Interplay between sign problem and Z_3 symmetry in three-dimensional Potts model

TL;DR

The work tackles the sign problem in finite-$\mu$ QCD by leveraging $Z_3$ symmetry through four constructed $Z_3$-symmetric 3-d Potts models with increasing site-state counts. By comparing these models to the ordinary Potts model, it maps how $Z_3$ symmetry affects the sign problem and how the deconfinement line in the $\mu$-$\kappa$ plane evolves with the number of states. The authors find that imposing exact $Z_3$ symmetry nearly cures the sign problem when only $Z_3$-element states are allowed, while introducing non-$Z_3$ states can reintroduce a sign problem (notably in the 13-state case under phase-quenched treatment). They also observe that the deconfinement transition becomes more sensitive to $\mu$ as the state space grows, highlighting a trade-off between symmetry constraints and dynamical richness. The results offer insights for Z3-QCD and potential flux-model reformulations and suggest directions for extending these ideas to address the sign problem in QCD-like theories.

Abstract

We construct four kinds of Z3-symmetric three-dimentional (3-d) Potts models, each with different number of states at each site on a 3-d lattice, by extending the 3-d three-state Potts model. Comparing the ordinary Potts model with the four Z3-symmetric Potts models, we investigate how Z3 symmetry affects the sign problem and see how the deconfinement transition line changes in the $μ-κ$ plane as the number of states increases, where $μ$ $(κ)$ plays a role of chemical potential (temperature) in the models. We find that the sign problem is almost cured by imposing Z3 symmetry. This mechanism may happen in Z3-symmetric QCD-like theory. We also show that the deconfinement transition line has stronger $μ$-dependence with respect to increasing the number of states.

Figures (11)

• Figure 1: The phase factor $\langle \cos{S_{\rm I}}\rangle ^\prime$ calculated with the phase quenched approximation in ordinary 3-d 3-state Potts model as a function of $\kappa$ and $\mu /M$.
• Figure 2: The expectation value $\langle |{\bar{\Phi}} |\rangle$ calculated with the reweighting method in ordinary 3-d 3-state Potts model as a function of $\kappa$ and $\mu /M$.
• Figure 3: Allowed regions of (a) $L({{\hbox{\boldmath $x$}}})$ and (b) $L^3({{\hbox{\boldmath $x$}}})$ in the complex plane.
• Figure 5: The expectation value $\langle{|{\bar{\Phi}}|}\rangle$ in $Z_3$-symmetric 3-d 3-state Potts model as a function of $\kappa$ and $\mu/M$.
• Figure 6: The expectation value $n$ in $Z_3$-symmetric 3-d 3-state Potts model as a function of $\kappa$ and $\mu/M$. Note that $n$ is dimensionless, since the dimensionless volume $V=6^3$ is used in calculations of (\ref{['n_density']}).