Violation of Cluster Decomposition and Absence of Light-Cones in Local Integer and Half-Integer Spin Chains

TL;DR

This paper investigates the ground state correlations in exactly solvable integer and half-integer spin chains, revealing violations of cluster decomposition and absence of light-cones for higher spins, contrasting with the s=1 case.

Contribution

It introduces a solvable model of half-integer spins (Fredkin spin chain) and analyzes the violation of cluster properties and light-cone behavior in integer spin chains.

Findings

Cluster decomposition violated for spins ≥2

Light-cone propagation present only for s=1

Higher spins show no cone formation and violate clustering

Abstract

We compute the ground state correlation functions of an exactly solvable chain of integer spins, recently introduced in [R. Movassagh and P. W. Shor, arXiv:1408.1657], whose ground-state can be expressed in terms of a uniform superposition of all colored Motzkin paths. Our analytical results show that for spin s$\ge$2 there is a violation of the cluster decomposition property. This has to be contrasted with s=1, where the cluster property holds. Correspondingly, for s=1 one gets a light-cone profile in the propagation of excitations after a local quench, while the cone is absent for s=2, as shown by time dependent density-matrix-renormalization-group. Moreover, we introduce an original solvable model of half-integer spins which we refer to as Fredkin spin chain, whose ground-state can be expressed in terms of superposition of all Dyck paths. For this model we exactly calculate the magnetization and correlation functions, finding that for s=1/2, a cone-like propagation occurs while for higher spins, s$\ge$3/2, the colors prevent any cone formation and clustering is violated, together with square root deviation from the area law for the entanglement entropy.