Flowing bosonization in the nonperturbative functional renormalization-group approach

1-Clear and detailed presentation
2-Methodological advance

1-Only minor new physical insight

This work discusses an implementation of the nonperturbative (or functional) renormalization group to one-dimensional quantum fluids within the partial bosonization approach. While previous works typically completely integrate out the phase field, resulting in an effective action for the density field alone, the approach discussed here keeps the phase field in the action. This allows one to study fluctuations of the phase field and might be applicable to more generic situations in which the action is no longer quadratic in the phase field. The approach is shown to reproduce the expected results for the Luttinger and Mott insulating phases, as long as a flowing (or dynamical) bosonization scheme is employed. Such schemes are by now widely used in the renormalization group literature in the context of high-energy, nuclear, and atomic physics problems, but have apparently so far not been employed in the description of one-dimensional quantum fluids. The paper is very well written and appears without obvious flaws. I believe that the result obtained here represent a significant advance as required for publication in SciPost Physics.

I do have two minimal comments that the authors might want to address before publication, see below.

1-As mentioned in the introduction and shown in Appendix B, the fact that the superfluid stiffness does not renormalize despite the presence of a periodic potential in the partially bosonized scheme without flowing bosonization, is a consequence of gauge invariance. It is also claimed that this statement holds independent of the approximation scheme used. I did not understand the latter statement. It seems to imply that it holds to arbitrary orders in the derivative expansion, while the actual calculations performed in Appendix B appear to make explicit use of the second-order truncation of the derivative expansion. Can the authors please elucidate this point?

2-It might be useful to repeat the definition of the RG time t in the captions of Fig. 1, as these are used in the labels of the plots.

1-original work
2-application of a method (dynamical bosonization) developed for other fields of physics to 1D quantum fluids.
3-manuscript well written and clear

1-potential applications of the method are not sufficiently presented in the manuscript

In this paper, the authors develop an extension of the nonperturbative functional renormalization group (FRG) for one-dimensional (1D) quantum fluids. In particular, they analyze a bosonized theory described by two fields: the "density" field $\varphi$ and its conjugate, the superfluid phase $\vartheta$. The main result is that the phase field has to be dynamically redefined for $\varphi$ and $\vartheta$ to be conjugate variables at any FRG scale $k$. This "flowing bosonization" procedure is then applied to the case of a Luttinger liquid in presence of a periodic potential.

I believe this paper to be well written, clear, and methodologically interesting.
I have nevertheless some minor comments I would ask the authors to address.

1- Page 6: the authors write "We construct [...] $R_k( Q)$ in the usual way." I believe that here a citation to other works where this type of regulator has been used is needed.

2- Page 6: "[...] we take $r(y)=\alpha/(e^y-1)$ with $\alpha$ of order unity." I think a brief discussion on how $\alpha$ is chosen is necessary.

3- Page 10, below Eq.(37): I believe, for clarity sake, that the authors should stress at this stage that $\bar{K}_k$ is obtained from the average of $Z_{1,k}(\phi)$, while $K_k$ from its value at zero field.

4- Sec. 3.1.1: In my opinion it is more sound to write Eq.(38) as
$\theta(Q)=-i\omega \alpha_k(Q) \phi(Q) + \beta_k(Q) \bar{\theta}(Q)$
as it is clear that $\theta$ must be a combination of $\bar{\theta}$ and $\partial_\tau \phi$. In this way it is more evident that the term in the second line of (39) is a $(\partial_\tau\phi)^2$ term and it has to vanish when summed with $Z_{1\tau,k}(\phi=0)(\partial_\tau\phi)^2$.

5- Page 13: The meaning of the paragraph from "It was pointed out in Ref. [3] ... " to "... preventing the superfluid stiffness to vanish when $k\to 0$." is unclear to me. I would ask the authors to clarify.

6- Sec. 3.2, below Eq.(57): the authors write "The coefficients $\alpha$ and $\beta$ are given by (41)." It is not clear whether, within an active frame perspective, one has to assume the coefficients to be given by Eq.(41) or this can be somewhat derived in a similar way to what is done for the passive frame approach.

7- In the conclusion the authors refer to the possibility of studying the Bose fluid by directly using the bosonic fields $\psi$ and $\psi^*$. I believe this discussion deserves to be at least briefly mentioned in the introduction as well, as it seems to be a little decontextualized.

8- I believe that the authors should improve their conclusion by further stressing (if possible) the potential applications of their method.

9- Appendix A.2: I think, for completeness sake, that if the authors do not want to show the explicit form of their flow equations, they should at least refer to other works where they appear.

In summary, I believe this paper can be published on SciPost Physics, after the authors have considered the changes listed above.