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Entanglement entropy and the large $N$ expansion of two-dimensional Yang-Mills theory

Two-dimensional Yang-Mills theory is a useful model of an exactly solvable gauge theory with a string theory dual at large $N$. We calculate entanglement entropy in the $1/N$ expansion by mapping the theory to a system of $N$ fermions interacting via a repulsive entropic force. The entropy is a sum of two terms: the "Boltzmann entropy", $\log \dim (R)$ per point of the entangling surface, which counts the number of distinct microstates, and the "Shannon entropy", $- \sum p_R \log p_R$, which captures fluctuations of the macroscopic state. We find that the entropy scales as $N^2$ in the large $N$ limit, and that at this order only the Boltzmann entropy contributes. We further show that the Shannon entropy scales linearly with $N$, and confirm this behaviour with numerical simulations. While the term of order $N$ is surprising from the point of view of the string dual - in which only even powers of $N$ appear in the partition function - we trace it to a breakdown of large $N$ counting caused by the replica trick. This mechanism could lead to corrections to holographic entanglement entropy larger than expected from semiclassical field theory.