Bogoliubov phonons in a Bose-Einstein condensate from the one-loop perturbative renormalization group

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The authors consider a 3D superfluid within a perturbative one-loop RG approach, focusing on the symmetry-broken phase. They show that the description of the low-energy behavior beyond the healing-length scale requires to include in the action a wavefunction renormalization of the temporal-derivative terms. The condensate depletion is shown to agree with Bogoliubov theory and the interaction-induced shift of the critical temperature is computed.

I do not see any new results in this paper. The problem of infrared divergences appearing in perturbative theory has been thoroughly discussed by Nepomnyashchii and Nepomnyashchii a long time ago and is now well understood in the framework of the functional renormalization group (FRG). In fact, the one-loop approximation proposed by the authors appears to be more complicated than the FRG approach proposed by Wetterich and others, at least when a simple truncation of the effective action is used. The authors justify their approach by the possibility to study universal and non-universal properties of more complex systems such as multi-component Bose gases. There is no doubt that the FRG approach (which has proven very efficient for many strongly-interacting boson systems) would be a much more powerful approach to study such systems.

The goal of the manuscript, which does not clarify any aspects of superfluidity in interacting boson systems, is therefore not clear. A possible goal could have been to show that the FRG flow equations, obtained from a simple truncation of the effective action by Wetterich and others, can be reproduced from a standard one-loop calculation in the symmetry-broken phase (even if I am not fully convinced that this would be enough to justify publication of a paper). Unfortunately, I cannot recommend publication of the manuscript.

More detailed comments are as follows:

1) In the second paragraph of Sec.III, the set of coupling constants $\bf g$ is not defined; it is defined only after Eq.(10).

2) There is a log missing in Eq.(15) and $Z_0[0]$ is missing in the rhs of (16).

3) After Eq.(40), the authors point out that "also the `hole' dispersion $\omega_k^-$ is gapless". Which other dispersion is gapless? Isn't the dispersion $\omega_k^+$ gapped?

4) The way the dimensionless couplings are defined is not always clear. In the FRG approach, dimensionless quantities are defined so as to remove any explicit dependence on the running momentum scale. It seems that a different choice is sometimes made and this obscures a little bit the physical meaning of the dimensionless quantities.

5) The authors ignore the renormalization of the coefficient $Z_x$ of the spatial derivative term, which amounts to neglecting the difference between the condensate density and the superfluid density (since $n_s=Z_x n_c$). This in turn leads to a violation of the Ward identity $n_s=n$ (the full density) associated with Galilean invariance.

6) After Eq.(55), the RG choice $Z_\tau'=1$ is justified by the fact that it preserves the canonical commutation relations of the Bose field. This, however, is not a sufficient condition since the second-order time-derivative term is incompatible with the canonical commutations relations of the Bose field.

7) I do not see how Eq.(63), which relates the dynamical critical exponent $z$ to $\eta_\tau=-\partial_l \ln Z_\tau^<$ can be correct. Since the dynamical critical exponent is defined by $z-1=d/dl c$ where the running sound-mode velocity $c$ depends on both $Z_\tau$ and $V$, the critical exponent $z$ must also depend on $dV/dl$. In fact, it is clear that the result $z=1$ follows from the two following properties: $Z_\tau$ vanishes in the infrared limit while $V$ takes a non-zero value.

8) The asymptotic scaling discussed in Sec.IV.C.1 should be compared with the scaling obtained in the FRG approach; see, e.g., Table I in [45]. More generally, one would like to know whether the one-loop flow equations are fully compatible (or even identical) with the FRG flow equations.

9) At the end of Sec.IV.C.2, the authors explain that they employ different schemes to solve the flow equations: for $l\leq l_1$, they set $V=0$, etc. Why not simply solving the flow equations with no further approximations?

10) The prefactor $\kappa$ of the interaction-induced shift of the critical temperature is an order of magnitude larger than the expected result. The fact that $\kappa$ cannot be reliably obtained from a one-loop approximation should not be a surprise since it requires the knowledge of the one-particle propagator at $T_c$ in a large momentum range including the crossover region between critical and non-critical modes. The calculation has been done using FRG in the BMW approximation with a result $\kappa\simeq 1.37$ which compares well with other methods [F. Benitez {\it et al.}, Phys. Rev. E 85, 026707 (2012)].

11) The coefficient $V_l$ of the second-order temporal derivative term is shown in Fig.7. Why doesn't $V_l$ reach a constant when $l\to\infty$? This behavior doesn't seem compatible with the sound-mode velocity taking a constant value in the infrared limit.

12) The authors never mention the Ginzburg scale $k_G$, whose crucial role has been emphasized by Pistolesi {\it et al.} [41]. Is this non-perturbative scale beyond the reach of the perturbative one-loop approach? Does this explain the apparent poor behavior of $V_l$ mentioned above?

Reject

1- The paper has a clearly defined scope and addresses an interesting research question.
2- The paper is clearly written and very pedagogical. The methods that are being applied are thoroughly explained. Derivations are presented in sufficient detail. Therefore, the paper is accessible also for non-experts.
3- The introduction explains the research context in great detail and provides a useful and extensive guide to the relevant literature.

The paper by Rasch et al. has a well-defined goal, which is stated clearly in the introduction: to present a simple perturbative RG approach to sound excitations in a Bose condensate. This approach is presented in a very clear, pedagogical, and accessible manner. In particular, the results obtained by integrating the RG flow equations demonstrate convincingly that the wave-function renormalization of the temporal derivative terms in the action are the crucial ingredient to describe sound modes and obtain convergent RG flows.

However, comparing the scope of the paper as stated by the authors to the criteria that should be met for a paper to be suitable for publication in SciPost Physics, it seems that the paper does not present the kind of groundbreaking research SciPost Physics is looking for. I believe that the paper is more suitably published in a less selective journal such as SciPost Core.

1- I do not quite understand the sentence below Eq. (7): "To satisfy ..., the linear term must vanish." Is not the vanishing of the linear term the condition that the expansion is around a saddle point?
2- What is the approximation in Eq. (75)? In particular, how does one get from Eq. (69) to Eq. (75)?
3- I believe there is a type below Eq. (75): it should be $1 + \partial_l$.

Accept in alternative Journal (see Report)