Phase separation in binary Bose mixtures at finite temperature

We thank the Referee for the positive assessment on the presentation, relevance, and timeliness of our manuscript. In the following and in the revised manuscript we answer the specific points raised by the them, providing the requested clarifications on our findings and on the adopted approach.

The referee writes:

i) “The authors don’t provide much of a physical explanation for their results. For example, the statement « the conditions for phase separation are not modified from the zero temperature case.”

Our response: This statement is based on the results shown in Fig.2 and in Fig.3. In fact, for all values of g_{12}<g, the dependence of the free energy on the polarization p is in very good agreement with the mean-field T=0 prediction for the magnetic susceptibility [see dotted blue line in panels a) and b) of Fig.2 and in Fig.3] and does not show any evidence of the ferromagnetic transition predicted by theories including beyond mean-field effects. Only when g_{12}>g [see panel d) in Fig.2] a minimum in free energy shows up at finite p, meaning that the mixture turns ferromagnetic. This is exactly the behavior expected at T=0, and it is recovered here at a high temperature not so far from the BEC transition point. Our conclusion is that we expect the same to be true also for lower temperatures, where thermal effects not captured by the mean-field description should play a minor role. In this respect one should also notice that higher order interaction effects at T=0 do not change the critical value g_{12}=g for the onset of ferromagnetism (see Ref.[16]). As an additional remark, we point out that our results do not exclude a non trivial interplay between ferromagnetic and critical fluctuations in the close vicinity of the transition point. Such studies would require a much deeper analysis of the shift of the transition point in interacting mixtures, which is clearly beyond the scope of this work. To better clarify the implications drawn from our results we have rewritten the paragraph at the end of the section where we discuss Fig.2 and Fig.3. For convenience, the new paragraph is reported hereafter: “From these results we conclude that, in contrast to HF and Popov predictions, the magnetic susceptibility depends very little on the temperature, and the conditions for phase separation seem to remain the same as at $T=0$. In fact, if $g_{12}<g$, our results indicate that the only thermodynamically stable phase is the paramagnetic state at $p=0$. A ferromagnetic state forms when $g_{12}>g$ and the effect of temperature is to reduce the equilibrium polarization from the $p=1$ value achieved only at zero temperature. This is found at a high temperature not far from the BEC transition point and we expect the same to be true also for lower temperatures, where thermal effects not captured by the mean-field description should play a minor role. In this respect one should also notice that higher order interaction effects at T=0 do not change the critical value g_{12}=g for the onset of ferromagnetism (see Ref.[16]). As an additional remark, we point out that our results do not exclude a non trivial interplay between ferromagnetic and critical fluctuations in the close vicinity of the transition point. To carefully investigate these effects would require a much deeper analysis of the shift of the transition point in interacting mixtures beyond the scope of this work. Furthermore, we expect the simple $T=0$ scenario to hold also at densities lower than $na^3=10^{-4}$. We checked…”

The referee writes:

“The choice of the parameters : N=128 and T=0.794 T_c. Is this choice arbitrary ? N looks small, what happens if one extends it to large values?” 

Our response: Indeed we mostly focus our analysis on the temperature T=0.794 T_c (we also consider different temperatures in the snapshots of particle configurations shown in Fig.4). However, the relevant behavior to establish the magnetic response concerns the polarization p at fixed temperature. The latter should be high enough to emphasize thermal effects, but not to close to the transition point to make sure that the mixture is still in the Bose condensed phase. Concerning the number of particles, we have checked that our results for the free energy do not change significantly by increasing N and can be considered well converged approaching the thermodynamic limit. For clarity we added two comments on the choice of the temperature and on the role of finite-size effects. For convenience, these two comments are reported hereafter: “This choice of parameters and, in particular, the choice of temperature emphasizes thermal effects in HF and Popov theories yielding important corrections to the $T=0$ magnetic susceptibility. We also note that finite-size effects in PIMC simulations of the free energy are negligible if one increases further the total number of particles.”

The referee writes:

iii) “The authors compared their Monte Carlo data for the chemical potential and free energy only with their own HF and Popov theories but they completely ignored the other theories such as the full HFB theory (see e.g. Phys. Rev. A 97, 033627 (2018) and Phys. Rev. A 104, 023310 (2021). The discrepancy between the Monte Carlo simulations and the HF and Popov theories couldn’t due to the missing of the anomalous correlations (pairing) in these perturbative theories?”

Our response: We thank the Referee for pointing out a possible misunderstanding on the content of what we refer to as HF and Popov theories, and for suggesting relevant references. The Popov theory we use here, which is explicitly outlined in Appendix C, consistently accounts for second order effects in the coupling constant. It is also known as second-order Beliaev theory extended to finite temperatures. In particular, at T=0, it provides the correct Lee-Huang-Yang terms in the equation of state for the single component and the similar terms arising from the zero-point motion of density and spin fluctuations in two-component mixtures. In the language of the Referee it accounts properly for both normal and anomalous correlations yielding at finite temperature a description of the thermodynamic behavior which extends to mixtures Popov’s result for the single-component gas (see Ref.[16]). The only difference compared to full HFB theory is that in the dispersion of elementary excitations at finite temperature we use the condensate density as calculated from lowest order theory instead of self-consistently. However, this approach is consistent up to second-order corrections and should be adequate in the dilute regime. We have tried to address this issue adding a clarifying comment at the beginning of Appendix C and making reference to the useful article pointed to by the Referee.  For convenience, the additional comment and references are reported hereafter: “The Hartree-Fock and Popov theories of repulsive binary Bose mixtures at finite temperature are described in details in Refs.~\cite{PhysRevLett.123.075301, PhysRevA.102.063303}. We note that Popov’s theory is also known as the finite temperature extension of Beliaev’s approach and includes the important contribution of anomalous fluctuations to thermodynamic quantities \cite{Phys. Rev. A 97, 033627 (2018); Phys. Rev. A 104, 023310 (2021)}. Here we report…”