Phase separation in binary Bose mixtures at finite temperature

We thank the Referee for the positive evaluation of our work and for the useful comments which help us improve the manuscript. We address below the points raised by the Referee, listing the corresponding changes made to the manuscript.

1. We agree with the referee that finite size effects are present. However, while they play a substantial role in the computation of correlation functions and susceptibilities, thermodynamic quantities such as chemical potential and free energy are less affected by them. This is especially true when looking at free energy differences. As mentioned in the text (pag. 6), we have checked that finite size effects are very small, and the overall picture does not change at larger sizes (chemical potential and free energy data have been checked against systems with double the size reported in the manuscript). For the computation of the thermodynamic properties of balanced mixtures we have instead performed the extrapolation to the thermodynamic limit considering sizes up to 512 particles. We have also looked at the particle positions snapshots for large systems (N=8000).

2. The gray vertical lines represent the critical polarizations for the ideal gas. The effect of interactions on the critical temperature is very small (see ref. [26]), therefore the BEC transition for the minority component is well approximated by the non-interacting critical densities that give the vertical lines in fig.1. The Monte Carlo results across the transition do appear smoother than what predicted by perturbative theories, this also happens for the single component case (see fig.9 in Appendix A.2), where the Monte Carlo data reproduces the results from the universal relations of ref.[26]. However we do not have the sufficient resolution around the transition to investigate the presence of a cusp.

3. We have changed the following sentences, softening the claims concerning the ferromagnetic transition as suggested by the Referee.

Third sentence of the abstract: "Results on the magnetic susceptibility show ..." has been changed to "Results on the magnetic susceptibility suggest ..."

Second sentence of the conclusions: "We can rule out a ferromagnetic transition predicted to occur at finite temperature by perturbative approaches and we find good agreement with the magnetic susceptibility from simple mean-field theory at zero temperature." has been changed to "For the values of the parameters considered in the simulations we do not find the ferromagnetic transition predicted to occur at finite temperature by perturbative approaches and we find good agreement with the magnetic susceptibility from simple mean-field theory at zero temperature. We further argue that a similar conclusion is expected to hold for lower values of the gas parameter."

1. Popov theory predicts a minimum in the free energy with a depth that scales as g^{3/2} as the gas parameter na^3 goes to zero. Moreover, this minimum in the free energy is shifted at higher temperatures when the gas parameter is reduced, suggesting that even in this regime critical fluctuations would invalidate the results from HF and Popov theories. We have improved the relevant text as follows:

Last sentence before Section 3.1: "We checked numerically that for vanishing gas parameter the free energy difference between the $p=0$ state and the stable minimum at finite $p$ predicted by Popov theory is suppressed as $g^{3/2}$ and furthermore the minimum is shifted towards higher temperatures. As a consequence, we expect critical fluctuations to play a major role in the magnetic response of the mixture also in the regime of extremely low densities." has been changed to "Numerical checks show that for vanishing gas parameter the free energy difference between the $p=0$ state and the stable minimum at finite $p$ predicted by Popov theory is suppressed as $g^{3/2}$ and furthermore the minimum is shifted towards higher temperatures occurring closer to the transition point. As a consequence, we expect critical fluctuations to play a major role in the magnetic response of the mixture also in the regime of extremely low densities, thereby invalidating the predictions of Popov theory."

1) Third sentence of the abstract:
"Results on the magnetic susceptibility show ..." has been changed to
"Results on the magnetic susceptibility suggest ..."

2) Second sentence of the conclusions:
"We can rule out a ferromagnetic transition predicted to occur at finite temperature by perturbative approaches and we find good agreement with the magnetic susceptibility from simple mean-field theory at zero temperature." has been changed to
"For the values of the parameters considered in the simulations we do not find the ferromagnetic transition predicted to occur at finite temperature by perturbative approaches and we find good agreement with the magnetic susceptibility from simple mean-field theory at zero temperature. We further argue that a similar conclusion is expected to hold for lower values of the gas parameter."

3) Last sentence before Section 3.1:
"We checked numerically that for vanishing gas parameter the free energy difference between the $p=0$ state and the stable minimum at finite $p$ predicted by Popov theory is suppressed as $g^{3/2}$ and furthermore the minimum is shifted towards higher temperatures. As a consequence, we expect critical fluctuations to play a major role in the magnetic response of the mixture also in the regime of extremely low densities." has been changed to
"Numerical checks show that for vanishing gas parameter the free energy difference between the $p=0$ state and the stable minimum at finite $p$ predicted by Popov theory is suppressed as $g^{3/2}$ and furthermore the minimum is shifted towards higher temperatures occurring closer to the transition point. As a consequence, we expect critical fluctuations to play a major role in the magnetic response of the mixture also in the regime of extremely low densities, thereby invalidating the predictions of Popov theory."