Multiversality of charge density wave onset in a Luttinger liquid

It has been recently realized that the universality class of a continuous transition between two phases of matter may not be unique: different regions of the phase boundary may realize distinct universality classes. We describe an example of such multiversality: a transition from a 1d Luttinger liquid to a charge density wave insulator. We show that while part of the boundary between these two phases realizes the Kosterlitz-Thouless transition (central charge $c = 1$, Luttinger parameter $K=1/2$), another part is descibed by an Ising transition coexisting with the Luttinger liquid (central charge $c = 1/2+1$ and variable Luttinger parameter). We demonstrate these conclusions using analytical renormalization group calculations, as well as numerical density martrix renormalization group simulations of a concrete microscopic model.