# Sign problem in $Z_3$-symmetric effective Polyakov-line model

## TL;DR

This paper constructs a $Z_3$-symmetric effective Polyakov-line model to study the sign problem in finite-temperature QCD-like theories, demonstrating that the symmetry reduces the severity of the sign problem via lattice simulations and a new reweighting method.

## Contribution

The paper introduces a $Z_3$-symmetric effective Polyakov-line model and shows it alleviates the sign problem compared to the conventional model.

## Key findings

- Sign problem is milder in $Z_3$-EPLM than in EPLM at low temperatures.
- The real action condition reduces the sign problem in $Z_3$-EPLM.
- A new reweighting method further weakens the sign problem in $Z_3$-EPLM.

## Abstract

As an effective model corresponding to $Z_3$-symmetric QCD ($Z_3$-QCD), we construct a $Z_3$-symmetric effective Polyakov-line model ($Z_3$-EPLM) by using the logarithmic fermion effective action. Since $Z_3$-QCD tends to QCD in the zero temperature limit, $Z_3$-EPLM also agrees with the ordinary effective Polyakov-line model (EPLM) there; note that ordinary EPLM does not possess $Z_3$ symmetry. Our main purpose is to discuss a sign problem appearing in $Z_3$-EPLM. The action of $Z_3$-EPLM is real, when the Polyakov line is not only real but also its $Z_3$ images. This suggests that the sign problem becomes milder in $Z_3$-EPLM than in EPLM. In order to confirm this suggestion, we do lattice simulations for both EPLM and $Z_3$-EPLM by using the reweighting method with the phase quenched approximation. In the low-temperature region, the sign problem is milder in $Z_3$-EPLM than in EPLM. We also propose a new reweighting method. This makes the sign problem very weak in $Z_3$-EPLM.