Nodal-antinodal dichotomy from anisotropic quantum critical continua in holographic models

Clear and rigorous holographic calculation establishing an interesting and novel effect in the probe fermion spectrum. Good analytical control in a field where most of the work currently done is condemned to almost purely numerical analysis.

Highly simplified model with no actual lattice. It is not clear how much of the results would remain valid in more realistic setups. One would like to understand better the meaning of the anisotropic positive-negative $z$ backgrounds and the relation to explicit holographic lattices.

This is a very precise and well written paper about the effect of certain anisotropic backgrounds on the holographic Fermi surfaces. The basic idea is that isotropy breaking in the background geometry, in addition to specific scaling properties in the infrared (IR), leads to Fermi surface vanishing in certain directions. Simple but very clear and convincing analytical results are a particular strength of the paper.

As I mentioned, the paper is very precise and I have no major objections to the text as it is, just a few nitpicks. On the other hand, I feel that a few things should be added in order to have a more complete and convincing story. A list of questions (from minor to moare significant) follows.

(1) In eq. (14) the bispinor components $\chi_\pm,\psi_\pm$ are strictly speaking functions of $\zeta$, not $r$.

(2) What are the units for the spectral weight plots? In Figs. 2, 3 and 9(a,b) the intensity scale varies a lot even though for fixed $m$ we expect roughly the same overall scale no matter what the other quantities are. For the $m=1/4$ plots indeed it is natural that the scale changes.

(3) What would happen if we had a Q-lattice along both axes? I guess the same conclusions would hold unless the periods are the same?

(4) The claim in the Discussion (p. 13) that dipole coupling will not destroy the quantum critical continuum is somewhat misleading -- it is true that it will not change $\mathcal{G}$ at leading order but it will shift the pole in $G$ so that it can move the quasiparticle outside the continuum or destroy the Fermi surface everywhere.

(5) What exactly is $\Gamma$ in Fig. 3 and in the corresponding parts of the text? The explanation in the text is that it is the full width at half maximum but is that the right quantity to consider knowing that we have a pole superimposed on the continuum? Sure the way $\Gamma$ is defined it will behave as in Figs. 3(d), 8(d) and 9(b,d) but when computing the \emph{total} width we dump together the two contributions -- the pole is still there and if you would plot the spectrum in the complex $\omega$ plane you would see it along both directions.

This actually boils down to a more general question -- can you claim that Fermi surface vanishes if the pole is still there just drowned in the continuum?

(6) One puzzling aspect is the appearance of negative Lifshitz exponent $z_y$. Negative $z$ values can be found in some of the papers on effective holographic theories but certainly are not easy to understand. This should be shortly discussed in the paper -- I understand that a full analysis goes beyond the scope of the current work but a few sentences should be devoted to this so that the readers are not left puzzling on whether and how this makes sense.

(7) Smearing of the Fermi surface was seen (though not studied in detail) for example in [1910.01542] also in absence of the Q-lattice contribution (in that paper the authors, including one author of the current paper, interpolate between the Q-lattice and the true scalar lattice). However the anisotropy is there all the time as both contributions are only along the $x$-axis. So was that the same effect as here? Does the configuration in [1910.01542] satisfy the same requirements as found in section 2 here? Or maybe anisotropy by itself is enough? If you look for example at MDC plots in [1807.11730] and [2208.05920] you see weakening or vanishing of the Fermi surface along one or both axes, even though only the explicit (IR-irrelevant!) lattice is present there.

The only relatively significant weakness of this paper is that one can doubt the relevance of the mechanism demonstrated here for real-world systems, even for true holographic lattices. The authors offer convincing arguments that the mechanism considered in this paper is genuine and does not \emph{require} an explicit lattice. But the catch is: does it \emph{remain valid}? How do we know that the effect will not disappear upon hybridization?

In particular, the statement in the Discussion (p. 12) that the nodal-antinodal dichotomy "... doesn’t have to be explained in terms of fermiology, for example by resonance scattering between the different Fermi surfaces" is misleading -- it's true but the big question is -- does it remain valid also with the fermiology included?

For the reasons stated above, the relation to strange metal experiments is a bit oversold in the abstract, it is fair to say that the paper is inspired by this problem but not to state that "... the nodal-antinodal dichotomy in underdoped cuprate superconductors, can be reproduced ... via a holographic dual".

Overall, this is a valuable result which will certainly be an important step in further work. The paper should definitely be published with just a minor revision. The question is only, whether in SciPost Physics or in SciPost Physics Core, since the paper deals with a relatively simplified model and provides more of a technical result than a complete description of a physical effect. I'm perhaps a little bit more in favor of SciPost Physics as the paper is well written and likely to stimulate further work so it should be visible. Either way, it will be a useful resource for the community.

This manuscript investigated the holographic fermion spectral function in presence of anisotropy in spatial dimensions. In particular, from the gravity perspective, the authors focus on how to engineer and understand the nodal-antinodal dichotomy on the Fermi surface in which the spectral function loses its quasi-particle peak in one of the spatial directions.

Using the exemplified gravity model, holographic Q-lattice model, the authors analyze the scaling behavior of IR geometry and find the analytic necessary conditions (or parameter range) relevant for the nodal-antinodal dichotomy in holography.

Furthermore, following the standard holographic method computing fermion spectral function in detail, the authors numerically verify that spectral function satisfying the necessary condition can mimic the nodal-antinodal behavior, which is also consistent with the given analytic argument from the IR green's function.

The holographic setup may be oversimplified in order to describe the real materials such as the one with the four-fold rotational symmetry or experimental results of the broken phase like underdoped cuprate superconductors. Nevertheless, the analysis in the manuscript is solid and pedagogical by the well organized explanations and calculations. Moreover, the results are novel, and allow for a further investigation of connection between the (anisotropy of) quantum critically and strongly correlated systems in a controllable setting. I believe the results are correct and of interest to the community, and so I can happily recommend publication in SciPost Physics with the suggestion of minor changes as follows.

1) On page 2: "..., as happens along the $k_a$ cut in the figure, ..." should be described with $k_n$ rather than $k_a$ in that the overlap happens in the $k_n$ direction in the figure 1.

2) On page 7: the argument of the functions ($\chi_{\pm}, \psi_{\pm}$) in Eq. (14) would be $\zeta$ rather than $r$.

3) On page 8: as demonstrated around Eq. (17), the chosen parameter $(z_x, z_y) = (-3, 3)$ corresponds to $\theta/\bar{z}=-1/3$ when both $\theta$ and $\bar{z}$ are divergent. Is it related with the semi-locally critical limit where the IR geometry is conformal to AdS ? If this is the case, I suspect that the given model, Eq. (16), may exhibit the linear resistivity in temperature, which is one of the major properties of the cuprate. Therfore, the semi-locally critical limit (if this is true in this model) may be another motivation to consider the model Eq.(16) in order to study the nodal-antinodal dichotomy in underdoped curate superconductors.

4) On page 24: $k_x$ in Eq. (67) may be replaced with $k_1$ in order to be consistent with the text below Eq. (69): "... the momentum of the lattice $k_1$ ...".

5) On page 27: when the fermion bulk mass $m$ is non-zero, the spectral function $\rho$ may not be a dimensionless quantity. Thus, the label for $\rho$ in Fig. 7 should be revised accordingly (for instance, $\frac{\rho}{\sqrt{\mu}}$). Also, in the same figure, the vertical dashed green lines seem to indicate $k_x=k_y=0.1 \mu$ contrary to the description in the caption.

The paper argues that the Nodal-antinodal dichotomy could be understood in models of strongly correlated electrons via the holographic duality. The main idea is that the quantum critical continuum with spatial anisotropy can get imprinted on the fermion spectral function, leading to washing out the quasiparticle peak in one direction while leaving it intact in the perpendicular one. Using a holographic Q-lattice model with suitable parameters, the authors demonstrate how this effect emerges due to the quantitatively different scaling of the self-energy in different directions.

Although it is not clear if this idea can be used to understand experimental data, this manuscript contains non-trivial results potentially of interest to the applied holography community and condensed matter community. The paper is clearly written and well-organized. Therefore, I recommend the manuscript for publication without change in SciPost Physics.