Particle spectrum of the 3-state Potts field theory: a numerical study

TL;DR

This work numerically analyzes the scaling limit of the two-dimensional three-state Potts field theory (perturbed D_4 minimal model) across temperature and magnetic-field perturbations using the truncated conformal space approach. It verifies that kink confinement yields both mesons and baryons, maps the Ising-like second-order transition at $\eta_-^c \approx 0.14$, and characterizes the disordered phase with confined neutral bound states, all in qualitative and quantitative agreement with theoretical predictions. The authors also determine the signs of the D_4 structure constants needed for the TCSA and compare high-temperature and weak-field analytic results with numerical data, providing a comprehensive nonperturbative view of confinement and phase structure in this 2D QFT. The results offer detailed mass trajectories for the lightest excitations and reinforce the use of TCSA in non-diagonal minimal models, with potential implications for understanding confinement mechanisms in related low-dimensional systems.

Abstract

The three-state Potts field theory in two dimensions with thermal and magnetic perturbations provides the simplest model of confinement allowing for both mesons and baryons, as well as for an extended phase with deconfined quarks. We study numerically the evolution of the mass spectrum of this model over its whole parameter range, obtaining a pattern of confinement, particle decay and phase transitions which confirms recent predictions.

Figures (8)

• Figure 1: Phase diagram of the field theory (\ref{['action']}). The thick lines correspond to the first-order (continuous line) and second-order (dashed line) phase transitions. Dots on the closed path $\tau^{14} + |h|^9 =\hbox{constant}$ mark the points corresponding to the spectra shown in figures \ref{['fig.ordered']}a--h. The ordered phase corresponds to the dotted part of the path.
• Figure 2: The low lying energy differences $E_i-E_0$ as functions of $R$ at the points a--h along the closed path of figure \ref{['sqr']}. Even levels are in red; odd levels are in green with dashed lines. Units refer to a fixed mass scale.
• Figure 3: $\frac{R}{2\pi}(E_i-E_0)$ as functions of $R$ for the lowest values of $i$ in the (a) even and (b) odd sectors at the second-order phase transition point. Degeneracies predicted by the critical Ising model (Table \ref{['tab1']}) are shown on the right hand side.
• Figure 4: The lowest energy difference $\frac{R}{2\pi}(E_i-E_0)$ in the two sectors as a function of $R$ at the second-order phase transition point. Data are shown with dots, the constants $2\Delta$ equal $1/8$ and $1$ predicted by the critical Ising model are shown with continuous lines.
• Figure 5: Masses of the lightest mesons $\pi_0^{(1)}$, $\pi_1^{(1)}$, $\pi_0^{(2)}$, $\pi_1^{(2)}$, $\pi_0^{(3)}$, the lightest baryon $p_+^{(1)}$, the elementary kinks $K_{ij}$ and the lightest mesonic kinks $\pi^{(1)}_{ij}$, along the closed path of figure \ref{['sqr']}. Even particles are shown in red, odd particles in green and the kinks in blue. Dashed lines show the stability thresholds for even particles (twice the mass of the lightest particle), odd particles (mass of the lightest even particle plus that of the lightest odd particle) and mesonic kinks (mass of $K_{ij}$ plus that of $\pi^{(1)}_{ij}$) in red, green and blue, respectively. The points $a$-$h$, having a correspondence in figures \ref{['sqr']} and \ref{['fig.ordered']}, are also marked.