Phase transition in the Rényi-Shannon entropy of Luttinger liquids

The R\'enyi-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 nc of the R\'enyi 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 J₁-J₂ 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<nc. From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.