Spectrum of the density matrix of a large block of spins of the XY model in one dimension

We consider 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 expression of Rényi entropy in terms of modular functions. The eigenvalues λn form exact geometric sequence. For example, for strong magnetic field λn = C exp(−πτ0n), here τ0 > 0 and C > 0 depend on anisotropy and magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but degeneracy gn increases sub-exponentially as eigenvalues diminish: $${g_{n}\sim \exp{(\pi \sqrt{n/3})}}$$. For weak magnetic field expressions are similar.