Photoemission "experiments" on holographic lattices

The manuscript "Photoemission 'experiments' on hologrphic lattices" is devoted to the study of fermionic correlation functions in holographic models which incorporate the tunable sclaing behavior (via the running dilaton) and the square crystal lattice (via the modulation of the chemical potential). The obtained spectral functions are inspected and some of their features are noted. The analytic treatment of the ferionic self energy induced by the modulated horizon is provided. The comparison to the Green's functions calculated in Hubbard model is performed.

I see several notable original features of this work:
1) To my knowledge this is the first time when the calcualation of hologrpahic fermionic spectra is performed on top of the 2D lattice. This is a significant technical achievement, since it requires noumerical solution of 3D partial differential equations
2) This is the first attempt to perform a quantitative comparison of the holographic calculations with a more conventional Hubbard framework
3) The analytical perturbative treatment of self energy is an interesting result as well, although this can be seen as a relatively straightforward generalization of [4].

The amount of effort invested in this work is remarkable and it is of no doubt that the authors demonstrate the ability to handle quite complex technical methods in holographic lattices. However the analysis of the results of these calculations performed in the manuscript doesn't really lead to any significant physical conclucsions. The features noted in the spectral functions are quite generic, the comparison to Hubbard model is qualitative at best and the authors admit themselves that the merit function which they use for the fitting isn't really selective in the space of model parameters, so it dosn't tell much. There are many unsupported claims in the manuscript, which require a much more detailed analysis to be truly justified, as I list below. The validity of several arguments is questionable, as I also describe below. Because of this lack of clear and justified results, I can't recommend the current manuscript for publication in SciPost.

The unstructured list of major issues in the manuscript.

1) The authors don't specify the units in which they measure temperature, chemical potential, frequency etc in the hologrpahic model. This makes it impossible to judge the results and compare with the existing literature. The authors use the statement "low temperature" what is the meaning of it?

2) Despite the fact that the lattice is listed as the main feature of the current study, its effects are never really discussed in the manuscript. The spectral functions in the (kx,ky) plane are never shown and the geometry of the Fermi surface is not discussed. It is unclear what is the geometry of the energy bends and to what extend is it due to the umklapp effects and the hybridisation of the neighbouring Brilluin zones. The relation between the lattice constant and the size of the Fermi surface is unclear. I wonder, whether the "Hubbard bands" which are the main focus of the manuscript would not be observed already in the homogeneous model in absence of the lattice. We know that the fermionic spectral functions can be quite complicated in holography, featuring the nested Fermi surfaces the gaps, the smeared continuous density and vague bumps even in the hologeneous models. So in order to claim that "Hubbard bands" are the feature of the lattice, it should be demonstrated that they disappear in absence of the lattice.

3) The fermionic calucalations are quite nontrivial in presence of the lattice, as the authors discuss on p.12. (Btw, the value of Bloch momentum k either appears or disappears in the argumens in eqs. (18)-(21), looks like there is a confusion here). I think more details on the actual calculation procedure should be provided, in order to evaluate what exaclty is measured here. How the numerical boundary conditions are imposed? To which Bloch momenta belong the sources and the responces in the fermionic computation and how is it related to the Green's function element $G^{nn}$? How the calcualtion setup is organized for the cases of different scaling dimensions of the fermionic probe? These details are valuable and should be included in the Appendix. It is also very interesting how the authors obtain the momentum integrated spectral functions in Figure 11. Naively evaluating spectral function at al momenta would require enourmous computational effort.

4) When analysing the EDC spectra the authors use very vague definitions of the features they are talking about. It really doesn't go beyond visual inspection. How exactly do we distinguish between quasiparticle and non-quasiparticle? What do we mean by "exponentially suppressed by temperature" for temperatures of order 1. On Fig.6 (bottom right) I can discern three soft bumps, but the authors claim the structure is "unimodal", how is this defined? All this analysis doesn't seem robust.

5) In the paragraph below Fig.8 the authors use the terminology "underdoped", "optimally doped" etc. without any definition. By no means these term have a commonly accepted meaning in holographic models. Similarly, the definition of "Mottness" is never provided.

6) In the last paragraph before 2.1.2 it is said that dilaton is zero at the horizon. I think this is a misprint.

7) On the bottom of Fig.3 and the related analysis the values of the metric functions at $\tilde{z} = 1$ are shown. (Note the mistake in AxesLabel there). This is the asymptotic boundary of AdS, where the Dirichlet boundary condition is imposed. So it has no relevance to the discussion about lattice being absent in IR. Something is confused here.

8) Before Fig.5 there is the statement that the asymmetric peak in EDC signals the non-Fermi liquid behavior. I don't think this is correct. This statement should rather be aplied to MDC (momentum distribution curve) where Fermi liquid does indeed always give a Lorentzian.

9) In point 3 in the end of Sec. 4.2 there is a claim that the local maxima in the self energy $\Sigma$ explain the appearence of the bumps in the spectral function ("Hubbard bands"). I don't see how it can be true, since the location and the number of the peaks in the spectral function is controlled by the poles of the Green's function, whose position in turn depend on the bulk physics. Self energy, evaluated in Sec. 4.2 at the horizon only affects the width of these peaks.

10) In point 2 in the end of Sec. 4.2 there is a discussion that at the Brilluin zone boundary the numerator (btw it reads "numerators" in the text) in (53) goes to zero. I don't see how it works, since at the BZ boundary the Bloch momentum $k$ must be finite, of course. What it really meant here in this paragraph?

11) I wonder, how the integral in (55) is evaluated using the holographic results? Is there a cutoff imposed at some $\omega$? This calculation deserves being mentioned in the Appendix.

The manuscript studies the single-electron spectral in a 2D holographic ionic lattice with hyperscaling-violating infrared geometry. As far as I know, this should be the first paper that computes the fermionic spectral function in a 2D holographic lattice. Some interesting results and robust features are observed. The structure of the spectral function is partially explained by a perturbative near-horizon analysis. Moreover, the authors compare the holographic results with the Hubbard model and find a good fit to the Hubbard Green's function, which brings AdS/CMT closer to the lab. The paper is clearly written and well-organized.

I have some questions and comments that are listed as follows.

1. The authors imposed the Dirichlet boundary condition for the dilaton field $\Phi$ at the horizon. As mentioned in the last paragraph in subsection 2.1.1, they required $\Phi$ to drop to zero at the horizon $\tilde{z}=0$. I find no reason to set $\Phi(\tilde{z}=0)=0$. Instead, for the coordinate systems in Eq.(5), one can impose the Neumann boundary condition for $\Phi$, i.e. $\partial_{\tilde{z}}\Phi=0$ at the event horizon $\tilde{z}=0$.

2. When introducing the spectral function $A(\omega, k)$, the authors disregarded all off-diagonal terms. But there are still an infinite number of diagonal terms. The authors should explain in detail how they define the spectral function in the holographic setup.

3. In a density plot, a legend is necessary to illustrate the value of the physical quantity one considers. In particular, in Figure 7 and Figure 10.

4. The authors claimed to set the source of the scalar operator to unity. More precisely, the UV expansion of $\Phi$ reads $\Phi(x,y,r)=r^{-\Delta_-}(\phi_s+\cdots)$ with $\phi_s$ the source of the dual scalar operator, and the authors wanted to set $\phi_s=1$. In terms of the new coordinate $\tilde{z}$ defined in Eq.(3), one has $\Phi(x,y,\tilde{z})=(1-\tilde{z}^2)^{\Delta_-}\left(\frac{\phi_s}{r_h^{\Delta_-}}+\cdots\right)$. Therefore, compared with Eq.(12), the authors actually fixed the scalar source $\phi_s= r_h^{\Delta_-}$ rather than $\phi_s=1$. Similarly, $\rho_(x,y)$ in the subleading term of $A_t$ in Eq.(12) should not be identified to be the charge density.

5. I suggest the authors providing more details about their numerical methods.

1) How did they deal with the gauge fixing? DeTurck method or others?
2) Did they use double precision numbers or higher precisions?
3) Did the authors use High-Performance Cluster or desktop?
4) In Figure 18, the authors showed the running time $t$ in function of the size of lattice $N$. What’s the unit of $t$?

I recommend this manuscript for publication in Sci.Post, after the authors address the points raised here.