 
Summary: CLASSIFICATION OF SOLVABLE MIRRORPERIODIC QUANTUM SPIN
CHAINS
CLAUDIO ALBANESE AND STEPHAN LAWI
Abstract. We present a classification scheme for mirror periodic quantum spin chains with
nearest neighbor couplings whose eigenstates can be expressed in analytically closed form
in terms of hypergeometric polynomials. These chains of arbitrary finite length exhibit a
strong state transfer property, according to which the mirror image of a state is periodically
reconstituted. We also construct their continuous space limit using the limit relations between
hypergeometric polynomials in the Askey scheme.
1. Introduction
Applications to quantum information theory have motivated the interest in the design of
quantum spin chains that implement interesting logic operations. In a recent paper [1], Albanese
et al. have considered the problem of engineering a chain able to execute the operation of mirror
inversion. More precisely, the authors considered a quantum spin chain with nearest neighbor
couplings whose Hamiltonian can be written as follows in the spin representation:
(1.1) H =
1
2
N1
x=0
