Finite temperature spin diffusion in the Hubbard model in the strong coupling limit

Reply to Referee 1

-We added a new figure, namely Fig. 2, which shows the temperature and magnetic field dependence of the spin Drude weight and the spin diffusion constant.

-The Drude weight for the spin transport is actually a simple free fermionic result, multiplied by a factor which depends on the magnetic field. Therefore, we did not add a discussion of the Mazur bounds. The most important result of our paper is in our point of view the spin diffusion constant, for which there have been no exact results yet. Therefore, we did not intend to discuss the multiple approaches for the ballistic part of the spin transport, in those cases when this part is non-zero. We hope the referee can agree with this choice.

-In the abstract we replaced the expression "absence of bias" with "absence of magnetic field".

Reply to Referee 2

-We have added the required citations to eq. (103). Also, we modified the text in the Introduction, where diffusion and GHD are mentioned. Let us add here, that it is not clear to us, what does "GHD" actually encompass. Perhaps those authors who worked on the theory from the beginning have a different view about this, than other researchers. To us the umbrella term "GHD" would include both the thermodynamic form factors, and hydro projections. In the current version we added a mention of both concrete methods to the introduction, whereas keeping the expression "GHD framework". We hope this is acceptable to the referee...

-In the paper we already have the specific components of the Onsager matrix, see eq. (85).
But in principle, this matrix can be computed directly from the current-current correlator.

-We can extract the finite time diffusion constant only numerically. We added a plot about it.

-Infinite temperature means here that the three local states have equal probability. One state is the vacuum, other two states are electron states with spin up and down. Therefore, the density is 2/3. Of course, if we think about the original Hubbard model, this picture applies only if T >> 1 but T << U. Because if the temperature is larger than the original coupling constant U, then the doublon degrees of freedom are also occupied. Right now we did not add this explanation to the text, but if the referee requires, we could add it.

-We added eq. (98) which shows the scattering kernel, derived directly from the Bethe equations.

-Unfortunately, we can not add anything about 1/U corrections. The limits U->infty and t->infty do not commute. This is why at any finite U the diffusion constant is infinite. We already commented about this in the
Conclusions. Unfortunately, we can not add more. The model is singular, and perhaps this is not very physical. Nevertheless, this is still the first quantum spin chain model, where the diffusion constant was calculated, so this gives a justification for the work we did.

-We replaced the title of that Section.