text:
\section{Introduction}
\label{Sec:Intro}

Quantum Hall (QH) droplets are mesoscopic two-dimensional (2D) electron gases placed in a strong perpendicular magnetic field and confined by some electrostatic potential: see Fig.~\ref{fiDrop}. They lie at the heart of the QH effect \cite{vonKlitzing, Tsui, LaughlinAnomalous} and provide a key benchmark for topological phases of matter as a whole. In practice, however, the vast majority of detailed analytical studies of QH droplets are limited to highly symmetric cases, typically involving isotropic traps or harmonic potentials that are translation-invariant in one direction \cite{DelplaceMontambaux:2010, Montambaux:2011}. This is especially troubling as far as edge properties are concerned, since these are sensitive to the shape of the trap and determine the system's low-energy excitations \cite{Bahcall:1991an, Frohlich, Wen:1990qp, Wen:1990se, Wen}.

The goal of this paper is to address this lack of analytical results by predicting universal aspects of many-body observables near the edge of essentially \i{any} anisotropic droplet. We achieve this by providing general, explicit, one-line formulas for the density, current, and correlations in the regime of strong magnetic fields. We also study the corresponding low-energy edge modes, which are described by a free-fermion chiral conformal field theory (CFT) whose Fermi velocity is constant provided distances along the boundary are measured by the canonical angle variable determined by the potential. These predictions are likely to be observable thanks to direct local imaging techniques in condensed matter systems \cite{Weitz, Ahlswede, Weis, Ilani, Steele, kamiyama_2022_fqhe} or quantum simulators \cite{Bretin, Schweikhard, Cooper, Dellabetta, Goldman1, Goldman2, Fletcher, Mukherjee:2021jjl}.

This is not the first time such questions appear in the literature. Indeed, random potentials with no symmetries are essential to model disorder, whose importance for the robustness of QH physics is hard to overstate \cite{Halperin, Trugman, Joynt}. An especially relevant series of works in that context is \cite{Champel1, Champel2}, which study the density and current of QH droplets with arbitrary potentials, at finite temperature, generally including Landau level mixing, in the semiclassical limit of strong magnetic fields and weak traps \cite{Trugman, Joynt}. However, the coherent states used in these references do not allow for any resolution at the single-particle level, precluding the computation of low-energy dynamics and long-range correlations along the boundary. Our objective here is instead to find explicit wave functions that depend on the shape of the edge, and use that as a starting point for many-body objects.

Regarding electronic edge correlations, similar questions have been addressed in the context of classical 2D Coulomb gases, where holomorphic methods play a key role \cite{ChoquardEtAl:1987, Jancovici:1995, ZabrodinWiegmann:2006}. More broadly, the results put forward here may be seen as microscopic, first-principles derivations of quantities that are normally studied within less controlled approximation schemes in the geometry of the QH effect \cite{Bradlyn:2012ea, Bradlyn, Abanov:2014ula, Can1, Can2, Can3, Klevtsov:2015eda}. Our hope is thus to build a bridge between these theoretical works and concrete observations that may soon be accessible in tabletop experiments with a high degree of control on the confining potential \cite{Fletcher, Mukherjee:2021jjl}.

Here is the plan of the paper. To begin, Sec.~\ref{semeth} summarizes our methods and results, avoiding all technical details. The next two sections are devoted to one-body physics in the lowest Landau level: Sec.~\ref{sesdiff} first discusses generalities on semiclassical holomorphic wave functions, while Sec.~\ref{sescale} presents a detailed computation of the semiclassical energy spectrum in a class of `edge-deformed' potentials of particular interest. This finally leads to Sec.~\ref{semany}, where we investigate many-body densities, currents, correlations, and low-energy edge modes in anisotropic traps. We conclude in Sec.~\ref{seccc} by discussing several follow-ups and open questions. To streamline the text, non-essential details are deferred to Apps.~\ref{appiso}--\ref{appexp}.

\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{fig1_potential_and_droplet.pdf}
\caption{2D electron droplet (shaded area) placed in a strong perpendicular magnetic field and confined by a typical anisotropic edge-deformed potential well \eqref{laviz}. At leading order in the thermodynamic limit, the droplet's boundary (thick black curve) coincides with the equipotential of the trap at the Fermi energy.}
\label{fiDrop}
\end{figure}

\section{Setup and main results}
\label{semeth}

This section summarizes our methods and key results. To start, we describe the general setup: a semiclassical limit (strong magnetic field, small magnetic length) in the lowest Landau level (LLL) \cite{Trugman, Joynt, GirvinLLL}. We then introduce edge-deformed potentials and present their approximate energy eigenstates, before finally giving simple formulas for the corresponding local many-body observables and dynamics in the vicinity of the edge. Some of these findings are depicted in Fig.~\ref{fipott}.

\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{fig2_mb_densities.pdf}
\caption{(a) A planar plot of the many-body density \eqref{ressum_rho_edge} along with several equipotentials (dashed lines), for a droplet of $N=100$ electrons confined by the edge-deformed trap of Fig.~\ref{fiDrop}. The constancy of the bulk density and its fall at the boundary are manifest. (b) The current's norm \eqref{ressum_J_edge} for the same droplet, together with the edge (black line) on which it is localized. (c) The correlation function \eqref{ressum_C_edge} for the same droplet, seen as a function of $\textbf x_2=(x,y)$ with $\textbf x_1=(\ell\sqrt{N\lambda(0)},0)$ denoted by a cross and fixed at the edge. Long-range correlations along the boundary are clearly visible.}
\label{fipott}
\end{figure*}

\subsection{Semi-classical limit in the LLL}

This work concerns spin-polarized non-interacting electrons of mass $M$ and charge $q$ in a 2D plane. Each electron is governed by a Landau Hamiltonian with an anisotropic potential $V(\textbf x)$,
\begin{equation}
\label{h}
H_{\text{1-body}}
=
\frac{1}{2M}(\textbf p-q\textbf A)^2+V(\textbf x),
\end{equation}
where $\textbf A$ is the vector potential of the magnetic field $\textbf B=\text{d}\textbf A$. (We view $\textbf A$ as a one-form, which simplifies some notation but is otherwise inconsequential.) We shall assume that $V(\textbf x)$ is `monotonous', by which we mean that it has a unique global minimum away from which it grows monotonously, but it is otherwise general. Consequently, the level curves or `equipotentials' of $V(\textbf x)$ are nested and take the form shown in Fig.~\ref{fipott}(a). We assume throughout that the potential is weak relative to the magnetic field \cite{Trugman, Joynt, Geller1, Geller2, Champel1, Champel2}, and that it is nearly constant on length scales comparable to the magnetic length
\begin{equation}
\label{ell}
\ell^2
\equiv
\frac{\hbar}{qB}>0.
\end{equation}
In that regime, the potential is a small perturbation of the pure Landau Hamiltonian $\propto(\textbf p-q\textbf A)^2$ and the eigenstates of \eqref{h} are expected to be well-approximated by wave functions in the LLL. For instance, if the potential $V(\textbf x)=V_0(r^2/2)$ is isotropic, the eigenstates of \eqref{h} have some definite angular momentum and reduce at strong $\textbf B$ to standard LLL wave functions in symmetric gauge:
\begin{equation}
\label{phim}
\phi_m(\textbf x)
=
\frac{1}{\sqrt{2\pi\ell^2}}\,\frac{z^m}{\sqrt{m!}}\,\mathrm{e}^{-|z|^2/2},
\end{equation}
where $m\geq0$ is an integer angular momentum and we have introduced the dimensionless complex coordinate
\begin{equation}
\label{zdef}
z
\equiv
\frac{x+\mathrm{i} y}{\sqrt{2}\ell}.
\end{equation}
Each wave function \eqref{phim} reaches its maximum on the circle $|z|=\sqrt{m}$, away from which it decays in a Gaussian manner within a magnetic length. Our goal will be to obtain similar approximate eigenstates in \i{anisotropic} traps, using the squared magnetic length \eqref{ell} as a small expansion parameter \cite{Note1}. Equivalently, we shall carry out a semiclassical (small $\hbar$), high field expansion (large $B$).

In practice, the projection to the LLL is implemented thanks to the (one-body) operator $P\equiv \sum_{m=0}^{\infty}|\phi_m\rangle\langle\phi_m|$, whose kernel can be read off from the wave functions \eqref{phim}:
\begin{equation}
\label{proj}
\langle z,\bar z|P|w,\bar w\rangle
=
\frac{1}{2\pi\ell^2}\,\mathrm{e}^{-(|z|^2+|w|^2)/2}\,\mathrm{e}^{z\bar w}.
\end{equation}
This is manifestly Gaussian and reduces to a delta function in the formal semiclassical limit $\ell\to0$. At small but finite $\ell$, the projection \eqref{proj} makes space non-commutative in the sense that the LLL-projected position operators $(x,y)$ satisfy the Heisenberg algebra
\begin{equation}
\label{pxp}
[PxP, PyP]
=
\mathrm{i}\ell^2.
\end{equation}
One can thus think of the plane $\mathbb{R}^2$ as a `phase space' whose canonical variables are $x,y$. This interpretation pervades much of the QH literature \cite{Bellissard, GirvinLLL, Girvin:1986zz, Pasquier:2007nda, Iso:1992aa, Susskind:2001fb, Polychronakos:2001mi, Hellerman:2001rj, Fradkin:2002qw, Dong:2020bkt, Du:2021hes} and will similarly affect our discussion. Indeed, projecting the Hamiltonian \eqref{h} to the LLL and looking for its spectrum leads to the eigenvalue equation
\begin{equation}
\label{claim}
P\,VP|\psi\rangle
=
E|\psi\rangle,
\end{equation}
where the unknowns are the energy $E$ and the quantum state $|\psi\rangle\in\text{LLL}$. Note that the kinetic term of \eqref{h} has disappeared in \eqref{claim}: the potential itself plays the role of an effective Hamiltonian in the non-commutative phase space $(x, y)$.

Exact solutions of Eq.\ \eqref{claim} are generally out of reach, so one has to resort to approximations. The semiclassical one that we shall use is well known in the QH context \cite{Trugman, Joynt, Champel1, Champel2, Charles:2003a, Charles:2003b}. Accordingly, we will look for solutions of Eq.\ \eqref{claim} labeled by a large quantum number $m\in\mathbb{N}$, seen as a generalization of angular momentum. This large $m$ limit is accompanied by a small $\ell$ limit, such that the area $2\pi\ell^2m$ remains finite. In that regime, the $m^{\text{th}}$ eigenstate is approximately Gaussian and localized on an equipotential $\gamma_m$ of $V(\textbf x)$, enclosing a quantized area such that the Bohr-Sommerfeld condition holds:
\begin{equation}
\label{oint}
\oint_{\gamma_m}x\,\text{d} y
=
2\pi\ell^2\,m.
\end{equation}
Equivalently, the flux of the magnetic field through the area enclosed by $\gamma_m$ is $m$ times the flux quantum. The energy of the $m^{\text{th}}$ state is then
\begin{equation}
\label{ev}
E_m
=
E^0_m
+\ell^2E^1_m
+O(\ell^4),
\end{equation}
where $E^0_m=V(\gamma_m)$ is the leading classical approximation and the quantum correction $E^1_m$ involves the Laplacian of the potential and the curvature of the equipotential $\gamma_m$ \cite{Charles:2003a, Charles:2003b}. Note that the more familiar Wentzel-Kramers-Brillouin (WKB) approximation of 1D quantum mechanics \cite{BenderOrszag:1999} includes (topological) Maslov corrections on the right-hand side of \eqref{oint}; we will find similar corrections below, although their interpretation as topological invariants is prevented by a subtle distinction between real and K\"{a}hlerian polarizations in geometric quantization \cite{Charles:2003a, Charles:2003b}.

\subsection{One-body results}

The semiclassical limit just outlined applies to any (monotonous) weak potential. In practice, our main concern is the physics of QH droplets near the edge, where the details of the bulk potential are irrelevant. Most of our explicit results will therefore be given for `edge-deformed' potentials, obtained as follows. Consider any monotonously increasing function $V_0(t)$ for $t\geq0$, and let $\lambda(\varphi)$ be any strictly positive $2\pi$-periodic function of the angle $\varphi\in[0,2\pi)$; normalize $\lambda$ so that $\oint\text{d}\varphi\,\lambda(\varphi)=4\pi$, where we write $\oint\text{d}\varphi$ as a shorthand for $\int_0^{2\pi}\text{d}\varphi$. Then adopt polar coordinates in the plane such that $x+\mathrm{i} y=r\,\mathrm{e}^{\mathrm{i}\varphi}$ and define the potential
\begin{equation}
\label{laviz}
V(r,\varphi)
=
V_0 \left(\frac{r^2}{\lambda(\varphi)}\right).
\end{equation}
We refer to this as an \i{edge-deformed} trap because it results from a deformation $r^2\mapsto r^2/\lambda(\varphi)$ that changes the shape of the boundary of isotropic droplets in a finite and smooth way, even in the thermodynamic limit where the droplet's area goes to infinity \cite{EstienneOblakSDiff}. In fact, the corresponding infinitesimal transformations are expected to be conformal transformations of the edge CFT \cite{Cappelli1, Cappelli:1992kf, Cappelli:1993ei, Cappelli:1994wb, Cappelli:1995yk, Cappelli:1996pg}. The class of potentials \eqref{laviz} is thus exhaustive, at least as far as edge effects are concerned.

The traps \eqref{laviz} turn out to allow for explicit calculations of the semiclassical energy spectrum, generalizing the known isotropic results reviewed in App.~\ref{appiso}. Indeed, we show in Sec.~\ref{sescale} that the corresponding eigenfunctions, solving \eqref{claim} in the LLL, are Gaussians localized on equipotentials $r=\ell\sqrt{m \lambda(\varphi)}$ at large quantum numbers $m$. They can be written in polar coordinates as
\begin{equation}
\label{e13}
\psi_m(\textbf x)
\sim
\frac{\mathrm{e}^{\mathrm{i}\Theta_{m}(\textbf x)}}{\sqrt{2\pi\ell^2 \sigma(\varphi)}}\,
\frac{\mathrm{e}^{-a^2/\sigma(\varphi)^2}}{(2\pi m)^{1/4}},
\end{equation}
where $\Theta_{m}(\textbf x)$ is a position-dependent phase, $a\equiv\bigl(r-\ell\sqrt{m\lambda(\varphi)}\bigr)\big/\ell \sqrt{\lambda(\varphi)}$ is a dimensionless radial coordinate that measures the distance from the equipotential, and
\begin{equation}
\label{sigma_lambda}
\sigma(\varphi)
\equiv \sqrt{ \frac{2}{\lambda(\varphi)} } \sqrt{1 + \left[ \frac{\lambda'(\varphi)}{2\lambda(\varphi)} \right]^2}
\end{equation}
is an angle-dependent width. We stress that this exhibits the expected `quantum smearing' of wave functions in a strong but finite magnetic field \cite{Champel1, Champel2}, which would be missed by the leading classical approximations $\ell^2=0$.

As for the energy of the state \eqref{e13}, its expansion \eqref{ev} up to neglected $O(\ell^4)$ contributions is
\begin{equation}
\label{ressum_E_m}
E_m
\sim
V(\gamma_{m})
+
\frac{\ell^2}{2}\Omega_{m}
\biggl[1
+
\Bigl(1+\frac{\Gamma_{m}}{\Omega_{m}}\Bigr)
\oint\frac{\text{d}\varphi}{4\pi}\lambda(\varphi)\sigma(\varphi)^2 \biggr],
\end{equation}
where $V(\gamma_{m})=V_0\bigl(\ell^2m\bigr)$ is the leading term, while the first quantum correction involves derivatives $\Omega_{m} \equiv V_0'(\ell^2m)>0$ and $\Gamma_{m} \equiv \ell^2m\,V_0''(\ell^2m)$. Each $m^{\text{th}}$ energy is thus determined by the potential and its derivatives at an equipotential that satisfies the quantization condition \eqref{oint}, in accordance with general theorems for holomorphic WKB theory \cite{Charles:2003a, Charles:2003b}.

\subsection{Many-body results}

Now consider the ground state of a large number $N\gg1$ of free spin-polarized electrons, each governed by the single-particle Hamiltonian \eqref{h}. This ground state is a Slater determinant of wave functions whose large $m$ behavior is the Gaussian \eqref{e13}. As we show in Sec.~\ref{semany}, the corresponding many-body density, current, correlations, and low-energy effective action can all be written in closed form in terms of $\lambda(\varphi)$ and the number $N$ of fermions. The density $\rho(\textbf x)=\sum_{m=0}^{N-1}|\psi_m(\textbf x)|^2$ thus satisfies the bulk behavior $\rho\sim\frac{1}{2\pi\ell^2}$, while its form near the edge is given by a complementary error function:
\begin{equation}
\label{ressum_rho_edge}
\rho(r,\varphi)
\sim
\frac{1}{4\pi\ell^2}
\erfc \biggl( \frac{\sqrt{2}\,a}{\sigma(\varphi)} \biggr),
\end{equation}
where $a\equiv\bigl( r - \ell \sqrt{N\lambda(\varphi)} \bigr) \big/ \ell \sqrt{\lambda(\varphi)}$ is again a dimensionless radial coordinate, now measuring the distance from the edge at $r_{\text{edge}}=\ell\sqrt{N\lambda(\varphi)}$. As a result, the ground state forms a star-shaped droplet whose boundary has an angle-dependent width \eqref{sigma_lambda} inherited from that of one-body wave functions.
Turning to the current $\textbf J=\sum_{m=0}^{N-1}\tfrac{1}{2i}(\psi_m^*\text{d}\psi_m-\psi_m\text{d}\psi_m^*-2iq\textbf A|\psi_m|^2)$, we write it as a one-form in polar coordinates to find
\begin{equation}
\label{ressum_J_edge}
\textbf J(r, \varphi)
\sim
-
\frac{\exp\left(-\frac{2a^2}{\sigma(\varphi)^2} \right)}{(2\pi\ell^2)^{3/2} \sigma(\varphi)}
\Bigl(
\ell\sqrt{N} \,\text{d}\varphi + \frac{\lambda'(\varphi)}{2\lambda(\varphi)^{3/2}} \,\text{d} r\Bigr).
\end{equation}
This is localized on the edge and tangent to it, missing the bulk behavior $J_i\propto\varepsilon_{ij}\partial_jV$ as expected in the LLL \cite{Geller1, Geller2}. Finally, the two-point correlation function $C(\textbf x_1,\textbf x_2)=\sum_{m=0}^{N-1}\psi_m^*(\textbf x_1)\psi_m(\textbf x_2)$ behaves near the edge as
\begin{equation}
\label{ressum_C_edge}
C(\textbf x_1,\textbf x_2)
\sim
\frac{\mathrm{e}^{\mathrm{i}\Theta_{N}(\textbf x_1,\textbf x_2)}}{4\pi\ell^2 \sqrt{\sigma(\varphi_1)\sigma(\varphi_2)}}
\frac{\mathrm{i}\exp \left( -\frac{a^2}{\sigma(\varphi_1)^2}-\frac{b^2}{\sigma(\varphi_2)^2} \right)}%
{\sqrt{2\pi N}\,\sin\bigl( \int_{\varphi_2}^{\varphi_1} \frac{\text{d}\theta}{4} \lambda(\theta) \bigr)}
\end{equation}
with $a\equiv\bigl( |\textbf x_1| - \ell \sqrt{N\lambda(\varphi_1)} \bigr) \big/ \ell \sqrt{\lambda(\varphi_1)}$
and similarly for $b$ in polar coordinates $(|\textbf x_1|,\varphi_1)$ and $(|\textbf x_2|,\varphi_2)$, respectively,
while $\Theta_{N}(\textbf x_1,\textbf x_2)$ is a complicated overall phase. Note again the Gaussian localization at the edge, as well as the long-range correlator $\propto\sin(...)^{-1}$ typical of gapless fermions. Indeed, we eventually confirm that the underlying low-energy edge modes are described by a chiral CFT of free fermions: see the action functional \eqref{lactose} below. The corresponding Fermi velocity is constant along the boundary when measured in terms of the angle variable of the potential \eqref{laviz}, namely $\theta(\varphi)\equiv\int_0^{\varphi}\text{d}\varphi'\,\lambda(\varphi')/2$.

\section{Anisotropic states from area-preserving maps}
\label{sesdiff}

This section presents the WKB ansatz [see \eqref{final}] that forms the basis of all our later considerations. The structure is ultimately quite simple: given a monotonous potential $V(\textbf x)$, we pick one of its equipotentials, $\gamma_m$, with quantized area \eqref{oint}. We then build a wave function with winding $m$, perfectly localized on $\gamma_m$, and finally project it to the LLL using the operator \eqref{proj}. General theorems \cite{Charles:2003a, Charles:2003b} ensure that LLL-projected eigenstates satisfying \eqref{claim} can indeed be built in this way. The detailed application of this method to edge-deformed traps \eqref{laviz} is given in Sec.~\ref{sescale}.

A note: what follows relies on the mathematics of area-preserving diffeomorphisms, which is not reviewed in detail. We refer instead to \cite{EstienneOblakSDiff} for an introduction whose language is similar to that adopted here. For more general discussions in the symplectic context, see \cite{IzKhMo:2016, DaSilva}.

\subsection{Potentials in action-angle variables}
\label{ssepac}

Let us be more precise about the geometry of the setup, remaining at the classical level for now.
We pick a smooth potential $V(\textbf x)$ and assume as in Sec.~\ref{semeth} that it is monotonous. Its unique global minimum is thus surrounded by nested level curves, and one can always find an area-preserving deformation of the plane that sends each equipotential on a circle \cite{IzKhMo:2016}. In other words, one can find an invertible smooth map $\textbf F:\mathbb{R}^2\to\mathbb{R}^2$ with unit Jacobian such that
\begin{equation}
\label{fivi}
V\big(\textbf F(\textbf x)\big)
=
V_0(r^2/2),
\end{equation}
where the trap on the right-hand side is isotropic (it only depends on $r=|\textbf x|$). If $\textbf F$ is the identity (or a rotation around the origin),
then $V$ was isotropic to begin with and its eigenstates satisfying \eqref{claim} are the standard wave functions \eqref{phim} with definite angular momentum. In the more general case of arbitrary $V$, Eq.\ \eqref{fivi} suggests using $\textbf F$ to map  the eigenstates \eqref{phim} on those corresponding to our general $V(\textbf x)$.

The existence of $\textbf F$ in \eqref{fivi} is guaranteed by the monotonicity of $V$, and is equivalent to the existence of globally well-defined action-angle variables. In fact, we can use this to write $\textbf F$ in a more explicit form that will be useful below. Let therefore $(\ell^2K,\theta)$ be action-angle coordinates for the potential $V(\textbf x)$ \cite{Note2}, where $K\geq0$ is dimensionless and $\theta\in[0,2\pi)$ is a genuine angle. They are normalized so that $\ell^2\text{d} K\wedge\text{d}\theta=\text{d} x\wedge\text{d} y$, which is to say that their Poisson bracket reads $\{\ell^2K, \theta\}=\ell^2$ in terms of the phase space $(x,y)$ with bracket \eqref{pxp}. Then the map $(x,y)\mapsto(\ell^2 K,\theta)$ is an area-preserving diffeomorphism in terms of which $V(\textbf x) = V_0(\ell^2K(\textbf x))$ is invariant under rotations of $\theta$. To be specific, write these coordinates as functions $K(x,y)$ and $\theta(x,y)$ and let the inverse be $x=F(K,\theta)$ and $y=G(K,\theta)$ for some functions $(F,G)$; this inverse is nothing but the deformation $\textbf F$ in \eqref{fivi}. In other words, knowing the action-angle variables of a potential $V$ allows us to map it on its (unique) isotropic cousin $V_0$, which in turn can be used to relate the corresponding anisotropic eigenstates to those in \eqref{phim}.

It should be clear that these considerations apply to any monotonous anisotropic trap, in which case one generally encounters intricate area-preserving maps with complicated action-angle variables. In Sec.~\ref{sescale}, we will argue that most of these difficulties wash away when focusing on edge physics, whereupon the only relevant maps are the `edge deformations' mentioned below Eq.\ \eqref{laviz}. For now, we remain general and turn to quantum considerations.

\subsection{Anisotropic eigenstates}
\label{sseniseg}

Using the action-angle variables $(\ell^2K,\theta)$ for $V(\textbf x)$, one can concretize the statements around Eqs.\ \eqref{oint}--\eqref{ev} into formulas and eventually obtain anisotropic eigenfunctions that satisfy \eqref{claim}.
Indeed, the Bohr-Sommerfeld quantization condition \eqref{oint} implies that the equipotential $\gamma_m$ is the set of points in $\mathbb{R}^2$ where $K = m$. Now consider the following quantum state, perfectly localized on $\gamma_m$:
\begin{equation}
\label{psim}
|\Psi_m\rangle
\equiv
2\pi\ell^2
\oint\text{d}\theta\,
n(\theta)
\,\mathrm{e}^{\mathrm{i} m\theta} \big|F(m,\theta),G(m,\theta)\big\rangle,
\end{equation}
where the normalization $2\pi\ell^2$ is included for later convenience, the `wave function' $\langle\textbf x|F(m,\theta),G(m,\theta)\rangle=\delta^2\big(\textbf x-\textbf F(m,\theta)\big)$ is a delta function,  and
$n(\theta)$ is some complex periodic function. The latter does not wind upon completing one turn in the plane along the equipotential, meaning that all the winding of \eqref{psim} is encoded in the phase $\mathrm{e}^{\mathrm{i} m\theta}$.

We stress that \eqref{psim} is analogous to the standard WKB ansatz $\psi(x)\sim\mathrm{e}^{\mathrm{i} S_0(x)/\hbar}\mathrm{e}^{\mathrm{i} S_1(x)}$ in 1D. Indeed, the phase $\mathrm{e}^{\mathrm{i} m\theta}$ is the leading classical contribution $\mathrm{e}^{\mathrm{i} S_0/\hbar}$ for $m \gg 1$, corresponding to the `geometrical optics' approximation of the wave function, while $n(\theta)$ is the `physical optics' quantum correction $\mathrm{e}^{\mathrm{i} S_1}$ that eventually needs to satisfy a transport equation in order for the Schr\"odinger equation to hold \cite{BenderOrszag:1999}. The only difference lies in the interpretation of areas in the plane as values of an `action', which ultimately stems from the non-commutative geometry \eqref{pxp} of LLL physics. Note that $n(\theta)$ is the only unknown in \eqref{psim}. Indeed, most of the WKB method below will concern the derivation of a transport equation for $n(\theta)$ from the requirement that \eqref{claim} be satisfied.

Starting from Eq.\ \eqref{psim}, it is straightforward to build a state in the LLL thanks to the projector \eqref{proj}: denoting
\begin{equation}
\label{e32}
\psi_m(z,\bar z)
\equiv
\langle z,\bar z|P|\Psi_m\rangle,
\end{equation}
one finds the wave function
\begin{multline}
\label{final}
\psi_m(z,\bar z)
=
\mathrm{e}^{-|z|^2/2}
\oint\text{d}\theta\,n(\theta)\,\mathrm{e}^{\mathrm{i} m\theta} \qquad\quad \\
\times\,\mathrm{e}^{-[F(m,\theta)^2 + G(m,\theta)^2] / 4\ell^2}\,
\mathrm{e}^{z[F(m,\theta) - \mathrm{i} G(m,\theta)] / \sqrt{2}\ell}.
\end{multline}
This is manifestly of the form $\mathrm{e}^{-|z|^2/2}$ times a holomorphic function that depends on the action variable $\ell^2m$ and the uniformizing map $\textbf F$ of Eq.\ \eqref{fivi}. It will be our starting point for the semiclassical solution of the eigenvalue equation \eqref{claim}.

As a consistency check, note that \eqref{final} simplifies for isotropic potentials. In that case, action-angle variables are just polar coordinates $\ell^2K=r^2/2$ and $\theta=\varphi$, and the map in \eqref{fivi} is $\textbf F(\textbf x)=\textbf x$, merely implementing a change from polar to Cartesian coordinates: $F(m,\theta)=\ell\sqrt{2m}\cos(\theta)$ and $G(m,\theta)=\ell\sqrt{2m}\sin(\theta)$. One can then verify that \eqref{final} with $n(\theta)=\text{const}$ coincides (up to normalization) with the standard LLL wave function \eqref{phim}. Similarly to that case, any projected wave function \eqref{final} reaches its maximum on the equipotential $\gamma_m$ and is approximately Gaussian close to it, as ensured by the kernel \eqref{proj}. This will be confirmed explicitly in Sec.~\ref{sescale} for edge deformations.

\subsection{Expanding the eigenvalue equation}
\label{ssexp}

None of what we wrote so far involves a manifest semiclassical expansion: it is hidden in the eigenvalue equation \eqref{claim} and the function $n(\theta)$ in \eqref{final}, since $n(\theta)$ should be expanded as a power series $n(\theta) = n_0(\theta) + \ell^2 n_1(\theta) + O(\ell^4)$ (as before, there are no odd powers of $\ell$ since $\ell^2 \propto \hbar$ is really the semiclassical parameter). It is therefore worth anticipating the first few terms of the semiclassical approximation of \eqref{claim}. We stress that the expansion below will eventually be limited to the leading order of the transport equation, so that only $n_0(\theta)$ will be calculated in the end. In principle, one could of course push the expansion to higher orders for more detailed results.

The semiclassical expansion of the right-hand side of \eqref{claim} is clear: it is given by the large $m$, small $\ell^2$ expansion of the projected wave function \eqref{final}, including an expansion of $n(\theta)$. As for the energy, its expansion was written in \eqref{ev}. The left-hand side of \eqref{claim} is more subtle, as its semiclassical expansion involves that of the operator $P\,VP$. The latter is a `Berezin-Toeplitz operator' \cite{Charles:2003a, Charles:2003b} that will play an important role for edge-deformed potentials, so we now explain its expansion in some detail. First, given Cartesian coordinates $(x,y)$, express the potential in complex coordinates \eqref{zdef} as $V(x,y)\equiv{\cal V}(z,\bar z)$ for some function ${\cal V}(z,\bar{w})$ which is holomorphic in its first argument and anti-holomorphic in the second. Then recalling that $P$ is the LLL projector with kernel \eqref{proj}, one finds
\begin{multline}
\label{interm}
\langle z,\bar{z}|
P\,VP
|w,\bar{w}\rangle
=
\frac{1}{2\pi\ell^2}\,
\mathrm{e}^{-\tfrac{1}{2}(|z|^2+|w|^2)} \\
\times
\int_{\mathbb{R}^2}\text{d} u\,\text{d} v\,V(u,v)\,
\mathrm{e}^{-|X|^2+z\bar{X}+\bar{w}X}
\end{multline}
with $X\equiv(u+\mathrm{i} v)/\sqrt{2}\ell$ similarly to \eqref{zdef}. Our task is to expand the integral on the right-hand side in the semiclassical limit. The key is to assume that the potential varies slowly on the scale of the magnetic length \cite{Trugman, Joynt, Champel1, Champel2}, {\it i.e.}\  we choose once and for all a smooth potential $V(\textbf x)$, independent of $\ell$, and let $\ell$ be small. In that regime, the integrals in \eqref{interm} are approximately Gaussian, which gives (see App.~\ref{App:PVP_expansion})
\begin{multline}
\label{topop}
\langle z,\bar z|
P\,VP
|w,\bar w\rangle
\stackrel{\ell \ll 1}{\sim}
\frac{1}{2\pi\ell^2}\,
\mathrm{e}^{-\frac{|z-w|^2}{2}}\,
\mathrm{e}^{\frac{z\bar w-\bar zw}{2}} \\
\times
\left[{\cal V}(z,\bar w)+\frac{\ell^2}{2}(\nabla^2V)(z,\bar w)\right]
\end{multline}
where $(\nabla^2V)(z,\bar w)$ is the bicomplex function that corresponds to the Laplacian of the potential, {\it i.e.}\  $(\nabla^2V)(z,\bar w) = \tfrac{4}{2\ell^2} \partial_z \partial_{\bar{w}}{\cal V}$. This is the standard semiclassical expansion of a Berezin-Toeplitz operator \cite{Charles:2003a, Charles:2003b}. Note the general structure: the entire $P\,VP$ operator boils down to $P$ itself, with kernel \eqref{proj}, multiplied by a function that coincides with $V$ at leading order but also includes quantum corrections. In the `zoomed-out' limit where the kernel of $P$ is a delta function, the first term of \eqref{topop} becomes ${\cal V}(z,\bar z)\delta^2(z-w,\bar z-\bar w)$ as expected. Moreover, for harmonic potentials, the truncated expression \eqref{topop} is actually \i{exact} since the next term $\nabla^4 {\cal V}$ and all subsequent ones vanish. This agrees with the common lore that `WKB is exact for quadratic Hamiltonians'.

\section{Edge-deformed anisotropic traps}
\label{sescale}

Here we apply the WKB ansatz of Sec.~\ref{sesdiff} to potentials \eqref{laviz} with scale-invariant level curves, obtained by acting with edge deformations \cite{EstienneOblakSDiff} on an isotropic trap. As we explain below, these are the most general traps one expects to find close to the edge of star-shaped QH droplets. The plan is as follows. First, we introduce edge deformations and give a few examples for later reference. Second, we apply Eq.\ \eqref{claim} to edge-deformed traps and expand it in the classical limit (large $m$, small $\ell^2$ with $\ell^2m = O(1)$ kept fixed). We keep track of all terms up to order $O(\ell^2)$, so as to capture the leading part of the transport equation for the function $n(\theta)$ in \eqref{psim}--\eqref{final}. This eventually yields an explicit energy spectrum [see \eqref{emi}] along with approximately Gaussian eigenfunctions [see \eqref{gaussbis}]. Lastly, we conclude with a consistency check by showing that our wave functions reproduce the asymptotic (large $m$) form of the known LLL-projected spectrum for anisotropic harmonic traps \cite{Jancovici:1981, ForresterJancovici:1996, Rider:2003, Bender:2010, LeeRiser:2016}.

\subsection{Edge deformations}

We have seen in Sec.~\ref{sesdiff} that area-preserving deformations play a key role for the semiclassical solution of the eigenvalue equation \eqref{claim}.
The group of all such deformations is obviously huge, so it is essential to identify the subset of transformations that are likely to be important for low-energy physics.
In fact, part of this work has already been carried out, at least implicitly, in the seminal series of papers \cite{Cappelli1, Cappelli:1992kf, Cappelli:1993ei, Cappelli:1994wb, Cappelli:1995yk, Cappelli:1996pg}, which we now use as a basis for the definition of edge deformations. (A similar motivation was put forward in \cite{EstienneOblakSDiff}.)

Label points on the plane by their polar coordinates $(r, \varphi)$, defined as usual by $x + \mathrm{i} y = r\,\mathrm{e}^{\mathrm{i}\varphi}$. Then, the boundary of any isotropic QH droplet is located at some fixed radius $r_{\text{edge}} = O(\ell\sqrt{N})$. What is the most general area-preserving deformation that preserves this order of magnitude? The answer is readily found by realizing that the constraint of preserving $r_{\text{edge}} = O(\ell\sqrt{N})$ is equivalent, at leading order in $1/N$, to the condition that the deformation commutes with overall dilations $r \mapsto \mathrm{const} \times r$. The most general deformation satisfying this criterion is an \i{edge deformation}
\begin{equation}
\label{edd}
\bigg(\frac{r^2}{2},\varphi\bigg)
\mapsto
\bigg(\frac{r^2}{2f'(\varphi)},f(\varphi)\bigg),
\end{equation}
where $f(\varphi)$ is an (orientation-preserving) deformation of the circle, {\it i.e.}\  any smooth map satisfying $f(\varphi+2\pi)=f(\varphi)+2\pi$ and $f'(\varphi)>0$. The angle-dependent rescaling of $r$ on the right-hand side ensures that the map preserves area, and reproduces the argument of the potential \eqref{laviz} with $\lambda=2f'$. Note that the set of maps \eqref{edd} is isomorphic to the group of diffeomorphisms of the circle, whose central extension famously leads to the Virasoro algebra encountered in CFT. Indeed, this motivates the statement in \cite{Cappelli:1993ei, Cappelli:1994wb} that generators of maps \eqref{edd} in the QH effect produce conformal transformations of edge modes.

We stress that the subset of transformations \eqref{edd} originates from an asymptotic analysis of the relevant orders of magnitude. One can undoubtedly consider other families of deformations, motivated by different considerations, but those are irrelevant for our purposes. For instance, the transformations $r^2\mapsto r^2+\alpha(\varphi)$ are crucial for the effective low-energy description of QH droplets \cite{Wen, Bahcall:1991an, Cappelli:1994wb}, but they are subleading compared to \eqref{edd} since they deform the radius $r_{\text{edge}}=O(\ell\sqrt{N})$ by terms of order $O(1/N)$ instead of $O(1)$. Conversely, one might consider `higher-spin transformations' \cite{Cappelli1, Cappelli:1993ei, Cappelli:1994wb} that change the radius in a dramatic way such as $r^2\mapsto\beta(\varphi)r^4[1 + O(1/r)]$, but these stretch QH droplets to an infinite extent in the thermodynamic limit, which is why we discard them.

Let us provide a few examples of edge deformations for future reference. First, \eqref{edd} includes rigid rotations around the origin given by $f(\varphi) = \varphi + \mathrm{const}$. A richer class is obtained by fixing some positive integer $k$ and considering all maps of the form
\begin{equation}
\label{boost}
\mathrm{e}^{\mathrm{i} kf(\varphi)}
=
\frac{\alpha\,\mathrm{e}^{\mathrm{i} k\varphi}+\beta}{\bar\beta\,\mathrm{e}^{\mathrm{i} k\varphi}+\bar\alpha}
\end{equation}
where $\alpha,\beta$ are complex and satisfy $|\alpha|^2-|\beta|^2 = 1$. For fixed $k$, such maps span a group locally isomorphic to $\mathrm{SL}(2,\mathbb{R})$, always containing a subgroup of rigid rotations. We will return to these deformations below, since they can be seen as Fourier modes for circle diffeomorphisms. In particular, setting $\alpha = \cosh\lambda$ and $\beta=\sinh\lambda$ for some real parameter $\lambda$ turns the map \eqref{boost} into an analogue of a Lorentz boost with rapidity $\lambda$. In terms of the bulk action \eqref{edd}, any deformation \eqref{boost} turns a circle into a `flower' with $k$ petals: see Fig.~\ref{fidens} for $k=3$. For $k = 2$, this maps the circle on an ellipse \cite{EstienneOblakSDiff}, which will be useful for anisotropic harmonic traps in Sec.~\ref{SubSec:Elliptic_comparison}.

\subsection{Edge-deformed potentials}

Given an isotropic potential $V_0(r^2/2)$, how is it affected by an edge deformation \eqref{edd}? The answer is provided by the anisotropic trap \eqref{laviz} with $\lambda(\varphi) = 2f'(\varphi)$:
\begin{equation}
\label{viviz}
V(r,\varphi)
\equiv
V_0\biggl(\frac{r^2}{2f'(\varphi)}\biggr).
\end{equation}
In what follows, we exclusively consider this class of potentials and refer to them as `edge-deformed traps', for the reasons stated above. Having fixed once and for all some circle deformation $f(\varphi)$, our goal is to solve the corresponding eigenvalue equation \eqref{claim} in the classical limit of high quantum numbers and small magnetic length.

We begin by listing the key classical data of the problem. The action-angle variables corresponding to \eqref{viviz} are $(\ell^2K, \theta) = \bigl(r^2\big/(2f'(\varphi)), f(\varphi) \bigr)$ with an inverse given by $(r^2/2,\varphi) = \big(\ell^2K/(f^{-1})'(\theta),f^{-1}(\theta)\big)$, where $f^{-1}$ denotes the 1D inverse of $f$. Points at constant $K$ are equipotentials, {\it i.e.}\  level curves of \eqref{viviz}, each of which is a set of points such that
\begin{equation}
\label{epot}
\frac{r^2}{2f'(\varphi)}=\ell^2 K
\end{equation}
with constant $K\geq0$. In Cartesian coordinates, this is the set of points $x = \sqrt{2\ell^2K f'(\varphi)}\,\cos(\varphi)$, $y = \sqrt{2\ell^2K f'(\varphi)}\,\sin(\varphi)$ for $\varphi \in [0, 2\pi]$. Equivalently, in terms of the angle variable $\theta=f(\varphi)\in [0, 2\pi]$, the equipotential is
\begin{equation}
\label{xypott}
\begin{aligned}
x
& = \sqrt{\frac{2\ell^2K}{(f^{-1})'(\theta)}}\,\cos(f^{-1}(\theta))\equiv F(K,\theta),\\
y
& = \sqrt{\frac{2\ell^2K}{(f^{-1})'(\theta)}}\,\sin(f^{-1}(\theta))\equiv G(K,\theta),
\end{aligned}
\end{equation}
where the notation $(F,G)$ was introduced in Sec.~\ref{ssepac}. Note that we will eventually focus on the regime where $K\gg1$ is very large in such a way that the dimensionful area $2\pi\ell^2 K$ be an $O(1)$ quantity as $\ell\to0$.

Moving just slightly away from the classical regime, we have seen in Sec.~\ref{sesdiff} that the expansion of the operator $P\,VP$ involves a bicomplex potential function ${\cal V}(z,\bar w)$. In the case of edge-deformed potentials \eqref{viviz}, with the conventions used there and above for complex coordinates, one finds
\begin{equation}
\label{vzw}
{\cal V}(z,\bar w)
=
V_0\left( \ell^2\frac{z\bar w}{f'\big(\tfrac{1}{2i}\log[z/\bar w]\big)} \right).
\end{equation}
Note that this only makes sense for $z$ and $w$ close to each other, otherwise taking $z \to \mathrm{e}^{2\pi\mathrm{i}}z$ affects the argument of $f'$ on the right-hand side. By contrast, when $z$ and $w$ remain close, taking $z \to \mathrm{e}^{2\pi\mathrm{i}}z$ also requires $w \to \mathrm{e}^{2\pi\mathrm{i}}w$, and this time the angle $\tfrac{1}{2i}\log[z/\bar w]$ is indeed invariant.

Finally, the expansion \eqref{topop} also involves the complexified Laplacian of the potential, but only its real value will be relevant at the order studied here. Let us therefore express the Laplacian of \eqref{viviz} in polar coordinates:
\begin{align}
\nabla^2V
& =
\frac{1}{f'}
\left(2-\frac{1}{2}\frac{f'''}{f'}+\frac{f''^2}{f'^2}\right)V_0'\left(r^2/2f'\right) \nonumber \\
& \quad +
\frac{r^2}{f'^2}\left(1+\frac{f''^2}{4f'^2}\right)V_0''\left(r^2/2f'\right).
  \label{lav}
\end{align}
Here the prime means differentiation with respect to the argument, namely $\varphi$ for $f(\varphi)$ and $r^2/2$ for $V_0(r^2/2)$. We shall rely on \eqref{vzw} and \eqref{lav} below, since they directly affect the eigenvalue equation \eqref{claim}.

\subsection{Eigenvalue equation and energy}

Having studied the potential \eqref{viviz}, let us turn to the quantum state meant to solve the eigenvalue equation \eqref{claim}. As in Sec.~\ref{sseniseg}, we begin by building a state \eqref{psim} that is perfectly localized on the equipotential, project to the LLL using the operator \eqref{proj}, and obtain the wave function \eqref{final} that now reads
\begin{multline}
\label{true}
\psi_m(z,\bar z)
=
\mathrm{e}^{-|z|^2/2}
\oint \text{d}\varphi\, f'(\varphi)\, n(f(\varphi)) \\
\times \exp\left[ \mathrm{i} mf(\varphi)-\tfrac{1}{2}mf'(\varphi)+z\,\sqrt{mf'(\varphi)}\,\mathrm{e}^{-\mathrm{i}\varphi} \right],
\end{multline}
where we changed variables using $\theta = f(\varphi)$. It remains to show that this solves the eigenvalue equation \eqref{claim} for edge-deformed traps \eqref{viviz} in the semiclassical regime, provided the function $n(\theta)$ satisfies a suitable transport equation. The latter is derived by expanding the energy \eqref{ev} and the potential \eqref{topop} to get
\begin{align}
0
& =
\oint \text{d}\varphi\, f'(\varphi)\, n(f(\varphi)) \nonumber \\
& \quad \times
\exp\Big[\mathrm{i} mf(\varphi)-\tfrac{1}{2}mf'(\varphi)+z\sqrt{mf'(\varphi)}\,\mathrm{e}^{-\mathrm{i}\varphi}\Big] \nonumber \\
& \quad \times
\Bigl( {\cal V}\big(z,\sqrt{mf'(\varphi)}\,\mathrm{e}^{-\mathrm{i}\varphi}\big)+\tfrac{\ell^2}{2}\nabla^2V-E^0_m-\ell^2E^1_m\Bigr)
  \label{egeqq}
\end{align}
where ${\cal V}(z,\bar w)$ is the bicomplex function \eqref{vzw} and the equation holds up to neglected $O(\ell^4)$ corrections. At leading order in the classical limit, the potential expansion \eqref{topop} boils down to $\langle z|P\,VP|w\rangle\sim{\cal V}(z,\bar z)\delta^2(z-w)$, so \eqref{egeqq} merely states that $E^0_m=V_0(\ell^2m)=V(\gamma_m)$. The issue is to find the two remaining unknowns: the function $n(f(\varphi))$ and the first-order energy correction $E^1_m$.

To determine these, the crucial step is to evaluate \eqref{egeqq} along the equipotential \eqref{epot} labeled by $K = m$, {\it i.e.}\  for $z=\sqrt{m f'(\alpha)}\,\mathrm{e}^{\mathrm{i}\alpha}$ with $\alpha \in [0,2\pi)$, where as usual we assume $m\gg1$. Indeed, if \eqref{egeqq} holds on a level curve, then it holds for all $z$ by holomorphicity. This is written in more detail in App.\ \ref{appexp}, where we show that the integrand of \eqref{egeqq} has a saddle point at $\varphi=\alpha$, eventually resulting in a transport equation for the unknown function $n$. Here we skip the computation and analyse separately the real and imaginary parts of the transport equation. We start with the real part, which will allow us to deduce the LLL-projected energy spectrum. The imaginary part is postponed to Sec.~\ref{SubSec:GWF}, where we also display the resulting nearly Gaussian wave functions.

Let $\Phi(\varphi)$ denote the phase of $n(f(\varphi))\equiv{\cal N}(\varphi)\,\mathrm{e}^{\mathrm{i}\Phi(\varphi)}$.
Then the real part of the transport equation [see \eqref{rtransport}] yields
\begin{multline}
\label{phiprime}
\Phi'(\varphi)
=
\frac{E^1_{m}}{\Omega_{m}} f'(\varphi)
-\frac{1}{2} \biggl(\frac{\Gamma_{m}}{\Omega_{m}}+1\biggr) \biggl(1+\frac{f''(\varphi)^2}{4f'(\varphi)^2}\biggr) \\
- \frac{1}{2}
+ \partial_{\varphi} \biggl(\frac{f''(\varphi)}{8f'(\varphi)}\biggr)
+ \frac{1}{2} \frac{\partial_{\varphi}[f''(\varphi)/2f'(\varphi)]}{1+f''(\varphi)^2/4f'(\varphi)^2},
\end{multline}
where $E^1_{m}$ is the first order correction to the energy \eqref{ev} and we introduced the parameters
\begin{align}
\label{OmegaGamma}
\Omega_{m} \equiv V_0'(\ell^2m)>0,
\qquad
\Gamma_{m} \equiv \ell^2m\,V_0''(\ell^2m).
\end{align}
In many-body droplets with $N$ electrons, these will respectively measure the Fermi velocity and the curvature of the potential at the Fermi surface when $m=N$. Note that all terms in \eqref{phiprime} are total derivatives save for the factor $1+[f''/2f']^2$, so the solution is
\begin{multline}
\label{phitap}
\Phi(\varphi)
=
\frac{E^1_{m}}{\Omega_{m}} f(\varphi)
-\frac{1}{2} \biggl( \frac{\Gamma_{m}}{\Omega_{m}}+1 \biggr) \int_{0}^{\varphi}\text{d}\theta\, \biggl(1+\frac{f''(\theta)^2}{4f'(\theta)^2}\biggr) \\
-\frac{\varphi}{2}
+\frac{f''(\varphi)}{8f'(\varphi)}
+\frac{1}{2}\arctan\biggl(\frac{f''(\varphi)}{2f'(\varphi)}\biggr) + \mathrm{const}.
\end{multline}
This turns out to imply a quantization condition for energy. Indeed, when we initially introduced the function $n(\theta)$ in \eqref{psim}, we mentioned that it must have a \i{vanishing} winding number along the equipotential, so that all the winding of the wave function is contained in the exponential factor $\mathrm{e}^{\mathrm{i} m\theta}$. The phase $\Phi(\varphi)$ must therefore be strictly $2\pi$-periodic, {\it i.e.}\  $\Phi(2\pi)=\Phi(0)$. Using \eqref{phitap}, this fixes the first quantum correction of the energy \eqref{ev}:
\begin{equation}
\label{equan}
\frac{E^1_{m}}{\Omega_{m}}
=
\frac{1}{2}
+
\biggl(\frac{\Gamma_{m}}{\Omega_{m}}+1\biggr)\oint
\frac{\text{d}\varphi}{4\pi}\, \biggl( 1+\frac{f''(\varphi)^2}{4f'(\varphi)^2} \biggr).
\end{equation}
The latter generally generally depends on $m$ through $\Gamma_{m}$ and $\Omega_{m}$ in \eqref{OmegaGamma}. A simplification only occurs in `harmonic' setups where $\Gamma_{m}=0$ and the right-hand side of \eqref{equan} is an $f$-dependent constant, for all $m$ \cite{Note3}. In any case, the full $m^{\text{th}}$ energy \eqref{ev} in the semiclassical limit can be written as
\begin{multline}
\label{etop}
E_m
\sim
V_0\bigl(\ell^2m\bigr) \\
+
\frac{\ell^2}{2}
\biggl[\Omega_m
+
\bigl(\Gamma_{m}+\Omega_{m}\bigr)
\oint
\frac{\text{d}\varphi}{2\pi} \biggl(1+\frac{f''(\varphi)^2}{4f'(\varphi)^2}\biggr)\biggr],
\end{multline}
reproducing the result announced in \eqref{sigma_lambda}--\eqref{ressum_E_m} with $\lambda=2f'$ and generalizing the isotropic value obtained for $f'=1$ [see \eqref{emma}]. The leading-order Bohr-Sommerfeld quantization condition \eqref{oint} is manifestly satisfied, while the first quantum correction can be expressed in terms of a Maslov-like shift and an integral of the Laplacian, confirming the general result in \cite{Charles:2003b}:
\begin{multline}
\label{emi}
E_m
=
V_0\Bigl(\ell^2\Bigl[m+\frac{1}{2}\Bigr]\Bigr) \\
+\frac{\ell^2}{4} \!
\oint\frac{\text{d}\varphi}{2\pi} f'(\varphi)\nabla^2V\big|_{r^2=2\ell^2(m+1/2)f'(\varphi)}
+O(\ell^4).
\end{multline}
(In the language of \cite{Charles:2003b}, our `Maslov-like' term actually stems from an integral of the curvature of $\gamma_m$.) It now remains to write the corresponding wave functions.

\subsection{Gaussian wave functions}
\label{SubSec:GWF}

As above, write $n(f(\varphi))={\cal N}(\varphi)\,\mathrm{e}^{\mathrm{i}\Phi(\varphi)}$ for the unknown function of the WKB ansatz, with a norm ${\cal N}(\varphi)=|n(f(\varphi))|$. Then the imaginary part of the transport equation [see \eqref{itransport}] can be recast into
\begin{equation}
\label{eqn}
\frac{{\cal N}'(\varphi)}{{\cal N}(\varphi)}
=
\frac{1}{4} \partial_{\varphi} \log \left[ \frac{1}{f'(\varphi)} \left(1+\frac{f''(\varphi)^2}{4f'(\varphi)^2}\right) \right],
\end{equation}
which remarkably has the form of an overall logarithmic derivative. The general solution is thus
\begin{equation}
\label{den}
\big|n\big(f(\varphi)\big)\big|
=
N_0
\left[\frac{1}{f'(\varphi)}\left(1+\frac{f''(\varphi)^2}{4f'(\varphi)^2}\right)\right]^{1/4},
\end{equation}
where the normalization $N_0$ will soon be fixed. Note the exponent ${1}/{4}$, typical of WKB approximations \cite{BenderOrszag:1999}.

We can now use \eqref{den} to evaluate approximate eigenfunctions \eqref{true} near their maximum, {\it i.e.}\  close to the equipotential \eqref{zalp}. To see this, zoom in on the equipotential by writing
\begin{equation}
z
\equiv
\bigl(\sqrt{m}+a\bigr) \sqrt{f'(\alpha)}\,\mathrm{e}^{\mathrm{i}\alpha}
\label{zalpp}
\end{equation}
for $m \gg 1$ and some finite $a$. The integral \eqref{true} then has a unique saddle point at $\varphi = \alpha -\mathrm{i}\delta_1/\sqrt{m} + O(1/m)$, with $\delta_1 = a \Bigl[ 1 - \mathrm{i} \tfrac{f''(\alpha)}{2f'(\alpha)} \Bigr]^{-1}$. The saddle-point approximation of the wave function \eqref{true} thus yields
\begin{align}
\psi_m(z,\bar z)
& \sim
\frac{1}{\sqrt{2\pi\ell^2}}\,
\frac{1}{(2\pi m)^{1/4}}\,
\mathrm{e}^{\mathrm{i} mf(\alpha) + \mathrm{i}\Phi(\alpha)} \nonumber \\
& \quad \times
\frac{1}{\sqrt{\sigma(\alpha)}}
\exp \left[ -\frac{f'(\alpha)\,a^2}{1-\mathrm{i}\frac{f''(\alpha)}{2f'(\alpha)}} \right],
  \label{aspt}
\end{align}
where we used Eqs.\ \eqref{phitap}--\eqref{den} for the phase and norm of $n(f(\alpha))$, fixed the integration constant in \eqref{den} to $N_0=\frac{1}{2\pi\ell}(\tfrac{m}{2\pi})^{1/4}$, and introduced the ubiquitous width
\begin{equation}
\label{sigma}
\sigma(\varphi)^2
\equiv
\frac{1}{f'(\varphi)}
\biggl(1+\frac{f''(\varphi)^2}{4f'(\varphi)^2}\biggr).
\end{equation}
This is the angle-dependent variance of the probability density of \eqref{aspt}, written in \eqref{sigma_lambda} in terms of $\lambda=2f'$. Indeed, one has
\begin{equation}
\label{gaussbis}
|\psi_m(z,\bar z)|^2
\sim
\frac{1}{2\pi\ell^2}\,
\frac{\displaystyle\mathrm{e}^{-2a^2/\sigma^2(\alpha)}}{\sqrt{2\pi m}\,\sigma(\alpha)},
\end{equation}
which satisfies the desired normalization condition $\int \text{d}^2\textbf x\, |\psi|^2 = 1$. Note that Eq.\ \eqref{aspt} reproduces the wave function stated in \eqref{e13}, once more with $\lambda=2f'$ and now involving the phase
\begin{equation}
\label{Theta_m_x}
\Theta_{m}(\textbf x)
=
mf(\varphi) + \Phi(\varphi) - \frac{a^2 f''(\varphi)}{2f'(\varphi) \sigma(\varphi)^2}.
\end{equation}
The Gaussian behavior of LLL-projected is thus manifest, as anticipated at the end of Sec.~\ref{sseniseg} for the general WKB ansatz \eqref{final}. It is illustrated in Fig.~\ref{fiwavefct} for two choices of the confining potential \eqref{viviz}. Finally, \eqref{aspt} generalizes the behavior of isotropic states \eqref{phim} [see \eqref{refas}], including the $O(1/\sqrt{m})$ contribution that we did not state here but that can be computed by incorporating the next order term $\delta_2/m$ for the saddle point and repeating the analysis \cite{Note4}.

\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{fig3_sp_wave_function.pdf}
\caption{The density of a wave function \eqref{aspt} at $m=30$ in an edge-deformed trap \eqref{viviz}. The Gaussian behavior is manifest, as is the angle-dependent `roller coaster' predicted by \eqref{gaussbis}. \i{Left:} Elliptic potential given by \eqref{viviz} with  $f$ of the form \eqref{boost} and $k=2$. \i{Right:} Same edge-deformed trap as in Figs.~\ref{fiDrop}--\ref{fipott}.}
\label{fiwavefct}
\end{figure}

\subsection{Comparison with elliptic wave functions}
\label{SubSec:Elliptic_comparison}

To conclude this section, we now focus on the `flower' deformations \eqref{boost} and show that the resulting transport equation is integrable: both the phase \eqref{phitap} and the norm \eqref{den} can be expressed in terms of elementary functions. These results are valuable in themselves since flower maps are the simplest edge deformations---they are analogues of Fourier modes for circle diffeomorphisms---, but also because their special case $k=2$ reproduces known wave functions in elliptic harmonic traps \cite{DiFrancesco:1994,ForresterJancovici:1996}, providing an important benchmark for our WKB approach.

Consider first the deformation \eqref{boost} with $\alpha=\cosh\lambda$ and $\beta=\sinh\lambda$ for an arbitrary integer $k$ and a real parameter $\lambda$. Then the energy quantization condition \eqref{equan} can be integrated exactly, yielding
\begin{equation}
\label{e50}
\frac{E^1_{m}}{\Omega_{m}}
=
\frac{1}{2}
+
\frac{1}{2}
\bigg(1+\frac{\Gamma_{m}}{\Omega_{m}}\bigg)
\bigg(1+\frac{k^2}{4} \bigl[\cosh(2\lambda)-1\bigr]\bigg).
\end{equation}
As for the solution of the transport equation, consisting of the phase \eqref{phitap} and norm \eqref{den}, it is found to be
\begin{align}
n(\theta)
& =
N_0\,\mathrm{e}^{\frac{\mathrm{i}}{8}\frac{\Gamma_{m}}{\Omega_{m}}k \sin (k\theta) \sinh(2\lambda)} \mathrm{e}^{\mathrm{i}\frac{\theta}{2}\left[\left(\frac{\Gamma_{m}}{\Omega_{m}}+1\right)\left(1-\frac{k^2}{4}\right)+1-\frac{k}{2}\right]} \nonumber \\
& \quad \times
\left(\frac{\cosh\lambda + \mathrm{e}^{\mathrm{i} k\theta}\sinh\lambda}{\mathrm{e}^{\mathrm{i} k\theta}\cosh\lambda + \sinh\lambda}\right)^{\frac{k^2-4}{8k}\left(1+\frac{\Gamma_{m}}{\Omega_{m}}\right)-\frac{1}{2k}+\frac{1}{4}} \nonumber \\
& \quad \times
\sqrt{\frac{-4\mathrm{e}^{\mathrm{i} k\theta} + (\mathrm{e}^{2\mathrm{i} k\theta}-1)k \sinh(2\lambda)}{\mathrm{e}^{\mathrm{i} k\theta}\cosh\lambda + \sinh\lambda}}
  \label{flosol}
\end{align}
up to an overall constant phase.

Note that \eqref{flosol} depends in a non-trivial way on the potential derivatives \eqref{OmegaGamma}, with some simplification in the `harmonic' regime $\Gamma_m=0$. Let us therefore apply Eqs.\ \eqref{e50}--\eqref{flosol} to the case of an elliptic potential $k = 2$ with constant stiffness $\Omega_m=\Omega>0$ (hence $\Gamma_m=0$). The corresponding edge deformation \eqref{edd} maps the isotropic harmonic potential $V_0(\textbf x)=\Omega\,r^2/2$ on its anisotropic cousin,
\begin{equation}
\label{ellipsepotential}
V(\textbf x)
=
\Omega\,\frac{\mathrm{e}^{-2\lambda}x^2+\mathrm{e}^{2\lambda}y^2}{2\cosh(2\lambda)},
\end{equation}
whose equipotentials are ellipses rather than circles. The energy correction \eqref{e50} then becomes $E^1_m=\frac{1}{2}\Omega[1+\cosh(2\lambda)]$ and the solution \eqref{flosol} of the transport equation can be written as
\begin{equation}
n(\theta)
=
N_0\sqrt{ \cosh\lambda-\mathrm{e}^{2\mathrm{i}\theta}\sinh\lambda}.
\end{equation}
It is straightforward to use this data to obtain the elliptic version of the normalized Gaussian wave function \eqref{aspt}:
\begin{align}
\psi_m(z,\bar{z})
& \sim
\left(\frac{2\pi}{m}\right)^{1/4}
\frac{1}{2\pi\ell}
\frac{\mathrm{e}^{\mathrm{i} m\theta}}{\sqrt{\cosh\lambda -\mathrm{e}^{-2\mathrm{i}\theta}\sinh\lambda}} \nonumber \\
& \quad\; \times
\exp\left( - \frac{\mathrm{e}^{2\mathrm{i}\theta}+\tanh(\lambda)}{\mathrm{e}^{2\mathrm{i}\theta}-\tanh(\lambda)} a^2 \right).
  \label{wave functions_ellipse}
\end{align}
Crucially, the latter coincides with the large $m$ approximation of the \i{exact} LLL-projected eigenstates of the harmonic potential \eqref{ellipsepotential}, as can be verified thanks to known asymptotic formulas for Hermite polynomials \cite{ForresterJancovici:1996}. This is actually true even at subleading order in $1/\sqrt{m}$, which we omit here for brevity.

\section{Many-body observables}
\label{semany}

This section finally applies the results of Secs.~\ref{sesdiff}--\ref{sescale} to fully-fledged QH droplets consisting of a large number $N\gg1$ of electrons. Specifically, we exploit our insights on near-Gaussian single-particle wave functions \eqref{aspt} to compute many-body observables and read off the universal shape-dependent effects implied by the width \eqref{sigma}. The density is treated first: it equals $\frac{1}{2\pi\ell^2}$ in the bulk, then drops to zero as an error function at the edge $r_{\text{edge}}=\ell\sqrt{2Nf'(\varphi)}$. We then turn to the current and show that it is localized as a Gaussian on the edge, to which it is tangent. Third, correlations near the edge are found to display the usual power-law behavior of free fermions, dressed by radial Gaussian factors. This reduces to known expressions in isotropic traps \cite{Cappelli:1992kf} and in the case of flower deformations \eqref{boost} with $k=2$, where one recovers the elliptic results of \cite{DiFrancesco:1994, ForresterJancovici:1996}. Finally, the radial behavior of correlations is shown to be consistent with the effective low-energy field theory of edge modes: we derive it microscopically and obtain a chiral CFT in terms of the angle variable on the boundary.

\subsection{Density}
\label{sedenss}

Consider a QH droplet of $N\gg1$ non-interacting 2D electrons subject to the Hamiltonian \eqref{h}, with a very strong magnetic field $\textbf B=\text{d}\textbf A$ and a weak edge-deformed potential \eqref{viviz}. The ground state of this many-body system is a Slater determinant of the wave functions $\psi_m$ for occupied states $m=0,1,\ldots,N-1$, where we recall that $m$ is a quantized action variable generalizing angular momentum (see the red dots in Fig.~\ref{figround}). Each $\psi_m$ yields a single-particle probability density $|\psi_m(\textbf x)|^2$, the sum of which gives the many-body density
\begin{equation}
\label{desum}
\rho(\textbf x)
=
\sum_{m=0}^{N-1}
|\psi_m(\textbf x)|^2.
\end{equation}
While WKB theory does not give access to the form of $\psi_m$ at low $m$, large values of $m$ should be correctly captured by the analysis of Sec.~\ref{sescale}, in which case the one-body density is approximately Gaussian and given by \eqref{gaussbis}. We now exploit this Gaussian form to evaluate the many-body density, both in the bulk and close to the edge. (Some technical details are highlighted along the way, as the same method will later allow us to study the many-body current and correlations.)

\begin{figure}[t]
\centering
\begin{tikzpicture}[scale=.85]
\draw[->] (-.5,0) -- (8,0);
\draw (8,0) node[above] {$m$};
\draw[->] (0,-0.5) -- (0,2.5);
\draw (0,2.5) node[right] {Energy};
\node at (0,0) {$\color{MyRed}\circ$};
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\draw[<->] (4.4,1.04) -- (6.6,1.04);
\draw (5.5,1.04) node[above] {$2\Lambda$};
\end{tikzpicture}
\caption{The one-body spectrum \eqref{emi}, where the states that are occupied in the many-body ground state are highlighted in red and those that contribute to the low-energy Hamiltonian \eqref{hato} are filled (black for `particles', red for `holes'). The cutoff $\Lambda$ is large but much smaller than $N$ in the sense that $\Lambda=O(1)$ in the thermodynamic limit.}
\label{figround}
\end{figure}

The key point is that each wave function \eqref{gaussbis} is localized on an equipotential of $V(\textbf x)$ with area $2\pi\ell^2m$, so the density close to some equipotential $|z| = \mathrm{const} \times \sqrt{f'(\varphi)}$ only receives sizeable contributions from wave functions whose quantum number is close to $|z|^2/f'(\varphi)$. Accordingly, the bulk density for $1\ll|z|\ll\sqrt{N}$ is obtained by letting the upper summation bound of \eqref{desum} go to infinity and writing the approximate density as
\begin{equation}
\rho(\textbf x)
\sim
\frac{1}{2\pi\ell^2}\,
\sum_{m=m_0}^{\infty}
\frac{\mathrm{e}^{-\frac{2}{\sigma^2(\varphi)}\left(\frac{|z|}{\sqrt{f'(\varphi)}}-\sqrt{m}\right)^2}}{\sqrt{2\pi m}\,\sigma(\varphi)},
\end{equation}
where the lower summation bound $m_0\gg1$ is irrelevant as long as it is much smaller than $|z|^2$, and $\sigma(\varphi)$ is the angle-dependent width \eqref{sigma}.
At large $|z|$, the Euler-Maclaurin formula allows us to approximate the sum over $m$ by a (Gaussian) integral over $\sqrt{m}$. This yields the expected density
\begin{equation}
\label{rhobulk}
\rho(\textbf x)
\sim
\frac{1}{2\pi\ell^2}
\end{equation}
in the bulk of a QH droplet with filling fraction $\nu=1$.

An analogous argument can be carried out close to the droplet's edge, with one key difference: the upper summation bound of \eqref{desum} is now crucial.
Thus, letting $|z|=\bigl(\sqrt{N}+a\bigr)f'(\varphi)$ with finite $a$ in the large $N$ limit and using once more the approximate Gaussian behavior \eqref{gaussbis}, the density \eqref{desum} near the edge behaves as
\begin{equation}
\label{edgerho}
\rho(\textbf x)
\sim
\frac{1}{2\pi\ell^2}\,
\sum_{k=1}^{\infty}
\frac{\mathrm{e}^{-\frac{2}{\sigma^2(\varphi)}\left(a+\frac{k}{2\sqrt{N}}\right)^2}}{\sqrt{2\pi N}\,\sigma(\varphi)},
\end{equation}
where we changed variables as $m \equiv N-k$ with $k = O(\sqrt{N})$ at large $N$ and only kept track of leading-order terms. For $N \gg 1$, the sum over $k$ can once more be converted into an integral, now over $k/2\sqrt{N}$. This yields the asymptotic behavior
\begin{equation}
\label{rhoo}
\rho(r,\varphi)
\sim
\frac{1}{4\pi\ell^2} \,
\erfc\left(
\frac{1}{\sigma(\varphi)}
\frac{r - \ell\sqrt{2Nf'(\varphi)}}{\ell\sqrt{f'(\varphi)}}
\right)
\end{equation}
where $\erfc$ denotes the complementary error function and the width \eqref{sigma} is inherited from that of LLL wave functions. This is a remarkably explicit result, announced in \eqref{ressum_rho_edge} with $\lambda=2f'$. It confirms that the density is roughly constant and equal to \eqref{rhobulk} in the bulk, then drops to zero within a distance of the order of the magnetic length \eqref{ell} around the edge at $r=\ell\sqrt{2N\,f'(\varphi)}$. See Figs.~\ref{fipott}(a)--\ref{fidens}(a).

\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{fig4_flower_droplet.pdf}
\caption{(a) The density \eqref{rhoo} for $N=100$ electrons and a `flower' edge deformation \eqref{boost} of order $k=3$. The constancy of density in the bulk and its sharp decay at the boundary are manifest. (b)  The current's norm \eqref{normj} for the same flower-shaped droplet. The localization on the edge equipotential (black line) is clearly visible, as is the position-dependent width of the Gaussian jump.  (c) The correlation function $C(\textbf x_1,\textbf x_2)$ \eqref{coco} for the same flower-shaped droplet, seen as a function of $\textbf x_2$ when $\textbf x_1=(\ell\sqrt{N\lambda(0)},0)$ is fixed close to the edge of the droplet.}
\label{fidens}
\end{figure*}

We stress that, in contrast to wave functions, the density \eqref{rhoo} only depends on the potential near the edge of the droplet: bulk deformations of the potential do not affect the quantized bulk density \eqref{rhobulk} in the limit of strong magnetic fields. In this sense, \eqref{rhoo} is a universal formula for the density of any QH droplet of LLL states whose edge traces an equipotential of the form $r^2=2\ell^2N\,f'(\varphi)$. It would be instructive to probe this local density in modern quantum simulators \cite{Goldman1, Goldman2, Fletcher, Mukherjee:2021jjl}.

Note that the leading-order formula \eqref{rhoo} receives a number of subleading corrections that can be systematically computed in our formalism; these are omitted here for brevity. A related comment is that the bulk value density \eqref{rhobulk} is only valid at extremely strong magnetic fields, which stems from the simplification provided by the LLL projection. The actual density profile depends on the gradient of the potential, but this involves higher Landau levels that are beyond our scope \cite{Champel1, Champel2}.

\subsection{Current}
\label{securr}

The current of a droplet of $N\gg1$ electrons can similarly be evaluated as a sum over single-particle currents. To this end, recall that the gauge-invariant one-body probability current of a charged wave function $\psi$ with mass $M$ is a one-form $\hbar\,\textbf j/M$ given by
\begin{equation}
\label{john}
\textbf j
=
\frac{1}{2\mathrm{i}}
\left( \psi^*\text{d}\psi - \psi\text{d}\psi^* - 2\mathrm{i}\frac{q}{\hbar}\textbf A|\psi|^2 \right),
\end{equation}
where the first term is only sensitive to the phase of $\psi$ and $\textbf A =\tfrac{1}{2}Br^2\,\text{d}\varphi=\tfrac{\hbar}{q}|z|^2\,\text{d}\varphi$ in symmetric gauge. Thus, the many-body current of a Slater determinant of the occupied states $\psi_m$ with $m = 0, \ldots, N-1$ is
\begin{equation}
\label{johnson}
\textbf J
=
\sum_{m=0}^{N-1}\textbf j_m,
\end{equation}
where $\textbf j_m$ is the single-particle current \eqref{john} of each $\psi_m$.

As before, the WKB approximation does not give access to wave functions for small $m$, but this is unimportant close to the edge. In that regime, we have already gathered all ingredients needed to evaluate the currents \eqref{john} up to small quantum corrections: the one-body density is given by \eqref{gaussbis}, while the derivative of the phase is contained in \eqref{aspt} and the phase transport equation \eqref{phiprime}. In practice, the WKB phase $\Phi$ turns out to be negligible at leading order, and the only relevant parts of the phase are those already visible in \eqref{aspt}: the (fast) phase $\mathrm{e}^{\mathrm{i} m f(\varphi)}$ together with the contribution from $\textbf A = \bigl(\hbar |z|^2/q\bigr) \text{d}\varphi$ eventually gives rise to the leading $\varphi$ component of the current, while the (slow) phase $\mathrm{e}^{-\mathrm{i} [ f''(\varphi)/2f'(\varphi)] a^2/\sigma^2}$ yields its radial component that is non-zero whenever $f''(\varphi) \neq 0$.

Starting from these facts, it is straightforward to adapt the method of Sec.~\ref{sedenss} to the many-body current \eqref{johnson}. Writing $|z| = \bigl(\sqrt{N}+a\bigr) \sqrt{f'(\varphi)}$, the sum over $m \equiv N-k$ becomes an integral over $k/(2\sqrt{N})=O(1)$, which yields the leading order result quoted in \eqref{ressum_J_edge} with $\lambda=2f'$:
\begin{equation}
\label{joo}
\textbf J(r,\varphi)
\sim
-\frac{\displaystyle\mathrm{e}^{-\frac{2a^2}{\sigma(\varphi)^2}}}{(2\pi \ell^2)^{3/2}\sigma(\varphi)}
\frac{\ell\sqrt{2Nf'(\varphi)}\,\text{d}\varphi+\tfrac{f''(\varphi)}{2f'(\varphi)}\,\text{d} r}{\sqrt{2f'(\varphi)}}
\end{equation}
where $a = \bigl( r - \ell\sqrt{2N f'(\varphi)} \bigr) \big/ \ell\sqrt{2f'(\varphi)}$ and $\sigma(\varphi)$ is given by \eqref{sigma}. Both components in \eqref{joo} receive subleading corrections that are omitted here. In particular, there is an $O(1)$ term in $J_{\varphi}$ that is non-zero on the edge, even in the isotropic case $f'=1$. The computation of that term requires the $O(1/\sqrt{m})$ correction that was omitted in \eqref{gaussbis}.

Using the metric $\text{d} s^2=\text{d} r^2+r^2\text{d}\varphi^2$, one verifies that the one-form $\ell\sqrt{2N\,f'}\,\text{d}\varphi + ({f''}/{2f'})\,\text{d} r$ in \eqref{joo} is the dual of a vector tangent to the equipotential at the droplet's edge \cite{Note5}. Moreover, the norm squared of \eqref{joo} is
\begin{equation}
\label{normj}
\|\textbf J(r,\varphi)\|^2
\sim
\frac{1}{2(2\pi\ell^2)^3} \exp\left[-\frac{2\bigl(r - \ell\sqrt{2N f'(\varphi)}\bigr)^2}{\ell^2 \sigma(\varphi)^2 f'(\varphi)}\right],
\end{equation}
showing that the current has a constant maximum along the edge but a varying width: see Fig.~\ref{fidens}.

Similarly to the density, it is important to remember that the LLL projection misses some important physics. Indeed, the actual bulk current is the symplectic gradient of the confining potential multiplied by the Hall conductance \cite{Geller1, Geller2, Champel1, Champel2}. No such effect occurs in \eqref{joo} because it requires higher Landau levels, which are beyond our scope.

\subsection{Correlations}
\label{secor}

The methods that we have applied to density and current can also be used to compute electronic correlations near the edge. Indeed, consider as before an anisotropic droplet whose occupied one-body states have quantum numbers $m=0,1,\ldots,N-1$. Then the correlation function between the points $\textbf x_1$ and $\textbf x_2$ is
\begin{equation}
C(\textbf x_1,\textbf x_2)
=
\sum_{m=0}^{N-1}
\psi_m^*(\textbf x_1)\,\psi_m(\textbf x_2),
\label{cofo}
\end{equation}
which reduces to the density \eqref{desum} when $\textbf x_1=\textbf x_2$. As before, we rename $m\equiv N-k$ and let the complex coordinates $z,w$ corresponding to $\textbf x_1,\textbf x_2$ be such that
\begin{equation}
\begin{aligned}
z
& =
\bigl(\sqrt{N}+a\bigr)\sqrt{f'(\varphi_{1\vphantom{2}})}\,\mathrm{e}^{\mathrm{i}\varphi_1}, \\
w
& =
\bigl(\sqrt{N}+b\bigr)\sqrt{f'(\varphi_2)}\,\mathrm{e}^{\mathrm{i}\varphi_2},
\end{aligned}
\end{equation}
where $a,b$ are finite at large $N$ and $\varphi_1,\varphi_2$ are the polar angles of $\textbf x_1,\textbf x_2$. One can then plug the Gaussian wave functions \eqref{aspt} into \eqref{cofo}, this time assuming $k$ finite, and performing the sum over $k$. The gradient expansion of the potential implies that the ratio $\Gamma_{m}/\Omega_{m}\sim\Gamma_N/\Omega_N+O(\ell^2)$ is nearly constant in this regime, so Eq.\ \eqref{cofo} becomes a geometric sum over $k$ that reproduces the result stated in \eqref{ressum_C_edge} with $\lambda=2f'$:
\begin{equation}
\label{coco}
\begin{aligned}
C(\textbf x_1,\textbf x_2)
& \sim
\frac{\mathrm{e}^{\mathrm{i}\Theta_{N}(\textbf x_1,\textbf x_2)}}{(2\pi)^{3/2}\ell^2\sqrt{N}}
\frac{1}{\sqrt{\sigma(\varphi_1)\sigma(\varphi_2)}} \\
& \quad \times
\frac{\mathrm{i}\,\mathrm{e}^{-\frac{a^2}{\sigma(\varphi_1)^2}-\frac{b^2}{\sigma(\varphi_2)^2}}}{2\sin\big([f(\varphi_1)-f(\varphi_2)]/2\big)},
\end{aligned}
\end{equation}
where $\sigma(\varphi)$ is the angle-dependent width \eqref{sigma}. The overall phase $\Theta_{N}(\textbf x_1,\textbf x_2) = \Theta_{N}(\textbf x_2) - \Theta_{N}(\textbf x_1)$ is given by \eqref{Theta_m_x}, and involves in particular the WKB phase \eqref{phitap}.

Some features of \eqref{coco} are worth emphasizing. First, note the Gaussian jump of power-law correlations near the edge, involving a free fermion correlation $\propto\sin([f(\varphi_1)-f(\varphi_2)]/2)^{-1}$ expressed in terms of $f(\varphi_1)$ and $f(\varphi_2)$; this is the standard static diagnostic of the presence of edge modes \cite{Cappelli:1992kf, Wen, Estienne:2021qqe}. A second striking aspect is the apparent lack of translation-invariance along the edge, caused not only by the argument $f(\varphi_1) - f(\varphi_2) = \int_{\varphi_2}^{\varphi_1} \text{d} \theta\, f'(\theta)$ but also by the widths $\sigma(\varphi_1)$ and $\sigma(\varphi_2)$. In particular, the product $\sigma(\varphi_1)^{-1/2}\sigma(\varphi_2)^{-1/2}$ is reminiscent of prefactors picked up by primary fields in CFT under conformal maps.

Since the boundary correlator \eqref{coco} holds in any edge-deformed trap \eqref{viviz}, it also applies to special cases of interest such as the anisotropic harmonic potential \eqref{ellipsepotential}. The corresponding correlations were actually computed long ago in \cite{ForresterJancovici:1996}, and they coincide with our result \eqref{coco} upon using the map \eqref{boost} with $k=2$, $\alpha=\cosh\lambda$, $\beta=\sinh\lambda$. In fact, this specific setup is also well known in the context of the Coulomb gas, since edge correlations can then be related by a conformal map to the standard Euclidean correlation function $(z_1-z_2)^{-1}$ of a free fermion CFT \cite{WiegmannZabrodin:2003}.

Finally, it is a simple matter to include time-dependence in the correlator \eqref{coco}. Indeed, the occupied one-particle states in \eqref{cofo} have definite energies $E_m$ given by \eqref{emi} at large $m$. This spectrum is approximately linear close to the Fermi energy: changing variables according to $m= N+k$ with $k$ finite at large $N$, one has
\begin{equation}
\label{enek}
E_{N+k}-E_N
\sim
\hbar\omega\,k
\end{equation}
where $\omega\equiv\ell^2\Omega_N/\hbar$ is the angular Fermi velocity given by the derivative \eqref{OmegaGamma} at $m=N$. Note that $\omega$ receives a number of subleading quantum corrections involving {\it e.g.}\  the curvature $\Gamma_N$ in \eqref{OmegaGamma} \cite{Wiegmann, Nardin_carusotto_2020, Nardin_carusotto_2022}; we neglect those. In the linear regime \eqref{enek}, one can repeat the asymptotic computation of the correlator and find once more a result of the form \eqref{coco}, now with a time-dependent overall phase and a time-dependent denominator $2\sin\big([f(\varphi_1)-f(\varphi_2)-\omega(t_1-t_2)]/2\big)$. This exhibits the standard ballistic propagation of correlations in a CFT, which we confirm below by deriving the effective low-energy dynamics of our droplet.

\subsection{Edge modes}
\label{Sec:MBO:Edge_modes}

The effective low-energy description of anisotropic QH droplets can be derived similarly to the isotropic case \cite{Cappelli:1992kf} inspired by Luttinger-liquid theory \cite{Haldane}. This has the advantage of circumventing topological field theory, at the cost of failing to apply in fractional QH states \cite{Bahcall:1991an, Frohlich, Wen:1990qp, Wen:1990se, Wen}. We now provide such a first-principles calculation, eventually concluding that edge modes span a free chiral CFT expressed in terms of the angle coordinate $f(\varphi)$ along the boundary. Aside from its intrinsic interest, this computation provides an independent check of the validity of the correlation \eqref{coco}.

Our starting point is the second-quantized Hamiltonian in a Fock space of free fermions,
\begin{equation}
\label{hham}
H
=
\sum_{m,n}(E_{m,n}-\mu)a^{\dagger}_{m,n}a^{\vphantom{\dagger}}_{m,n}.
\end{equation}
Here the sum runs over eigenstates of the one-body Hamiltonian \eqref{h}, labeled by the Landau level $n \in \mathbb{N}$ and the `action variable' quantum number $m \in \mathbb{N}$ within each level. Their energies are denoted $E_{m,n}$, and $\mu$ is some chemical potential to be fixed below. For each pair $(m,n)$, the operator $a^{\dagger}_{m,n}$ creates the corresponding eigenstate. The exact energy spectrum is unknown, but this is not an issue since low-energy excitations all belong to the LLL, with an approximately linear dispersion \eqref{enek} within a window $[-\Lambda,\Lambda]$ around the Fermi momentum. Then choosing the chemical potential of \eqref{hham} as $\mu = \mathrm{const} - \tfrac{1}{2}\hbar\omega$, the low-energy approximation of the many-body Hamiltonian \eqref{hham} becomes
\begin{equation}
\label{hato}
H
\sim
\sum_{k\in[-\Lambda,\Lambda]}
\hbar\omega\,(k+1/2)\,
a^{\dagger}_{N+k}a^{\vphantom{\dagger}}_{N+k},
\end{equation}
where the sum is over all integers $k$ in the specified window.
Since $a_{N+k}$ are fermionic Fock space operators, the $\Lambda\to\infty$ limit of \eqref{hato} yields a gapless Hamiltonian for free fermions written in Fourier modes, and the chemical potential enforces antiperiodic (Neveu-Schwarz) boundary conditions. It now only remains to relate the edge CFT to bulk wave functions.

To this end, note that the Fock space operators in \eqref{hato} create states in the LLL that are given by the Gaussian wave function \eqref{aspt}. One can therefore express them in terms of 2D creation operators $c^{\dagger}(\textbf x)$:
\begin{equation}
a^{\dagger}_{N+k}
\sim
\oint\frac{f'(\varphi)\,\text{d}\varphi}{\sqrt{2\pi}}\,\mathrm{e}^{\mathrm{i} (k+1/2)f(\varphi)}\,
\Psi^{\dagger}(f(\varphi)),
\end{equation}
where the 1D fermionic field $\Psi$ is defined as a radial integral involving the wave function \eqref{aspt},
\begin{multline}
\label{efild}
\Psi^{\dagger}(f(\varphi))
\equiv
\frac{\mathrm{e}^{\mathrm{i} (N-1/2)f(\varphi)}\,
\mathrm{e}^{\mathrm{i}\Phi(\varphi)}}{f'(\varphi)\sqrt{\sigma(\varphi)}} \\
\times \int_{0}^{\infty}
\frac{r\,\text{d} r}{\ell(2\pi N)^{1/4}}\,
c^{\dagger}(\textbf x)\,
\mathrm{e}^{-\frac{1}{\sigma^2(\varphi)}\Bigl(\frac{r}{\sqrt{2\ell^2f'(\varphi)}}-\sqrt{N}\Bigr)^2}
\end{multline}
with $\sigma$ the width \eqref{sigma} and $\Phi$ the WKB phase \eqref{phitap}. Note that in writing \eqref{efild} we assumed that the momentum index $k$ and the cutoff $\Lambda$ are of order $O(1)$ in the thermodynamic limit. It is then clear that the operator $\Psi^{\dagger}(\varphi)$ creates an electron at the position $\varphi$ on the edge.

From this point onward, the derivation of the low-energy effective field theory is essentially done. Indeed, the Hamiltonian \eqref{hato} expressed in terms of the edge field \eqref{efild} reads
\begin{equation}
\label{efeham}
H
=
\hbar\omega\oint \text{d}\theta\,\Psi^{\dagger}(\theta)\,(-\mathrm{i}\partial_{\theta})\Psi(\theta).
\end{equation}
Furthermore, the normalization of the field $\Psi$ defined by \eqref{efild} is canonical in angle variables: using the standard anticommutator $\{c(\textbf x_1),c^{\dagger}(\textbf x_2)\}=\delta^{(2)}(\textbf x_1-\textbf x_2)$, one similarly finds $\{\Psi(f(\varphi_1)),\Psi^{\dagger}(f(\varphi_2))\}
=\delta(f(\varphi_1)-f(\varphi_2))$ in terms of the Dirac delta function on a circle. This determines the kinetic term of the action functional for $\Psi$, which is thus a canonical fermionic expression $\sim\Psi^{\dagger}\partial_t\Psi$. It can be combined with the Hamiltonian \eqref{hato} to write down the fermionic action functional of edge modes,
\begin{equation}
\label{lactose}
S[\Psi,\Psi^{\dagger}]
=
\hbar
\int\text{d} t\,\text{d}\theta\,\mathrm{i}\Psi^{\dagger}(\theta)
\big(\partial_t+\omega\partial_{\theta}\big)\Psi(\theta),
\end{equation}
where we recall that the angular Fermi velocity is $\omega=\ell^2\Omega_{N}/\hbar$, given by \eqref{OmegaGamma}. This is manifestly a (1+1)D free chiral CFT in terms of the angle variable $\theta=f(\varphi)$.

The low-energy effective theory \eqref{lactose} is universal: for any trapping potential, edge modes are described by a chiral fermionic CFT expressed in terms of the angle coordinate of the trap at the boundary, as could have been guessed from the dynamics of electronic guiding centers induced by the potential $V$ in the non-commutative plane \eqref{pxp}. In the present case, the angle coordinate was just $\theta=f(\varphi)$; more general cases involve more complicated action-angle variables. We stress that the angle variable generally has nothing to do with other obvious position coordinates, such as the polar angle $\varphi$ or the arc length
\begin{equation}
s(\varphi)
=
\ell\sqrt{2N}
\int_0^{\varphi}\text{d}\alpha\,\sqrt{f'(\alpha)+\frac{f''(\alpha)^2}{4f'(\alpha)}}.
\end{equation}
Any such `wrong' coordinate makes the apparent Fermi velocity of edge modes position-dependent. This is reminiscent of inhomogeneous CFTs whose light-cones are curved owing to the presence of a non-zero space-time curvature \cite{AllegraEtAl2016, DSVC:2017, DSC:2017curvedlightcones, GLM:2018, LangmannMoosavi:2019, Moosavi:2021iCFT, GMS:2022, Moosavi:2022iTLL}. However, one should keep in mind that our edge modes sense a \i{flat} metric $\omega^2\text{d} t^2-\text{d}\theta^2=\omega^2\text{d} t^2-f'(\varphi)^2\text{d}\varphi^2$ whose light-cones are straight lines in terms of the angle variable $\theta=f(\varphi)$.

Let us conclude this section by showing that the action \eqref{lactose} is consistent with the seemingly complicated correlator \eqref{coco}. We start from the definition \eqref{efild} to write the 1D correlation function $\langle \Psi^{\dagger}(\theta_1)\Psi(\theta_2) \rangle$ as a double radial integral of the 2D quantity $\langle c^{\dagger}(\textbf x_1)c(\textbf x_2) \rangle$.
Now using the asymptotic relation \eqref{coco}, one finds that all normalizations simplify and the correlator of the edge field boils down to
\begin{equation}
\label{intcor}
\langle\Psi^{\dagger}(\theta_1)\Psi(\theta_2)\rangle
=
\frac{1}{2\pi}\,
\frac{\mathrm{i}}{2\sin\big([\theta_1-\theta_2]/2\big)}.
\end{equation}
The same result would have been obtained directly from the low-energy action \eqref{lactose}: it is a correlation function of free gapless fermions written in terms of the angle coordinates $\theta_1=f(\varphi_1)$ and $\theta_2=f(\varphi_2)$. As a bonus, time-dependent correlations automatically satisfy the behaviour $\propto\sin\big([\theta_1-\theta_2-\omega(t_1-t_2)]/2\big)^{-1}$ stated at the end of Sec.~\ref{secor}.

\section{Conclusion and outlook}
\label{seccc}

This work was devoted to a detailed study of mesoscopic droplets of non-interacting planar electrons placed in a perpendicular magnetic field and confined by star-shaped anisotropic traps with scale-invariant level curves. In particular, we provided explicit formulas for the corresponding wave functions and energy spectrum, allowing us to compute the many-body density, current, and correlations of an entire droplet. The resulting low-energy edge modes were eventually shown to behave as a chiral CFT in terms of the angle variable along the boundary. This was all achieved in a regime of high magnetic fields, ultimately equivalent to a semiclassical limit.

These results pave the way for a number of applications and follow-ups. Indeed, recent advances  in quantum simulation suggest the tantalizing possibility of probing local properties of QH-like droplets in the lab \cite{Goldman1, Goldman2, Fletcher, Mukherjee:2021jjl}, both for static ground states and their dynamical edge excitations. The density \eqref{rhoo} or the current \eqref{joo} then predict observable shape-dependent widths, while the low-energy theory \eqref{lactose} predicts the ballistic propagation of local boundary disturbances with a position-dependent velocity. More generally, the geometry of the QH effect \cite{Bradlyn:2012ea, Bradlyn, Abanov:2014ula, Can1, Can2, Can3, Klevtsov:2015eda} could soon become relevant for experiments involving ultracold atoms or photonics. Our framework provides a bridge between this field of mathematical physics and concrete observables in mesoscopic quantum physics. Verifying the predictions put forward here, through linear response experiments or direct imaging, would be a fascinating example of many-body quantum mechanics at work.

Turning now to theory, the link between our formalism and QH symmetries deserves further study: following the series of works \cite{Cappelli1, Cappelli:1992kf, Cappelli:1993ei, Cappelli:1994wb, Cappelli:1995yk, Cappelli:1996pg}, one can think of edge deformations as approximately unitary operators acting on many-body QH states. It is then natural to wonder how these operators get composed together; do they span a Virasoro group as in CFT? If yes, how to derive the central charge $c=1$ in terms of microscopic wave functions? More broadly, what are the operators implementing area-preserving deformations in the sense of the WKB ansatz \eqref{final}? One expects these operators to provide a finite (exponentiated) form of the operators studied in \cite{Cappelli1, Cappelli:1993ei, Cappelli:1994wb}, with non-commutative composition laws consistent with the geometry \eqref{pxp} of LLL physics.

Note that the discussion above was mostly focused on leading-order properties. For instance, one may wonder what are irrelevant corrections to the edge field theory \eqref{lactose}, especially following the recent numerical observation \cite{Nardin_carusotto_2020} that density waves on the edge satisfy a non-linear Korteweg-de Vries equation at late times. This regime is presumably described by \i{small} droplet deformations of the form $r^2\mapsto r^2+\alpha(\varphi)$, spanning a U(1) Kac-Moody algebra whose level is sensitive to the filling fraction \cite{Cappelli1, Cappelli:1993ei, Cappelli:1994wb}. The resulting non-linear edge waves would then be described by an evolution equation in an infinite-dimensional group manifold. This perspective is standard in hydrodynamics \cite{ArnoldOrigin, ArnoldKhesin, Oblak:2020jek}, but it has only recently come to be appreciated in condensed matter physics \cite{Delacretaz_son_2022}. The geometric study initiated here provides a basis for considerations of this kind in the QH effect, including the possibility of inhomogeneous (position-dependent) corrections in anisotropic traps.

Another obvious extension of this work is the \i{fractional} QH regime. In that context, no single-particle description is available, but many-body predictions such as the edge density \eqref{rhoo} or the current \eqref{joo} conceivably display universal geometric features that would remain true in interacting many-body ground states \cite{Geller2}. It would be thrilling to derive such predictions from the family of edge transformations studied here, either from a microscopic analysis of the Laughlin wave function, or thanks to the reformulation of fractional QH states as CFT correlation functions \cite{Moore:1991ks}.

\acknowledgments{We are grateful to Laurent Charles for illuminating discussions on semiclassical methods in K\"ahlerian geometric quantization. B.O.\ also thanks Mathieu Beauvillain, Nathan Goldman, and Marios Petropoulos for collaboration on closely related subjects. Finally, we acknowledge useful and motivating interactions with Jean Dalibard, Benoit Dou\c{c}ot, Jean-No\"el Fuchs, Marc Geiller, Gian Michele Graf, Semyon Klevtsov, Titus Neupert and Nicolas Regnault. The work of B.O.\ is supported by the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowska-Curie grant agreement No. 846244. B.L.\ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (Grant No. ERC-StG-Neupert-757867-PARATOP).
P.M.\ gratefully acknowledges financial support from the Wenner-Gren Foundations (No.\ WGF2019-0061). B.E.\ is supported by the ANR grant \i{TopO} No.\ ANR-17-CE30-0013-01.}

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The beginning of a PID is usually no different to any other formal document you will be expected to write and covers Document Change History, Related Documents and Purpose.
As you will see above, this section is merely where you should note any updates made to the document, and what caused those updates.
This section is where all the documents and information which you have used in writing the PID need to be stated so they can be reference checked later if required.
How and when the arrangements covered in this document will be put into place.
When approved by the Project Board it will provide the baseline for the project and will become 'locked'. The PID will not be updated between stages unless major changes are made to the project which will require prior approval from the Project Board. It is anticipated that any revisions which need to be made will be done so at the end of each stage to incorporate detailed milestones and plans for the next phase, together with any organisational/ team changes required.
The PID will form the basis of decisions taken for the project and be used at the conclusion of the project as a basis for post project review. Updates between stages to the approved project PID will take the form of addendums.
Ensure you keep on top of these sections as there is nothing worse than a Project PID being queried at a later stage when no record has been kept of when and why changes to a PID were made.


text:
Round Lake is located southeast of Kalurah, New York. Fish species present in the lake are brook trout, and black bullhead. Access via old railroad bed from the west and a trail from Aldric from the east.

References

Lakes of St. Lawrence County, New York
Lakes of New York (state)

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text:
Festival Review: Top Ten Sets at Osheaga 2012
Gilles LeBlanc and Stewart Wiseman
August 6, 2012 | 4:13pm ET
The theme of this year's Osheaga Music and Arts Festival could have been titled "The Price of Success." What had been building as a niche event for musically aware fans since 2006 turned into a juggernaut in 2012, seemingly overnight. For the first time in their history, Osheaga organizers had to deal with sellout crowds, even with the grounds at Parc Jean-Drapeau on Saint Helen's Island having been expanded considerably. The lineup was that good, featuring a mix of artists parachuting in and out of Lollapalooza, as well as some band grabs exclusive to Montreal that had the fest community buzzing in the months leading up to it.
Try as she might, Mother Nature could not put on a damper on the enthusiasm that spread throughout the entire weekend. Extreme heat and the threat of thunderstorms? Pfft, bring it on was the mantra chanted. Nothing was going to get in the way of anyone's enjoyment. Considering the insane level of festival coverage devoted during these three days, Consequence of Sound decided to focus on what were deemed the absolute best, most unique and special sets during Osheaga 2012.
-Gilles LeBlanc
Friday, August 3rd – Virgin Mobile River Stage – 5:50 p.m.
When Osheaga's lineup was first announced, Franz Ferdinand turned plenty of heads. Everyone's favorite band named after the Austrian archduke responsible for World War I have been shrouded in a little mystery in recent years. There still hasn't been a formal announcement of a fourth studio album, which might explain why their set time was curiously early. However, once they finally materialized in Montreal for their first North American show in almost three years, scrutiny over Franz Ferdinand quickly turned to blame. Their scorchingly danceable set also introduced two new songs: "Right Thoughts! Right Words! Right Action!" and "Scarlet and Blue". The latter is classic-sounding Franz Ferdinand, with revving guitars supplemented by sparse notes played on an electronic organ. We were also treated to the infamous Franz Ferdinand drum circle on "Outsiders". And I have to add that while I'm sure no woman in attendance Friday would ever kick him out for eating kräckers, if you know what I'm saying, what's with the new stache Mr. Kapranos? -Gilles LeBlanc
Pages: 1234567891011
City and Colour
Osheaga Music and Arts Festival
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text:
ALA-CBC Joint Committee
CBC Diversity Committee
Mid-Level Career Committee
Graphic Novel Committee
Early Career Committee
Home > Blog > Women's National Book Association Announces…
February 19, 2013 On Our Radar
Women's National Book Association Announces Nominees for the 2013 WNBA Pannell Award
Since 1983, the Women's National Book Association has awarded one of the most prestigious honors in children's bookselling. Given annually at BookExpo America's Children's Book and Author Breakfast, the WNBA Pannell Award recognizes bookstores that enhance their communities by bringing exceptional creativity to foster a love of reading in their young patrons.
Every year a panel of publishing professionals selects two winners of the award—one a general bookstore and one a children's specialty bookstore. The store nominations come from customers, sales reps, store personnel, or anyone who has been impressed with the work of a particular independent bookstore. This year's nominees are:
General Bookstore
Avid Book Shop, Athens, GA
Byrd's Books, Bethel, CT
Nicola's Books, Ann Arbor, MI
Newtonville Books, Newtonville, MA
Porter Square Books, Cambridge, MA
Main Street Books, Davidson, NC
Vroman's Bookstore, Pasadena, CA
Park Road Books, Charlotte, NC
Children's Specialty Store
Books and Cookies, Santa Monica, CA
4 Kids Books & Toys, Zionsville, IN
Hooray for Books, Alexandria, VA
The Bookbug, Kalamazoo, MI
The Voracious Reader, Larchmont, NY
Children's Book World, Los Angeles, CA
The nominated store puts together an electronic submission with a description of activities, goals, or any contribution to the local community that involves young people and books. Photos, media coverage, letters from customers, or anything else that transmits the degree of contribution can be included in the submission. The deadline for submissions is March 31, 2013.
The Pannell Award jurors for 2013 are:
Cheryl Willis Hudson, vice president and editorial director of Just Us Books
Andrew Karre, editorial director of Carolrhoda, an imprint of the Lerner Publishing Group
Lisa von Drasek, curator of Children's Literature Research Collections, Univ. of Minnesota
Emma D. Dryden, children's editorial and publishing consultant, Dryden Books
Kelli Chipponeri, executive editor/children's, Chronicle Books
The jurors will make their decision by late April, and a phone call will notify the winners, as well as all stores sending submissions. Each of the two winners will receive a $1,000 check and a framed signed original piece of art by a children's illustrator. The presentation of the award will be in New York at the BEA/ABA Children's Book and Author Breakfast, which draws more than 1,000 attendees.
Along with WNBA, Penguin Young Readers Group co-sponsors the award. Founded in 1917, WNBA is a national organization of women and men who promote the value to the written word by championing the role of women in the book community and by providing a forum for the exchange of ideas and information relating to the written word. www.wnba-books.org.
History is Lit


text:
Q: XML mapping with @XmlAnyElement I would like to achieve something like this.
<zoo>
    <lion> ... </lion>
    <dog> ... </dog>
</zoo>

I have this class here.
public class MainGroup {
    private List<Widget> widgets;

    @XmlAnyElement
    public List<Widget> getWidgets() {
        return widgets;
    }
    public void setWidgets(List<Widget> widgets) {
        this.widgets = widgets;
    }
}

And this Widget superclass has got subclasses such as Button, Combobox...
I would like to achieve something like this.
<MainGroup>
    <Button>...</Button>
    <Combo>...</Combo>
</MainGroup>

I am having this exception
[com.sun.istack.internal.SAXException2: class com.test.Button nor any of its super
  class is known to this context.

I tried adding @XmlElementRef but it is still not working.
@XmlElementRefs({
     @XmlElementRef(name="Button", type=Button.class),
     @XmlElementRef(name="Combo", type=Combo.class)
})

A: Mapping your Use Case
My answer is based on information gathered from one of your related questions:

*

*Why doesn't JAXB writes out SWT Widgets?
Since you are mapping classes for which you do not have the source (and therefore can't add JAXB annotations), I would recommend using the @XmlElements mapping.
@XmlElements({
     @XmlElement(name="Button", type=Button.class),
     @XmlElement(name="Combo", type=Combo.class)
})
public List<Widget> getWidgets() {
    return widgets;
}

@XmlElements corresponds to the XML Schema concept of xsd:choice.

*

*http://blog.bdoughan.com/2010/10/jaxb-and-xsd-choice-xmlelements.html
About @XmlRootElement

Ok, I am missing quite a lot of things out here. It seems like I add
  to add this @XmlRootElement annotation to my subclasses of Button and
  Combo to achieve that.
Can anyone explain to me why I need that annotation in my
  subclasses... I am confused, I thought an XML would only have a
  @XmlRootElement which in my case should be in MainGroup class.

@XmlRootElement corresponds to global elements in the XML schema, which involves more that just the root element in the document you are unmarshalling.  I'll describe a couple of the roles below:
@XmlElementRef
@XmlElementRef corresponds to the concept of substitution groups.  In an XML Schema you can specify that one global element is substitutable for another.  In JAXB @XmlRootElement (and @XmlElementDecl) is leveraged to specify global elements:

*

*http://blog.bdoughan.com/2010/11/jaxb-and-inheritance-using-substitution.html
@XmlAnyElement
@XmlAnyElement corresponds to the concept of xs:any in XML Schena.  This is part of the document that is pretty free form.  In JAXB when you map a property with @XmlAnyElement(lax=true) it will convert elements matching @XmlRootElement declarations into the corresponding domain objects.

*

*http://blog.bdoughan.com/2010/08/using-xmlanyelement-to-build-generic.html

A: Ok, I am missing quite a lot of things out here.
It seems like I add to add this @XmlRootElement annotation to my subclasses of Button and Combo to achieve that.
Can anyone explain to me why I need that annotation in my subclasses... I am confused, I thought an XML would only have a @XmlRootElement which in my case should be in MainGroup class.

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