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Effect of nextnearest neighbor hopping on the singleparticle excitations at finite temperature
by Harun Al Rashid and Dheeraj Kumar Singh
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Authors (as registered SciPost users):  Dheeraj Kumar Singh 
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Preprint Link:  scipost_202310_00028v1 (pdf) 
Date submitted:  20231024 05:53 
Submitted by:  Singh, Dheeraj Kumar 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approaches:  Theoretical, Computational 
Abstract
In the halffilled oneorbital Hubbard model on a square lattice, we study the effect of nextnearest neighbor hopping on the singleparticle spectral function at finite temperature using an exactdiagonalization + MonteCarlo based approach to the simulation process. We find that the pseudogaplike dip, existing in the density of states in between the N\'{e}el temperature $T_N$ and a relatively higher temperature $T^*$, is accompanied with a significant asymmetry in the hole and particleexcitation energy along the highsymmetry directions as well as along the normalstate Fermi surface. On moving from ($\pi/2, \pi/2$) toward $(\pi, 0)$ along the normal state Fermi surface, the holeexcitation energy increases, a behavior remarkably similar to what is observed in the $d$wave state and pseudogap phase of high$T_c$ cuprates, whereas the particleexcitation energy decreases. The quasiparticle peak height is the largest near ($\pi/2, \pi/2$) whereas it is the smallest near $(\pi, 0)$. These spectral features survive beyond $T_N$. The temperature window $T_N \lesssim T \lesssim T^*$ shrinks with an increase in the nextnearest neighbor hopping, which indicates that the nextnearest neighbor hopping may not be supportive to the pseudogaplike features.
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Reports on this Submission
Anonymous Report 3 on 20231223 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202310_00028v1, delivered 20231223, doi: 10.21468/SciPost.Report.8335
Report
In the present manuscript, the authors discuss the effect of nextnearest neighbor (NNN) hopping t' on the spectral function of the halffilled singleband Hubbard model on the square lattice at finite temperature, with a special focus devoted to 'pseuodogaplike' features.
Their method of choice is an approach that corresponds to a meanfield treatment of the interaction term, combining exact diagonalization for the electronic part with a Monte Carlo sampling of the magnetic vector fields of the magnetization. At low temperatures, the technique reduces to standard HartreeFock approximation. It notably ignores spatial and thermal fluctuations in the charge channel and also does not include any temporal fluctuations. These limitations restrict its applicability to the halffilled case of the model. The technique has been described in the authors' previous publication, Ref. 34, where they also provide a very similar analysis of the same quantities of interest for the case t'=0.
Overall, the paper is providing some new insights on the modifications of the spectral function of the Hubbard model with respect to t' via this meanfield method. However, the pseudogap phase of the doped and undoped Hubbard model has been studied in detail over the last decades. A detailed comparison to known results for the 'pseudogap' phase at the halffilled tt' Hubbard model as well as on the relation to other meanfield techniques is missing. It is therefore not clear for the readers, where the paper contributes novel findings and where it is simply reproducing results that are already wellknown, even though now accessed with a potentially different numerical technique. Furthermore, some statements in the paper are inprecise or misleading and definitely need to be corrected, see below.
For all these reasons, I am not sure whether the manuscript meets the acceptance criteria of the journal and cannot recommend the publication of the manuscript in its current form in SciPost Physics. Some of the most urgent points that would need revision before an eventual resubmission are listed below.
Requested changes
1) A detailed comparison with the existing literature at halffilling is missing. Only stating that the findings are similar to those of cluster perturbation theory is not sufficient. It is unclear which features are in (qualitative or quantitative) agreement with CPT. Furthermore, there are many techniques which provided results that allow for comparison, some of them going beyond the limitations of the meanfield technique used here, some being at a comparable level. These differences need to be discussed. In particular, the authors might want to compare their results to numerically exact results on the PG phase of the Hubbard model, e.g. using lattice Quantum Monte Carlo (QMC) , determinant Quantum Monte Carlo (detQMC) or Variational quantum Monte Carlo (VMC).
2) Also the technique itself should be discussed in view of other techniques, in particular of meanfield type. How does it differ from standard meanfield techniques, slave bosons, slave spins, Hubbard1 approximation etc.
3) It is unclear why the discussion focusses on the highsymmetry path $(\pi/2,\pi/2)(\pi,0)$. Changing t' results in a change of the dispersion on the noninteracting level, $U=0$, and reshapes the Fermi surface. It is thereby clear that some changes of the spectral function are simply due to the modifications of the dispersion, the shift of vanHove singularities etc. In order to assess the effects of interactions, it would be useful to provide the noninteracting dispersion or the Fermi surface. To make statements on the 'pseudogaplike' features of the spectral function, its behavior should be studied along the Fermi surface with respect to the noninteracting reference for each t'.
4) Referring to the dip in the density of states (DOS) as a 'pseudogaplike feature' can be misleading. The pseudogap is a kdependent phenomenon that does not necessarily need to be related to a dip in the DOS. Such a dip can in principle also stem from a (fully) gapped system at low temperature, whose gap is progressively filled by thermal excitations at higher temperature. The authors' technique includes spatial fluctuations in the spin sector, thereby they should have access to the kdependent selfenergy which can give them a precise (and direct) answer on the existence or absence of a pseudogap.
5) Figure 1: A legend should be added to make clear which quantities are plotted in panels a) and b)  the spin structure factor and the NN spinspin correlator.
6) Figures 4 & 5 seem to show the very same data, plotted in a color map and a line plot respectively. Figure 5 should therefore be removed or replaced by a plot which shows the amplitude of the spectral function along the high symmetry path for the different temperatures.
7) The presentation of Figures 6 & 7 is slightly confusing and needs to be improved, in particular depicting the kpath more clearly. Also, it is not clear why the path $(\pi/2,\pi/2)(\pi,0)$ can be compared in this way as a function of t', which even modifies the dispersion of the noninteracting model.
8) The whole study is done at $U=4t$, which is less than half the bandwidth of the system. Nevertheless, the authors seem to suggest that they can safely interpret the broadening of the spectral function in therms of the Heisenberg spin exchange $J=4t^2/U$. This is problematic for different reasons: i) $U=4t$ is far from the Heisenberg limit; ii) the NNN spin exchange J' is not taken into account, despite the presence of the NNN hopping t'; iii) the broadening of the spectral function should be rather described by (the imaginary part of) the selfenergy, which the authors do not show.
9) Claims on agreement with experiment need to be substantiated. Which cuprates, which measurement techniques and which studies do the authors have in mind? What does it mean to be "in good agreement with experiment"?
10) In the discussion and conclusion, the authors should be careful in assessing the transferability of their results to doped cuprates: 1) Their technique cannot be applied in a straightforward way to doped systems since it neglects several types of fluctuations, which are known to be important in these systems. 2) The pseudogap of doped systems is not necessarily the same pseudogap that the authors study here at halffilling with a meanfield approach that is taylored to capture the physics in the strongcoupling limit.
11) Given the vast literature on the tt' Hubbard model at halffilling, the authors should explain more carefully what they mean when saying that their study fills a 'longstanding gap'.
Anonymous Report 2 on 20231212 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202310_00028v1, delivered 20231212, doi: 10.21468/SciPost.Report.8261
Report
In the present manuscript the authors provide an analysis of the effects induced by a nearestneighbor hopping amplitude on the half filled Hubbard model. In particular, they focus on the spectral properties along the node and antinode in the pseudogap regime. The present studies extends a previous work of the same authors (Ref. 34) to a finite t'. They conclude that larga values of t' are unfavorable to the pseudogap behavior.
The method they used combines several approaches, and involves a series of limitations. Specifically, the method can not be extended to finite doping because the charge degrees of freedom are frozen (both spatial and thermal fluctuations are neglected). Similarly, the treatment of dwave superconducting fluctuations is not included. In the conclusions, the authors state that additional auxiliary fields can be introduced, details are however missing. Given the restricted applicability of the approach, I would have appreciated a paragraph or short section about a concrete route that allows to overcome the above restrictions. I also think the limitations of the method should be mentioned in the methods section and not only in the discussion at the end.
The findings on the t' dependence are interesting and worth to be published, but given the limitations of the approach I would rather recommend SciPost Physics Core as suitable journal, after the points raised have been addressed.
Author: Dheeraj Kumar Singh on 20240118 [id 4261]
(in reply to Report 2 on 20231212)
General comment: In the present manuscript the authors provide an analysis of the effects induced by a nearestneighbor hopping amplitude on the half filled Hubbard model. In particular, they focus on the spectral properties along the node and antinode in the pseudogap regime. The present studies extends a previous work of the same authors (Ref. 34) to a finite $t^\prime$. They conclude that large values of $t^\prime$ are unfavorable to the pseudogap behavior.
Reply: We are thankful to the referee for various constructive comments. In the following, we reply to specific comments by listing them pointwise.
Comment: The method they used combines several approaches, and involves a series of limitations. Specifically, the method can not be extended to finite doping because the charge degrees of freedom are frozen (both spatial and thermal fluctuations are neglected).
Reply: (i) We would like to clarify that the method can be extended to finite doping, however, the effective model should be modified accordingly. An appropriate effective model will be $tJ$ model, which is a derived from the Hubbard model. The $d$wave superconductivity can arise from the nextnearest neighbor attractive interactions present in the $tJ$ model and magnetic order will arise from the nearest neighbor exchange interaction. Alternatively, one can use $tUV$ model as well to investigate the consequence of competing $d$wave superconductivity, where $U$ and $V$ are onsite repulsive and nextnearest neighbor attractive interaction parameters.
(ii) In the current work, we were primarily interested in investigating the role of nextnearest neighbor hopping on the singleparticle spectrum at half filling, because, to the best of our knowledge, this problem has remained largely unexplored due to the challenges arising from (a) small cluster size limiting the momentum resolution and (b) famous sign problem in the implementation of simulations based on quantum Monte Carlo (QMC) or determinant QMC. Notably, these methods go beyond the meanfield approximation, and can incorporate the thermal and spatial fluctuations.
(iii) Furthermore, at half filling, the effect of the thermal fluctuations in the orderparameter field, which corresponds to the magnetic moments in the current work, is expected to be significantly large on the singleparticle spectral function in comparison to the charge fluctuations. This is because a small rotation of magnetic moment does not require as much energy as involved in the charge fluctuations corresponding to the double occupancy of a site, unless the system is doped with the charge carriers.
Comment: Similarly, the treatment of $d$wave superconducting fluctuations is not included. In the conclusions, the authors state that additional auxiliary fields can be introduced, details are however missing.
Reply: As we mentioned earlier that we were primarily focused on the role of nextnearest neighbor hopping on the singleparticle spectrum. It would be of strong interest to know how the presence of two competing order such as magnetic and $d$wave superconductivity affect the spectral features, particularly, the pseudogaplike features. In the discussion, we have a added a brief discussion as to how the additional auxiliary fields can be introduced via $tJ$ model, which is a derived from the Hubbard model for small dopings. The $d$wave superconductivity can arise from the nextnearest neighbor attractive interactions present in the $tJ$ model. Alternatively, one can use $tUV$ model as well to investigate the consequence of competing $d$wave superconductivity, where $U$ and $V$ are onsite repulsive and nextnearest neighbor attractive interaction parameters.
Comment: Given the restricted applicability of the approach, I would have appreciated a paragraph or short section about a concrete route that allows to overcome the above restrictions. I also think the limitations of the method should be mentioned in the methods section and not only in the discussion at the end.
Reply: We have incorporated suggested discussion as well as mentioned the limitations of the method in the method section.
Comment: The findings on the $t^\prime$ dependence are interesting and worth to be published, but given the limitations of the approach I would rather recommend SciPost Physics Core as suitable journal, after the points raised have been addressed.
Reply: Our earlier work was mostly focused on the development and testing of the new method which combined three techniques namely, travelingcluster approximation, parallelization of the update process and twistedboundary conditions. The sole aim of combining these three techniques is the accessibility to a largesystem size critical for a better momentum resolution for the spectral function so that any conclusion on the nature of spectral gap may be free from any finite size effect. Then, we demonstrated this idea with the help of halffilled Hubbard model with only nearestneighbor hopping.
However, the actual correlatedelectron system described by an effective oneband model may involve longrange hopping. The system of high$T_c$ cuprates is one such standard example. Presence of the longrange hopping can have significant impact not only on the phase diagram but also on the spectral properties etc. which deserves a separate treatment. Several studies in the past have attempted to investigate the role of $t^{\prime}$. However, the approaches mainly include HartreeFock approximation based meanfield theory, dynamical meanfield theory (DMFT) etc., which do not incorporate the spatial fluctuations in the orderparameter field. The former does not even take into account the thermal fluctuations in the orderparameter field.
On the other hand, the methods such as QMC, detQMC etc., which go beyond meanfield theories, also use finite cluster size and suffer from famous sign problem in the absence of the particlehole symmetry, \textit{i. e.} when the longrange hopping is considered. Even if doping is introduced, the use of these methods may be restricted to a certain temperature range.
In our simulation, in the absence of particlehole symmetry, the chemical potential may fluctuate with temperature, therefore, the simulation requires to check the chemical potential and update the same in accordance with half filling at intermediate steps. This enhances the computational cost in the simulation process. Another factor which is responsible for raising the computational cost is the frustration introduced by the nextnearest neighbor hopping in the system, which raises the equiliberation time.
To the best of our knowledge, the current work may be the first study, which has examined the temperature dependent role of nextnearest neighbor hopping in the halffilled Hubbard model on the spectral features with a momentum resolution which is almost free from finitesize effect. For all the above reasons, we believe that the manuscript may deserve the visibility associated with SciPost Physics.
Anonymous on 20240118 [id 4263]
(in reply to Dheeraj Kumar Singh on 20240118 [id 4261])With these explanations, I now more clearly understand the points made by the authors and am convinced of the relevance of the present contribution. The manuscript appears clearly improved by the changes and the additional discussions. I recommend publication.
Anonymous Report 1 on 20231116 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202310_00028v1, delivered 20231116, doi: 10.21468/SciPost.Report.8131
Report
In this theoretical work the authors analyze the singleorbital Hubbard model at halffilling using a theoretical scheme exploiting exactdiagonalization and MonteCarlo. Their focus is to investigate the role of nearestneighbor (nn) hopping on the single particle spectral function at different temperature. The analysis reveals a strong asymmetry of the hole/particleexitation energy along the highsymmetry direction. The coherence features present a strong dependence from the momentum and appear to be compatible with the pseudogap phase of cuprates. The range of temperature in which those are observed within the model shrinks with increasing the nn hopping, thus nn may not be supportive to these pseudogaplike features.
The results are discussed at halffilling and considering only the magnetic fluctuating channel. The conclusion of the work well discuss what to expect by doping the system and point out the importance of taking into account multiple interacting channel in order to explore the role of the nn hopping in a more realistic model.
The topic of the research is of high interest for the community working on correlated systems and superconductivity. It provides interesting insights and hints that could be used in the future to further explore the role of nn hopping e.g. in doped systems, in the presence of other interacting channels and so on. The quality of the research is very high and the presentation of the results is very good. The technical aspects of the computation are explicitly discussed highlighting the advantage of the choose procedure and the approximations involved in the calculation.
I find the paper fulfilling the criteria for publication, however I invite the authors to consider a few comments before resubmitting.
Requested changes
1) The introduction is currently organized via a back and forth from theory and experiments to discuss the two main aspects that this work wants to address, i.e. the asymmetry of the phase diagram and the temperature and momentumdependence of the quasiparticle excitations.
I think it would be helpful to separate the two so that the reader will have first a clear idea of the experimental aspects that will be considered followed by an overview of the theoretical state of the art in this context. I invite the authors to consider such a reorganization of the introduction.
2) The description of Fig. 7 is misleading. In the text the authors claim that the raising of energy moving from (pi/2, pi/2) to (pi, 0) is almost linear, however the plots show a more complex behavior (those curve are not even close straight lines). The only temperature at which the energy curves are lines is at T~ TN were the grows is also quite small. I think that what describe the plot saying that the increasing is monotonic would generate less confusion.
3) Again in Fig 7. Can the author find a xlabel easier to understand looking at the plot withour reading the caption? One option would be putting explicitly (pi/2, pi/2) and (pi, 0) as stating and ending point.
Author: Dheeraj Kumar Singh on 20240118 [id 4260]
(in reply to Report 1 on 20231116)
We are thankful to the referee for the encouraging comments on the current work. We also appreciate specific comments, that have been helpful in improving the manuscript further.
Comment: The introduction is currently organized via a back and forth from theory and experiments to discuss the two main aspects that this work wants to address, i.e. the asymmetry of the phase diagram and the temperature and momentumdependence of the quasiparticle excitations. I think it would be helpful to separate the two so that the reader will have first a clear idea of the experimental aspects that will be considered followed by an overview of the theoretical state of the art in this context. I invite the authors to consider such a reorganization of the introduction.
Reply: We have now restructured the introduction so that the parts giving the theoretical and experimental backgrounds are now separate, begining with the experimental aspects first, and then followed by an overview of the theoretical status.
Comment: The description of Fig. 7 is misleading. In the text the authors claim that the raising of energy moving from $(\pi/2,\pi/2)$ to $(\pi,0)$ is almost linear, however the plots show a more complex behavior (those curve are not even close straight lines). The only temperature at which the energy curves are lines is at $T~ T_N$ were the grows is also quite small. I think that what describe the plot saying that the increasing is monotonic would generate less confusion.
Reply: We have modified the corresponding sentence in the manuscript so that there is no confusion. Now, instead, it is mentioned that the holeexcitation energy increases monotonically upon moving from $(\pi/2,\pi/2)$ toward $(\pi,0)$ and it varies almost linearly in the vicinity of $T_N$.
Comment: Again in Fig 7. Can the author find a xlabel easier to understand looking at the plot without reading the caption? One option would be putting explicitly $(\pi/2,\pi/2)$ and $(\pi,0)$ as stating and ending point.} \\
Reply: We have now used symbols, which are frequently employed to represent the highsymmetry points in the Brillouin zone. In particular, it may be noted that $(\pi/2,\pi/2)$ is also one of the high symmetry points in the reduced or magnetic Brillouin zone. Thus, $(\pi/2,\pi/2)$ and $(\pi,0)$ are represented by X$^{\prime}$ and M$^{\prime}$, respectively. We hope that with this change, the $x$label will now be easier to understand.
Author: Dheeraj Kumar Singh on 20240118 [id 4262]
(in reply to Report 3 on 20231223)We are thankful to the referee for various constructive comments. Below, we respond to those comments pointwise while the corresponding changes in the revised manuscript is also mentioned.
Comment: A detailed comparison with the existing literature at halffilling is missing. Only stating that the findings are similar to those of cluster perturbation theory is not sufficient. It is unclear which features are in (qualitative or quantitative) agreement with CPT. Furthermore, there are many techniques which provided results that allow for comparison, some of them going beyond the limitations of the meanfield technique used here, some being at a comparable level. These differences need to be discussed. In particular, the authors might want to compare their results to numerically exact results on the PG phase of the Hubbard model, e.g. using lattice Quantum Monte Carlo (QMC), determinant Quantum Monte Carlo (detQMC) or Variational quantum Monte Carlo (VMC).
Reply: We would like to clarify that the exactdiagonalization + MonteCarlo (ED+MC) based method used in the current work incorporates all types of thermal and spatial fluctuations in the orderparameter fields associated with the antiferromagnetic order. This is in contrast with the standard meanfield theories which ignore both thermal and spatial quantum fluctuations. Thus the magnetic moments, unlike the static meanfield theory, does not melt in the method used here when the system approaches $T_N$. Similarly, in the HartreeFock approximation based approaches, the spectral function can develop a gap only below $T_N$. On the contrary, we find that the gap persists beyond $T_N$, which we refer to pseudogaplike gap. Another advantage of the current approach is that as the temperature increases, thermal fluctuations start dominating over the quantum fluctuations, therefore, the accuracy of ED + MC increases in capturing essential physics.
A variety of techniques have been utilized which goes beyond standard meanfield theories, which mainly include densitymatrix renormalization group (DMRG), dynamicalmeanfield approximation (DMFT), cluster DMFT, QMC, detQMC, VMC, etc. The applicability of DMRG has largely been restricted to the quasione dimensional systems. Although, the DMFTbased methods do capture the Mott transition, the selfenergy correlation incorporated is independent of momentum rendering it not suitable for the study of momentumdependent spectral features. The QMC and detQMC were used extensively to study finitesize system with the difficulty that, at low temperatures, the correlations length is greater than the lattice size. Therefore, the correlations are overestimated for smaller clusters because they are artificially closer to criticality than a system in the thermodynamic limit. This may result in the disentanglement of the MI and AFM transitions. Another important consequence of the finitesize effect is a good momentum resolution for the singleparticle spectral function, which has remained challenging till now.
Another important issue with QMC and DetQMC is the sign problem, which severly restricts its range of applicability especially when the hopping beyond nearestneighbors is considered, which introduces particlehole asymmetry. Even when doping away from halffilling is considered, the DetQMC can used only in a limited temperature range.
For these reasons, despite a significant volume of work at half filling, most of them were restricted to only nearestneighbor hopping, particularly, the studies based QMC, detQMC, etc. The effect of $t^{\prime} \ne 0$ has been mainly studied using clusterperturbation theory (CPT). CPT combines the solutions of small individual clusters of an infinite lattice system with Block theory of conventional band description to provide an approximation for the Green's function in the thermodynamic limit. In particular, at intermediate interaction strength, it may be challenging to make accurate prediction for the Hubbard gap mainly constrained by finitesize induced level splitting. We have incorporated the summary of all the points discussed above in the revised manuscript.
Comment:Also the technique itself should be discussed in view of other techniques, in particular of meanfield type. How does it differ from standard meanfield techniques, slave bosons, slave spins, Hubbard1 approximation etc.
Reply: Slavebosons or spins meanfield theoretic approaches have been quite successful in studying the Mott transition. Recent developments also show their uses in studying the magnetically ordered phases. However, again inclusion of magnetic order ignores the spatial and thermal fluctuation, which is contrastingly different from the method we adopt in this work.
Comment: It is unclear why the discussion focuses on the highsymmetry path $(\pi/2,\pi/2)(\pi,0)$. Changing $t^\prime$ results in a change of the dispersion on the noninteracting level, $U=0$, and reshapes the Fermi surface. It is thereby clear that some changes of the spectral function are simply due to the modifications of the dispersion, the shift of vanHove singularities etc. In order to assess the effects of interactions, it would be useful to provide the noninteracting dispersion or the Fermi surface. To make statements on the 'pseudogaplike' features of the spectral function, its behavior should be studied along the Fermi surface with respect to the noninteracting reference for each $t^\prime$.
Reply: We were primarily focused on how the gap opens along the Fermi surface. While the Fig. 4 and 5 showed the dispersion with quasiparticle momentumresolved amplitude, Fig. 6 and 7 showed how the gap evolves along the Fermi surface. In Fig. 7, there is typo, the highsymmetry direction $(\pi/2,\pi/2)(\pi,0)$ should be replaced with Fermi surface along $(\pi/2,\pi/2)(\pi,0)$. Along the Fermi surface, gapless excitation is present, therefore, we did not plot it. We have corrected the typo, that is, replaced the phrase ``highsymmetry direction''
by ``Fermi surface $(\pi/2,\pi/2) \rightarrow (\pi,0)$''.
Comment: Referring to the dip in the density of states (DOS) as a 'pseudogaplike feature' can be misleading. The pseudogap is a $k$dependent phenomenon that does not necessarily need to be related to a dip in the DOS. Such a dip can in principle also stem from a (fully) gapped system at low temperature, whose gap is progressively filled by thermal excitations at higher temperature. The authors' technique includes spatial fluctuations in the spin sector, thereby they should have access to the $k$dependent selfenergy which can give them a precise (and direct) answer on the existence or absence of a pseudogap.
Reply: In using the phrase 'pseudogaplike feature' for the current context, we have followed the convention used for the holedoped cuprates. In other words, with rising temperature, the gap should be filled up. Conventionally, it is expected that the gap should disappear as the temperatures increases up to $T_N$. As seen from Fig. 2, the gap continues to persist despite the loss of longrange order, which is also reflected in the momentumresolved spectral function. In such a scenario, we needed a terminology to describe this phenomenon. For this reason, we have used the phrase.
Comment:Figure 1: A legend should be added to make clear which quantities are plotted in panels a) and b)  the spin structure factor and the NN spinspin correlator.
Reply: In the revised manuscript, we have added the legends separately to the Fig. 1 (a) and (b) to avoid any confusion.
Comment: Figures 4 \& 5 seem to show the very same data, plotted in a color map and a line plot respectively. Figure 5 should therefore be removed or replaced by a plot which shows the amplitude of the spectral function along the high symmetry path for the different temperatures.
Reply: We have removed Fig. 5 and included a Figure showing the amplitude of the spectral function along the high symmetry path at different temperatures.
Comment:The presentation of Figures 6 \& 7 is slightly confusing and needs to be improved, in particular depicting the $k$path more clearly. Also, it is not clear why the path $(\pi/2,\pi/2)(\pi,0)$ can be compared in this way as a function of $t^\prime$, which even modifies the dispersion of the noninteracting model.
Reply: The confusion here arises because of a typo. Actually, the ${\bf k}$ path in both the figures are along the Fermi surface along the path $(\pi/2,\pi/2)(\pi,0)$ instead along the highsymmetry direction along $(\pi/2,\pi/2)(\pi,0)$. We have modified the caption to remove this confusion.
Comment: The whole study is done at $U=4t$, which is less than half the bandwidth of the system. Nevertheless, the authors seem to suggest that they can safely interpret the broadening of the spectral function in therms of the Heisenberg spin exchange $J = 4t^2\/U$. This is problematic for different reasons: i) $U=4t$ is far from the Heisenberg limit; ii) the NNN spin exchange $J^\prime$ is not taken into account, despite the presence of the NNN hopping $t^\prime$; iii) the broadening of the spectral function should be rather described by (the imaginary part of) the selfenergy, which the authors do not show.
Reply: It is true that $U=4t$ is far from the strong coupling limit where the Hubbard model can be mapped to the Heisenberg model. Here, we would like to clarify that though the results are presented in the manuscript only for $U = 4t$, we did check the calculations for higher value of $U$ also and found that the thermal broadening increases with increasing $U$. For this reason, we wanted to comment as to what is expected when $U$ is increased. When incorporated NNN spin exchange $J^\prime$ because of NNN hopping $t^\prime$, the broadening will be enhanced further because of the frustration introduced.
Comment: Claims on agreement with experiment need to be substantiated. Which cuprates, which measurement techniques and which studies do the authors have in mind? What does it mean to be "in good agreement with experiment"?
Reply: In the current work, we are primarily interested in the effect of nextnearest neighbor hopping on the singleparticle excitation of halffilled Hubbard model. Therefore, the phrase "in good agreement with experiment" was used in reference to the ARPES measurements carried out in the undoped cuprates, particularly, the momentumdependent singleparticle gap sructures (PRL 74, 964 (1995), PRL 80, 4245 (1998), PRB 70, 092503
(2004)).
Comment: In the discussion and conclusion, the authors should be careful in assessing the transferability of their results to doped cuprates: 1) Their technique cannot be applied in a straightforward way to doped systems since it neglects several types of fluctuations, which are known to be important in these systems. 2) The pseudogap of doped systems is not necessarily the same pseudogap that the authors study here at halffilling with a meanfield approach that is tailored to capture the physics in the strongcoupling limit.
Reply: We agree that our results are not transferable to the doped cuprates because then several types of orderparameter fields will come into picture and the $d$wave superconducting order parameter is prominent amongst them. It is indeed true that the pseudogap feature that we discuss may be entirely different from the one arising as a result of multiple competing orders in the doped cuprates. Precisely, for this reason, we have used the phrase ``pseudogaplike'' instead of psuedogap at various points in the manuscript.
Comment: Given the vast literature on the $tt^\prime$ Hubbard model at halffilling, the authors should explain more carefully what they mean when saying that their study fills a 'longstanding gap'.
Reply: As discussed in the reply to earlier comments, to the best of our understanding, most of the earlier studies, which go beyond meanfield theory, have focused on the halffilled Hubbard model without nextnearest neighbor hopping mainly because of particlehole asymmetry induced sign problem. Furthermore, CDMFT or CPT may provide a picture corresponding only to small clusters, thus suffering from finitesize induced level splitting. On the other hand, the method that we have used for the simulation, beside being free from sign problem, has the advantage of accessing a largesized system, thus able to obtain a momentumresolution never obtained before. A very good momentum resolution, which is free from finitesize effect, is absolutely necessary to conclusively establish the existence of small gap as found in the pseudogap or pseudogaplike phases along the Fermi surface. It is in this respect, we used the phrase 'longstanding gap'. We have modified the phrase in the revised manuscript to avoid any emphasis on the ``longstanding gap''.