Perfect state transfer in inhomogeneous XX model of q-Racah type

TL;DR

This work addresses perfect state transfer in open XX spin chains by constructing exactly solvable models whose inhomogeneous couplings and fields are dictated by the three-term recurrence relations of $q$-Racah and para $q$-Racah orthogonal polynomials. The one-excitation sector reduces to a Jacobi-matrix problem, enabling analytic PST criteria via persymmetry and odd eigenvalue spacings, and connecting chain design to Favard's theorem. It presents two PST families: a first model from $q$-Racah polynomials with explicit parameter constraints expressed through Chebyshev polynomials and an integer-spacing condition, and a second model from para $q$-Racah polynomials obtained through a limiting procedure that handles a pole, yielding another set of eigenvalue-spacing constraints. The results include explicit formulas for couplings and fields, recover a known PST case as a special instance, and suggest avenues for extending PST to almost perfect transfer, fractional revival, and multivariate generalizations in higher dimensions.

Abstract

New exactly solvable one-dimensional XX spin chain models that exhibit perfect state transfer are defined. These models have inhomogeneous couplings and magnetic fields determined from the three-term recurrence relations satisfied by the q-Racah and para q-Racah polynomials. Due to this connection with orthogonal polynomials, the one-excitation sector can be solved analytically. This allows us to provide explicit sets of conditions on the polynomial parameters that guarantee the occurrence of perfect state transfer across these spin chains.