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 a of the density matrix of the block. We calculate the asymptotic for L -> infinity 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 rescaling, 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 -> alpha(-1) and show that the entropy becomes an elementary function of the magnetic field and the anisotropy when alpha is an integer power of 2; this includes the purity rho(2). We also analyze the behavior of the entropy as alpha -> 0 and 8 and at the critical magnetic field and in the isotropic limit (XX model).