Photoemission "experiments" on holographic lattices

Dear Editors, dear Referees,

We present now the updated version of the paper. We have done our best to address the questions in the reports. We have also included numerous minor clarifications and corrected a few typos. The overall methodology and message of the paper however stay pretty much the same as in the first version.

Response to Referee 1:

01) Indeed, the dilaton does not drop to zero at the thermal horizon, this is only true of $A_t$, as can be seen also from our Eq. (8) and the Appendix A. In the previous version we had a misprint attributing this behavior also to $\Phi$ (the actual IR boundary behavior is correct, both in Eq. (8) and further in the Appendix). In general, we obtain the boundary behavior of the dilaton from the IR expansion given in the main text and Appendix, and in the $\tilde{z}$ coordinate that indeed gives the Neumann condition, i.e. vanishing derivative. This is now corrected at the end of Section 2.1.2.

02) We have now expanded the Section 3.2 to include some more details of the Green function calculation (see also the point 03 in the response to the other referee).

03) We have now included a legend in all density plots, and in a few other (line) plots where we found it useful. We have not included a legend in the many EDC plots at six standard momenta because these momenta are introduced at the beginning and stay the same in all figures; we think including this one and the same legend every time would unnecessarily cram the plots. Also, we do not include the color legend in 3D plots in Fig. 1 and 3 as the colors are there just to guide the eye and the z-axis values can be directly read off from the frame ticks. We have mentioned this in the figure captions.

04) True, in the usual $r$ coordinate the source is rescaled by $r_h^{\Delta_-}$. The text is now more precise on this matter (it says we set to unity the leading term, which is proportional but not equal to source). This is not a problem however as the important point is to set a nonzero source so we do not form a condensate; this remains true no matter how large the source is.

It is also true that the subleading term of $A_t$ in Eq. (12) is not the charge density because the presence of the lattice makes the expression for charge density in terms of the derivatives of $A_t$ more complicated. We have thus renamed the coefficient to $a(x,y)$ and do not call it charge density anymore. Importantly, this change does not affect anything in the paper as we only tune the leading term ($\mu$) and do not work with the subleading term at all.

05) We have slightly expanded Appendix B so we discuss some more details of the numerics, in particular (1) we use the Einstein-DeTurck trick (2) we use double precision (3) for most runs we have used a cluster; for testing the code and for low-resolution spectra we have used a desktop computer (4) the running time of the test example is in seconds (now mentioned in the caption of Fig. 18).

In addition to above, we have also corrected a number of small inaccuracies and typos and clarified the text in a few places.

Response to Referee 2:

01) All dimensionful quantities are expressed in terms of the lattice wavevector $Q$. This is now stated at the very beginning of Section 4.

02) Concerning the Hubbard-like bands, we have now shown numerically (by computing the spectra for the same EMD model in homogeneous space) in Appendix E.3 that the Hubbard bands vanish in absence of lattice. This also follows from the analysis in point 3 of Section 4.2 (see also point 09 in this response).

Concerning MDCs, we agree that this is the more interesting part, however it requires substantial additional calculations and we believe it requires a separate work, in particular since the most interesting results seem to happen for strong lattices for which the numerics is very difficult (and goes beyond the scope of this paper where we work exclusively with weak lattices). We include a new Appendix (Appendix D) where we present a few examples of MDC curves, however no attempt is made for a systematic discussion.

03) Indeed the $\omega$- and $k$-dependence of the Bloch functions was not always spelled out explicitly in Eqs. (18-21), this is now corrected: we write the subscripts $\omega$ and $\mathbf{k}$ whenever the wavefunctions depend on them, whereas coordinate dependence is written as argument of a function (e.g. $\psi_{\omega\mathbf{k}}(x,y,z)$).

In Section 2.2 we have added a paragraph explaining in more detail the calculation of the Green function. In short, the source is a plane wave and from the response we read off the components from different zones so we can construct the diagonal component of the Green function (see also the point 02 in the response to the other referee).

04) Concerning the quasiparticle detection, the ultimate criterion (in absence of good analytic insight) is to compute the spectral function, i.e. the retarded propagator in the bottom half of the complex $\omega$ plane -- then one can directly see the pole and differentiate it from a branch cut. However, such a calculation was too demanding for our present work (it requires a large number of points and the convergence becomes progressively worse as we increase the magnitude of the imaginary part of $\omega$). Therefore, we have simply scanned the peak in real $\omega$ with increasing resolution, and decided we have the quasiparticle if the peak becomes ever sharper with resolution until its width drops below $O(T)$ at temperature $T$; if it is broader than the temperature times a factor of order unity then it is not a quasiparticle. This is not quite satisfying but is apparently the only choice in absence of analytic results. This is now explained in footnote 14.

Concerning the exponential suppression of the spectral weight inside the gap, more precise statement is that the gap is exponentially small compared to the ratio of the peak maximum to the typical "background" value of the spectral function, i.e. its value away from the peak (which in general grows with temperature). We have included this clarification in the text.

Concerning the "unimodality" of the high-temperature spectrum in Figs. 6 and 7, we agree it is not the right word. What we meant is really that the side bumps (bands) also melt away with $T$, not only the quasiparticle. This is now explicitly said in the text.

05) We agree that the "underdoped/overdoped" terminology is not uniquely defined and can be confusing. For that reason these terms are not used anymore. Instead, in relation to Fig. 11 (the only place where this terminology was used) we just directly describe the situation: the peak of the spectral weight shifts from $\omega<0$ (electrons) to $\omega\sim 0$ (excitations near the Fermi surface) to $\omega>0$ (holes). Mottness is defined as the shift of the spectral weight from high to low frequencies when the system is doped, without closing the gap, i.e. without entering the metalic phase.

06) Indeed, we had a typo, the dilaton does not drop to zero at the horizon. This is now corrected (see also the point 01 in the response to the first referee).

07) Indeed the axes label was wrong in the bottom panels of Fig. 3 (instead of x-z the axes are really x-y). Also, the text in the figure caption weas misleading, this is now corrected. The bottom figures are really just sanity checks that we get AdS asymptotics.

08) While in general it is indeed the asymmetric MDC which provides a definite proof of non-Fermi-liquid behavior, in EMD models of this type (with scaling solutions at zero temperature), it is known from the literature that non-Fermi-liquid phases always have asymmetric self-energies. This extends to our weak-binding analysis too. So while not true in general, it is true for this model. Also, in Appendix D we include a few MDCs which indeed show asymmetric structure. This is now explained (and the Appendix D referred to) in footnote 15.

09) It is true that the (imaginary part of) self-energy does not affect the existence nor the position of the poles (quasiparticle peaks), but the wide bumps/bands are not quasiparticle peaks and do not correspond to poles. Higher values (bands) and lower values (the minima between bands and the quasiarticle) come solely from the finite part of the propagator, i.e. from the self-energy. We briefly discuss this at the end of point 3 of Section 4.2.

10) Indeed this point was not clear in the original text. Actually it has nothing to do specifically with the edge of the BZ. The point is that the numerator of I (Eq. 53) can fall off faster than the denominator (if $\vert\mathbf{k}\vert<1$) so even for small $\omega$ the self-energy (proportional roughly to $\exp(-2I)$) can stay of order unity, which explains the weakening of the peak in some cases. The text of point 2 in Section 4.2 is now reformulated accordingly.

11) We now discuss in some more detail the integral (55), mainly in a newly written section in Appendix D. In short, a cutoff is indeed imposed at large $\omega$ since we cannot numerically integrate to infinity, however we show that the result is very stable and practically independent of the cutoff thanks to the fact that our spectral weight falls off exponentially at large $\omega$, in accordance with the dynamical boundary source motivated in Section 3.3.

In addition to above, we have also corrected a number of small inaccuracies and typos and clarified the text in a few places.

We thank both referees for their instructive and insightful remarks and comments.

The authors

We have introduced numerous small changes, clarifications and corrections. We do not list here every single change, only more substantial ones.

01) Corrected typo in Eq. (11).
02) Improved discussion of UV boundary conditions in Section 2.1.2.
03) Corrected the axis labels and captions in Fig. 3.
04) Improved notation in Section 3.1, in particular Eqs. (18-22).
05) Expanded discussion of the boundary conditions for $G_R$ in Section 3.2.1.
06) Added a short discussion on detection of quasiparticles and gaps in Section 4.1 (p. 18).
07) Added a short discussion on symmetry of quasiparticle peaks in Section 4.1 (p. 19).
08) Clarified discussion of Fig. 11 in its caption and in the text of Section 4.2.3 (p. 23).
09) Improved interpretation of results in Section 4.2 (p. 28-29).
10) Newly written Appendix D and Figs. 21-22 with some very basic remarks on MDCs.
11) Added Section E.1 and Fig. 23 on the numerical implementation of the Hilbert transform.
12) Added Section E.3 and Fig. 25 on absence of Hubbard-like bands in absence of lattice.

xx) Added a legend in Figs. 7, 10, 17.