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Beyond universality in repulsive SU(N) Fermi gases
by Jordi Pera, Joaquim Casulleras, Jordi Boronat
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Submission summary
| Authors (as registered SciPost users): | Jordi Boronat |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2206.06932v4 (pdf) |
| Date accepted: | 2024-06-27 |
| Date submitted: | 2024-05-31 08:46 |
| Submitted by: | Boronat, Jordi |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| 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.
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- Provide a novel and synergetic link between different research areas.
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- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
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:
Firstly, the authors do not specify how they determine the magnetization displayed on Fig. 5. If it is evaluated by using the value of the order parameter P that minimizes the energy E for a given value of $x$, as shown on Fig. 3, then it should be explicitly stated in the manuscript. Similarly, the authors do not provide the values chosen for the parameters $a_1$ and $r_0$ on the softcore results shown on Fig. 10.
Our response: The Referee is right. The polarization comes from the minimization of the energy. We have added a comment on that.
Concerning the second point, the values of $r_0$ and $a_1$ used in Fig. 10 are the same as the ones used in Fig. 2. We have mentioned that explicitly in the caption of Fig. 10 and also in the discussion of the results. Anyway, we have introduced a new sentence in the text to avoid any misunderstanding.
Concern 2:
Secondly, based on the behavior of the magnetization on Fig 5 and the susceptibility on Fig 7, it seems that for S=7/2 and S=9/2 the Fermi gas undergoes two phase transitions, namely a second-order phase transition from an unpolarized system and then a first-order phase transition towards a polarized phase. This interpretation of the results, that is also further supported by the two singularities observed in the magnetic susceptibility, hints at the existence of an intermediate phase with some sort of magnetic ordering. Indeed, several examples can be found in the literature of SU(N) systems featuring rich phase diagrams, such as in Nucl. Phys. B 996 (2023) 116353. Could the authors comment on this remark?
Our response: We thank the Referee for this observation. We have thus included a new comment about what happens between the peaks in the magnetic susceptibility.
Concern 3:
Top of right column of p2: "require of a combination" -> "require a combination".
Our response: Done.
Concern 4:
Left column on p3: "s=1/2" -> "S=1/2".
Our response: Done.
Concern 5:
Right column on p6: "$k_Fa_0=0,85$"-> "$k_Fa_0=0.85$" and "$k_Fa_0=0,9$"-> "$k_Fa_0=0.9$".
Our response: Done.
Concern 6:
On p6, the authors claim that the use plane-waves Slater determinant in Diffusion Monte Carlo describe more efficiently polarized Fermi gases. I would ask the authors to add a reference to support this statement.
Our response: Diffusion Monte Carlo works in the so called Fixed Node approximation where the nodes are fixed and correspond to the ones of the variational wave function used for importance sampling. Pandharipande and others proved that the first correction to the nodes, coming from a plane-wave Slater determinant, are the backflow correlations that change the position of the nodes due to the interparticle interactions. Pandharipande proved that backflow correlations are more relevant in normal liquid $^3$He than in polarized $^3$He because in the polarized phase s-wave collisions are forbidden. This is the reason behind the statement that the plane-wave model is worse in normal than in fully polarized Fermi liquids. In the new version of the manuscript, we have added two references on this particular issue.
Concern 7:
On top of the right column of p7, could the authors clarify what they mean by "partial discontinuous transitions"?
Our response: We have added a comment on that.
List of changes
1) In response to Concern 1 we have introduced the following sentences:
In order to gain a deeper understanding of the phase transition, we plot in Fig. 5 the order parameter (in
this instance, the polarization) as a function of the gas parameter $x$. The polarization we plot is the one that minimizes the energy at a given $x$.
We report the critical gas parameters obtained using both the hard-sphere and soft-sphere
potentials. The soft-sphere potential is the same we used in Fig. 2, this means $r_0=0.424\,a_0$ and $a_1=1.1333\,a_0$.
2) In response to Concern 2 we have introduced the following sentences:
For spins 7/2 and 9/2, we see the singular double transition (two peaks) that we have mentioned before. With increasing density, the first peak corresponds to the truncated continuous transition, and the second peak to the latter first-order phase transition. The two peaks point to the existence of an intermediate phase between the non-polarized phase and the fully-polarized one. This phase would be located around the bottom that lies between peaks in Fig.~7. And, according to Fig.~5, this intermediate phase would have a partial polarization, hence, the Fermi gas would exhibit some kind of magnetic ordering. The rich phase diagram that appears in SU(N) Fermi systems have also been pointed out in Ref.~[45].
3) In response to Concern 7 we have introduced the following sentence:
For spin $3/2$ and $5/2$, we have partial discontinuous transitions, as there is a polarization jump, but it does not reach $P=1$, hence, the label 'partial'. If it reached $P=1$ directly, it would be a total discontinuous transition.
Published as SciPost Phys. 17, 030 (2024)