Entanglement Entropy in 2D Non-abelian Pure Gauge Theory

TL;DR

This work provides a comprehensive entanglement-entropy analysis for 2D pure YM theory with gauge group $U(N)$ on arbitrary Riemann surfaces. Using the replica trick and the exact YM$_2$ partition function, it derives a general EE formula $S = 2 l v + \ln Z - \langle \ln(d_R^{\chi(\Sigma)-2l} e^{- A C_2(R)/(2N)}) \rangle$, showing EE depends only on the number of disjoint intervals $l$ because of area-preserving diffeomorphisms. In the large-$N$ limit, all Renyi entropies are computed on the sphere, revealing a Douglas–Kazakov phase transition that imprints a non-analyticity in the EE, with a calculable low-temperature exponential approach $S(T)-S(0) \sim O\left( \frac{m_{gap}}{T} e^{-m_{gap}/T} \right)$ and a topological (zero-area) limit $S^{top}_l$. The results illuminate how geometry and topology control entanglement in gauge theories, provide a benchmark for gauge-field EE without local degrees of freedom, and suggest avenues for including matter (e.g., Schwinger model) in future work.

Abstract

We compute the Entanglement Entropy (EE) of a bipartition in 2D pure non-abelian $U(N)$ gauge theory. We obtain a general expression for EE on an arbitrary Riemann surface. We find that due to area-preserving diffeomorphism symmetry EE does not depend on the size of the subsystem, but only on the number of disjoint intervals defining the bipartition. In the strong coupling limit on a torus we find that the scaling of the EE at small temperature is given by $S(T) - S(0) = O\left(\frac{m_{gap}}{T}e^{-\frac{m_{gap}}{T}}\right)$, which is similar to the scaling for the matter fields recently derived in literature. In the large $N$ limit we compute all of the Renyi entropies and identify the Douglas-Kazakov phase transition.