Ellipses of constant entropy in the XY spin chain

Entanglement in the ground state of the XY model on the infinite chain can be measured by the von Neumann entropy of a block of neighbouring spins. We study a double scaling limit: the size of the block is much larger than 1 but much smaller than the length of the whole chain. The entropy of the block has an asymptotic limit in the gapped regimes. We study this limiting entropy as a function of the anisotropy and of the magnetic field. We identify its minima at product states and its divergencies at the quantum phase transitions. We find that the curves of constant entropy are ellipses and hyperbolas, and that they all meet at one point (essential critical point). Depending on the approach to the essential critical point, the entropy can take any value between 0 and ∞. In the vicinity of this point, small changes in the parameters cause large change of the entropy.