Entanglement Spectrum for the XY Model in One Dimension

Abstract

We consider the reduced density matrix of a large block of consecutive spins in the ground states of the XY spin chain on an infinite lattice. We derive the spectrum of the density matrix using the expression of the Renyi entropy in terms of modular functions. The eigenvalues \lambda_n form an exact geometric sequence. For example, for strong magnetic field \lambda_n = C \exp{(-\pi \tau_0 n)}, here \tau_0>0 and C > 0 depend on the anisotropy and the magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but the degeneracy g_n increases sub-exponentially as eigenvalues diminish: g_n \sim \exp{(\pi \sqrt{n/3})}. For weak magnetic field expressions are similar.