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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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Submission summary

Authors (as registered SciPost users): Jordi Boronat · Stefano Giorgini · Sebastiano Pilati · Gabriele Spada
Submission information
Preprint Link: scipost_202302_00011v2  (pdf)
Date submitted: 2023-06-23 15:33
Submitted by: Pilati, Sebastiano
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational


We investigate the magnetic behavior of finite-temperature repulsive two-component Bose mixtures by means of exact path-integral Monte-Carlo 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.

Current status:
Has been resubmitted

Author comments upon resubmission

We thank the First Referee for their essentially positive assessment on the relevance and validity of our manuscript. In the revised version and in the reply to the Referee, we address the three comments raised by them.

List of changes

To address the first comment made by the First Referee, we have rewritten the paragraph at the end of the section where we discuss Fig.2 and Fig.3., as follows:

> “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…”

To address the second comment raised by the First Referee, we have included the following comment:

> “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.”

To address the third comment raised by the First Referee, we included the following statement with the related additional references:

> “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…”

Reports on this Submission

Anonymous Report 2 on 2023-8-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202302_00011v2, delivered 2023-08-28, doi: 10.21468/SciPost.Report.7729


1-Long standing questions are probed, such as the validity of certain perturbative theories, and the question of whether the ferromagnetic transition can be traversed by varying temperature.

2- the results provide an important and interesting contribution to better understanding the role of critical fluctuations on the properties of bose-bose mixtures

3-The manuscript is well written.


1- focus is given to only a limited range of temperature and interaction strengths

2-The error bars on the Monte Carlo results are relatively large


This manuscript investigates phase separation and thermodynamic properties of Bose mixtures using path integral Monte Carlo simulations. The focus is on the effects of temperature, and to what extent Hartree Fock and Popov theories are inaccurate in the vicinity of the BEC phase transition.

By calculating the free energy and chemical potentials they find ferromagnetic phases for which the minority component is in the normal phase, suggesting that in this regime, lowering the temperature would transition to the zero temperature prediction for a paramagnetic phase. A main result is that in this regime the Monte Carlo theory does not predict this same temperature-driven ferromagnetic transition. Also, near the transition there are significant shortcomings of the HF and Popov theories.

These results are interesting and provide an important piece for a long standing puzzle, which has continued to be of interest over the years. For this reason I would be in favor of supporting publication if the authors are able to address my technical concerns/questions below.

Requested changes

1-Thermal fluctuations are expected to diverge leading up to the ferromagnetic transition, due to the diverging susceptibility. Can the authors justify and comment on why the small size of their systems does not qualitatively affect their Monte Carlo results by artificially suppressing these long-wavelength fluctuations due to the small size of the numerical box?

2-Figure 1 shows the predicted critical polarisation as vertical lines. This closely matches the positions of the cusps for the HF and Popov theories, signalling the BEC transition for the minority component. However, the Monte Carlo data in this figure looks smooth. Can the authors comment whether this is also approximately the transition point for the Monte Carlo simulations, and what evidence do they have for this?

3-Some of the statements made seem a little strong to me, for example in the conclusion "We can rule out a ferromagnetic transition predicted to occur at finite temperature by perturbative approaches" or in the abstract "Results on the magnetic susceptibility show that the conditions for phase separation are not modified from the zero temperature case.". While I do agree that the results presented show clear differences between the theories, and are consistent with these statements, I am not convinced that they have definitely proven them. I say this partly because only a perfectly balanced mixture is considered, with a focus mostly on a single temperature, and some of the error bars on the Monte Carlo are quite large.

4-a little before section 3.1 begins there is the sentence "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.". Could the authors clarify what is meant, for example I do not understand which minimum is shifted to higher temperatures.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2023-7-4 (Invited Report)


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.

  • validity: good
  • significance: good
  • originality: high
  • clarity: good
  • formatting: excellent
  • grammar: excellent

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