Renyi Entropy of the XY Spin Chain

We consider the one-dimensional XY quantum spin chain in a transverse magnetic field. We are interested in the Renyi entropy of a block of L neighboring spins at zero temperature on an infinite lattice. The Renyi entropy is essentially the trace of some power $\alpha$ of the density matrix of the block. We calculate the asymptotic for $L \to \infty$ analytically in terms of Klein's elliptic $\lambda$ - function. We study the limiting entropy as a function of its parameter $\alpha$. We show that up to the trivial addition terms and multiplicative factors, and after a proper re-scaling, the Renyi entropy is an automorphic function with respect to a certain subgroup of the modular group; moreover, the subgroup depends on whether the magnetic field is above or below its critical value. Using this fact, we derive the transformation properties of the Renyi entropy under the map $\alpha \to \alpha^{-1}$ and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when $\alpha$ is a integer power of 2, this includes the purity $tr \rho^2$. We also analyze the behavior of the entropy as $\alpha \to 0$ and $\infty$ and at the critical magnetic field and in the isotropic limit [XX model].