Effect of next-nearest neighbor hopping on the single-particle excitations at finite temperature

Reply to the comments of Referees
Comments of referee 1 (anonymous report 1)

Comment: After considering the revised version of the manuscript and the reply from the authors to the first round of report I can recommend the article for publication.
As I already discussed in my previous report, the manuscript 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 results of the analysis are discussed in a convincing way. Technical aspects of the computation are explicitly discussed highlighting the advantage of the procedure used and the approximations involved in the calculation.
The current version of the manuscript presents a more logic introduction and better explain some aspects of the results that were unclear in the previous version of the manuscript

Reply: We are thankful to the referee for several comments in the previous round of refereeing, which led to us to clarify many points by adding to the discussion part. This has improved the manuscript significangly and enhanced its readability.

Comments of referee 2 (anonymous report 2)

Comment: In their responses, the authors addressed all the points raised by the reviewers, almost all of them satisfactorily. In the revised version, most of the points have been taken into account and lead to changes that have improved the manuscript.
Nevertheless, I would have liked to have seen a representation of the self-energy that would have made it possible to compare the nature of the approximations of the authors' technique with other approaches based on Green's functions. Similarly, I still feel that plotting the Fermi surfaces of the non-interacting and interacting systems for different values of t' would have improved readability and made reading Figures 6 and 7 easier. Since the manuscript has been much improved, including the correction of some misleading passages, adding a discussion of the limitations of the technique and providing a more complete reference to existing techniques, I recommend the article for publication in SciPost Physics.

Reply: We express our thankfulness to the referee for several critical comments in the previous round of refereeing for the improvement of the manuscript. Accordingy, the revised version was significantly improved.

In the version, we are about to submit, we have incorporated the self energy into the Fig. 4 while the Fermi surface for interacting and non-interacting system is presented as Fig. 6. The real part of the self energy $\Sigma ({\bf k}, \omega)$ is plotted along the high-symmetry direction. The Fermi surface for the paramagnetic state with pseudogap-like feature has been plotted with the help of $A({\bf k}, 0)$ for temperature $T \ge T_N$ as it does not exist below $T_N$. A significant broadening of $A({\bf k}, 0)$ can be seen because of large thermal/spatial fluctuations in the order parameter fields at higher temperatures. It may be noted the Fermi surface is obtained with the help of $A({\bf k}, 0)$ with quasiparticle excitation energy being zero. On the other hand, a more general quantity such as $A({\bf k}, \omega)$ discussed in Fig. 6 and Fig. 7 of the previous version of the manuscript provides more detailed spectral features, especially the gap structures, of the single-particle excitation. Therefore, the consequences of variation of $t^{\prime}$ is mostly captured in Fig. 6 and Fig. 7 (Fig. 7 and Fig. 8 of the version to be submitted). A change of $t^{\prime}$ on the thermally broadened $A({\bf k}, 0)$ is expected to modify the Fermi surface as in the non-interacting case while retaining the interaction intruduced modification as in the case of $t^{\prime} = 0.3$. For this reason, we have restricted ourselves to the case of $t^{\prime} = 0.3$ and shown the temperature dependence $A({\bf k}, 0)$ instead.