Critical Q=1 Potts Model and Temperley-Lieb Stochastic Processes

TL;DR

This paper explores the connection between the critical Q=1 Potts model, the XXZ quantum spin chain, and Temperley-Lieb stochastic processes, revealing insights into their ground states, spectra, and interface growth models.

Contribution

It establishes a link between the Q=1 Potts model, XXZ chain, and Temperley-Lieb stochastic processes, highlighting their shared stationary states and spectral properties.

Findings

The ground state wave function of the XXZ chain is trivial.

The ket ground state corresponds to the stationary distribution of a stochastic process.

Allowing defects leads to models with the same stationary states and spectra as the XXZ chain.

Abstract

We consider the groundstate wave function and spectra of the $L$-site XXZ $U_q[s\ell(2)]$ invariant quantum spin chain with $q=\exp(\pi i/3)$. This chain is related to the critical Q=1 Potts model and exhibits $c=0$ conformal invariance. We show that the problem is related to Hamiltonians describing one-dimensional stochastic processes defined on a Temperley-Lieb algebra. The bra groundstate wave function is trivial and the ket groundstate wave function gives the probabilty distribution of the stationary state. The stochastic processes can be understood as interface RSOS growth models with nonlocal rates. Allowing defects which can hop on the interface one obtains stochastic models having the same stationary state and spectra (but not degeracies) as the XXZ chain.