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Phase separation in binary Bose mixtures at finite temperature
by Gabriele Spada, Luca Parisi, Gerard Pascual, Nicholas G. Parker, Thomas P. Billam, Sebastiano Pilati, Jordi Boronat, Stefano Giorgini
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Authors (as registered SciPost users):  Jordi Boronat · Stefano Giorgini · Sebastiano Pilati · Gabriele Spada 
Submission information  

Preprint Link:  scipost_202302_00011v1 (pdf) 
Date submitted:  20230207 12:27 
Submitted by:  Spada, Gabriele 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approaches:  Theoretical, Computational 
Abstract
We investigate the magnetic behavior of finitetemperature repulsive twocomponent Bose mixtures by means of exact pathintegral MonteCarlo simulations. Novel algorithms are implemented for the free energy and the chemical potential of the two components. Results on the magnetic susceptibility show that the conditions for phase separation are not modified from the zero temperature case. This contradicts previous predictions based on approximate theories. We also determine the temperature dependence of the chemical potential and the contact parameters for experimentally relevant balanced mixtures.
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Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 202351 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202302_00011v1, delivered 20230501, doi: 10.21468/SciPost.Report.7124
Report
In their manuscript, the authors study the magnetic behavior of finitetemperature repulsive twocomponent Bose mixtures using exact pathintegral MonteCarlo simulations. The authors implement a new algorithms for the free energy, the chemical potential, and the magnetic susceptibility of the two components. The main result is that conditions for phase separation remain unchanged from the zero temperature case. The calculation of the chemical potential and contact parameters as a function of temperature, show their deviations in the critical region from the results of perturbative methods.
Overall, the manuscript is well presented, and can be considered as a timely contribution to an interesting topic. However there are some mysterious points that should be addressed more carefully.
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 ». 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?
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? Effects of thermal fluctuations on the phase separation in Bose mixtures has been also discussed in Phys. Rev. A 97, 033627 (2018). The above references should be cited and commented on.
In conclusion, before giving my definitive approval, I recommend that the authors provide more details into the physical reasons behind the obtained results. I think this would be of great benefit not only to the manuscript and also to the reader.
Author: Sebastiano Pilati on 20230622 [id 3753]
(in reply to Report 1 on 20230501)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:
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 meanfield 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 meanfield 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 meanfield 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 meanfield 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:
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 finitesize 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 finitesize effects in PIMC simulations of the free energy are negligible if one increases further the total number of particles.”
The referee writes:
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 secondorder Beliaev theory extended to finite temperatures. In particular, at T=0, it provides the correct LeeHuangYang terms in the equation of state for the single component and the similar terms arising from the zeropoint motion of density and spin fluctuations in twocomponent 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 singlecomponent 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 selfconsistently. However, this approach is consistent up to secondorder 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 HartreeFock 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…”
Anonymous on 20230704 [id 3778]
(in reply to Sebastiano Pilati on 20230622 [id 3753])In this revised version, the authors have improved the manuscript with
respect to the previously submitted version. They answer to almost of my comments. I therefore recommend acceptance of this manuscript for publication in SciPost.