Generalized entropy of the Heisenberg spin chain

We consider the XY quantum spin chain in a transverse magnetic field. We consider the Rényi entropy of a block of neighboring spins at zero temperature on an infinite lattice. the Rényi entropy is essentially the trace of some power $\alpha$ of the density matrix of the block. We calculate the entropy of the large block in terms of Klein's elliptic $\lambda$-function. We study the limit entropy as a function of its parameter $\alpha$. We show that the Rényi entropy is essentially an automorphic function with respect to a certain subgroup of the modular group. Using this, we derive the transformation properties of the Rényi entropy under the map $\alpha\to\alpha^{-1}$.