SciPost Submission Page
The Quantum Perfect Fluid in 2D
by Aurélien Dersy, Andrei Khmelnitsky, Riccardo Rattazzi
Submission summary
| Authors (as registered SciPost users): | Aurélien Dersy |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2211.09820v1 (pdf) |
| Date submitted: | 2023-03-06 18:12 |
| Submitted by: | Dersy, Aurélien |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider the field theory that defines a perfect incompressible 2D fluid. One distinctive property of this system is that the quadratic action for fluctuations around the ground state features neither mass nor gradient term. Quantum mechanically this poses a technical puzzle, as it implies the Hilbert space of fluctuations is not a Fock space and perturbation theory is useless. As we show, the proper treatment must instead use that the configuration space is the area preserving Lie group $S\mathrm{Diff}$. Quantum mechanics on Lie groups is basically a group theory problem, but a harder one in our case, since $S\mathrm{Diff}$ is infinite dimensional. Focusing on a fluid on the 2-torus $T^2$, we could however exploit the well known result $S\mathrm{Diff}(T^2)\sim SU(N)$ for $N\to \infty$, reducing for finite $N$ to a tractable case. $SU(N)$ offers a UV-regulation, but physical quantities can be robustly defined in the continuum limit $N\to\infty$. The main result of our study is the existence of ungapped localized excitations, the vortons, satisfying a dispersion $\omega \propto k^2$ and carrying a vorticity dipole. The vortons are also characterized by very distinctive derivative interactions whose structure is fixed by symmetry. Departing from the original incompressible fluid, we constructed a class of field theories where the vortons appear, right from the start, as the quanta of either bosonic or fermionic local fields.
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Please see the attached PDF file.
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I should start this report with a confession: I'm not an expert in this area, and was only vaguely aware of some of the issues that arise when quantizing a perfect fluid. So this report should be taken in the spirit of someone with a general interest in quantumy and fluidsy things.
Having said that, I enjoyed this "long and winding" paper very much. The writing is admirably clear, explaining the important issues, giving some toy examples of quantum mechanics on group manifolds, and then showing how these very same toy examples can solve the problem at hand, at least in 2d, by replacing the infinite dimensional group of diffeos with SU(N) and taking the large N limit. This latter procedure has long been invoked in te study of incompressible quantum Hall fluids, starting with the work of Susskind, and it is pleasing to see it used here in the more general context. The key punchline of the paper is a detailed study of the vorton dynamics.
I have only very minor comments. First, there is some kind of glitch above equation (9), around footnote 4, where a ctrl-c, ctrl-v went awry.
Second, there are many other places in physics where a flat band, with $\omega=0$, arises and we don't have any problem quantising. The most famous example is a Landau level. Typically, this simply gives rise to a very large degeneracy of states which are then lifted by arbitrarily small interactions which often push the system into some interesting phase. I wasn't entirely sure why the transverse modes in a perfect fluid, which also form a flat band, are so very different from this. Is it because the flat band is protected by a symmetry and so cannot be lifted by perturbations? Or for some other reason?
Finally, I was puzzled by the discussion on page 21 about the (rough) doubling of the degrees of freedom and the various options, A, B1 and B2, to circumvent this. Option A was to live with the doubling. Why doesn't this give rise to the wrong value of the heat capacity for this fluid?
In summary, I think that this is a well written and extremely interesting paper. I recommend publication in SciPost.
Report 1 by Slava Rychkov on 2023-4-19 (Invited Report)
Report
This paper makes progress on the problem of constructing a quantum perfect fluid in two spatial dimensions - a conjectural quantum state of matter which should be different in superfluid in that (some) transverse degrees of freedom are not gapped.
Working in the incompressible regime, which this paper focuses on, classical perfect fluid can be associated with motion on the group of volume-preserving diffeomorphisms. A key proposal of this paper is to recover a quantum theory based on this infinite-dimensional group from a sequence of finite-dimensional approximations by $SU(N)$ groups, $N\to\infty$. This is based on an old observation that $SDiff(T^2) = SU(\infty)$.
For a reasonable cutoff action, Eq. (120), the authors find that most states are gapped, but states in the adjoint representations of $SU(N)$, which they dub vorton, and which have a non-relativistic dispersion relation $E= p^2/2 M$. The vorton mass has the form of fluid mass contained in a region of the size of the UV cutoff $\Lambda$. (The limit $N\to\infty$ is taken with $\Lambda$ fixed.)
These findings are quite interesting and I recommend publication.
The paper is well written and includes pedagogical explanations of classical and quantum mechanics on finite-dimensional group manifolds, before passing on to the infinite-dimensional case.
Various puzzling issues remain but they are honestly admitted and can justly be postponed to future work.
I have only a few minor comments:
- p.7 l.8 "whenevr we" - please check this phrase
- p.8 "the results of this paper<...>do not seem to match those of ref. [28], though honestly we did not investigate that in full detail." It would be very helpful for the readers to have a discussion, if only short, perhaps in the discussion section, of how the results of this work compare to the results of [28] and what does not seem to match. (Currently only the difference of methodologies is discussed.) Who but the authors are in a better position to make such a discussion? I am sorry if this has already been included and I did not notice it.