Phase diagram and two-particle structure of the $Z_3$-chiral Potts model

TL;DR

This paper investigates the phase diagram and two-particle structure of the $Z_3$-chiral Potts model using numerical methods, revealing oscillatory correlations, particle scattering states, and parity violation effects.

Contribution

It provides the first detailed numerical analysis of the low-lying spectrum and particle structure of the $Z_3$-chiral Potts model, including scattering states and phase behavior.

Findings

Massive phases exhibit oscillatory correlation functions.

Two-particle scattering states with $Z_3$ charges are identified.

In the superintegrable case, the $Q=2$ particle has twice the mass of the $Q=1$ particle.

Abstract

We calculate the low-lying part of the spectrum of the $Z_3$-symmetrical chiral Potts quantum chain in its self-dual and integrable versions, using numerical diagonalisation of the hamiltonian for $N \leq 12$ sites and extrapolation $N \ra \infty$. From the sequences of levels crossing we show that the massive phases have oscillatory correlation functions. We calculate the wave vector scaling exponent. In the high-temperature massive phase the pattern of the low-lying levels can be explained assuming the existence of two particles, with $Z_3$-charge $Q\!=\!1$ and $Q\!=\!2$, and their scattering states. In the superintegrable case the $Q\!=\!2$-particle has twice the mass of the $Q\!=\!1$-particle. Exponential convergence in $N$ is observed for the single particle gaps, while power convergence is seen for the scattering levels. In the high temperature limit of the self-dual model the parity violation in the particle dispersion relation is equivalent to the presence of a macroscopic momentum $P_m = \pm \vph/3$, where $\vph$ is the chiral angle.