# Quantum Hall Edges Beyond the Plasma Analogy

Per Moosavi, Blagoje Oblak, Bastien Lapierre, Benoit Estienne, Jean-Marie Stéphan

► To cite this version:

Per Moosavi, Blagoje Oblak, Bastien Lapierre, Benoit Estienne, Jean-Marie Stéphan. Quantum Hall Edges Beyond the Plasma Analogy. Phys.Rev.Res., 2025, 7 (2), pp.023134. 10.1103/PhysRevResearch.7.023134. hal-04891895

HAL Id: hal-04891895

https://hal.science/hal-04891895v1

Submitted on 19 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Quantum Hall edges beyond the plasma analogy

Per Moosavi,<fnref n="1" />,*, Blagoje Oblak,<fnref n="2" />,† Bastien Lapierre,<fnref n="3" />,‡ Benoit Estienne,<fnref n="4" />,§ and Jean-Marie Stéphan,<fnref n="5" />,||

<figure><img src="image_2.png" /></figure>

(Received 22 November 2024; accepted 18 March 2025; published 12 May 2025)

We demonstrate that the widely used plasma analogy is unreliable at predicting edge properties of quantum Hall states. This discrepancy arises from a fundamental difference between quantum Hall droplets and plasmas (Coulomb gases): The former are incompressible liquids subject to area-preserving deformations, while the latter are governed by electrostatics and thus involve conformal transformations. Consequently, the plasma analogy fails at the edge, except in fine-tuned geometries, as it does not account for the emergent local edge velocity. We quantitatively show how the analogy’s failure affects physical quantities, such as fluctuations of local observables and absorption rates in microwave spectroscopy, measurable in both solid-state experiments and quantum simulators.

DOI: 10.1103/PhysRevResearch.7.023134

I. INTRODUCTION

This paper critically contrasts two commonly related phys-
ical systems. The first is a topological phase of matter
famously responsible for the quantum Hall (QH) effect [1–5]:
A two-dimensional (2D) incompressible quantum liquid of
electrons in a strong perpendicular magnetic field. The second
is a 2D Coulomb gas (CG) [6–15]: A classical statistical sys-
tem of planar point charges interacting via a 2D electrostatic
potential. Both systems are confined into droplets of finite area
by trapping potentials, inherently present in any experiment.

Our investigation is motivated by what is known as the
plasma analogy, stating that QH systems are equivalent to
suitable CGs. This correspondence underlies Laughlin’s the-
ory of the fractional QH effect [5] and provides a handle
on microscopic aspects that are otherwise out of reach due
to strong electronic correlations. The analogy naturally also
holds for integer QH droplets of free electrons in harmonic
potentials, whose one-body wave functions coincide with
those of a random matrix ensemble—the (elliptic) Ginibre
ensemble [16,17]. This includes the simplest, familiar case

of isotropic QH droplets, whose wave functions have definite
angular momentum. However, the plasma analogy remains
untested for arbitrary potentials. This is a significant gap in the
literature, given that realistic QH potentials are always disor-
dered and anisotropic [18–20]. Additionally, even in quantum
simulators, where the potential can be controlled [21–27], it
is desirable to know how much of the plasma analogy may be
harnessed to model QH physics.

In the present paper, we put the plasma analogy to the
test. This relies on a recent semiclassical treatment of integer
QH droplets in general anisotropic potentials [28], allowing
for direct comparisons with CG predictions. The question
is: Does the plasma analogy work for generic QH droplets?
Strikingly, while reproducing bulk properties, the answer for

<figure><img src="image_3.png" /><figcaption>FIG. 1. QH droplets and CGs, shown in yellow along with a few level curves of the confining potential in the QH case. Both involve maps that send them on a disk $D$, but with different geometric constraints, namely symplectic (area-preserving) in QH and conformal (angle-preserving) in CG. Given the same initial droplet $\Omega$, such maps do not generally coincide, leading to different angle coordinates governing the emergent long-range edge correlations: The coordinate is $\theta$ for QH, $\alpha$ for CG, and $\varphi$ in the isotropic case.</figcaption></figure>

<footnote n="1">¹Department of Physics, Stockholm University, 10691 Stockholm, Sweden</footnote>

<footnote n="2">²Université Claude Bernard Lyon 1, ICJ UMR5208, CNRS, 69622 Villeurbanne, France</footnote>

<footnote n="3">³Department of Physics, Princeton University, Princeton, New Jersey 08544, USA</footnote>

<footnote n="4">⁴Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE, F-75005 Paris, France</footnote>

<footnote n="5">⁵ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France</footnote>

<footnote>*Contact author: per.moosavi@fysik.su.se</footnote>

<footnote>†Contact author: oblak@math.univ-lyon1.fr</footnote>

<footnote>‡Contact author: blapierre@princeton.edu</footnote>

<footnote>§Contact author: estienne@lpthe.jussieu.fr</footnote>

<footnote>||Contact author: jean-marie.stephan@ens-lyon.fr</footnote>
edge phenomena is negative: QH droplets and CGs of identical shape almost always have different electronic correlations along the edge, resulting, e.g., in unequal values for fluctuations of local observables. Such fluctuations can be measured, at least indirectly, through microwave absorption experiments [29–33], so the implications of the QH/CG mismatch are both theoretical and experimental.

These differences between QH droplets and CGs should not come as a surprise. Since CGs ultimately rely on electrostatics, their physics is entirely dictated by their shape—albeit in a nonlocal manner. By contrast, QH physics is that of an incompressible fluid, so it depends not only on the droplet’s shape but also on the local slope of the potential at the edge. This is the core of our argument. Abstractly, it can be phrased in terms of maps that send anisotropic droplets on their isotropic cousins, as in Fig. 1: The map is symplectic (area-preserving) for QH but conformal (angle-preserving) for CG. It is still true that QH and CG predictions sometimes coincide, but this only occurs for fine-tuned setups, such as harmonic potentials [9]. Needless to say, no such fine-tuning is expected in solid-state experiments, where even just determining the confining potential is a challenge.

The paper is organized as follows. Section II briefly reviews the links between QH physics and symplectic geometry on one hand, and CGs and conformal geometry on the other. This highlights the QH/CG discrepancy, whose implications are then examined in the remaining sections. Indeed, Sec. III illustrates the difference between QH and CG edge correlations through various explicit examples, and Sec. IV focuses on the consequences for fluctuations of local observables, some of which are evaluated using Monte Carlo numerics on the CG side. Finally, we conclude in Sec. V. Further details on the mismatch, including brief reviews of QH droplets and CGs, are deferred to the Supplemental Material (SM) [34].

## II. QUANTUM HALL IS SYMPLECTIC, COULOMB GAS IS CONFORMAL

Here, we review the basics of QH and CG physics and outline their respective links with symplectic and conformal geometries.

### A. Quantum Hall droplets

To begin, consider a classical particle with charge $q$ in the plane, subjected to a uniform perpendicular magnetic field **B** and a confining potential **V**. Assuming the magnetic field is strong, the particle traces high-frequency cyclotron orbits whose guiding center **x**(*t*) slowly drifts along equipotentials (level curves) of **V**(**x**). The drift velocity is

$$ \dot{x}^i = -\epsilon^{ij} \frac{\partial_j V}{qB} \qquad (1) $$

in Cartesian coordinates **x** = (x¹, x²) = (x, y), using the Levi-Civita symbol $\epsilon^{ij}$. Viewing Eq. (1) as an equation of motion for the coordinates (x, y) with Hamiltonian $V(x, y)$, the space where the particle lives effectively becomes the phase space of its guiding center. Thus, guiding-center coordinates do not commute; they are instead canonically conjugate, satisfying {$x, y$} = 1/qB.

This symplectic structure has important implications. Namely, the guiding-center velocity (1) is tangent to equipotentials of **V**, and its norm

$$ v(\mathbf{x}) \equiv \sqrt{\dot{x}^2 + \dot{y}^2} = \frac{1}{qB} |\nabla V(\mathbf{x})| \qquad (2) $$

is generally nonconstant. One can nevertheless make the motion look isotropic in terms of well-chosen coordinates, at least when the equipotential has the topology of a circle. Indeed, let the equipotential have length *L* and label its points by the arc length *s* to define an angle

$$ \theta(s) \equiv 2\pi \int_{0}^{s} \frac{ds'}{v(s')} \int_{0}^{L} \frac{ds'}{v(s')}, \qquad (3) $$

where the lower integration limit is arbitrary and $v(s)$ is the norm (2) at $s$. This is the canonical angle coordinate along the equipotential, unique up to rotation. It is conjugate to a canonical action coordinate $\mathcal{A}$, with dimensions of area, such that $d\mathcal{A} \wedge d\theta = 2\pi dx \wedge dy$ [35]. Each value of $\mathcal{A}$ labels a level curve of $V$, so the potential can be written as $V(\mathbf{x}) = V_0(\mathcal{A}(\mathbf{x}))$ for a strictly increasing function $V_0(\cdot)$ of $\mathcal{A}$ only, at least in a neighborhood of the equipotential [36]. These coordinates make the guiding center motion in Eq. (1) uniform in the sense that

$$ \dot{\mathcal{A}} = 0, \qquad \dot{\theta} = \frac{2\pi}{qB} V_0'(\mathcal{A}) > 0. \qquad (4) $$

When $V$ has a unique minimum surrounded by nested level curves, action-angle coordinates are globally well defined on the plane, and $\mathcal{A}$ can be chosen to be the actual area enclosed by an equipotential. For example, if $V$ is isotropic, then $\mathcal{A} = \pi r^2$ and $\theta = \varphi$ in polar coordinates.

The problem of solving the equation of motion (1) thus boils down to finding a canonical (area-preserving) map

$$ F: \mathbb{R}^2 \to \mathbb{R}^2 : (x, y) \mapsto (\mathcal{A}, \theta) \qquad (5) $$

that converts the anisotropic potential $V$ into its isotropic cousin $V_0(\mathcal{A})$, as in Fig. 1. Here, we write the map as if it was globally well defined, which is generally neither true nor required: All one needs is that it sends a neighborhood of an equipotential on a neighborhood of a circle in a smooth, bijective, and area-preserving manner. In our case, the equipotential of interest will be the edge of a droplet, whose topology is always that of a circle no matter how disordered the bulk potential is.

Turning to quantum physics, the corresponding problem of an electron in a strong magnetic field with a harmonic potential dates back to Refs. [37,38] and is solvable using special functions. More generally, for any confining potential $V$, the above classical structures carry over to the quantum case, including many-body states at low energy. Position operators projected to the lowest Landau level (LLL) fail to commute, and the position of a coherent state in the LLL satisfies Eq. (1) in the semiclassical limit of weak potentials and strong magnetic fields [39–41]. Relatedly, semiclassical eigenstates in the LLL are localized on equipotentials, each enclosing a quantized area $\mathcal{A} = 2\pi m\ell^2$, where $\ell = \sqrt{\hbar/|qB|}$ is the magnetic length and $m$ is a (large) positive integer [28,42,43]. Extrapolating to a many-body system of $N \gg 1$ free electrons in the LLL is straightforward: They occupy
the domain $\Omega$ where $V(\mathbf{x}) \le E_F$ for some Fermi energy $E_F$, forming a quantum Hall (QH) droplet with area $A = 2\pi Nl^2$ and uniform bulk density $\rho = 1/2\pi l^2$. A similar picture is expected for fractional QH droplets confined by a weak potential [2,5], then with total area $A = 2\pi Nl^2/\nu$ and bulk density $\rho = \nu/2\pi l^2$, both depending on the filling fraction $\nu$.

One of the hallmarks of topology in QH droplets is that they are bulk insulators but admit gapless edge modes, responsible for long-range correlations along the boundary. In particular, density-density edge correlations universally behave as

$$ \frac{1}{\sin^2([\theta_1 - \theta_2]/2)} \quad (\text{QH}) \qquad (6) $$

in terms of the canonical angle (3) along the edge [28,44]. (The actual correlator includes prefactors that are irrelevant here; cf. Eq. (26) or the SM [34] for details.) Equivalently, low-energy edge modes propagate at a constant angular velocity in the $\theta$ coordinate [28], similar to that of guiding centers (4). Note that this is also expected to hold for fractional QH droplets, underscoring once more the significance of the area-preserving map in Eq. (5). The latter determines all the edge properties, hence the low-energy physics, of QH systems in the semiclassical limit.

## B. Coulomb gases

Consider now $N$ classical point charges in the plane with positions $\mathbf{x}_1, \dots, \mathbf{x}_N$, interacting through a pairwise repulsive logarithmic potential, with no kinetic energy. The system's total energy is

$$ E(\mathbf{x}_1, \dots, \mathbf{x}_N) = - \sum_{1 \le i < j \le N} \log |\mathbf{x}_i - \mathbf{x}_j| + \frac{N}{2} \sum_{j=1}^{N} W(\mathbf{x}_j), \qquad (7) $$

where $W$ is a confining potential that grows faster than $\log|\mathbf{x}|$ as $|\mathbf{x}| \to \infty$. When such a system is prepared at a finite temperature $\beta^{-1}$, it is known as a 2D Coulomb gas (CG) or log-gas. Its link with QH droplets stems from a coincidence, known as the plasma analogy: When $W(\mathbf{x}) = |\mathbf{x}|^2$ is isotropic and harmonic, the probability density of a configuration $\mathbf{x}_1, \dots, \mathbf{x}_N$ at temperature $\beta^{-1}$ coincides with the $N$-body probability density of the Laughlin wave function at filling fraction $\nu = 2/\beta$ [5]. As we shall argue, the analogy generally fails to extend to anisotropic states, at least as far as edge properties are concerned.

In the thermodynamic limit $N \gg 1$, the CG ground state is the configuration that minimizes Helmholtz free energy. In practice, energy dominates entropy due to the long-range interaction in Eq. (7) [45], so it suffices to minimize energy alone, regardless of temperature. This is most easily described in terms of the (normalized) mean density $\rho(\mathbf{x}) \equiv \frac{1}{N} \langle \sum_{j=1}^N \delta(\mathbf{x} - \mathbf{x}_j) \rangle$, where the expectation value $\langle \cdot \rangle$ is taken in the canonical ensemble. When $N \to \infty$, the density is supported in some domain $\Omega$ [46]. Minimizing the energy (7) over all density profiles yields the Euler-Lagrange equation [34]

$$ \frac{W(\mathbf{x})}{2} - \int_{\Omega} dy \rho(\mathbf{y}) \log |\mathbf{x} - \mathbf{y}| = \text{const}, \qquad (8) $$

valid for all $\mathbf{x} \in \Omega$, where $dy \equiv dy^1 dy^2$. Taking the Laplacian of both sides gives $\rho = \nabla^2 W/4\pi$ inside the droplet and

<figure><img src="image_4.png" /><figcaption>FIG. 2. Monte Carlo simulation of the norm of the total electric field $\mathbf{E}(\mathbf{y}) = -\nabla_y((\Phi(\mathbf{y}))^x - \langle \Phi(\mathbf{y}) \rangle - \log|\mathbf{x} - \mathbf{y}|)$ for $N = 256$ particles and potential $W(x, y) = x^2 + y^2 - (x^3 - 3xy^2)/10$, shown as a function of $\mathbf{y} = (y^1, y^2)$. (The potential is polynomial so as to give an anisotropic CG at minimal numerical cost, but it is otherwise unimportant.) Left: Fixed unit charge at $\mathbf{x} \in \Omega$; its effect is screened over short distances. Right: Fixed unit charge at $\mathbf{x} \notin \Omega$, in which case the electric potential is a long-range Green's function. In both plots, the electric field vanishes inside $\Omega$ and is discontinuous on $\partial\Omega$, signaling charge accumulation on the boundary.</figcaption></figure>

$\rho = 0$ outside, which fixes the density. In turn, plugging $\rho = \nabla^2 W/4\pi$ back into Eq. (8) implicitly determines $\Omega$ through a (generally complicated) inverse problem. The droplet's shape is thus determined by the potential $W$, but in contrast to QH droplets, it is also strongly affected by Coulomb interactions. (This is why our use of $V$ vs $W$ is important to distinguish QH and CG potentials.) A much easier task is to first choose $\Omega$ and a normalized $\rho$ therein, then reverse-engineer the potential $W$ in $\Omega$ through Eq. (8). In fact, this is precisely how CGs with uniform bulk density are meant to be used in the plasma analogy.

Note that the left-hand side of Eq. (8) is the total electrostatic potential at $\mathbf{x}$: It is the sum of the external potential $W$ and the mean potential $\langle \Phi(\mathbf{x}) \rangle$ created by the CG itself, with

$$ \Phi(\mathbf{x}) = - \sum_{j=1}^{N} \log |\mathbf{x} - \mathbf{x}_j|. \qquad (9) $$

In that sense, Eq. (8) is a screening condition: The CG behaves as a perfect conductor. It implies that CGs are governed by 2D electrostatics in the thermodynamic limit.

A simple way to understand the relevance of conformal geometry outside CGs and on their edge is to study correlations. In that context, the fundamental object is the (connected) two-point function of the field in Eq. (9), which can be determined from linear response. Indeed, suppose a (small) charge $q$ is inserted at some position $\mathbf{x}$, so that the energy $E$ in Eq. (7) is modified to $E + q\Phi(\mathbf{x})$. Denoting by $\langle \cdot \rangle^x$ the corresponding expectation value, linear response yields

$$ \langle \Phi(\mathbf{y}) \rangle^x - \langle \Phi(\mathbf{y}) \rangle \sim -q\beta \langle \Phi(\mathbf{x}) \Phi(\mathbf{y}) \rangle_c. \qquad (10) $$

Here, $\langle \cdot \rangle_c$ denotes the connected correlator, and the left-hand side is the potential indirectly created by the charge $q$ at some other point $\mathbf{y}$, without including the potential $-q \log|\mathbf{x} - \mathbf{y}|$ due to the point charge itself. By screening, the CG acts as a perfect conductor, so computing the left-hand side of Eq. (10) boils down to an electrostatic problem whose solution depends on whether $\mathbf{x}, \mathbf{y}$ are inside or outside the droplet; see Fig. 2 and the SM [34] for details. When $\mathbf{x}, \mathbf{y}$ are inside, the total potential $\langle \Phi(\mathbf{y}) \rangle^x - \langle \Phi(\mathbf{y}) \rangle - q \log|\mathbf{x} - \mathbf{y}|$ is constant and
its gradient, the total electric field, vanishes in $\Omega$ as in any conductor. When $\mathbf{x}$ is inside and $\mathbf{y}$ outside (or vice versa), screening implies that the total field only depends on the point outside. However, when both $\mathbf{x}, \mathbf{y}$ are outside the droplet, the total potential is the Green's function $K(\mathbf{x}, \mathbf{y})$ of the Laplacian $-(1/q)\nabla^2$ outside $\Omega$, with Dirichlet boundary conditions on $\partial\Omega$.

If $\Omega$ is the unit disk, the corresponding Green's function $K_0(\mathbf{x}, \mathbf{y})$ can be found explicitly through the method of images. Green's functions in more complicated geometries are then related to $K_0$ by conformal maps [34]. Specifically, the relevant conformal transformation

$$G: \mathbb{R}^2 \setminus \Omega \to \mathbb{R}^2 \setminus \mathcal{D}: \mathbf{x} \mapsto G(\mathbf{x}) \quad (11)$$

maps the exterior of the droplet on the exterior of the disk [47], whereupon the Green's function outside $\Omega$ is $K(\mathbf{x}, \mathbf{y}) = K_0(G(\mathbf{x}), G(\mathbf{y}))$. This provides a natural conformal angle coordinate, which we denote $\alpha$: For $\mathbf{x} \in \partial\Omega$, it is the polar angle of the vector $G(\mathbf{x})$.

We stress that conformal maps such as in Eq. (11) are extremely rigid since they must be invertible and holomorphic (in the coordinate $x^1 + ix^2$) everywhere outside $\Omega$. Indeed, provided the map (11) sends infinity on itself, the Riemann mapping theorem ensures that it is unique up to an overall rotation. Put differently, all correlations in a CG are fully determined by the shape of the droplet $\Omega$.

An important application of the conformal map (11) is provided by density-density correlations, especially since they can be compared with their counterparts in QH droplets. While bulk correlations are short-ranged due to screening [6], edge correlations are long-ranged. They can be predicted from electrostatics: They are, in fact, correlations of the edge charge density produced at the boundary of the CG by the discontinuous electric field; see, again, Fig. 2. The resulting correlator behaves as

$$\frac{1}{\sin^2([\alpha_1 - \alpha_2]/2)} \quad (\text{CG}) \quad (12)$$

in terms of the conformal angle $\alpha$ along the edge, determined by the map (11) [8,12,48,49]. (As in Eq. (6), prefactors are omitted here; see the SM [34] for details.)

### C. Mismatch

The difference between QH droplets and CGs can be summarized in terms of the maps in Eqs. (5) and (11). Consider a (simply connected) domain $\Omega$ in the plane, and compare two scenarios: One where it is occupied by a QH droplet in a potential $V$, and another where it is occupied by a CG in a potential $W$. Both systems share the same boundary $\partial\Omega$. The density correlator between points $\mathbf{x}_1$ and $\mathbf{x}_2$ on $\partial\Omega$ is given by Eq. (6) with $\theta_i = \theta(\mathbf{x}_i)$ for QH, and by Eq. (12) with $\alpha_i = \alpha(\mathbf{x}_i)$ for CG. The coordinates $\theta_i$ and $\alpha_i$ are, respectively, derived from the area-preserving map (5) and the angle-preserving map (11). These maps usually differ, except in a few highly specific cases, causing the discrepancy between Eqs. (6) and (12) despite their superficial similarity. Physically, this is because electrostatics is only sensitive to the shape of a CG, while QH physics is sensitive to both the

<figure><img src="image_5.png" /><figcaption>FIG. 3. Two confining potentials, with their level curves as insets. The one in (a) is isotropic, so the norm (2) of the edge velocity is uniform in the resulting QH droplet. The one in (b) is anisotropic, but at least one of its equipotentials is a circle, so it gives rise to a QH droplet whose edge modes have an inhomogeneous velocity (2) despite the droplet's isotropic shape. QH physics distinguishes these two situations. By contrast, from the CG perspective, the two systems are equivalent since edge properties are determined by the droplet's shape alone.</figcaption></figure>

shape and the local gradient of the confining potential at the edge.

This mismatch is our main statement, so let us rephrase it for later reference. Namely, the maps in Eqs. (5) and (11) yield two parameterizations of the edge:

$$\theta: \partial\Omega \to S^1: \mathbf{x} \mapsto \theta(\mathbf{x}) \quad (\text{QH}), \quad (13)$$

$$\alpha: \partial\Omega \to S^1: \mathbf{x} \mapsto \alpha(\mathbf{x}) \quad (\text{CG}). \quad (14)$$

These typically differ, in the sense that the one-dimensional diffeomorphism $\alpha \circ \theta^{-1}: S^1 \to S^1$ is not just a rotation $\varphi \mapsto \varphi + \text{const}$ (except in rare special cases). In the remainder of this paper, we verify this claim in several situations of interest and point out its experimental ramifications. We stress that this does not affect the universal correspondence between bulk topology and edge modes [50]: Indeed, as shown in Ref. [28], the emergent effective edge theory is unaffected by our deformations of the sample's geometry (at least to leading order in $N$).

### III. MISMATCHED EDGE CORRELATIONS IN ANISOTROPIC DROPLETS

Below, we provide detailed examples of the difference between the angle coordinates in Eqs. (13) and (14). We begin with disk-shaped QH droplets whose confining potential is anisotropic, then consider anisotropic droplets of the form introduced in Ref. [28], including square droplets where CG predictions are known analytically.

#### A. Anisotropic disks

Consider a QH droplet whose domain $\Omega$ is a disk but whose confining potential is anisotropic as in Fig. 3(b) (so its level curves are not concentric circles). Edge modes then propagate with a velocity whose norm (2) is nonconstant, leading to a canonical angle (3). When the droplet $\Omega$ is a disk, the arc length is $s = L\varphi/2\pi$, and the angle (3) becomes a nontrivial function $\theta(\varphi)$, expressed as a functional of the local norm $v(\varphi)$ of the edge velocity. The resulting map (13) is a nontrivial diffeomorphism of the circle since $\partial\Omega = S^1$.
Now suppose one were to use the plasma analogy to predict edge correlations in this setup. Consider therefore a disk-shaped CG with uniform bulk density. The corresponding potential in the screening condition (8) is necessarily isotropic, and the conformal map (11) is just the identity (up to rotation), so the coordinate $\alpha$ in the edge correlator (12) coincides with the polar angle: $\alpha = \varphi$. This manifestly differs from the canonical angle (3), exemplifying the announced mismatch.

Note that potentials giving rise to disk-shaped droplets with anisotropic edge velocities can be built explicitly. Indeed, let $\theta(\varphi)$ be any smooth function such that [51]

$$ \theta(\varphi + 2\pi) = \theta(\varphi) + 2\pi, \quad \theta'(\varphi) > 0. \quad (15) $$

Then, choose a radius $R$ for the QH droplet, and let the potential near the edge take the form

$$ V(r, \varphi) = V_0(\pi R^2 + \pi[r^2 - R^2]/\theta'(\varphi)) \quad (16) $$

for some monotonically increasing function $V_0(\cdot)$. The corresponding action-angle coordinates in a neighborhood of the edge are

$$ \mathcal{A} = \pi R^2 + \pi[r^2 - R^2]/\theta'(\varphi), \quad \theta = \theta(\varphi). \quad (17) $$

Fixing the Fermi energy at $V_0(\pi R^2)$, the edge of the droplet is the circle $r = R$, but the norm (2) of the edge velocity is generally anisotropic since $v(\varphi) = \text{const}/\theta'(\varphi)$.

## B. Flower droplets

Let us turn to anisotropic QH droplets confined by edge-deformed potentials as in Ref. [28]: Given any smooth function $\theta(\varphi)$ satisfying Eq. (15), define

$$ V(r, \varphi) \equiv V_0(\pi r^2 / \theta'(\varphi)), \quad (18) $$

where $V_0(\cdot)$ is, again, a monotonically increasing function. The corresponding action-angle coordinates are given by Eq. (17) with $R=0$, and the equipotentials

$$ r^2 = \frac{\mathcal{A}}{\pi} \theta'(\varphi) \quad (19) $$

are nested as in Fig. 1. An advantage of this family of potentials is that their semiclassical eigenstates in the LLL can be found analytically, for any $\theta(\varphi)$ [28].

A subclass of edge-deformed QH droplets is provided by flower deformations with an integer number $k$ of petals, for which the function $\theta(\varphi)$ in the potential (18) is given by

$$ e^{ik\theta(\varphi)} = \frac{\cosh(\lambda)e^{ik\varphi} + \sinh(\lambda)}{\sinh(\lambda)e^{ik\varphi} + \cosh(\lambda)}, \quad (20) $$

where $\lambda \in \mathbb{R}$ is a deformation parameter. (The cases $k=2$ and $3$ are shown in the inset of Fig. 4.) It is then straightforward to express the QH density-density correlator (6) in terms of the polar angle $\varphi$, for any $k$ and $\lambda$. For $k=2$ and $V_0(\mathcal{A}) \propto \mathcal{A}$, the potential (18) is actually harmonic [and anisotropic if $\lambda \neq 0$ in Eq. (20)].

Now suppose, again, one uses the plasma analogy to describe this situation. To this end, consider a CG with uniform bulk density, whose shape $\Omega$ is a flower droplet with a boundary given by Eq. (19) in polar coordinates, with some fixed area $\mathcal{A}$. The resulting edge density correlator takes the form

<figure><img src="image_6.png" /><figcaption>FIG. 4. Differences between coordinates $\theta(\varphi)$ and $\alpha(\varphi)$ for edge-deformed droplets (19) given by flower deformations (20), with $(k, \lambda) = (3, 0.2)$ (purple), $(k, \lambda) = (3, 0.4)$ (yellow), $(k, \lambda) = (2, 0.4)$ (blue), and a square (21) (magenta). Equipotentials of the respective QH droplets are shown on the side.</figcaption></figure>

(12) in terms of the restriction of the conformal map (11) to the boundary $\partial\Omega$. The problem is to find this map $\mathbf{x} \mapsto G(\mathbf{x})$. As before, $G(\mathbf{x})$ is uniquely fixed by the shape of the droplet through the Riemann mapping theorem. It admits no known analytical expression (unless $k=2$), but it can be built numerically using Fekete points [52]. This is how Fig. 4 was obtained: In it, we show the difference $\theta(\varphi) - \alpha(\varphi)$ for flower droplets with $k=2$ or 3 petals. The difference is manifestly nonconstant when $k=3$, confirming that $\theta(\varphi)$ and $\alpha(\varphi)$ differ by more than a rotation. In fact, the mismatch worsens when $\lambda$ increases. Similar conclusions hold for all droplets with $k \neq 2$ petals and generally for the vast majority of QH droplets confined by edge-deformed potentials (18).

The exception $k=2$ corresponds to droplets whose boundary, given by Eq. (19), is an ellipse. In that case, the conformal map (11) can be found analytically [34], and the CG edge correlator (12) coincides with the correlator (6) of a QH droplet in the potential (18); see the straight line where $\theta(\varphi) = \alpha(\varphi)$ in Fig. 4. There is a reason for this coincidence: The distribution of particles in an elliptic CG coincides with the distribution of eigenvalues of random matrices in the elliptic Ginibre ensemble [9,17], which in turn coincides with the density of LLL wave functions in a harmonic potential.

A geometric way to understand why $k=2$ is exceptional is to bluntly impose $\theta(\varphi) = \alpha(\varphi)$ on the edge given by Eq. (19), then attempt to view $\alpha(\varphi)$ as the boundary value of a bijective conformal map (11). One readily verifies that this fails for any flower deformation (20), unless $k=2$. The failure is due to zeros or poles of the would-be conformal extension, which always appear outside $\Omega$ unless $k=2$; see the SM [34] for details. More generally, for any edge-deformed droplet, imposing $\theta(\varphi) = \alpha(\varphi)$ on the curve (19) makes it impossible to view $\alpha(\varphi)$ as the restriction of a bijective conformal map (11), except in exceedingly fine-tuned cases. This underscores, again, that the plasma analogy is typically unreliable on the edge of anisotropic QH droplets.

## C. Square droplets

We finally turn to a case where both QH and CG can be treated analytically, namely a square droplet. On the QH side, we build the droplet by choosing an edge-deformed potential (18) whose equipotentials are squares. The corresponding
canonical angle coordinate is

$$ \theta(\varphi) = q(\varphi) + \frac{\pi}{4} \tan(\varphi - q(\varphi)), \quad (21) $$

$$ q(\varphi) \equiv \frac{\pi}{2} \left\lfloor \frac{2\varphi}{\pi} + \frac{1}{2} \right\rfloor, $$

using the floor function ⌊·⌋. On the CG side, the problem is to find the conformal map (11) that sends the outside of a square on the outside of a disk, thereby fixing the CG angle (14). The solution is well known: It is given by a Schwarz-Christoffel map [53] whose explicit form involves a hypergeometric function [34]. The ensuing CG angle can be expressed as

$$ \alpha(\varphi) = -i \log \left[ G \left( \frac{\sqrt{\pi}}{2} \frac{e^{i\varphi}}{|\cos(\varphi + q(\varphi))|} \right) \right] \quad (22) $$

in terms of the polar angle $\varphi$ and the map $G$ in Eq. (11), which is holomorphic in the complex coordinate $x + iy$. One can, again, compare the QH and CG angles in Eqs. (21) and (22): Their difference $\theta(\varphi) - \alpha(\varphi)$, shown in Fig. 4, fails to be a constant, exhibiting once more the QH/CG mismatch.

## IV. CONSEQUENCES FOR FLUCTUATIONS AND ADMITTANCE

Differences between QH droplets and their would-be plasma analogues affect the computation of physical quantities. Indeed, the mismatch between the correlators in Eqs. (6) and (12) entails different predictions for the contribution of low-energy edge modes to, say, linear-response coefficients of anisotropic droplets. Here, we demonstrate its consequences for fluctuations of local operators, known as linear statistics in the CG literature [12,54,55]. As we shall explain, the fluctuation-dissipation theorem makes it possible to probe such fluctuations through microwave absorption experiments that are within reach using current technologies [29–33].

### A. Fluctuations in quantum Hall droplets

To begin, consider the QH side. Given a smooth real function $U(\mathbf{x})$, define the Fock-space operator

$$ \hat{U} \equiv \int d\mathbf{x} U(\mathbf{x}) \hat{c}^{\dagger}(\mathbf{x}) \hat{c}(\mathbf{x}), \quad (23) $$

where $\hat{c}^{(\dagger)}(\mathbf{x})$ are local fermionic annihilation (creation) operators satisfying $\{\hat{c}(\mathbf{x}_1), \hat{c}^{\dagger}(\mathbf{x}_2)\} = \delta^{(2)}(\mathbf{x}_1 - \mathbf{x}_2)$. We wish to evaluate the variance of $\hat{U}$ in the $\nu = 1$ QH ground state. By Wick's theorem, one can express the variance as

$$ \mathrm{Var}[\hat{U}]_{\mathrm{QH}} = \frac{1}{2} \iint d\mathbf{x} \, d\mathbf{y} |C(\mathbf{x}, \mathbf{y})|^2 [U(\mathbf{x}) - U(\mathbf{y})]^2, \quad (24) $$

where $C(\mathbf{x}, \mathbf{y})$ is the electronic two-point correlation function. The latter is short-ranged in the bulk, where

$$ |C(\mathbf{x}, \mathbf{y})|_{\text{bulk}} \sim \frac{1}{2\pi l^2} \exp \left( -\frac{|\mathbf{x} - \mathbf{y}|^2}{4l^2} \right) \quad (25) $$

with $l$ the magnetic length introduced below Eq. (5). By contrast, correlations are long-ranged along the edge, as mentioned around Eq. (6). A more precise statement, derived in

Ref. [28] for edge-deformed potentials, is

$$ |C(\mathbf{x}, \mathbf{y})|_{\text{edge}} \sim \frac{\mathcal{N}}{\sqrt{v(\theta_{\mathbf{x}})v(\theta_{\mathbf{y}})}} \frac{\exp\left(-\frac{d_{\mathbf{x}}^2}{2l^2} - \frac{d_{\mathbf{y}}^2}{2l^2}\right)}{2 \sin\left(|\theta_{\mathbf{x}} - \theta_{\mathbf{y}}|/2\right)} \quad (26) $$

up to an omitted normalization $\mathcal{N}$ [34], where $\theta_{\mathbf{x}}$ is the canonical angle coordinate (13) of $\mathbf{x}$, $d_{\mathbf{x}}$ is the Euclidean distance between $\mathbf{x}$ and the edge, $v(\theta_{\mathbf{x}})$ is the norm (2) of the edge velocity at $\theta_{\mathbf{x}}$, and similarly for $\mathbf{y}$. Plugging this and Eq. (25) into the variance (24), and assuming that the function $U(\mathbf{x})$ varies slowly on length scales of the order of $l$, yields

$$ \mathrm{Var}[\hat{U}]_{\mathrm{QH}} \sim \int_{\Omega} \frac{d\mathbf{x}}{4\pi} |\nabla U|^2 + \oint \frac{d\theta_1 d\theta_2 (U_{\theta_1} - U_{\theta_2})^2}{32\pi^2 \sin^2\left(\frac{\theta_1 - \theta_2}{2}\right)}, \quad (27) $$

where $U_{\theta}$ is the function of $\theta$ obtained by restricting $U(\mathbf{x})$ to the edge and labeling points by their canonical angle. The result (27) holds at leading order in the thermodynamic limit $N \to \infty$ with $Nl^2$ kept finite. It exhibits a splitting between bulk and edge contributions, the latter being sensitive to the norm (2) of the local edge velocity through the coordinate $\theta$.

### B. Fluctuations in Coulomb gases

On the CG side, the plasma analogue of the quantum variance (24) is the thermal fluctuation of the linear statistics $\sum_{j=1}^N U(\mathbf{x}_j)$, where $U$ is the same function as in Eq. (23). One is usually interested in the thermodynamic limit,

$$ \mathrm{Var}[U]_{\mathrm{CG}} \equiv \lim_{N \to \infty} \mathrm{Var} \left[ \sum_{j=1}^{N} U(\mathbf{x}_j) \right]. \quad (28) $$

The latter can be split into bulk and edge correlations in direct analogy to Eq. (27). Indeed, it was shown in Refs. [12,54,56] that the CG variance (28) is [34]

$$ \mathrm{Var}[U]_{\mathrm{CG}} = \int_{\Omega} \frac{d\mathbf{x}}{2\pi\beta} |\nabla U|^2 + \oint \frac{d\alpha_1 d\alpha_2 (U_{\alpha_1} - U_{\alpha_2})^2}{16\pi^2\beta \sin^2\left(\frac{\alpha_1 - \alpha_2}{2}\right)}, \quad (29) $$

where $U_{\alpha}$ is defined analogously to $U_{\theta}$ in Eq. (27), but using the conformal coordinate (14) instead of the canonical one (13). For the free-fermion value $\beta = 2$, Eq. (29) looks identical to Eq. (27), up to one key difference: The angles $\theta$, $\alpha$ stem from different maps—(5) vs (11)—and thus lead to distinct functions $U_{\theta}$, $U_{\alpha}$.

### C. Mismatch

Returning to the examples of Sec. III, it is straightforward to illustrate the quantitative mismatch between the fluctuations in Eqs. (27) and (29). Indeed, take for simplicity the function $U(x, y) = x$ for which $|\nabla U|^2 = 1$, so that the bulk terms in Eqs. (27) and (29) are proportional to the droplet's area. The resulting total variances are plotted in Fig. 5 for flowers with $k = 3$ petals in Eqs. (19)–(20), showing that QH and CG predictions disagree and that the mismatch worsens as the anisotropy increases. The same conclusions can be drawn for the square droplet: In that case, the variances in Eqs. (27) and (29) turn out to be

$$ \mathrm{Var}[\hat{U}]_{\mathrm{QH}} \approx 0.5183083, \quad \mathrm{Var}[U]_{\mathrm{CG}} \approx 0.5469944, \quad (30) $$
<figure><img src="image_7.png" /><figcaption>FIG. 5. The total variance of $U(x, y) = x$ for flower droplets with $k = 3$ petals and $\lambda$ ranging from 0 to 0.4. Main figure: The variance (27) for QH droplets is compared with the CG result (29) obtained from a numerical evaluation of the conformal map (11), including an extrapolation based on Monte Carlo (MC) data for the CG at $\lambda = 0, 0.1, 0.2, 0.3$, and 0.4 [34]. The two match for $\lambda = 0$ but not $\lambda > 0$. Inset: Details of the extrapolation for $\lambda = 0.3$. The variance is extrapolated by a fit of the form $a + b/\sqrt{N}$ for $N = 64, 128$, and 256 particles. The intercept with the y axis yields the coefficient $a$, which is shown as an empty green circle and differs significantly from its QH counterpart. To further illustrate this, we include an MC point for $N = 1024$, which has a larger error bar but remains consistent with a leading $1/\sqrt{N}$ correction.</figcaption></figure>

which manifestly differ. Their exact values can be expressed using an infinite series and a generalized hypergeometric function, respectively [34].

### D. Microwave absorption

One way to measure the edge contribution to the QH fluctuation (27) is through microwave absorption experiments [29–33]. Indeed, consider a QH droplet perturbed by an electrostatic potential energy $U(x) \cos(\omega t)$, with driving frequency $\omega > 0$ close to the angular Fermi velocity $\omega_F$ of edge modes [57]. Time-dependent perturbation theory applied to low-energy eigenstates then predicts that the droplet absorbs this radiation at a rate [28]

$$ \Gamma(\omega) \sim \frac{1}{2\hbar^2} \sum_{p=1}^{\infty} p |U_p|^2 \delta(\omega - p\omega_F), \quad (31) $$

where $U_p = \int \frac{d\theta}{2\pi} e^{ip\theta} U_\theta$ is the $p$th Fourier mode of the function $U_\theta$ in Eq. (27). [Just as Eq. (27), the result (31) relies on the assumption that $U(x)$ is almost constant on length scales of order $\ell$.] Integrating $\Gamma(\omega)$ over all frequencies yields

$$ \begin{aligned} \int_0^\infty d\omega \, \Gamma(\omega) &= \frac{1}{2\hbar^2} \sum_{p=1}^\infty p |U_p|^2 \\ &= \oint \frac{d\theta_1 d\theta_2 (U_{\theta_1} - U_{\theta_2})^2}{2\hbar^2 \sin^2\left(\frac{\theta_1 - \theta_2}{2}\right)}, \end{aligned} \quad (32) $$

[1] K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980).

which is nothing but the edge contribution to the variance (27). In this sense, the latter is measurable in QH experiments, allowing direct verification of the deviation between QH and CG physics.

Note that the link in Eq. (32) between absorption and variance was to be expected, given the fluctuation-dissipation theorem. In fact, the latter ensures that the full variance (27)—including the bulk term—is the integral of the full absorption rate, including effects of higher Landau levels. No such bulk term appears in Eq. (32) since the rate (31) is limited to low-energy transitions within the LLL.

### V. CONCLUSIONS

In this paper, the plasma analogy was tested on integer QH droplets formed by anisotropic confining potentials. While it reproduces QH data in the bulk, the analogy generally fails at the edge—apart from exceptional, fine-tuned scenarios. Such discrepancies are expected to be directly relevant for experiments, both because of advances in measurement techniques for solid-state systems [32,33,59,60] and increases in the precision of quantum simulators [21–27]. In each case, deviations from leading-order results are becoming visible, which is where the conclusions of this paper are expected to be most striking.

We showed that the edge discrepancies stem from an elementary mismatch between QH and CG physics: The former is incompressible hydrodynamics, the latter is electrostatics. In geometric terms, QH is symplectic, while CG is conformal. This distinction entails different angular dependencies of the corresponding correlators in Eqs. (6) and (12), which in turn affects, e.g., fluctuations of local observables—quantities proposed to be measurable in the near future [29–33]. Such a breakdown of the plasma analogy is similarly expected near the edge of fractional QH states, as electron interactions are unlikely to improve the situation. It would be interesting to study this question for fractional droplets and other related systems where interactions play an important role, such as quantum dots or artificial atoms [61–63] and QH droplets with quasihole excitations [64].

### ACKNOWLEDGMENTS

We are grateful to Alexander Abanov, Yacin Ameur, Mathieu Beauvillain, Gabriel Cardoso, Laurent Charles, Benoit Doucot, Nathan Goldman, Gian Michele Graf, Hans Hansson, Leonardo Mazza, Gaultier Lambert, Alberto Nardin, Marios Petropoulos, Nicolas Rougerie, Gregory Schehr, Ood Shabtai, and Paul Wiegmann for fruitful discussions on closely related subjects. P.M. acknowledges financial support from the Wenner-Gren Foundations (Grant No. FT2022-0002). B.L. acknowledges financial support from the Swiss National Science Foundation (Postdoc.Mobility Grant No. 214461).

[2] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982).
[3] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982).

[4] R. B. Laughlin, Quantized Hall conductivity in two dimensions, Phys. Rev. B 23, 5632 (1981).

[5] R. B. Laughlin, Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50, 1395 (1983).

[6] B. Jancovici, Exact results for the two-dimensional one-component plasma, Phys. Rev. Lett. 46, 386 (1981).

[7] B. Jancovici, J. Lebowitz, and G. Manificat, Large charge fluctuations in classical Coulomb systems, J. Stat. Phys. 72, 773 (1993).

[8] B. Jancovici, Classical Coulomb systems: Screening and correlations revisited, J. Stat. Phys. 80, 445 (1995).

[9] P. J. Forrester and B. Jancovici, Two-dimensional one-component plasma in a quadrupolar field, Int. J. Mod. Phys. A 11, 941 (1996).

[10] P. Di Francesco, M. Gaudin, C. Itzykson, and F. Lesage, Laughlin's wave functions, Coulomb gases and expansions of the discriminant, Int. J. Mod. Phys. A 09, 4257 (1994).

[11] P. Wiegmann and A. Zabrodin, Large scale correlations in normal non-Hermitian matrix ensembles, J. Phys. A: Math. Gen. 36, 3411 (2003).

[12] T. Leblé and S. Serfaty, Fluctuations of two dimensional Coulomb gases, Geom. Funct. Anal. 28, 443 (2018).

[13] B. Jancovici, Classical Coulomb systems near a plane wall. II, J. Stat. Phys. 29, 263 (1982).

[14] A. Alastuey and B. Jancovici, On potential and field fluctuations in two-dimensional classical charged systems, J. Stat. Phys. 34, 557 (1984).

[15] Ph. Choquard, B. Piller, and R. Rentsch, On the dielectric susceptibility of classical Coulomb systems. II, J. Stat. Phys. 46, 599 (1987).

[16] J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, J. Math. Phys. 6, 440 (1965).

[17] V. L. Girko, Elliptic law, Theory Probab. Appl. 30, 677 (1986).

[18] B. I. Halperin, Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25, 2185 (1982).

[19] T. Champel and S. Florens, Quantum transport properties of two-dimensional electron gases under high magnetic fields, Phys. Rev. B 75, 245326 (2007).

[20] T. Champel, S. Florens, and L. Canet, Microscopics of disordered two-dimensional electron gases under high magnetic fields: Equilibrium properties and dissipation in the hydrodynamic regime, Phys. Rev. B 78, 125302 (2008).

[21] R. J. Fletcher, A. Shaffer, C. C. Wilson, P. B. Patel, Z. Yan, V. Crépel, B. Mukherjee, and M. W. Zwierlein, Geometric squeezing into the lowest Landau level, Science 372, 1318 (2021).

[22] B. Mukherjee, A. Shaffer, P. B. Patel, Z. Yan, C. C. Wilson, V. Crépel, R. J. Fletcher, and M. Zwierlein, Crystallization of bosonic quantum Hall states in a rotating quantum gas, Nature (London) 601, 58 (2022).

[23] R. Yao, S. Chi, B. Mukherjee, A. Shaffer, M. Zwierlein, and R. J. Fletcher, Observation of chiral edge transport in a rapidly rotating quantum gas, Nat. Phys. 20, 1726 (2024).

[24] V. Crépel, R. Yao, B. Mukherjee, R. Fletcher, and M. Zwierlein, Geometric squeezing of rotating quantum gases into the lowest Landau level, Comptes Rendus. Physique 24, 241 (2023).

[25] J. Léonard, S. Kim, J. Kwan, P. Segura, F. Grusdt, C. Repellin, N. Goldman, and M. Greiner, Realization of a fractional quantum Hall state with ultracold atoms, Nature (London) 619, 495 (2023).

[26] F. Binanti, N. Goldman, and C. Repellin, Spectroscopy of edge and bulk collective modes in fractional Chern insulators, Phys. Rev. Res. 6, L012054 (2024).

[27] C. Braun, R. Saint-Jalm, A. Hesse, J. Arceri, I. Bloch, and M. Aidelsburger, Real-space detection and manipulation of topological edge modes with ultracold atoms, Nat. Phys. 20, 1306 (2024).

[28] B. Oblak, B. Lapierre, P. Moosavi, J.-M. Stéphan, and B. Estienne, Anisotropic quantum Hall droplets, Phys. Rev. X 14, 011030 (2024).

[29] V. I. Talyanskii, A. V. Polisski, D. D. Arnone, M. Pepper, C. G. Smith, D. A. Ritchie, J. E. Frost, and G. A. C. Jones, Spectroscopy of a two-dimensional electron gas in the quantum-Hall-effect regime by use of low-frequency edge magnetoplasmons, Phys. Rev. B 46, 12427 (1992).

[30] I. V. Andreev, V. M. Muravev, D. V. Smetnev, and I. V. Kukushkin, Acoustic magnetoplasmons in a two-dimensional electron system with a smooth edge, Phys. Rev. B 86, 125315 (2012).

[31] J. Cano, A. C. Doherty, C. Nayak, and D. J. Reilly, Microwave absorption by a mesoscopic quantum Hall droplet, Phys. Rev. B 88, 165305 (2013).

[32] A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner, M. J. Manfra, A. C. Doherty, and D. J. Reilly, On-chip microwave quantum Hall circulator, Phys. Rev. X 7, 011007 (2017).

[33] E. Frigerio, G. Rebora, M. Ruelle, H. Souquet-Basiège, Y. Jin, U. Gennser, A. Cavanna, B. Plaçais, E. Baudin, J.-M. Berroir et al., Gate tunable edge magnetoplasmon resonators, Commun. Phys. 7, 314 (2024).

[34] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.7.023134 for additional information. It has three parts: Part A reviews QH droplets, and Part B reviews CGs. Part C complements the main text by showing that the canonical angle along a QH edge generally fails to extend to the global conformal map required in the corresponding CG. The Supplemental Material also includes Refs. [65,66].

[35] In Hamiltonian mechanics, A measures the phase-space area enclosed by an energy level. Here, phase space is the noncommutative space of guiding-center positions, so action and area are synonymous.

[36] Our normalization of the action coordinate A differs from that of the action coordinate K in Ref. [28]. The two are related by $A = 2\pi e^2 K$. Relatedly, $V_0(\cdot)$ here differs from that in Ref. [28] by a factor $2\pi$ in the argument.

[37] V. Fock, Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld, Z. Phys. 47, 446 (1928).

[38] C. G. Darwin, The diamagnetism of the free electron, Math. Proc. Camb. Philos. Soc. 27, 86 (1931).

[39] S. A. Trugman, Localization, percolation, and the quantum Hall effect, Phys. Rev. B 27, 7539 (1983).

[40] R. Joynt and R. E. Prange, Conditions for the quantum Hall effect, Phys. Rev. B 29, 3303 (1984).

[41] T. Champel and S. Florens, Electron quantum dynamics in closed and open potentials at high magnetic fields: Quantization
and lifetime effects unified by semicoherent states, Phys. Rev. B 80, 125322 (2009).

[42] L. Charles, Berezin-Toeplitz operators, a semi-classical approach, Commun. Math. Phys. 239, 1 (2003).

[43] L. Charles, Quasimodes and Bohr-Sommerfeld conditions for the Toeplitz operators, Commun. Partial Differ. Equ. 28, 1527 (2003).

[44] O. Shabtai, Off-diagonal estimates of partial Bergman kernels on $S^1$-symmetric Kähler manifolds, J. Geom. Anal. 35, 166 (2025).

[45] F. J. Dyson, Statistical theory of the energy levels of complex systems. II, J. Math. Phys. 3, 157 (1962).

[46] We use the same letter $\Omega$ to denote both the domain occupied by a QH droplet and that occupied by a CG, even though they are a priori unrelated. The plasma analogy ultimately requires choosing the domains to be identical.

[47] For comparison with QH droplets, we assume that the disk $D$ and the droplet $\Omega$ in Eq. (11) have the same area. This entails no loss of generality, since it can be ensured by composing $G(x)$ with an overall dilation that does not affect the angle $\alpha$ introduced below the map (11).

[48] H. Hedenmalm and A. Wennman, Planar orthogonal polynomials and boundary universality in the random normal matrix model, Acta Math. 227, 309 (2021).

[49] Y. Ameur and J. Cronvall, Szegő type asymptotics for the reproducing kernel in spaces of full-plane weighted polynomials, Commun. Math. Phys. 398, 1291 (2023).

[50] Y. Hatsugai, Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett. 71, 3697 (1993).

[51] Geometrically, any smooth real function that satisfies Eq. (15) defines an (orientation-preserving) diffeomorphism of the circle.

[52] E. B. Saff and V. Totik, Logarithmic Potentials with External Fields (Springer, Berlin, Heidelberg, 1997).

[53] T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mapping, Cambridge Monographs on Applied and Computational Mathematics (Cambridge University Press, Cambridge, 2002).

[54] A. Zabrodin and P. Wiegmann, Large-$N$ expansion for the 2D Dyson gas, J. Phys. A: Math. Gen. 39, 8933 (2006).

[55] B. D. Bruyne, P. L. Doussal, S. N. Majumdar, and G. Schehr, Linear statistics for Coulomb gases: Higher order cumulants, J. Phys. A: Math. Theor. 57, 155002 (2024).

[56] Y. Ameur, H. Hedenmalm, and N. Makarov, Random normal matrices and Ward identities, Ann. Probab. 43, 1157 (2015).

[57] The angular Fermi velocity $\omega_F$ typically lies in the microwave range in experiments where this has been carried out; see, e.g., Ref. [58].

[58] H. Kamata, T. Ota, K. Muraki, and T. Fujisawa, Voltage-controlled group velocity of edge magnetoplasmon in the quantum Hall regime, Phys. Rev. B 81, 085329 (2010).

[59] K. Hashimoto, T. Champel, S. Florens, C. Sohrmann, J. Wiebe, Y. Hirayama, R. A. Römer, R. Wiesendanger, and M. Morgenstern, Robust nodal structure of Landau level wave functions revealed by Fourier transform scanning tunneling spectroscopy, Phys. Rev. Lett. 109, 116805 (2012).

[60] K. Hashimoto, C. Sohrmann, J. Wiebe, T. Inaoka, F. Meier, Y. Hirayama, R. A. Römer, R. Wiesendanger, and M. Morgenstern, Quantum Hall transition in real space: From localized to extended states, Phys. Rev. Lett. 101, 256802 (2008).

[61] P. A. Maksym and T. Chakraborty, Quantum dots in a magnetic field: Role of electron-electron interactions, Phys. Rev. Lett. 65, 108 (1990).

[62] S.-R. Eric Yang and A. H. MacDonald, Coupling between edge and bulk in strong-field quantum dots, Phys. Rev. B 66, 041304(R) (2002).

[63] E. Lieb, J. Solovej, and J. Yngvason, Ground states of large quantum dots in magnetic fields, in *The Stability of Matter: From Atoms to Stars*, edited by W. Thirring (Springer, Berlin, Heidelberg, 2001), p. 171.

[64] G. Lambert, D. Lundholm, and N. Rougerie, Quantum statistics transmutation via magnetic flux attachment, Prob. Math. Phys. 4, 803 (2023).

[65] G. Cardoso, J.-M. Stéphan, and A. G. Abanov, The boundary density profile of a Coulomb droplet. Freezing at the edge, J. Phys. A: Math. Theor. 54, 015002 (2020).

[66] F. Stillinger and R. Lovett, Ion-pair theory of concentrated electrolytes. I. Basic concepts, J. Chem. Phys. 48, 3858 (1968).