We thank the editors for their very rapid handling of our manuscript and both referees for their timely and helpful reports. Neither referee found serious issues in our work; thus, both recommend publication in APS journals. Referee A also applauds our "clear and thorough analysis". However, the initial submission apparently overemphasized the connection to a previous paper (Gorelik et al., PRL 2010), so that the referees could not fully appreciate the important and unforeseen new physics established in our work (Referee A: "worried whether there is sufficient general interest for ... PRL"; Referee B: "seems to me not of wide enough interest ... innovation [for PRL]"). Thankfully, both referees made the reasons for their reservations very explicit, giving us a chance to properly address all of them. The corresponding changes have certainly greatly improved the manuscript. 

Since the question of PRL level interest and relevance of our paper is the only one under debate, we will address it first, before responding in detail to all comments and suggestions by the referees.

1) We think it is consensus that the prospect of building "quantum simulators" (for condensed matter systems) based on trapped cold atoms is of enormous interest and potential relevance. Several prominent experimental groups (including those led by Nobel laureate W. Ketterle and T. Esslinger) have done their best for years in order to reach the milestone of detecting antiferromagnetism (AF) in cold fermions - without any success so far. In such a situation, game-changing theoretical proposals appear particularly relevant.

2) We offer such a game-changing proposal: using suitable probes, the relevant AF physics can be detected in an entropy range s<0.7 which has already been touched experimentally. Contrary to common conception, radical breakthroughs in cooling technology (in order to reach s<0.34) are not needed. In comparison to the (presently unsurmountable) cooling hurdle, the precision requirement of our proposal seems fairly benign.

3) We also offer a surprising fundamental insight relevant at least for the magnetism and statistical physics community as well as for everybody interested in nano- and mesoscopic physics: if one carefully distinguishes between intrinsic and indirect effects of dimensionality, the intrinsic effects (including long-range order in 3d) are really subtle; in contrast, short-range AF order (well-defined also in mesoscopic systems such as cold atoms) is nearly universal.

4) This is "bleeding edge" research, demanding computational resources that have to be limited to very special occasions. The point is that (i) not many groups in the world could have done this study and (ii) the main authors of this paper essentially "bet the farm" on this project (in spite of being involved in other PRL level projects).

We hope that the revised version of our manuscript makes these points more transparent. In response to a request by Referee B, we have also significantly extended our previous finite-size analysis of QMC data and have made statements concerning their accuracy more precise and explicit. While this extended analysis ultimately confirmed our expectations, our rigorous treatment proved a bit difficult and required excessive amounts of computer time, which delayed this resubmission. We hope that the referees appreciate this extra effort; for the benefit of the readers, we have prepared a corresponding supplement. 

We should also stress that all relevant limitations of DMFT have already been pointed out in Gorelik et al. (PRL 2010); this paper remains valid without any corrections. We also started the present "beyond DMFT" collaboration immediately after submission of above PRL. So outside input was not needed for guidance of our research. However, Refs. 18 and 19 clearly underline that relevant (at PRL level) physical questions were still open in 2010. We should also note that our preprint has already been cited twice, including a PRL submission of the Scarola group (arXiv:1107.4349) with similar physical focus (but less elaborate methods).

Since our resubmission resolves also all other points raised by the referees, as detailed below, we are confident that the manuscript is now suitable for publication in Physical Review Letters.

Detailed reply and list of changes:

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Report of Referee A -- LE13343/Gorelik
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> In their manuscript, Gorelik et al. provide a comparison of different numerical simulations (DMFT, QMC, DCA, Bethe Ansatz) to assess the double occupancy of ultracold fermionic atom mixtures in optical lattices. The authors are to be applauded for such a clear and thorough analysis of the numerical methods that establish the double occupancy as a possible signature of local magnetic correlations in the Fermi gas. 

> While I found the manuscript in general very readable, I found both the abstract and the introduction to be a bit technical. 

We have rewritten the abstract and corrected the introduction to exclude technical details and to focus on the physical site of the discussed phenomena.

> In general, I am also worried whether there is sufficient general interest for a publication of the manuscript in PRL. The main result had been published by the authors before (Gorelik et al. PRL 105, 065301 (2010)), which showed that local antiferromagentic correlations of the Fermi gas lead to an increase in the fraction of doubly occupied sites. The number of doubly occupied sites could thus be used as a measure of temperature or magnetic correlations in the gas. 

The main result presented in this manuscript is the universality of the entropy scale for the local antiferromagnetic physics for the whole range of dimensions from d=1 to d=infinity. We also discuss the features in double occupancy for the weak coupling case, which is also a new aspect. 
We are grateful to the Referee for drawing attention to these wrong accents in the initial version of manuscript. We have adjusted the text accordingly.

> Subsequent numerical simulations have questioned this interpretation (refs. [18,19]), which lead to the authors pursuing a detailed comparison between different numerical methods. Their finding that the general result of DMFT put forth in their first paper remains true, but that e.g. kink structures are rounded off by more exact numerical methods (QMC 3d,2d or Bethe Ansatz in 1d), is valuable, but in my opinion does not warrant publication of the present manuscript in PRL. It does not seem to be a major advance that would be interesting for the general reader, but it seems more a result that will be valuable for evaluating the quality of different numerical methods. 

The brief discussion of how the DMFT results will be modified by the non-local correlation is already present in our initial work [Gorelik et al. PRL 105, 065301 (2010)] and was not initiated by [Fuchs et al. PRL 106, 030401 (2011)] or [De Leo et al. PRA 83, 023606 (2011)]. Indeed, none of these papers were addressing the temperatures low enough to capture the discussed phenomena. We have reduced this technical discussion, and also removed the part of Figure 1 that showed the comparison of already published D(T) data for simple cubic lattice. 

> The role of double occupancy as an experimental probe for magnetic correlations seems borderline, too. The absolute signal predicted by the authors is just within the measurement limit of the best experiments in this respect by the Zurich group of T. Esslinger. 

The absolute signal in double occupancy at strong coupling is indeed within the measurement limit (as of 2008), i.e. it should be possible to measure if the low enough temperature will be achieved. In the weak coupling regime (see Figure 2), the significantly higher absolute value of double occupancy as well as the abrupt decrease of D due to the AF correlations are less demanding to the detection accuracy, but would require lower temperature.

> In conclusion, the authors have provided a detailed numerical comparison of the double occupancy during the buildup of magnetic correlations. The main results had been predicted and published before. I therefore cannot recommend publication in PRL, but would recommend consideration for publication in PRA or PRB. 
> Finally, I have some minor comments on the manuscript, which the authors might want to take into account when resubmitting the manuscript. 

> 1) Page 1, second column. The authors mention that for current experiments, long range magnetic order is not relevant, as it is not probed in modulation spectroscopy or superlattice experiments. In this context the authors neglect the prospect of using light scattering or noise correlations that have been proposed (and partly demonstrated). Both could reveal the presence of longer ranged antiferromagnetic correlations. It is therefore certainly not true that long-range order is not relevant in current experiments. 

Momentum-space based techniques, e.g. Bragg spectroscopy, would have to deal with the fact that (nearly) saturated AF long-range order in an infinite system is reached only far below the continuous Neel transition. So a reduction in the entropy by a factor of 3 or 4 would be required for a strong signal, as well as global equilibration within experimental time scales. Moreover, the concept of long-range order is not rigorously defined and might be misleading in the finite and inhomogeneous systems realized in fermionic experiments, with a shortest experimental length scale of about 10 lattice spacings. In contrast, short-range correlations remain well-defined.
Corresponding discussion is added to the revised manuscript.

> 2) Caption Fig. 2. For self-consistency the authors should define what they mean by a 'spin-crossover' temperature. 

We have added this definition to the text of manuscript.

> 3) Page 2, first column. I am a bit skeptical that one can really talk about a 'significant enhancement' of the double occupancy. In absolute numbers the enhancement is quite small, only relative to the almost zero value of D for higher temperature the enhancement is larger.

We have already addressed this aspect above while discussing the double occupancy as an experimental probe for magnetic correlations.

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Report of Referee B -- LE13343/Gorelik
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> This manuscript discusses double occupancy as a probe for antiferromagnetic correlations in optical lattice experiments. Specifically, it compares predictions for double occupancy and entropy of dynamical mean-field theory (DMFT) and Quantum Monte-Carlo (QMC) calculations.
> Already in Ref. 17, PRL 105, 065301 (2010), written in part by a subset of the current authors, has argued that an upturn of double occupancy at low temperatures is a signature of the onset of antiferromagnetic correlations. This was an important contribution. Later works (Ref.18) have questioned the sharp features observed in DMFT just above the Neel transition. The present manuscript thus compares DMFT to QMC predictions in various dimensions. The conclusion drawn from these comparisons is that DMFT, known to be exact only in infinite dimensions, predicts the double occupancy "more accurately than expected" in D=2 and D=3. Also, it is proposed to use the double occupancy versus entropy curve for thermometry.
>In my view, this paper is rather a "response" to the "comment" in other works (Ref.18) that DMFT predicts unphysically sharp features around T_Neel, and defends DMFT a posteriori via comparison to QMC. This seems to me not of wide enough interest, nor of high enough innovation to merit publication in Physical Review Letters. It was clear before that DMFT will only be exact in infinite dimensions. The fact that it seems to be not too far off the mark in D=2 and D=3 for the double occupancy and entropy is good news but it seems to be accidental. For any new problem tackled by DMFT one will not be sure whether it can be trusted - there is no way to predict the "error bar", i.e. the discrepancy between D = infinity and D=2 and 3. One would, as in this paper, rather trust QMC calculations.

The main result of this paper is not the justification of DMFT for calculating properties of 3D systems, but rather the conclusion of fundamental interest about the universality of the entropy scale for the local antiferromagnetic physics for the whole range of dimensions from d=1 to d=infinity.
The absence of sharp features at DMFT Neel temperature in finite dimensions was already discussed in our original paper [Gorelik et al. PRL 105, 065301 (2010)], so that the discussion of this aspect by [Fuchs et al. PRL 106, 030401 (2011)] and [De Leo et al. PRA 83, 023606 (2011)] brought no any new aspects except for the inability of DCA to enter the AF phase.
The comparison with QMC results is indeed very useful to judge the accuracy of DMFT prediction in finite dimensions. This is especially important, since the simulation of experimentally relevant systems, i.e. trapped atomic clouds with about 10^5 particle, is not possible within QMC due to the large size of the system, and sign problem.

We have rewritten the paper to put forward the interesting physical phenomena, and to reduce technical aspects.

> One comment on the QMC calculations: They are here taken to be exact solutions of the problem. Is this not an overstatement? What are the error bars? If only 6^3 and 8^3 clusters have been calculated, how can one extrapolate to infinite cluster size with confidence? It seems no reliable error bars are available. The authors should comment on this.

The statistical error in D is less then 5x10^(-4), being smaller than 10^(-5) in most cases; Trotter errors have been eliminated by extrapolation; finite size effects are negligible in the strong coupling case. Thus, the QMC results, as presented in the manuscript, could be considered accurate within about half symbol size.
Corresponding discussion of the accuracy of DMFT and QMC results was added to the revised version.

> The proposal of plotting double occupancy versus entropy is valid and interesting. The paper should certainly be published, after slight revisions and commenting on error bars in the QMC calculation, in Physical Review A.

We are grateful to the Referee for acknowledging the quality of our results.

> A small point:
> - TNeel (for example from QMC) should be represented in Fig. 2 with an arrow.

We have adjusted the style of Figure 2 for better readability.

