Phase transition in the Renyi-Shannon entropy of Luttinger liquids

The Renyi-Shannon entropy allows extraction of some universal information about many-body wave functions. For a critical spin chain with central charge c = 1, we show that it exhibits a phase transition at some value n(c) of the Renyi parameter n which depends on the Luttinger parameter R. A replica-free formulation establishes a connection to boundary entropies in conformal field theory and reveals that the transition is triggered by a vertex operator which becomes relevant at the boundary. Our numerical results (XXZ and J(1)-J(2) spin chains) match the continuum limit prediction, confirming its universal character. The replica approach used in previous works turns out to be correct only for n < n(c). From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.