions for 1-excitation dynamics on a triangular domain.

Contribution

It introduces a new class of exactly solvable spin lattices and links their dynamics to bivariate Krawtchouk polynomials derived from harmonic oscillator representations.

Findings

Identified spin lattices with exact 1-excitation solutions

Connected state transfer properties to bivariate Krawtchouk polynomials

Provided explicit mathematical framework for these systems

Abstract

The quantum state transfer properties of a class of two-dimensional spin lattices on a triangular domain are investigated. Systems for which the 1-excitation dynamics is exactly solvable are identified. The exact solutions are expressed in terms of the bivariate Krawtchouk polynomials that arise as matrix elements of the unitary representations of the rotation group on the states of the three-dimensional harmonic oscillator.