Deconfinement from Thermal Tensor Networks: Universal CFT signature in (2+1)-dimensional $\mathbb{Z}_N$ lattice gauge theory

TL;DR

This paper uses thermal tensor networks to study the deconfinement transition in (2+1)D $bZ_N$ lattice gauge theories, extracting universal CFT data and confirming the Svetitsky-Yaffe conjecture for various N.

Contribution

It introduces a tensor network approach to analyze the universal features of deconfinement transitions in lattice gauge theories, including emergent symmetries and precise critical point determination.

Findings

Verified the universality class matches Svetitsky-Yaffe predictions for N=2,3,5.

Identified an intermediate phase with emergent U(1) symmetry in the $bZ_5$ theory.

Determined critical couplings consistent with Monte Carlo results and extrapolated zero-temperature transition points.

Abstract

Tensor networks offer a sign-problem-free approach to study lattice gauge theories, but extracting precise universal information associated with the deconfinement transition remains challenging. In this work, we study the deconfinement transition of (2+1)-dimensional $\mathbb{Z}_N$ lattice gauge theories at finite temperature using a thermal tensor network approach, where the partition functions at finite temperature are formulated as three-dimensional tensor networks. These tensor networks are first contracted in the temporal direction, and the subsequent coarse-graining in the spatial directions yields a renormalized transfer matrix, the spectrum of which directly encodes the universal conformal field theory data. In particular, by numerically extracting the central charge and scaling dimensions, we verify that the universality class of the thermal deconfinement transition matches the prediction of the Svetitsky-Yaffe conjecture for $N=2,3,5$. Moreover, we show that the $\mathbb{Z}_5$ theory at finite temperature exhibits an intermediate phase with an emergent U(1) symmetry. Critical couplings are determined via Gu-Wen ratios and agree with existing Monte Carlo simulations. Finally, extrapolating these critical couplings at finite temperature enables us to determine the deconfinement transition points for $N=2,3$ at zero temperature.