Displaced Drude peak and bad metal from the interaction with slow fluctuations.

Dear Editor, we acknowledge receipt of the referee reports.

We thank both referes for their contructive reports, and we provide here a revised version of the manuscript and figures which includes all the suggested changes. We have made an extra effort to both improve the manuscript readability and to improve the statistics of the data shown in Fig. 4. We also acknowledge that both reviewers have asked for a more complete description of the algorithm used in the calculations and we now provide more details both in the body of the manuscript and in a devoted appendix of our paper. Other issues raised separately by the referees are addressed below.

Referee 1

We thank the referee for their positive assessment of the manuscript and for their constructive suggestions. We have made the required changes and reworded parts of the manuscript in order to discuss more extensively the modelling of the bosonic degrees of freedom (generality and limitations). We have also briefly addressed electronic interacion effects in the conclusions, which remain as an open question.

• The authors seem to never explicitly discuss some of the details of the numerical calculations, like the number of sites/levels used in the exact diagonalization algorithm, whether they are relying on the full spectrum or just its extremal part (Lanczos), etc. This is useful to establish the accuracy of the calculation with respect to the discretization or possible finite size effects, (which I assume are mitigated by the statistical average).

More details on the algorithm used in the calculations are provided both in the body of the manuscript, in the figure captions and in appendix A.

• The basic modelling used in this work is essentially a Holstein system. One can reasonably expect the presence of electronic interaction to partially modify the results, either quantitatively or even qualitatively. Can the authors briefly comment on this aspect? This seems to be particularly relevant to make more adherent comparison with materials which are known to be strongly correlated, e.g. Cuprates.

This question is now briefly addressed in the "discussion and conclusions" Section.

• In fig.2 and fig.3 can the author report the specific values of \omega_0 (and g) determining the adimensional \lambda term?

As now discussed after Eq. (1), the model in the slow boson limit $\omega_0\to 0$ is univocally determined by the sole parameter $\lambda$, in terms of which both the static-ED and the finite-density DMFT calculations are performed. This is defined as $\lambda=g^2/2M\omega^2_0 D$, that we keep finite as $\omega_0\rightarrow 0$. Notice that working at finite $k=M\omega^2_0$, as we did, the $\omega_0\rightarrow 0$ limit is equivalent to the large ion mass limit, as is generally assumed for the adiabatic limit. To obtain the optical response shown in Figs. 2 and 3 we fix the parameter $g$ as per Eq. (3) of Appendix A: by setting $k=1$ and hopping parameter $t=1$ on the square lattice ($D=4t$) we obtain $g=\sqrt{8\lambda}$ in dimensionless units.

• Concerning fig.4:

  i) What do indicate open and filled symbols in panel a)

  ii) as a rule of thumb the scaling of some quantity usually requires it to span one or even two order of magnitude variation fo the independent variable (here the Temperature). While most of the data at intermediate values of \lambda do fulfil this "rule", other results for smaller or larger \lambda are much more limited in their variation. Why?

i) this has now been clarified in the text

ii) the deviations from scaling in Fig. 4 are now discussed in the manuscript

As a general rule, different intervals of s correspond to a given interval of temperatures (Fig. 4a), as the values of s increase with $\lambda$. This explains why the data at different lambda cover different portions of the scaling curve. Moreover, in our calculations the temperature interval itself is limited at very strong $\lambda$ by the appearance of the polaron peak, that completely masks the DDP at low temperature (cf. Fig. 3a, the green curve at T=0.5); it is also limited at very small $\lambda$ by finite size effects: at the lowest temperatures, the localization length becomes comparable with the cluster size.

• At pag.5, third line the authors discuss the strong coupling regime and write "Because of bosonic disorder is now strong...". However, this sentence (although correct) somehow anticipates the forthcoming results concerning P(x) and s/s_0 and the many-body enhancement of the disorder.

We have reworded the paragraph to restore the correct logic of causality.

• At pag.6, last paragraph the authors discuss the limiting case g=0 (absence of electron-boson coupling). I guess the resulting expression for s_0 is valid for \lambda-->0 not strictly for g=0.

We have now corrected this statement.

Referee 2

We thank the referee for their positive assessment of the manuscript and for their constructive suggestions. We have made the required changes, adding a more complete description of the algorithm used in the calculations: details are now provided both in the body of the manuscript and in the devoted appendix A.

1. In pages 2-3, I cannot find what is “T”. I think (from what I found in the appendices) that T is the temperature and that maybe in figures 2-3 the adimensional T values reported in the panels are in terms of “t” (the coupling), i.e. T=0.2t and so on, but I think you should discuss this point in the main text (not only the appendices).

   The temperature T is now properly introduced in the manuscript, and the units of t are explicitly reported in the legend of Figs. 2-3.

2. Caption of fig. 2. Even though it is said in the text, I think you should say also in the caption that the dashed lines represent the weak localization.

   All the different lines are now properly defined in the caption.

3. Figure 4. Why some points are filled circles, while other are open circles? What is the difference?

   The meaning of the filled/open circles is now explained in the manuscript and in the caption of Fig. 4.

4. The acronym CDW should be defined.

   The acronym CDW is now properly defined.

5. Maybe I am missing something, but I cannot find details of the numerical calculations (at least for figures in the main text), e.g. the size of the finite cluster (which is on the contrary provided for fig 8 in the appendices) or the algorithm used for the diagonalization part.

   Full details on the algorithm used in the calculations are now provided in the body of the manuscript, in the figure captions and in the devoted appendix A.

6. Just a personal opinion: the figures appear a bit difficult to read as the colours are not well chosen. For example

   • in fig. 2 and 3 the brown solid line (DMFT results) for all three curves is a bit confusing, you could for example replace it with dotted lines of the color of each ED curve.

   • in fig. 3 it is quite difficult to visualize the yellow peak under the broad red shaded area.

   We have changed the colors and line types for a better readability.

7. There are some typos in the references, please check them. For example, in ref. 30 one author is “S.F.” (I assume it stands for Simone Fratini)

   The typo has been corrected (a missing comma in the bibtex file).

List of changes:

- caption of Fig. 1 reworded for clarity
- discussion on the general applicability of the model added after Eq. (1)
- line types and colors modified in Fig.2 and Fig. 3; calculation details added in the captions
- Fig. 4: improved data with increased statistics; meaning of open/filled symbols explained
- short paragraph added on p.7 discussing the deviations from scaling observed in Fig. 4(b)
- discussion of the disorder variance parameter s on p.7 corrected
- details on the determination of the phase diagram Fig. 5 added on pages 7 and 8; paragraph added discussing the range of validity of the classical boson approximation
- Fig. 6, discussion on the observed low-temperature saturation behavior added in the concluding section p.9
- extended calculations details in Appendix A: added full description of the single-polaron calculations including sample figures for the optical conductivity in this case.
- some sentences in the manuscript have been reworded for clarity