Confinement in the q-state Potts model: an RG-TCSA study

TL;DR

This work investigates confinement in the scaling $q$-state Potts field theory under a weak magnetic field using RG-TCSA, focusing on both Ising ($q=2$) and three-state Potts ($q=3$). It benchmarks meson spectra against semiclassical WKB, Airy-function, and Bethe–Salpeter predictions, and confirms recent three-body baryon predictions in the $q=3$ case. The results show that WKB provides near-perfect meson masses across the parameter range, while the three-body Potts baryons are well described by a quantum mechanical three-kink model, validating RG-TCSA as a precise tool for non-integrable theories. These findings highlight the practical utility of RG-improved truncation methods for accessing confinement phenomena in lattice-inspired quantum field theories with multiple vacua and bound-state spectra.

Abstract

In the ferromagnetic phase of the q-state Potts model, switching on an external magnetic field induces confinement of the domain wall excitations. For the Ising model (q = 2) the spectrum consists of kink-antikink states which are the analogues of mesonic states in QCD, while for q = 3, depending on the sign of the field, the spectrum may also contain three-kink bound states which are the analogues of the baryons. In recent years the resulting "hadron" spectrum was described using several different approaches, such as quantum mechanics in the confining linear potential, WKB methods and also the Bethe-Salpeter equation. Here we compare the available predictions to numerical results from renormalization group improved truncated conformal space approach (RG-TCSA). While mesonic states in the Ising model have already been considered in a different truncated Hamiltonian approach, this is the first time that a precision numerical study is performed for the 3-state Potts model. We find that the semiclassical approach provides a very accurate description for the mesonic spectrum in all the parameter regime for weak magnetic field, while the low-energy expansion from the Bethe-Salpeter equation is only valid for very weak fields where it gives a slight improvement over the semiclassical results. In addition, we confirm the validity of the recent predictions for the baryon spectrum obtained from solving the quantum mechanical three-body problem.

Figures (7)

• Figure 3.1: Extrapolation fits of the relative energy levels in the Ising model in the ferromagnetic phase with magnetic field $\tilde{h}=0.008$ for the first four excited states at dimensionless volume $mR=10$.
• Figure 3.2: Extrapolation fits of the relative energy levels in the Potts model in the ferromagnetic phase with magnetic field $\tilde{h}=-0.05$ for the first four excited states at dimensionless volume $mR=10$.
• Figure 4.1: Energy of the false vacuum for different values of the magnetic field in the Ising model. Lines are the theoretical predictions: blue is for (\ref{['eq:fv_endensity']}), while purple one takes into account the string tension renormalization (the two lines are almost indistinguishable in the graphs). Dotted lines for different cut-off data, red dots are extrapolated (average of even and odd cut-off extrapolations)
• Figure 4.2: Energy of the false vacuum for different values of the magnetic field in the three state Potts model. Continuous lines are the theoretical predictions from (\ref{['eq:fv_endensity']}), dashed lines are data with coupling constant renormalization before extrapolation for some values of the cut-off. Red dots are extrapolated data in the $\mathcal{C}$-even sector, while black squares are extrapolated data from the $\mathcal{C}$-odd sector.
• Figure 4.3: Effect of the extrapolation for the first three mesonic state in the $\mathcal{C}$-even and odd sectors with magnetic field $\tilde{h}=-0.02$. Dashed lines for data with running coupling for some values of the cut off, while large dots with dotted lines are the extrapolated data. We also plotted the energy of the even sector false vacuum (both before and after extrapolation, the latter marked with green) shifted up by $2m$ to demonstrate that the wavy feature corresponds to the "false meson" resonance (meson configuration over the false vacuum).