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Entanglement in the ground state of the XY model on the infinite chain can bemeasured by the von Neumann entropy of a block of neighboring spins. We study adouble scaling limit: the size of the block is much larger then 1 but muchsmaller then the length of the whole chain. The entropy of the block has anasymptotic limit. We study this limiting entropy as a function of theanisotropy and of the magnetic field. We identify its minima at product statesand its divergencies at the quantum phase transitions. We find that the curvesof constant entropy are ellipses and hyperbolas and that they all meet at onepoint (essential critical point). Depending on the approach to the essentialcritical point the entropy can take any value between 0 and infinity. In thevicinity of this point small changes in the parameters cause large change ofthe entropy.Comment: Revised Version, 20 pages, 8 figures, 1 tabl

Ellipses of constant entropy in theXYspin chain