Entanglement entropy for pure gauge theories in 1+1 dimensions using the lattice regularization

We study the entanglement entropy (EE) for pure gauge theories in 1+1 dimensions with the lattice regularization. Using the definition of the EE for lattice gauge theories proposed in a previous paper (1) (S. Aoki, T. Iritani, M. Nozaki, T. Numasawa, N. Shiba and H. Tasaki, JHEP 1506 (2015) 187), we calculate the EE for arbitrary pure as well as mixed states in terms of eigenstates of the transfer matrix in 1+1 dimensional lattice gauge theory. We find that the EE of an arbitrary pure state does not depend on the lattice spacing, thus giving the EE in the continuum limit, and show that the EE for an arbitrary pure state is independent of the real (Minkowski) time evolution. We also explicitly demonstrate the dependence of EE on the gauge fixing at the boundaries between two subspaces, which was pointed out for general cases in the paper (1). In addition, we calculate the EE at zero as well as finite temperature by the replica method, and show that our result in the continuum limit corresponds to the result obtained before in the continuum theory, with a specific value of the counter term, which is otherwise arbitrary in the continuum calculation. We confirm the gauge dependence of the EE also for the replica method.