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Beyond universality in repulsive SU(N) Fermi gases
by Jordi Pera, Joaquim Casulleras, Jordi Boronat
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Jordi Boronat |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2206.06932v2 (pdf) |
| Date submitted: | 2023-06-01 08:29 |
| Submitted by: | Boronat, Jordi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Itinerant ferromagnetism in dilute Fermi gases is predicted to emerge at values of the gas parameter where second-order perturbation theory is not accurate enough to properly describe the system. We have revisited perturbation theory for SU(N) fermions and derived its generalization up to third order both in terms of the gas parameter and the polarization. Our results agree satisfactorily with quantum Monte Carlo results for hard-sphere and soft-sphere potentials for $S = 1/2$. Although the nature of the phase transition depends on the interaction potential, we find that for a hard-sphere potential a phase transition is guaranteed to occur. While for $S= 1/2$ we observe a quasi-continuous transition, for spins $3/2$ and $5/2$, a first-order phase transition is found. For larger spins, a double transition (combination of continuous and discontinuous) occurs. The critical density reduces drastically when the spin increases, making the phase transition more accessible to experiments with ultracold dilute Fermi gases. Estimations for Fermi gases of Yb and Sr with spin $5/2$ and $9/2$, respectively, are reported.
Current status:
Reports on this Submission
Strengths
1) The manuscript addresses a central problem in condensed matter physics, namely, the equation of state of interacting Fermi gases, extending the body of knowledge by including third-order perturbative corrections for general spin values and arbitrary polarization.
2) The obtained results are compared with previous perturbative theories and with quantum Monte Carlo simulations. This provides the community with a sound benchmark for many-body theories.
3) The manuscript investigates the effects of different spin values on the ferromagnetic transition, finding interesting qualitative changes in the type of transition, including quasi-continuous, partial discontinuous transitions, and truncated continuous transitions.
4) The above finding are relevant for on-going cold-atom experiment performed with Yb or Sr atoms.
5) The results are well presented, in a format that favors future comparisons with other theories.
Weaknesses
1) The role of intra-species interaction is not fully discussed.
2) Some of the terms in the perturbative expansion had to be computed numerically, but the numerical accuracy is perfectly adequate.
Report
The authors investigate the zero-temperature equation of state of interacting Fermi gases using perturbation theories. They extend the body of knowledge by including third-order corrections for different spin values and arbitrary polarization. The reported equation of state is not universal in term of the s-wave scattering length, and it also depends on s-wave effective range and p-wave scattering length. The transition to itinerant ferromagnetism is investigated, analyzing the role of the spin value and of the scattering parameters. Interestingly, different types of transitions are found for increasing spin, from quasi-continuous, to partial discontinuous transitions, to truncated continuous transitions. Comparisons with variational quantum Monte Carlo data are discussed, showing quite good agreement.
Computing the equation of state of interacting Fermi gases has been and still is a central problem in many-body theory. This manuscript presents novel sound results which significant extend the body of knowledge in the field. It also discusses novel phase transitions which could be observed in on-going cold-atom experiments. Therefore, I find that the manuscript is suitable for publication in SciPost Physics. Below, I report a few comments and suggestion to be considered by the authors before publication.
Comments/suggestions:
-) The definition of quasi-continuous transition could be stated more explicitly. For example, do the author refer to a very sharp continuous transition, or to a discontinuous transition with a tiny jump in the order parameter? While the two cases might not be distinguishable numerically and in experiments, it would be useful to know if the author refer to one of the two cases only, or if the two cases cannot be disherned due to grid spacings of the independent variable or due to the numerical integration.
-) With the scale adopted in figure 6, the susceptibility data are not well visible. Maybe a semi-log scale would help. Furthermore, the authors mention that the finite value at criticality is due to the numerical integration. Did the authors verify that the critical value increases when improving the numerical accuracy?
-) Some integrals in the third-order terms are numerically computed. In the main text, the authors mention the use of an "accurate adaptative Monte Carlo integration", making reference to the Appendix. However, I do not find a clear explanation of what is meant for "adaptive".
-) It is worth mentioning that high-order perturbative results, but for spin balanced Fermi gases only, have been reported also in Physical Review C 104, 014003 (2021).
-) The authors do not explicitly discuss the results including also intra-species interactions. However, their formalism does in fact lead to terms corresponding to intra-species interactions. Would it be possible to show, e.g. in Figure 1, the equation of state without excluding intra-species interactions? Notice that quantum Monte Carlo data including also such interactions have been reported in Phys. Rev. A 107, 053305 (2023). See also , Phys. Rev. Lett. 123, 070404 (2019) for perturbative results including intra-species interactions in a single component gas.
Requested changes
1) Improve the definition of quasi-continuous transitions.
2) Improve visualization of susceptibility data in Fig. 6.
3) Better discuss the divergence of the susceptibility at criticality.
Author: Jordi Boronat on 2023-10-20 [id 4049]
(in reply to Report 1 on 2023-08-30)We kindly thank the Referee for his/her positive comments and for finding our manuscript appropriate for SciPost Phys. In the following, we address the comments and suggestions pointed out by the Referee and the changes introduced in the manuscript accordingly.
Concern 1: The definition of quasi-continuous transition could be stated more explicitly. For example, do the author refer to a very sharp continuous transition, or to a discontinuous transition with a tiny jump in the order parameter? While the two cases might not be distinguishable numerically and in experiments, it would be useful to know if the author refer to one of the two cases only, or if the two cases cannot be disherned due to grid spacings of the independent variable or due to the numerical integration.
A: We mean the second case, that is, a discontinuous transition with a very tiny jump (if any). Due to the fact that our results come from a numerical treatement, we prefer to be cautious and describe the transition as quasi-continuous. We cannot exclude that a continuous transition could be finally the case. To make this point more clear, we have added the following comment in the Results section:
" By quasi-continuous, we mean that the transition could be discontinuous, but with a tiny jump. Apparently our results show a continuous transition but at third order we no longer have a fully analytical expression and, hence, our prediction has the limits of our numerical accuracy"
Concern 2: With the scale adopted in figure 6, the susceptibility data are not well visible. Maybe a semi-log scale would help. Furthermore, the authors mention that the finite value at criticality is due to the numerical integration. Did the authors verify that the critical value increases when improving the numerical accuracy?
A: Following the suggestion of the Referee, we have modified figure 6 using a semi-log scale. And yes, we have checked that the susceptibility at the critical point increases with better numerical accuracy. At least, until where we can reach.
Concern 3: Some integrals in the third-order terms are numerically computed. In the main text, the authors mention the use of an "accurate adaptative Monte Carlo integration", making reference to the Appendix. However, I do not find a clear explanation of what is meant for "adaptive".
A: The adaptative Monte Carlo method is not explained in the Appendix. In the new version of the manuscript, we have added two references [1] and [2]. This method is a Monte Carlo multidimensional integration based on "Las Vegas" algorithm. This algorithm is adaptative in the sense that it explores the shape of the function to be integrated, adapts the grid depending on it, and samples the relevant regions using this irregular grid.
Concern 4: It is worth mentioning that high-order perturbative results, but for spin balanced Fermi gases only, have been reported also in Physical Review C 104, 014003 (2021).
A: We thank the Referee for pointing us to this interesting reference.
We have included a sentence in the Introduction and the suggested reference:
" It is worth mentioning that, for a spin balanced gas, the fourth order term has been fully derived in Ref. [25]."
Concern 5: The authors do not explicitly discuss the results including also intra-species interactions. However, their formalism does in fact lead to terms corresponding to intra-species interactions. Would it be possible to show, e.g. in Figure 1, the equation of state without excluding intra-species interactions? Notice that quantum Monte Carlo data including also such interactions have been reported in Phys. Rev. A 107, 053305 (2023). See also , Phys. Rev. Lett. 123, 070404 (2019) for perturbative results including intra-species interactions in a single component gas.
A: Following the advice of the Referee, we have modified Figure 1. In the new figure, we have included the Monte Carlo data of the unpolarized gas with intra-species interactions and compared it with our third-order expansion. We have also modified the comments on the results reported in Figure 1
to discuss the case of intra-species interactions. The two provided references have been cited. The new paragraph reads:
" Having a perturbative prediction at our disposal, we can make a comparison between our results and the existing quantum Monte Carlo data. We compare two sets of values in Fig. 1. The black points are diffusion Monte Carlo (DMC) data for spin 1/2 ~[9]. This set does not include $p$-wave scattering terms between particles with the same $z$-spin component. The brown points in the same figure are DMC data by Bertaina et al.~[41] which include intraspecies interaction. The blue and purple lines are our theoretical predictions for these two cases, respectively. Both results correspond to non-polarized gases. As we can see, the energy is higher when the intraspecies interaction is considered. Moreover, although it is not shown here, intraspecies interaction in the
fully-polarized gas make the energy increase with the density [41,42]. In the rest of our results, we do not include this contribution. The orange line corresponds to the fully-polarized gas, the green one stands for the configuration of minimum energy, that is, at each value of the density, we select the polarization that minimizes the energy. And finally, the red line corresponds to the second-order energy (universal expansion) to show the differences with the third-order expansion. We can see how the second-order and the third-order expansions reproduce the same energy for values of $k_Fa_0 \lesssim 0.4$. Beyond that, the difference intensifies with increasing density~[14]. Concerning the DMC points, although they are upper-bounds to the exact energy due to the sign problem, they fit pretty well the third-order curves."
Concern 6: Improve the definition of quasi-continuous transitions.
A: Done.
Concern 7: Improve visualization of susceptibility data in Fig. 6.
A: Done.
Concern 8: Better discuss the divergence of the susceptibility at criticality.
A: We have added the following comment:
"We recall that, for second-order phase transitions, the susceptibility must diverge. Notice, however, that we obtain a very large value~($\sim2e3$) at $k_Fa_0=0,85$ and not a real divergence due to our finite numerical precision. If the accuracy is improved, the critical value increases."
References: [1] G. P. Lepage, Adaptive multidimensional integration: vegas enhanced, Journal of Computational Physics 439, 110386 (2021).
[2] P. Lepage, gplepage/vegas: vegas version 5.4.2 (v5.4.2), Zenodo, https://doi.org/10.5281/zenodo.8175999 (2023).