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

TL;DR

This study uses the RG-TCSA numerical method to analyze confinement phenomena in the q-state Potts model, comparing predictions from various approaches with high-precision results, especially for the 3-state case.

Contribution

First numerical high-precision analysis of the 3-state Potts model confinement spectrum using RG-TCSA, validating semiclassical and Bethe-Salpeter predictions.

Findings

Semiclassical approach accurately describes mesonic spectrum for weak magnetic fields.

Bethe-Salpeter low-energy expansion is valid only for very weak fields.

Confirmed recent quantum mechanical predictions for baryon spectrum.

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.