How to construct spin chains with perfect state transfer

TL;DR

This paper provides a systematic method to construct and characterize $XX$ quantum spin chains with perfect state transfer using orthogonal polynomials, offering new models and an efficient algorithm for Hamiltonian design.

Contribution

It introduces a complete characterization of PST spin chains via orthogonal polynomials and presents a constructive algorithm for Hamiltonian construction, including new models linked to symmetric $q$-Racah polynomials.

Findings

Complete characterization of PST models with mirror symmetry.

An efficient algorithm for Hamiltonian construction.

Introduction of a new model related to symmetric $q$-Racah polynomials.

Abstract

It is shown how to systematically construct the $XX$ quantum spin chains with nearest-neighbor interactions that allow perfect state transfer (PST). Sets of orthogonal polynomials (OPs) are in correspondence with such systems. The key observation is that for any admissible one-excitation energy spectrum, the weight function of the associated OPs is uniquely prescribed. This entails the complete characterization of these PST models with the mirror symmetry property arising as a corollary. A simple and efficient algorithm to obtain the corresponding Hamiltonians is presented. A new model connected to a special case of the symmetric $q$-Racah polynomials is offered. It is also explained how additional models with PST can be derived from a parent system by removing energy levels from the one-excitation spectrum of the latter. This is achieved through Christoffel transformations and is also completely constructive in regards to the Hamiltonians.