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

TL;DR

This work advances the study of deconfinement in (2+1)-D $ Z_N$ lattice gauge theories by formulating finite-temperature partition functions as three-dimensional tensor networks and extracting universal CFT data from a renormalized transfer matrix. By employing thermal tensor-network renormalization (TTNR) with optimal temporal projectors and subsequent 2D TRG/TNR coarse-graining, it confirms the Svetitsky–Yaffe universality for $N=2$ and $N=3$ via central charges and scaling dimensions, and reveals an intermediate emergent U(1) phase in the $N=5$ case described by a Tomonaga–Luttinger liquid. The study also uses Gu–Wen ratio analysis to illustrate the duality between gauge theories and dual clock models and demonstrates that finite-temperature results can be extrapolated to zero temperature, yielding critical couplings in agreement with Monte Carlo benchmarks. Overall, the results showcase the power of thermal tensor networks to access universal deconfinement data and motivate extensions to non-Abelian and higher-dimensional gauge theories, with potential impacts on lattice QCD simulations and quantum simulations of gauge theories.

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.

Figures (16)

• Figure 1: Graphical representation of two types of PEPOs constructing Eq. \ref{['eq:TN_AC']}. Blue tensors refer to the link constraint tensors $A^{(n, \mu)}$, and red square tensors refer to plaquette tensors $C^{(n, \mu, \nu)}$.
• Figure 2: Electric PEPO is removed by imposing the temporal gauge condition.
• Figure 3: Schematic illustration of the tensor-network structure along the temporal direction at each spatial site when the temporal gauge is imposed. Yellow and red tensors denote Eqs. \ref{['eq:T_ele']} and \ref{['eq:T_mag']}, respectively. Note that the top and bottom indices are contracted due to the periodic boundary condition.
• Figure 4: (a) Partition function in Eq. \ref{['eq:TN_dualS']} at height $L_z$, consisting of columns of six-legged tensors. (b) Sequential coarse-graining of the column along the vertical direction using projectors in the TTNR method, whose details are given in Appendix \ref{['appendix:projector']}, resulting in a single effective tensor. (c) Tracing out the vertical indices reduces the three-dimensional tensor network of height $L_z$ to an effective two-dimensional infinite network, as shown in (d). (e) Coarse-graining of the two-dimensional tensor network using TRG/TNR methods. Since the geometry of the tensor network in (f) is identical to that in (d), the coarse-graining procedure can be iterated to evaluate the partition function. As the number of tensors reduces by half, the network of $2^s$ tensors becomes a single tensor after $s$ steps. The procedure is identical for the partition function in Eq. \ref{['eq:TN_ele_mag']}.
• Figure 5: Central charge of (a) $N=2$ and (b) $N=3$ dual spin models with $(\chi_{{\rm TTNR}},\chi_{\text{Loop-TNR}})=(48, 36)$. The central charges are obtained from the fixed-point tensor using the prescription in Ref. Gu:2009dr. The values $c=0.5$ and $c=0.8$ are consistent with the two-dimensional Ising and three-state Potts CFTs, respectively. Dashed vertical lines indicate the critical inverse gauge couplings $\beta_c$ at vanishing temperature ($L_z \to \infty$) obtained using MC simulations Borisenko:2013xna.