Exact valence bond entanglement entropy and probability distribution in the XXX spin chain and the Potts model

TL;DR

This paper derives exact formulas for valence bond entanglement entropy and probability distributions in the XXX spin chain and Potts model, challenging previous conjectures and connecting statistical mechanics with quantum entanglement measures.

Contribution

It provides closed-form expressions for valence bond entanglement entropy and distributions in these models, using a boundary Coulomb gas approach, and clarifies their scaling behavior.

Findings

Number of valence bonds scales as (4/pi^2) ln L in the XXX chain.

Disproves the conjecture linking valence bonds to von Neumann entropy.

Results extend to the Q-state Potts model.

Abstract

By relating the ground state of Temperley-Lieb hamiltonians to partition functions of 2D statistical mechanics systems on a half plane, and using a boundary Coulomb gas formalism, we obtain in closed form the valence bond entanglement entropy as well as the valence bond probability distribution in these ground states. We find in particular that for the XXX spin chain, the number N_c of valence bonds connecting a subsystem of size L to the outside goes, in the thermodynamic limit, as <N_c> = (4/pi^2) ln L, disproving a recent conjecture that this should be related with the von Neumann entropy, and thus equal to 1/(3 ln 2) ln L. Our results generalize to the Q-state Potts model.