Flowing bosonization in the nonperturbative functional renormalization-group approach

Bosonization allows one to describe the low-energy physics of one-dimensional quantum fluids within a bosonic effective field theory formulated in terms of two fields: the "density" field phi and its conjugate partner, the phase theta of the superfluid order parameter. We discuss the implementation of the nonperturbative functional renormalization group in this formalism, considering a Luttinger liquid in a periodic potential as an example. We show that in order for theta and phi to remain conjugate variables at all energy scales, one must dynamically redefine the field theta along the renormalization-group flow. We derive explicit flow equations using a derivative expansion of the scale-dependent effective action to second order and show that they reproduce the flow equations of the sine-Gordon model (obtained by integrating out the field theta from the outset) derived within the same approximation. Only with the scale-dependent (flowing) reparametrization of the phase field theta do we obtain the standard phenomenology of the Luttinger liquid (when the periodic potential is sufficiently weak so as to avoid the Mott-insulating phase) characterized by two low-energy parameters, the velocity of the sound mode and the renormalized Luttinger parameter.