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Magnetic Thomas-Fermi theory for 2D abelian anyons
by Antoine Levitt, Douglas Lundholm, Nicolas Rougerie
Submission summary
| Authors (as registered SciPost users): | Nicolas Rougerie |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2504.13481v1 (pdf) |
| Date submitted: | May 21, 2025, 4:25 p.m. |
| Submitted by: | Nicolas Rougerie |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Two-dimensional abelian anyons are, in the magnetic gauge picture, represented as fermions coupled to magnetic flux tubes. For the ground state of such a system in a trapping potential, we theoretically and numerically investigate a Hartree approximate model, obtained by restricting trial states to Slater determinants and introducing a self-consistent magnetic field, locally proportional to matter density. This leads to a fermionic variant of the Chern-Simons-Schr{\"o}dinger system. We find that for dense systems, a semi-classical approximation yields qualitatively good results. Namely, we derive a density functional theory of magnetic Thomas-Fermi type, which correctly captures the trends of our numerical results. In particular, we explore the subtle dependence of the ground state with respect to the fraction of magnetic flux units attached to particles.
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Strengths
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A few comments first:
(1) This manuscript unfortunately ignores some very relevant literature. Very similar Hartree evaluation of chern-simons fermion systems has been done in the context of the composite fermion descriptions of quantum Hall systems (in the spirit of Halperin-Lee-Read, Lopez-Fradkin) with many different environments and geometries and background fields. Some of these works have been compared to experiment, (and possibly some have been compared to exact diagonalizaton or DMRG). There has also been quite a bit of thought about what this approximation gets right or does not get right. Possibly there has been comparison to DMRG or exact diagonalization as well (similar things have certainly been done many times to test self-consistent hartree, but in translationally invariant systems --- I'm not sure if it has been done in the presence of an external potential). Related to this, is the Jain-style, wavefunction approach that gives similar but more quantitative results for quantum hall systems (one reference is given, but only referenced tangentially.) I realize that these are not exactly the same problem being considered here, but they are closely related and given the large body of work in that context it is very much worth seeking the right connections to it.
(2) I would really like to see more discussion of what limits does this thomas-fermi theory become accurate --- and in what limit does (if any) does the original hartree become accurate. There is one limit mentioned in section 3, but I suspect there are other limits that can be discussed.
(3) A third issue is whether the anyon model presented, even before the hartree approximation, is an accurate depiction of ANY known physical system (outside of the above mentioned Halperin-Lee-Read context, which it does not match perfectly). If you take, for example, a short-ranged parent Hamiltonian that gives the Laughlin state as its exact ground state, and consider the quasi-hole anyons of this state (using the short ranged parent Hamiltonian removes the long ranged coulomb interaction between quasi-holes). The Hamiltonian presented in the paper is not an accurate depiction of this case, since the quasi-holes have exactly zero kinetic energy. The quasielectrons are more complicated, but it still does not match. It would be interesting to try starting with electrons, or bosons, or spins, with some hamiltonian that we might actually be able to make in a lab, and try to find some reasonable way to engineer the Hamiltonian being studied.
The above comments are things that I would like to see added to the manuscript if possible and I hope these comments are helpful to the authors for improving the paper.
All of these comments notwithstanding, my main concern with the manuscript (and the reason I recommend SciPost Core rather than SciPost) is simply the level of interest there will be in the community. While the work as it stands is solid, I think it is not likely to be of much interest to many people outside of the authors and a very small number of others who have been looking similar problems. (Point 3 above is perhaps the most relevant to this statement).
Requested changes
Not required changes, but discussions of points 1-3 above would be useful.
Recommendation
Accept in alternative Journal (see Report)
Strengths
Weaknesses
Report
Recommendation
Accept in alternative Journal (see Report)
We thank very much the referee for his/her careful reading and his/her remarks. We react to a few points below:
Reply to "Strengths" field:
We thank the referee for this summary. We emphasize that the "expected oscillatory dependence on \alpha" is itself a new finding of the paper, derived from magnetic Thomas-Fermi theory. Unless we have missed a reference, no prediction corresponding to our formulae (4.3)-(4.9) and Figure 1 appeared in the literature before.
Reply to "Weaknesses" field:
We are not sure of what is meant by "incremental" here. As noted above, the theory predictions (in particular the oscillatory behavior in \alpha) are new findings. The numerics are not merely a check, since they reveal the momentum distribution, that we suggest as an experimental probe. Besides, it seems to us that the only previous in depth numerical study comparable to ours is Ref. 35 (Hu-Murthy-Rao-Jain 2021).
The Hartree model is certainly an approximation, but the full anyon Hamiltonian is way out of reach of numerical simulations for the particle numbers we need to consider (up to N=100 in the results presented in the paper). In fact, we do not think there is any way of attacking this problem from the basic model, not any more than one could compute the ground state of a large molecule without resorting to DFT. True, we neglect some exchange and two-body terms, but it is extremely unlikely that those would modify the qualitative findings significantly. They would however blur features that are quite challenging to catch numerically, as discussed in the paper.
Author: Nicolas Rougerie on 2025-10-16 [id 5940]
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We warmly thank the referee for his/her hard work on our manuscript.
Several remarks of the referee will lead to a revision of the paper, whether or not it gets accepted for publication. Before eventually posting this, we thought some quick reactions would be in order.
We feel that the referee's opinion that the paper is 'not of broad interest' is at least partially based
See below for more details.
We thank the referee for his/her mention of literature that we should connect to, but it would be quite helpful to be given more precise references. We already cite Lopez-Fradkin and can certainly add a mention of Halperin-Lee-Read. We can also think of e.g.
Kohn-Sham Theory of the Fractional Quantum Hall Effect Yayun Hu and J. K. Jain PRL, 2019
that we thought we had cited in connection with our Ref 35. The absence of this citation was a mistake. We have spotted some other bibliographical adjustments that we will perform in a future revision. Eg Ref 37 should have been to the book
Composite Fermions J. K. Jain
where many other references can be found in Section 5.16, that we will quote more precisely. We note that Hu-Jain 2019 mentioned above is a relatively recent contribution, published in a broad-audience journal. It seems to us that this fact contradicts the impression of the referee that the questions we adress are not of general interest to the community.
Besides, the issues one needs to adress for a trapped inhomogeneous anyon system are quite different than those arising in the composite fermion description of FQHE droplets, as shown by the predictions we make regarding density profiles in position and momentum space. As for comparisons of the Hartree approach with others (DMRG, Jain-Style wave functions), several are made in Ref 35, that we could quote more extensively. But the main point of our paper is to reveal effects that require rather large particle numbers (up to N=100 in the simulations we present). We do not think comparisons with more resource-consuming methods are feasible for such numbers.
Concerning the specific comment:
(2) I would really like to see more discussion of what limits does this thomas-fermi theory become accurate --- and in what limit does (if any) does the original hartree become accurate. There is one limit mentioned in section 3, but I suspect there are other limits that can be discussed.
--> Section 4 does discuss another limit, the proper magnetic Thomas-Fermi one Eq (4.1), following Lieb-Solovej-Yngvason Ref 45. Indeed, the limit of Section 3 is less general than this one. In a future revision we will insist on these points some more.
Regariding point (3) of the report : There are in fact several physical situations for which our basic model (before Hartree approximation) is an accurate description. Let us insist some more on what is perhaps the most emblematic. The model is indeed the correct one for the physics of quantum impurities strongly coupled to a Laughlin liquid, as two of us demonstrated in Ref 50 (there are also related papers that we will discuss appropriately in a revision)
Emergence of fractional statistics for tracer particles in a Laughlin liquid D. Lundholm and N. Rougerie PRL 2016
Briefly, in this context the quantum impurities couple to Laughlin quasi-holes, giving the latter an effective kinetic energy and thus lifting the referee's main objection to the applicability of our Hamiltonian to the physics of Laughlin quasi-holes.
Another direction (see e.g. refs 21 and 63) is to directly engineer the model by coupling cold fermionic atoms to an artificial density-dependent gauge field. We note that a scheme similar to that of Ref 21 has been experimentally realized in Ref 24. The experiment is not directly related to the model we consider, but it gives hope to realize in the future experiments more in the spirit of the scheme of Ref 63, which yields hamiltonians very close to ours.
For the reasons mentioned above, we do not agree about the level of broad interest of our paper. There is in particular quite some activity towards realizing density-dependent gauge fields in a cold atoms context, see e.g.
Observing anyonization of bosons in a quantum gas Nature volume 642, pages 53–57 (2025
Realization of one-dimensional anyons with arbitrary statistical phase, Science, 386 (2024), pp. 1055–1060.
Realizing a 1d topological gauge theory in an optically dressed BEC, Nature, 608 (2022), pp. 293–297.
All the above are 1D realizations of density-dependent gauge fields, but proposals for 2D realizations do exist, cf again Ref 63. An obvious issue is that of possible experimental signatures. We provide in particular a clean one in the momentum density of the gas, accessible with cold atoms through time-of-flight experiments.