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We consider the Renyi alpha-entropies for Luttinger liquids (LL). For largeblock lengths l these are known to grow like ln l. We show that there aresubleading terms that oscillate with frequency 2k_F (the Fermi wave number ofthe LL) and exhibit a universal power-law decay with l. The new criticalexponent is equal to K/(2 alpha), where K is the LL parameter. We presentnumerical results for the anisotropic XXZ model and the full analytic solutionfor the free fermion (XX) point.

Parity effects in the scaling of block entanglement in gapless spin chains