Flowing bosonization in the nonperturbative functional renormalization-group approach

TL;DR

This paper develops a nonperturbative functional renormalization group method for bosonization in one-dimensional quantum fluids, introducing a dynamic redefinition of the phase field to accurately capture low-energy physics.

Contribution

It presents a novel implementation of the functional renormalization group with a dynamic phase field redefinition, reproducing known models and phenomenology of Luttinger liquids.

Findings

Derived explicit flow equations matching the sine-Gordon model.

Demonstrated the necessity of a scale-dependent phase redefinition.

Reproduced Luttinger liquid phenomenology with two key parameters.

Abstract

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 $\varphi$ and its conjugate partner, the phase $\vartheta$ 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 $\vartheta$ and $\varphi$ to remain conjugate variables at all energy scales, one must dynamically redefine the field $\vartheta$ 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 $\vartheta$ from the outset) derived within the same approximation. Only with the scale-dependent (flowing) reparametrization of the phase field $\vartheta$ 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.