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

A bstract 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,” − Σ 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.