Functional Renormalization Group for fermions on a one dimensional lattice at arbitrary filling

1. The paper extends the fRG analysis to away-from-half-filling conditions in 1D lattice models for interacting fermions, addressing the lack of particle-hole symmetry and its consequences, which is a valuable extension of the research.

2. This fRG approach allows for systematic consideration of lattice effects on the effective continuum field theory at low energies.

3. The detailed explanation of how the flow of coupling constants is conducted in each case is very informative and beneficial.

1. The paper could improve by delineating its advancements over existing fRG methods more explicitly.

2. The approach presented in this paper may be highly technical and complex, which could limit its accessibility to readers who are not specialists in the field, although this is somewhat unavoidable.

This manuscript introduces a weak-coupling functional renormalization group (fRG) approach for one-dimensional (1D) interacting fermion lattice models, applicable away from half filling, using the extended Hubbard Model (EHM) to show how lattice effects influence low-energy effective field theories. It finds that at half filling, irrelevant interactions affect phase transitions, with a notable impact on BOW, SDW, and superconducting states. Away from half filling, the loss of particle-hole symmetry alters high-energy state contributions to phase stability and transitions, leading to deviations from traditional continuum theory predictions. The findings of this paper align with previous research and suggest broader applicability to various lattice models, with potential extensions to explore doped fermionic systems.

Applying the fRG approach to cases away from commensurate fillings in interacting electron models is challenging, yet this paper presents one such method and meticulously explains its effectiveness when applied to the EHM. This provides valuable information for the future development of this technique. Therefore, if the manuscript is sufficiently improved by addressing the questions/suggestions I have listed below as much as possible, I would recommend its publication in SciPost Physics.

(1) The authors state in the introduction that 'These RG results were strictly speaking limited to the EHM model at half-filling.' Does this mean that traditional RG methods were not applicable when deviating from half-filling, or that they could be applied but did not yield good results? Clarifying these points and further highlighting the technical contributions developed in this paper would enhance the understanding of its significance.
(2) This may be a trivial question on Fig.1: Why is the density of states not symmetric with respect to the band center (\epsilon=0) for the tight-binding model? Is there already some form of interaction incorporated in a circular manner?
(3) It is unclear if the flow equations Eqs.(49-51) for channels at q=\pm2k_F are still applicable to the case away from half filling.
(4) While the qualitative agreement with previous works at half-filling is acknowledged, it is not clear if the results at half filling are also improved by the presented fRG approach?
(5) At small doping levels, how can it be determined that the charge gapless CDW, BOW, and SDW phases are incommensurate? It is presumed that charge and/or spin gapped phases are commensurate, but can commensurate CDW or SDW phases still occur when the system deviates from half-filling? Moreover, while the concept of incommensurate CDW and SDW is comprehensible, what constitutes an incommensurate BOW state because BOW state is characterized by a two-site unit cell?
(6) Related to question (5), when investigating the EHM lattice model using numerical methods such as DMRG or QMC, it is anticipated that away from half-filling, the charge gap in the repulsive region would close immediately. To put it plainly, at V=0, the Bethe Ansatz indicates that only at n=1 is there a charge gap due to a singularity, but it is evident that deviating from n=1 leads to a charge gapless state. However, Fig. 9 still shows a charge gapped state for U>1.5. How can this discrepancy be understood?
(7) It would be helpful to the general reader if the paper could provide a clearer explanation of how the gapful and gapless states in the charge and spin sectors are determined.
(8) Some figures on flow of the vertices and couplings are missing titles on the vertical axis. It would be beneficial for clarity and completeness if these could be added.
(9) The boundary between the SDW and TS phases is indicated to be in the second quadrant, while the boundary between the CDW and SS phases is stated to be in the fourth quadrant. Could there be a possibility that this is opposite?
(10) The 1D doped EHM has been studied by an fRG method in a relatively recent paper: Y.-Y. Xiang et al., J. Phys.: Condens. Matter 31 (2019). A brief statement about discrepancy to this paper may be useful.
(11) Please correct some typos, e.g., abbreviation `EHM' is twice defined, `TL' is defined in the second appearance of Tomonaga-Luttinger, there is no vector notation (arrow) in |n|\neq0, etc.

1. The work presents interesting results on so far not or little explored regions of the phase diagram, in particular on but not restricted to the aspect of a bond-order wave that appears "on top" of the transition region defined by U=2V between charge-density and spin-density waves for U,V>0. The evolution of this peculiar behaviour when moving away from half filling is outlined and discussed in detail, along with several other aspects. The underlying data on which the deduction of the phase diagram is based is discussed and presented in detail.

2. The manuscript is well structured and gives a systematic presentation of model, method and results.

3. Results are validated against previous/exisiting works in detail, before new results are presented.

1. At some stages I felt some more information could be included on the formal level as well as concerning auxiliary parameters.

2. Some physical aspects may deserve to be outlined with a little more clarity. I tried to make some suggestions on these matters in the comments below.

The manuscript "Functional renormalization group for fermions on a one-dimensional lattice at arbitrary filling" presents a new application of functional renormalization group methods to the extended Hubbard model in one dimension. In particular, the authors consider the non-linearity of the dispersion as well as the momentum dependence of the flowing couplings. The scheme is first outlined and developed, and then applied numerically. It is found to reproduce previous results at half filling for the phase diagram and is then applied to non-half-filled (i.e. "doped") cases, for which it uncovers several new aspects and observations.

In summary, I consider the manuscript to be suitable for publication in SciPost Physics. I do offer some remarks, questions and suggestions below, the main intention being to provide constructive input.

My comments below are above all meant as questions, suggestions and remarks, rather than firmly requested changes. In case I might ask for information that is actually given in the manuscript, please ignore the respective comment(s). I cannot rule out that I may have overlooked something.

General comment:

i) It is somewhat implied in abstract and introduction, that the matter of interest are ground state properties. This is however not made very explicit, and the numerical calculations are then actually done at a small but finite temperature, if I understand correctly. It seems appropriate to me to outline and justify this procedure more explicitly, also in light of the generically delicate situation of long-range order - or rather its absence - in 1d (quantum) systems even at T=0.

a) I would find it helpful to clarify this aspect early in the manuscript and to state the actual temperature value that was chosen for the numerics. While a value T=10^(-7) is stated once in the caption of Figure 10, it is however not clear to me if this is the general value chosen for all numerical calculations.

b) In the conclusion it is stated that "the nature of ground states" was checked. A comment in how far and for which type of quantities/observables a small but finite temperature in this numerical approach allows conclusions about the ground state would be beneficial to corroborate the conclusions.

c) Technically, (1-PI) fRG computations can be done at T=0, as employed in previous works, sometimes even being a preferred choice. What is the reason for not doing this here? Are there singular contributions at T=0 that are not regularised by the chosen momentum cut-off? -> A brief sentence stating the reason for the actual value of T that was chosen in contrast to T=0 seems helpful to me.

Section 2:

ii) Equation (6): The quantity 'L' could/should be defined here already.

iii) Equation (13): Extracting a factor 'T' from the coupling function is unfamiliar to me. In particular, it makes the limit T->0 appear awkward. Is there a necessity or deeper reason to do this? It also seems to collide with equation (8), if I'm not mistaken.

iv) Equation (14) - concerning the regulator:

a) I would already at this stage briefly but explicitly state the choice of the actual regulator that is used, or at least point to the appendix.

b) Eq. (23) suggest a sharp cut-off by virtue of Theta functions, while in appendix A.3 it is outlined in detail that it is actually a smooth cut-off. Adding the parameter 'a' to these Theta functions in the equations/definition of the main body could avoid this possible misinterpretation.

c) For completeness, the chosen value of the "smoothness" parameter 'a' could/should be explicitly specified and be related to other quantities with which it 'numerically interferes', such as the temperature and the lowest cut-off value that is reached in the computations.

v) Related to this, in Figures 2,3: The dashed/"derived" propagators are somewhat loosely defined in the caption as "line in the outer shell". In 1-PI fRG they are more generally defined as "single-scale" propagators (e.g. Ref. 30) and it may be worthwhile to define them as such briefly but more precisely in the main body, in particular since the scheme employs a slightly softened cut-off.
Overall, a more explicit statement on the elements that are depicted in the two figures would improve clarity, it need not be long.

vi) Equation (20): Again, the factor 'T' puzzles me, c.f. comment iii).

vii) Section 2.3: The paragraph between equation (20) and (21) is somewhat unclear to me. Formally, the 1-PI fRG equations are exact, with a regulator being implemented in the quadratic part of the bare action (only). The coupling function, in turn, is always defined everywhere for all momenta and (usually) not subject to a separate, additional cut-off. Also, since the non-derived propagators (for a momentum cut-off) live above the cut-off energy, I would not expect "unavailable states" in the low-energy section of the flow, in contrast to the reasoning provided in the text by "...namely above the scaled energy \Lambda of integrated degrees of freedom".
It is unclear to me if such a function is also used in the numerics or only required to make contact with the g-ology continuum model.
Maybe this part can be made clearer, potentially also by adding a reference. It could however also be that this question is due to my personal (lack of) understanding, and that I am simply not familiar with this type of procedure. My feeling is that this modification might owe to procedures that are common in other types of g-ology RG treatments.

viii) Equation (23): Here, it is implicitly suggested that self-energy effects are neglected, since bare propagators are used. Yet, it is later stated that "some" self-energy corrections are actually implemented by means of a renormalised Fermi velocity, below Eq. (33). Thus, the propagators inside the loops are not really bare G^0 entities, if I understand correctly. This could be outlined earlier, c.f. comment v), also since in 1-PI fRG the loop contributions when going beyond G^0 and including a flowing self-energy (not done here) cannot generally be written as \Lambda-derivatives of bubbles - c.f. comment v) about single-scale propagators.

ix) Section 2.3, second last sentence before section 2.3.1 - "... we do not consider uniform q->0 responses": I assume this is essentially due to the fact that potentially relevant forward scattering contributions in a scheme with a (nearly) sharp momentum cut-off only enter the flow asymptotically for \Lambda->0, and thus cannot properly compete with the contributions of interest here, which begin to flow to strong coupling at higher values of the cut-off. That of course is a subtle matter and may also depend on the ratios of the different "smoothness" parameters that are used, see also comment i)a) and iv)c).
-> Maybe it could be mentioned which kinds of ordering tendencies are thereby a priori "deselected" and in how far this constitutes a restriction - or not - on what follows. Are there parameter regions in the phase diagram where q->0 responses could play a relevant role, like it is the case e.g. for ferromagnetism in 2d?

x) Section 2.3.1: The bubble "intensities" are stated. I would find it useful to include an explicit computation, which should be rather concise but could be helpful. It could be included/added in Appendix A.2.

xi) Section 2.3.2: The first sentence mentions the "low-temperature" limit. Can this be better quantified/specified, c.f. comment i)? Is it generally the value given in the caption of Figure 10, i.e. 10^(-7)?

xii) Fig. 5 lacks the indications '(a)' and '(b)' - it is obvious what is meant, though.

xiii) A general remark on the graphical presentation of the phase diagrams: They appear as continuous diagrams with sharply defined and continuous transition/separation lines. I assume that in practice numerous fRG runs have been conducted for a number of parameter sets to map this out. Maybe it is possible to indicate this somehow, at least as an example in one of the diagrams, to be able to relate the continuously depicted diagrams to the actual set of results from which they are deduced, similar to e.g. Fig. 8 in DOI 10.1103/PhysRevB.61.7364 .

xiv): Section 2.4 - concerning the self-energy corrections: They are included in terms of a renormalised Fermi velocity v_F which is calculated "in the scaling limit".

a) How is "scaling limit" meant here? In the scope of 1-PI fRG there is no rescaling involved in the formalism.

b) Would it be feasible (and worthwhile at all) to include v_F as a flowing quantity, and at what cost and effort? C.f. comment viii).

c) Why is it better or maybe even necessary to work with the "final" v_F rather than the initial/bare value? Does this lead to relevant qualitative/quantitative changes in the results?

xv) On the general strategy to expand the effective interaction in \xi: Would it also be possible/feasible to work with a more direct and sufficiently fine "brute-force" patching of the effective interaction in momentum space, such as e.g. in various other (2d) fRG works? This would of course increase the number of couplings constants that have to be treated numerically, but in light of 2d calculations based on that technique I would (naively) expect this to be feasible.

Section 3:

xvi) First paragraph - "The calculations are limited to the weak-coupling sector": It might be worth briefly stating the reason, assumably the truncation of the 1-PI fRG hierarchy. This could also be mentioned earlier in the text, c.f. comments iv) and v).


xvii) Figure 7,8:

a) The y-axes could be labelled explicitly in the plots.

b) Explicit quantitative information on the actual value of the low-energy scale at which the flow is stopped would be nice, and how this relates to the temperature that is chosen for the numerics, to better understand the mutual relevance of the various low-energy scales - c.f. comment i).

c) Fig. 8 is discussed in the text before Fig. 7. Both figures show results for specific points in Fig. 6.
-> Merging the two Figures into one might be an option for better readability and a more direct view of the underlying results.


xviii) Figure 11: I find the legends to be a little small.


xix) Section 3.2 - third paragraph - "... we can write N(\xi)=1/(Pi*v_F)": Since v_F is renormalised, c.f. comment xiv), shouldn't this be the renormalised value? This would then differ somewhat from g-ology, wouldn't it?


xx) Section 3.2.1 - last paragraph: "... in the second quadrant": It was not clear to me which one is the "second". I'd suggest to use "lower left", "lower right", etc., to avoid ambiguities. Or it might be an option to add thin lines to separate the quadrants and to label them explicitly as e.g. I, II, III and IV. The mutually distinct nature of the physics would justify that, in my opinion.


xxi) A general question and a mere matter of interest: With increasing temperature I would expect the flow to become regularised at some stage, and this should define a "one-loop T_c", i.e. a physical "cross-over/binding/short-range-ordering" temperature. Has this been looked at in this context?


xxii) Section 3.2.3 - page 27 lower part: "Calculations carried out ... namely up to \mu=sqrt(2) (3/4 filling)": Why is this value chosen as the upper limit for the calculations? What would happen beyond? Is the method still applicable then? If not why not? Or would it simply go beyond the scope of this work? C.f. comment xxiii) -> quarter filling.

Section 4:

xxiii) It is mentioned that various previous results could be confirmed. The new results are then presented stating that "We have also carried out ... away from half filling": To my personal taste it would well be worth emphasising a bit more that these are new results, by which unexplored terrain is being entered. These are in my view the most striking results contained in the manuscript, drawing their solidity of course from the fact that the method is in line with previous results.
That said, there is some previous work on the special case of quarter filling, e.g. DOI 10.1103/PhysRevB.75.113103, and likely/possibly others (I did not manage to do a comprehensive research on this). If possible and sensible, it might be useful to compare against such prior results, too. It would be interesting if and how quarter filling might emerge as a special case within the formalism presented here. That can of course also be left for future work.

xxiv) Some words on the specifics of the numerical implementation and the computational costs would be of interest, also to have an idea of what is possible and feasible in such a set-up. Ideally, the underlying code might be worth being developed further and even be published as a result in its own right.

xxv) As part of the outlook, anisotropic systems are mentioned. These systems may actually permit to compare the extension of typical 1d (f)RG schemes to extensions of typical 2d (f)RG schemes, the two approaches often being quite different in nature. I wonder how the authors would feel about this aspect.

xxv) Appendix A.3, c.f. also comment iv): There are two parameters 'a', but for two different purposes. Also, the actual values that are used in the numerics are not specified (or I did not find them). I suggest to explicitly name them as two separate parameters and to provide the numerical values that were used.