Adiabatic Deformations of Quantum Hall Droplets
Blagoje Oblaka,b and Benoit Estienneb
<EMAIL_ADDRESS><EMAIL_ADDRESS>
a CPHT, CNRS, École polytechnique, Institut Polytechnique de Paris, 91120
Palaiseau, France.
b Laboratoire de Physique Théorique et Hautes Energies,
Sorbonne Université and CNRS UMR 7589, F-75005 Paris, France.
Abstract We consider area-preserving deformations of the plane, acting on
electronic wave functions through ‘quantomorphisms’ that change both the
underlying metric and the confining potential. We show that adiabatic
sequences of such transformations produce Berry phases that can be written in
closed form in terms of the many-body current and density, even in the
presence of interactions. For a large class of deformations that generalize
squeezing and shearing, the leading piece of the phase is a super-extensive
Aharonov-Bohm term ($\propto N^{2}$ for $N$ electrons) in the thermodynamic
limit. Its gauge-invariant subleading partner only measures the current, whose
dominant contribution to the phase stems from a jump at the edge in the limit
of strong magnetic fields. This results in a finite Berry curvature per unit
area, reminiscent of the Hall viscosity. We show that the latter is in fact
included in our formalism, bypassing its standard derivation on a torus and
suggesting realistic experimental setups for its observation in quantum
simulators.

###### Contents

  1. 1 Introduction and motivation
  2. 2 Adiabatic deformations of quantum wires
    1. 2.1 Berry phases from group actions
    2. 2.2 Berry phases from circle deformations
    3. 2.3 Examples of adiabatic deformations
  3. 3 Quantum area-preserving deformations
    1. 3.1 Area-preserving deformations
    2. 3.2 Divergence-free vector fields and incompressible flows
    3. 3.3 Unitary deformations are quantomorphisms
  4. 4 Adiabatic deformations of planar droplets
    1. 4.1 Invitation: Berry phases of neutral states
    2. 4.2 Berry phases of charged states
    3. 4.3 Examples of adiabatic deformations
    4. 4.4 Deformations of Hall droplets and Hall viscosity
  5. 5 Conclusion and outlook
  6. A Quantomorphisms on closed surfaces
    1. A.1 Line bundles, connections and holonomies
    2. A.2 Lifting diffeomorphisms to line bundles
    3. A.3 Quantomorphisms from parallel transport
    4. A.4 Hamiltonian vector fields and infinitesimal quantomorphisms
  7. B Cocycle contribution to the Berry phase

##  1 Introduction and motivation

One of the defining characteristics of topological phases of matter is their
robustness under smooth changes of the Hamiltonian. This notably includes
geometric deformations of the sample supporting the system; after all, the
low-energy field theory of any such phase is topological, hence insensitive to
bulk diffeomorphisms [1, 2, 3]. Crucially, topological invariance entails the
presence of chiral edge modes propagating along the boundary of the system—and
these generally do react to deformations in a non-trivial manner, one that
actually determines the low-energy physics [4, 5, 6]. It is therefore of
interest to model the effects of geometric deformations on topological phases
of matter, including their boundary.

The present work provides just such an analysis in the paradigmatic case of
the quantum Hall (QH) effect [7, 8]. In that context, the importance of
diffeomorphisms has been recognized since the early days, as the symplectic
structure of position operators projected to the lowest Landau level is
essential both for magneto-rotons [9] and for the non-commutative approach to
QH physics [10]. The key actors for these lines of thought are area-preserving
deformations spanning the Girvin-MacDonald-Platzmann algebra [9], i.e. a
$w_{1+\infty}$ algebra [11, 12, 13, 14, 15, 16, 17, 18]. In particular, a
staple of the seminal works [12, 13, 14] was the sketch of a relation between
bulk deformations and conformal transformations of gapless edge modes. We
shall partly rely on these insights to focus on area-preserving maps of
physical interest, which we dub ‘edge deformations’ for reasons that will
become clear below (see fig. 1).

Figure 1: An initially isotropic planar electron droplet deformed by a typical
edge deformation (applied using eq. (25) below). The shape of the edge changes
in an arbitrary manner while preserving the droplet’s area, even in the
thermodynamic limit.

An apparently unrelated class of deformations leads to the Hall viscosity [19,
20], also known as ‘odd viscosity’ [21] or ‘Lorentz shear modulus’ [22, 23],
where a QH sample on a torus reacts to adiabatic changes of the modular
parameter. In that case, one reads off the Berry curvature [24] associated
with linear reparametrizations of the metric, finding an extensive result
(proportional to the number $N\gg 1$ of electrons) whose coefficient is
quantized and robust against disorder [25], similarly to the Hall conductance
[26]. The resulting geometric response and relation to hydrodynamics has been
studied in great detail in recent years [27, 28, 29, 30, 31, 32, 33, 34, 35,
36, 37], but concrete observations seem elusive despite recent encouraging
results in graphene [38]. As we shall show here, the Hall viscosity can in
fact be derived on a plane, as a response to a simple subset of the
aforementioned edge deformations. This bypasses the complicated toroidal wave
functions typically needed in standard computations of the Hall viscosity,
suggesting that the latter can be measured as a response to specific time-
dependent perturbations in tabletop quantum simulators [39, 40, 41, 42, 43].

Our approach is based on a one-body formalism and is then extended to many-
body droplets. Concretely, we consider area-preserving deformations of the
metric and potential in a Landau Hamiltonian (see eq. (37) below). This is
done in a gauge-invariant and unitary manner through so-called quantomorphisms
that may be seen as local generalizations of magnetic translations [44, 45].
One can then choose a set of slow time-dependent deformations, forcing the
Hamiltonian to become time-dependent as well. Provided these deformations
start and end at the same configuration, and assuming the initial wave
function was an energy eigenstate, the final wave function coincides with the
initial one up to an uninteresting dynamical phase and a crucial Berry phase
[46, 24, 47]. The latter can in fact be written as a fairly simple expectation
value when parameter variations are unitary [48, 49, 50], as will be the case
here. Indeed, we show in this way that adiabatic quantomorphisms produce Berry
phases containing two separately gauge-invariant terms: the first is a
contribution of the current that appears universally whenever diffeomorphisms
act on wave functions, and the second is an Aharonov-Bohm (AB) phase weighted
by the density [51]. Schematically,

$\text{Berry
phase}=\int\text{d}^{2}\textbf{x}\,\Big{(}\text{current}\times\text{velocity}\,+\,\text{density}\times\text{AB
phase}\Big{)}$ (1)

where $\text{d}^{2}\textbf{x}=\text{d}x\,\text{d}y$ in Cartesian coordinate,
and we refer to eq. (50) below for the detailed expression. We stress that
this can all be written in terms of explicit formulas, applying both to one-
body states and fully-fledged droplets of $N\gg 1$ electrons, interacting or
not.

While the computations that we carry out hold for any charged quantum state in
the plane, a regime of particular interest is that of weak potentials and
strong magnetic fields. The one-body energy spectrum then splits into familiar
Landau levels, resulting in a many-body density that is roughly constant and
quantized in the bulk of a droplet, but zero outside. The many-body current,
on the other hand, is typically small in the bulk but jumps in a Gaussian
fashion near the edge (see fig. 10 below) [52, 53]. This entails a distinction
between the two pieces of the Berry phase (1): the AB phase is sensitive to
bulk deformations, while the current measures both bulk and edge effects. We
eventually illustrate this by restricting attention to a specific class of
planar diffeomorphisms, namely the aforementioned ‘edge deformations’ that
contain, in particular, all linear maps

$\begin{pmatrix}x\\\ y\end{pmatrix}\mapsto\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}x\\\ y\end{pmatrix}\qquad\text{with}\qquad ad-
bc=1.$ (2)

Acting on a Hall droplet with an adiabatic sequence of such transformations
produces a Berry phase (1) whose AB term is super-extensive ($\propto N^{2}$
for $N$ electrons). Discarding the latter leaves out the gauge-invariant
current contribution, which turns out to be extensive in the limit of strong
magnetic fields. The corresponding Berry curvature per unit area provides an
infinite-parameter analogue of the Hall viscosity, to which it reduces up to
an overall factor 2 in the case of linear maps (2) applied to integer QH
states. For fractional QH states, the Berry curvature that we find is again
very similar to Hall viscosity, but differs from it by a factor $2\nu$ in
terms of the filling fraction $\nu<1$. We eventually show that this mismatch
is due to the fact that the current term in (1) measures variations of both
metric and potential, while the Hall viscosity is defined as a response to
metric variations alone. In this sense, there was no reason for the Berry
phase (1) to be related to viscosity at all; it just so happens that its
extensive piece is proportional to the Hall viscosity.111The proportionality
could have been guessed on geometric grounds: the parameter space for linear
maps (2) is a hyperbolic plane, which has a unique
$\text{SL}(2,\mathbb{R})$-invariant Berry curvature up to normalization.

$\big{|}\Psi(g_{t})\big{>}$$g_{t}$Deformations Figure 2: A base space,
consisting of all area-preserving planar deformations, supports a line bundle
(so the fibers are complex planes). Given some deformation $g$, the fiber
above it is the ray of some quantum state $|\Psi(g)\rangle$. The bundle is
endowed with a Berry connection. Parallel transport of a quantum state along
some closed loop $g_{t}$ generally results in a net holonomy—a Berry phase.
The latter measures an ‘area’ (highlighted in yellow) in the group of
deformations. Adapted from [50, fig. 1].

As usual, statements on Berry phases can be phrased either in terms of finite
holonomies as in fig. 2, or in terms of local Berry curvatures and linear
response.222Holonomies actually contain more information, as they also know
about Berry phases along non-contractible cycles—think e.g. of the two cycles
of a flat torus. In the case of planar deformations of isotropic states,
parameter space is homotopic to a point so this subtlety plays no role. In the
case at hand, connections and curvatures are differential forms on an
infinite-dimensional parameter space consisting of all area-preserving
diffeomorphisms, so they are somewhat unwieldy. We have therefore chosen to
present most of our results in terms of (conceptually simpler) finite phases,
omitting for now the detailed study of infinite-dimensional quantum geometry.
The latter will be crucial in practice, as the likeliest avenue to observe the
effects of the phase (1) is to study adiabatic linear response in quantum
simulators, where the high degree of control over microscopic details may help
overcome issues of decoherence and disorder.

The remainder of this work is organized as follows. We begin with preliminary
material in sec. 2, showing first how unitary group actions give rise to
parameter-dependent Hamiltonians, then applying the idea to adiabatic
diffeomorphisms of one-dimensional (1D) quantum wires. Sec. 3 then introduces
area-preserving diffeomorphisms and their unitary, gauge-invariant action on
charged wave functions through quantomorphisms [44, 45]. We present this
concept in detail and show that diffeomorphisms are represented in a
projective manner, with a specific central extension that we compute. Our key
results are then derived in sec. 4. This includes the formula (50) for Berry
phases of 2D droplets subjected to adiabatic quantomorphisms, its value (63)
in the special case of edge deformations, and its application to QH states and
the Hall viscosity (79). Finally, the conclusion is devoted to a brief
discussion of follow-ups, while app. A provides a detailed review of the
geometry of quantomorphisms in the case of compact surfaces such as a torus,
where subtleties occur that do not affect the plane. App. B is a technical
aside useful for sec. 4.

One last minor warning before we proceed: this work involves tools in
differential geometry that are not reviewed in detail. We therefore refer to
[54, chaps. 1–2] for useful background on diffeomorphisms, vector fields and
flows, and to [55, part I] or [56] for a pedagogical introduction to Lie
groups and symplectic geometry. Finally, various aspects of groups of
diffeomorphisms, along with some symplectic geometry, are reviewed in [57,
chaps. 5–6] in a language that may be closer to the habits of physicists.

##  2 Adiabatic deformations of quantum wires

This section provides crucial background for later use. Specifically, we start
by showing that unitary Lie group actions give rise to geometric phases [24,
48] when some group elements fail to commute with the Hamiltonian. This
general fact is then applied to diffeomorphisms of wave functions on a circle,
with potential implications in quantum wires with persistent currents [58, 59,
60]. The resulting phases mimic those produced by adiabatic conformal maps
[50], and will eventually appear on the edge of Hall droplets in sec. 4.4.

###  2.1 Berry phases from group actions

Here we sketch a general derivation of Berry phases associated with adiabatic
group actions in quantum mechanics, obtained upon seeing group elements as
labels for parameter-dependent Hamiltonians [50, sec. 2].

#### Parameter-dependence from group theory.

We are interested in transformations that act on a quantum system and span a
continuous group. Accordingly, let $G$ be a (connected) Lie group acting on
some Hilbert space ${\cal H}$, with $\,{\cal U}[g]$ the unitary operator
implementing the transformation $g\in G$. We assume that the assignment
$g\to\,{\cal U}[g]$ furnishes a representation of $G$, i.e. it is compatible
with multiplication in $G$; for later convenience we choose a right
representation, so

$\,{\cal U}[f]\,{\cal U}[g]=\,{\cal U}[gf]$ (3)

for all $f,g\in G$. We stress that this is merely a matter of convention: it
makes some formulas look simpler, but it does not affect the physics.

Now let $H$ be an ‘unperturbed’ Hamiltonian operator, and force the system to
undergo a transformation due to $g\in G$; imagine e.g. a spin in a rotating
magnetic field, where $g$ is a rotation. Then the transformed Hamiltonian is

$H[g]=\,{\cal U}[g]H\,{\cal U}[g]^{{\dagger}}.$ (4)

If some $\,{\cal U}[g]$’s fail to commute with $H$, the operators $H[g]$ are
‘deformed’ Hamiltonians, each depending parametrically on a point $g\in G$.
Their energy spectrum coincides with that of $H$: $|\Psi\rangle\in{\cal H}$ is
an eigenstate of $H$ if and only if $\,{\cal U}[g]|\Psi\rangle$ is an
eigenstate of $H[g]$ with the same eigenvalue. Importantly, eigenstates depend
on $g$ even though their energy does not, which is ultimately why Berry phases
appear in this context. Examples include the usual action of 3D rotations on a
qubit [24, 61], or that of Lorentz boosts on relativistic states [62, 63, 64],
or conformal maps in conformal field theory [50]. Starting in sec. 3, $G$ will
consist of area-preserving deformations acting on fermionic wave functions.

#### Berry phases.

Parameter-dependence entails the presence of geometric phases [24]. In the
case at hand, let $g_{t}$ be some path in $G$ where $t\in[0,T]$ is a time
variable, giving rise to a time-dependent Hamiltonian $\,{\cal
U}[g_{t}]H\,{\cal U}[g_{t}]^{\dagger}$. The problem is to solve the ensuing
Schrödinger equation. If the path $g_{t}$ is traced very slowly,333‘Slowly’
normally means that the operator norm of $\hbar\partial_{t}(\,{\cal
U}[g_{t}]H\,{\cal U}[g_{t}]^{\dagger})$ is much smaller than the square of the
energy gap separating $|\Psi\rangle$ from other eigenstates, but this is too
strong since the adiabatic theorem even holds for gapless ground states [65].
Indeed, this is the situation we’ll encounter in sec. 4. and provided the
initial state vector is an eigenstate of $\,{\cal U}[g_{0}]H\,{\cal
U}[g_{0}]^{\dagger}$ with isolated and non-degenerate energy,444Non-degeneracy
eventually implies that the Berry connection is Abelian; degenerate, non-
Abelian cases are not treated here. In sec. 4, the degeneracy of Landau levels
will be lifted by a weak confining potential. the solution of the Schrödinger
equation is itself a time-dependent energy eigenstate. In formulas, if
$|\Psi\rangle$ is a (time-independent) normalized eigenstate of $H$ with
energy $E$ and the initial condition is $\,{\cal U}[g_{0}]|\Psi\rangle$, then
the adiabatic theorem [46, 65] ensures that the state vector at time $t$ is

$|\psi(t)\rangle\sim\exp\left[-i\frac{Et}{\hbar}-\int_{0}^{t}\text{d}\tau\,\langle\Psi|\,{\cal
U}[g_{\tau}]^{\dagger}\partial_{\tau}\,{\cal
U}[g_{\tau}]|\Psi\rangle\right]\,{\cal U}[g_{t}]|\Psi\rangle$ (5)

up to corrections that vanish as the rate of change of $g_{t}$ goes to zero.
It follows that any closed path $g_{t}$ in $G$, with period $T$ say, leads to
a final state vector $|\psi(T)\rangle\sim e^{i\theta}|\psi(0)\rangle$ that
coincides with the initial one up to a phase $\theta$. The latter is the sum
of a dynamical phase $-ET/\hbar$ and a Berry phase [24, 48, 50]

${\cal B}_{\Psi}[g_{t}]=i\oint\text{d}t\,\langle\Psi|\,{\cal
U}[g_{t}]^{\dagger}\partial_{t}\,{\cal
U}[g_{t}]|\Psi\rangle=-\oint\text{d}t\,\langle\Psi|\mathfrak{u}\big{[}(\partial_{t}g_{t})g_{t}^{-1}\big{]}|\Psi\rangle,$
(6)

where the second equality follows from eq. (3) and $\mathfrak{u}[v]$ is the
Hermitian operator obtained by differentiating $\,{\cal U}[g]$ at the
identity, for any Lie algebra element $v$:

$\mathfrak{u}[v]\equiv-i\partial_{\epsilon}\big{|}_{0}\,{\cal U}[e^{\epsilon
v}]\qquad\text{for any $v\in\mathfrak{g}$}.$ (7)

The phase (6) is thus a functional of the curve $g_{t}$ and depends
parametrically on the reference state $|\Psi\rangle$. Famous examples of this
kind include the Berry phases of a spin in a rotating magnetic field [24, 61]
and Thomas precession [62, 63, 64], respectively stemming from unitary
rotations and Poincaré transformations.

Note that transformations $g_{t}$ that belong to the stabilizer of
$|\Psi\rangle$ in the sense that $\,{\cal
U}[g_{t}]|\Psi\rangle\propto|\Psi\rangle$ give rise to vanishing phases
(modulo $2\pi$). As a result, the actual parameter space responsible for the
phases (6) is not quite the group manifold $G$, but its quotient by the
stabilizer of $H$. For example, many of the cases treated below will involve
isotropic Hamiltonians whose eigenstates have definite angular momentum; these
are invariant under rotations, so their parameter space will be a quotient
$G/\text{U(1)}$. This subtlety is not crucial in practice, as one can just
compute Berry phases and notice, after the fact, that they vanish for certain
families of deformations (namely those in the stabilizer).

#### Central extensions.

To conclude this general group-theoretic presentation, let us provide a
generalization of eq. (6) that will be of the utmost importance later. Suppose
indeed that the unitary operators $\,{\cal U}[g]$ furnish a projective
representation of $G$, i.e. that eq. (3) is corrected by a phase factor:

$\,{\cal U}[f]\,{\cal U}[g]=e^{i\mathsf{C}(g,f)}\,\,{\cal U}[gf].$ (8)

Here $\mathsf{C}(g,f)$ is some non-zero real function, known as a central
extension, that satisfies the following identity in order to preserve the
associativity of composition:555Eq. (9) says that $\mathsf{C}$ is a two-
cocycle in the sense of group cohomology; see e.g. [57, sec. 2.1] for details.

$\mathsf{C}(g,f)+\mathsf{C}(h,gf)=\mathsf{C}(h,g)+\mathsf{C}(hg,f).$ (9)

Then the second equality in the phase (6) no longer holds, since it relied on
the condition that $\,{\cal U}$ be an exact representation (with
$\mathsf{C}=0$ in (8)). Instead, when $\,{\cal U}$ is projective, the cocycle
$\mathsf{C}$ adds an extra term to the Berry phase: assuming without loss of
generality that the neutral element $e\in G$ acts trivially (i.e. $\,{\cal
U}(e)=\mathbb{I}$) so that $\mathsf{C}(e,g)=0$, one finds

${\cal
B}_{\Psi}[g_{t}]=-\oint\text{d}t\,\langle\Psi|\mathfrak{u}\big{[}(\partial_{t}g_{t})g_{t}^{-1}\big{]}|\Psi\rangle-\oint\text{d}t\,\partial_{\tau}\big{|}_{t}\mathsf{C}(g_{\tau},g_{t}^{-1}).$
(10)

Such an extension will affect area-preserving maps and their Berry phases in
secs. 3–4, essentially due to non-commuting magnetic translations.

###  2.2 Berry phases from circle deformations

Having reviewed how unitary group actions yield geometric phases, we now turn
to the phases produced by adiabatic deformations of a 1D quantum system on a
circle. This is a key toy model for future reference: first because the
unitary action of quantomorphisms in sec. 3.3 is inspired by the 1D setup, and
second because deformations of the edge of Hall droplets effectively produce
1D phases in sec. 4.3. Note that our discussion of diffeomorphisms of the
circle is kept to a minimum; we refer e.g. to [57, sec. 6.1] for a smoother
introduction. See also [50] for a derivation of Berry phases nearly identical
to the ones studied here, albeit in the context of 1D conformal field theory.

#### Unitary circle diffeomorphisms.

Consider a particle confined to a 1D ring, i.e. a unit circle $S^{1}$. Let its
quantum state be described by a $2\pi$-periodic wave function $\Psi(\varphi)$
in the Hilbert space $L^{2}(S^{1})$. We wish to deform this wave function via
diffeomorphisms of the circle, i.e. invertible smooth maps $g:S^{1}\to S^{1}$
whose inverse $g^{-1}$ is also smooth. Since we are ultimately interested in
continuous paths of diffeomorphisms connected to the identity, we focus on
orientation-preserving maps; these span a group denoted $\text{Diff}\,S^{1}$,
with a ‘multiplication’ given by the composition of functions. The simplest
way to describe such maps is to lift them to $\mathbb{R}$, yielding smooth
functions $g:\mathbb{R}\to\mathbb{R}:\varphi\mapsto g(\varphi)$ that satisfy
$g^{\prime}(\varphi)>0$ and $g(\varphi+2\pi)=g(\varphi)+2\pi$ for all
$\varphi$.666The lift is not unique since $g(\varphi)$ and $g(\varphi)+2\pi n$
describe the same diffeomorphism for any integer $n$, but this ambiguity is
harmless since it does not affect space-time derivatives of time-dependent
deformations. For instance, rotations lift to $g(\varphi)=\varphi+\theta$;
more general diffeomorphisms can be seen as ‘wiggly’ versions of rotations
(see fig. 3). Other examples of circle diffeomorphisms are given in eq. (21)
below.

Figure 3: Several generic diffeomorphisms of the circle, all chosen to fix the
origin so that $g(0)=0$ and $g(2\pi)=2\pi$. Loosely speaking, any such
function can be seen as an identity map $g(\varphi)=\varphi$ with extra
‘wiggles’. (The sharp version of this statement is that any orientation-
preserving diffeomorphism of the circle is homotopic to the identity, i.e.
that the group $\text{Diff}\,S^{1}$ is connected.)

How should circle deformations act on wave functions? A straightforward answer
is provided by the fact that wave functions are half-densities—their norm
squared is a measure. Thus a natural action of $\text{Diff}\,S^{1}$ on
$L^{2}(S^{1})$ is

$\big{(}\,{\cal
U}[g]\Psi\big{)}(\varphi)\equiv\sqrt{g^{\prime}(\varphi)}\,\Psi(g(\varphi))$
(11)

for any $g\in\text{Diff}\,S^{1}$, where the square root on the right-hand side
ensures unitarity.777Thus deformations are not mere changes of variables
$\Psi\to\Psi\circ g$, which would not be unitary! One readily verifies that
this is indeed a right representation in the sense of eq. (3). Intuitively, it
mimics the fact that deformations of wave functions are induced by those of a
sample, and that the local height of a wave function follows the local density
of points; see fig. 4 for a cartoon. One can also deduce from (11) the
expression of Lie algebra operators (7) that implement infinitesimal
diffeomorphisms, i.e. vector fields. Indeed, writing
$g(\varphi)=\varphi+\epsilon\,v(\varphi)$ with a $2\pi$-periodic function
$v(\varphi)$ and expanding (11) up to first order in $\epsilon$ yields

$\mathfrak{u}[v]\Psi\stackrel{{\scriptstyle\text{\eqref{ux}}}}{{\equiv}}-i\lim_{\epsilon\to
0}\tfrac{1}{\epsilon}(\,{\cal
U}[g]\Psi-\Psi)=-i\big{(}v(\varphi)\partial_{\varphi}+\tfrac{1}{2}v^{\prime}(\varphi)\big{)}\Psi(\varphi).$
(12)

This will be useful to apply the Berry phase formula (6) to time-dependent
diffeomorphisms. Note that the operator (12) can be recast in terms of
standard position and momentum: using $p=-(i\hbar/R)\partial_{\varphi}$ on a
circle of radius $R$, one has
$(\hbar/R)\mathfrak{u}[v]=v(\varphi)p-i(\hbar/2R)v^{\prime}(\varphi)$, where
the second term is a ‘correction’ ensuring that $\mathfrak{u}[v]$ is
Hermitian. One can thus think of $\mathfrak{u}[v]$ as a position-dependent
translation, as should indeed be the case for diffeomorphisms.

Figure 4: A circle (black, dashed points) supports a wave function with
initially uniform density (blue-yellow density plot). The circle is acted upon
by two deformations: the first is a local contraction towards $\varphi=0$
(i.e. a dilation away from $\varphi=\pi$); the second is a rotation
$\varphi\mapsto\varphi+\theta$. The effect on the wave function is manifest:
the initial contraction affects its density through the square root in (11),
while the rotation rigidly rotates its argument according to the change of
argument in (11). Note that the plot shows the procedure one would actually
carry out in a lab: if $g_{1}$, $g_{2}$ are the two transformations acting on
the wave function according to eq. (11), then the transformations of the
circle that are actually shown are the inverses $g_{1}^{-1}$ and $g_{2}^{-1}$.
This minor subtlety is due to our choice to use right group representations
(3), and would not occur if all $g$’s in eq. (11) were replaced by $g^{-1}$’s.

As explained in sec. 2.1, group elements acting on a reference Hamiltonian $H$
produce deformed Hamiltonians $\,{\cal U}[g]H\,{\cal U}[g]^{\dagger}$ that
depend parametrically on $g$. To develop some intuition on the operators (11),
it helps to ask how they actually modify a typical one-body Hamiltonian with
some $2\pi$-periodic potential $V$,

$H=\frac{p^{2}}{2M}+V(\varphi).$ (13)

Using the definition (11) and working again on a circle of radius $R$, one
finds

$\,{\cal U}[g]H\,{\cal
U}[g]^{\dagger}=\frac{1}{2M}\,p\Big{(}\frac{1}{g^{\prime}(\varphi)^{2}}\Big{)}p+V\big{(}g(\varphi)\big{)}+\frac{\hbar^{2}}{4MR^{2}\,g^{\prime}(\varphi)^{2}}\bigg{(}\frac{g^{\prime\prime\prime}}{g^{\prime}}-\frac{5}{2}\Big{(}\frac{g^{\prime\prime}}{g^{\prime}}\Big{)}\\!\\!\bigg{.}^{2}\,\bigg{)}.$
(14)

Here the deformed kinetic term involves a non-Euclidean metric
$g^{\prime}(\varphi)^{2}\text{d}\varphi^{2}$, as was to be expected; the
ordering makes it manifest that the Hamiltonian is Hermitian. As for the
potential term, it is deformed from $\varphi$ to $g(\varphi)$, but also
receives an ‘anomalous’ potential contribution reminiscent of the Schwarzian
derivative [66, sec. 4].

By the way, note that the deformation-dependent Hamiltonians (14) suffice for
the derivation of Berry phases, without requiring any digression on unitarity.
An experimenter could indeed start from the operators (14) and make $g_{t}$
slowly time-dependent, which would eventually lead to Berry phases in the
usual way. It just so happens that the $g$-dependence of wave functions is
given, in the present case, by unitary operators (11).

#### Berry phases.

Unitary diffeomorphisms (11) give rise to Hamiltonians (14) labelled
parametrically by a point $g$ in the $\text{Diff}\,S^{1}$ group manifold.
Following sec. 2.1, one may then study Berry phases resulting from adiabatic,
cyclic parameter variations. Let therefore $g_{t}$ be some closed path in
$\text{Diff}\,S^{1}$, and pick some normalized eigenstate $\Psi$ (with non-
degenerate, isolated energy) of the undeformed Hamiltonian $H$. Adiabatic time
evolution takes the form (5) with a Berry phase (6) that now
reads888Throughout this work, holonomies and time integrals over periodic
paths are written as $\oint$. By contrast, spatial integrals, be they on a
circle or on a plane, are denoted by the symbol $\int$.

${\cal
B}_{\Psi}[g_{t}]=i\oint\text{d}t\int_{0}^{2\pi}\text{d}\varphi\,\sqrt{g_{t}^{\prime}(\varphi)}\,\Psi^{*}(g_{t}(\varphi))\partial_{t}\Big{(}\sqrt{g_{t}^{\prime}(\varphi)}\,\Psi\big{(}g_{t}(\varphi)\big{)}\Big{)}$
(15)

owing to the definition (11). This can be made explicit by evaluating the time
derivative, integrating by parts and changing the integration variable from
$\varphi$ to $g_{t}(\varphi)$:

${\cal
B}_{\Psi}[g_{t}]=-\frac{M}{\hbar}\oint\text{d}t\,\text{d}\varphi\,j(\varphi)\,\dot{g}_{t}\big{(}g_{t}^{-1}(\varphi)\big{)},$
(16)

where the dot denotes a partial time derivative, $M$ is the mass of the
particle and $j(\varphi)$ is the probability current of $\Psi$,

$j\equiv\frac{\hbar}{2Mi}\big{(}\Psi^{*}\partial_{\varphi}\Psi-\Psi\partial_{\varphi}\Psi^{*}\big{)}.$
(17)

We stress that eq. (16) is an explicit functional of the path of deformations
$g_{t}(\varphi)$, and otherwise only depends on the state $\Psi$ through its
probability current $j(\varphi)$. This is a general phenomenon: as we confirm
in sec. 4, Berry phases produced by adiabatic diffeomorphisms measure currents
of quantum states. Also note that eq. (16) was derived in a one-body context,
but it is equally valid in thermodynamically large systems provided the
probability current $j(\varphi)$ is replaced by the many-body current density
$J(\varphi)$. One may therefore expect the phases (16), or the corresponding
linear response, to be observable in 1D quantum wires with persistent currents
[58, 59, 60].

Our argument so far was a brute-force computation based on eq. (15), but there
exists an equivalent derivation in terms of Lie-algebraic data, owing to the
expression on the far right-hand side of eq. (6). Indeed, the group
$\text{Diff}\,S^{1}$ consists of deformations of a circle, so its algebra
consists of infinitesimal diffeomorphisms $\varphi\mapsto\varphi+v(\varphi)$,
i.e. vector fields $v(\varphi)\partial_{\varphi}$. The Lie algebra element
$(\partial_{t}g_{t})g_{t}^{-1}$ in (6) is thus a time-dependent vector field

$v_{t}(\varphi)\equiv\tfrac{\partial}{\partial\tau}g_{\tau}\big{(}g_{t}^{-1}(\varphi)\big{)}\big{|}_{\tau=t}=\dot{g}_{t}\big{(}g_{t}^{-1}(\varphi)\big{)},$
(18)

whose flow is the one-parameter family of diffeomorphisms $g_{t}$. In
hydrodynamics, $v_{t}$ would be the actual velocity vector field of the fluid
flow $g_{t}$ (see e.g. [67]). Abstractly, one can also think of (18) as the
logarithmic time derivative of the path of deformations $g_{t}(\varphi)$. One
can then plug (18) in the Lie algebra operator (12) and use the Berry phase

${\cal
B}_{\Psi}[g_{t}]=-\frac{M}{\hbar}\oint\text{d}t\,\text{d}\varphi\,j(\varphi)\,v_{t}(\varphi)$
(19)

to reproduce eq. (16). This reformulation is no surprise, but it will be worth
keeping in mind once we turn to 2D diffeomorphisms.

###  2.3 Examples of adiabatic deformations

Let us exhibit time-dependent diffeomorphisms whose Berry phase (16) takes a
manageable form. Consider first the simplest case, namely time-dependent rigid
rotations $g_{t}(\varphi)=\varphi+\theta_{t}$ such that
$\theta_{T}=\theta_{0}+2\pi n$ for some integer $n$. Then eq. (16) yields

${\cal B}_{\Psi}[g_{t}]=-2\pi
n\,\tfrac{M}{\hbar}\int\text{d}\varphi\,j(\varphi),$ (20)

so the Berry phase measures the average current. In particular, if
$\Psi=\tfrac{1}{\sqrt{2\pi}}e^{is\varphi}$ is a plane wave with integer
angular momentum $s$, then the current $j(\varphi)=\hbar s/(2\pi M)$ is a
quantized constant and the Berry phase vanishes modulo $2\pi$. This was
anticipated below eq. (7): the parameter space of isotropic Hamiltonians is
not quite the whole group of diffeomorphisms, but its quotient
$\text{Diff}\,S^{1}/S^{1}$.

Let us now turn to less elementary diffeomorphisms that will turn out to be
crucial for the Hall viscosity in sec. 4.4. Namely, given any positive integer
$k$, consider a map $\varphi\mapsto g(\varphi)$ defined by

$e^{ikg(\varphi)}=\frac{\alpha\,e^{ik\varphi}+\beta}{\beta^{*}e^{ik\varphi}+\alpha^{*}}\qquad\text{with}\qquad\alpha,\beta\in\mathbb{C}\text{
such that }|\alpha|^{2}-|\beta|^{2}=1.$ (21)

At fixed $k$, such maps span a group locally isomorphic to SL(2,$\mathbb{R}$).
A one-parameter family of deformations of this kind is depicted in fig. 5;
they stretch the circle by ‘pinching it’ at $k$ equally distributed points.
What happens when these perturbations become time-dependent and act on a
rotation-invariant Hamiltonian? In that case the reference state $\Psi$ is a
plane wave and the ensuing Berry phase takes the form (16) with constant
$j(\varphi)=\hbar s/(2\pi M)$ in terms of an integer angular momentum $s$.
Making $\alpha,\beta$ time-dependent in (21) and using eq. (16) then yields

${\cal
B}_{\Psi}[g_{t}]=-\frac{2s}{k}\oint\text{d}t\,\text{Im}(\alpha^{*}\dot{\alpha}-\beta^{*}\dot{\beta}).$
(22)

This is explicit, but not especially illuminating. A more striking result is
obtained thanks to SL(2,$\mathbb{R}$) coordinates $(\lambda,\theta,\chi)$
defined via $\alpha\equiv e^{i(\chi+\theta)}\cosh\lambda$ and $\beta\equiv
e^{i(\chi-\theta)}\sinh\lambda$, so that eq. (22) becomes

${\cal
B}_{\Psi}[g_{t}]=-\frac{2s}{k}\oint\text{d}t\,\dot{\theta}_{t}\,\cosh(2\lambda_{t})$
(23)

where the contribution of $\chi_{t}$ drops out since it merely corresponds to
adiabatic rotations of an isotropic system. Note that (23) is nothing but a
hyperbolic area written in ‘polar coordinates’ $(\lambda,\theta)$. Indeed,
writing (23) as a surface integral ${\cal B}=\int{\cal F}$, the corresponding
Berry curvature

${\cal F}=-\frac{4s}{k}\,\sinh(2\lambda)\,\text{d}\lambda\wedge\text{d}\theta$
(24)

is proportional to the area form on a hyperbolic plane. This is no
coincidence: it lies at the root of both Thomas precession [50, 63, 64] and
the Hall viscosity [19, 20], so we return to it in much greater detail in sec.
4.4, where (24) is recast in terms of coordinates $\tau_{1},\tau_{2}$ on an
upper half-plane with area form
$\text{d}\tau_{1}\wedge\text{d}\tau_{2}/\tau_{2}^{2}$.

Figure 5: A sequence of diffeomorphisms (21) with $k=3$ and
$(\alpha,\beta)=(\cosh\lambda,\sinh\lambda)$, where $\lambda$ ranges from $0$
(innermost circle) to $0.6$ (outermost circle). The innermost circle consists
of 50 uniformly distributed points; successive deformations spoil uniformity,
with manifest density maxima at $\varphi=0\mod{2\pi/3}$ and minima at
$\varphi=\pi/3\mod{2\pi/3}$. (Inner points are smaller than outer ones for
readability.) The same kind of deformation, albeit with $k=1$, was used to
produce the contraction in fig. 4.

##  3 Quantum area-preserving deformations

Having introduced unitary diffeomorphisms in 1D, we now turn to their area-
preserving peers in 2D. This is a key preliminary for Berry phases such as
(6), which require detailed knowledge of the operators $\,{\cal U}[g]$.
Accordingly, this section begins with basic facts on area-preserving
diffeomorphisms and their infinitesimal cousins, namely divergence-free
(‘symplectic’) vector fields [56]. We then turn to quantum mechanics and
introduce gauge-invariant unitary deformations of wave functions, which are
found to coincide with the ‘quantomorphisms’ normally encountered in geometric
quantization [44, 45]. Along the way, we show that the resulting
representation is projective in the sense of eq. (8). Numerous examples are
provided throughout, notably including ‘edge deformations’.

###  3.1 Area-preserving deformations

Area-preserving diffeomorphisms arise naturally in analytical mechanics and
symplectic geometry [68, 56], where they are seen as symmetries of phase
space. Here we list their basic properties and provide a few examples for
future reference, including the key notion of ‘edge deformations’ inspired by
[14]. The presentation is purely classical for now: all quantum aspects are
relegated to sec. 3.3.

Consider a plane $\mathbb{R}^{2}$ supporting a uniform magnetic field B; in
Cartesian coordinates $(x,y)$, one can write
$\textbf{B}=B\,\text{d}x\wedge\text{d}y$ as an area form.999Bold fonts are
used for all objects that carry spatial indices: x is a position ‘vector’,
$\bm{g}$ is a ‘vector’-valued map, A is a one-form, B is a two-form, $\bm{v}$
is a vector field, etc. By definition, a diffeomorphism of the plane is an
invertible smooth map
$\bm{g}:\mathbb{R}^{2}\to\mathbb{R}^{2}:\textbf{x}\mapsto\bm{g}(\textbf{x})$
whose inverse $\bm{g}^{-1}$ is also smooth. We say that $\bm{g}$ is area-
preserving if it leaves the area form invariant
($\bm{g}^{*}\textbf{B}=\textbf{B}$), i.e. if it has unit Jacobian. The set of
all such maps is a group under composition, denoted
$\text{SDiff}\,\mathbb{R}^{2}$ for ‘special’ diffeomorphisms, analogously to
the special linear groups $\text{SL}(n)$.

Let us exhibit some classes of diffeomorphisms that will appear below. A first
example, somewhat trivial but still important, is given by translations
$\textbf{x}\mapsto\textbf{x}+\textbf{a}$; these form an (Abelian) subgroup
$\mathbb{R}^{2}$ of $\text{SDiff}\,\mathbb{R}^{2}$. Secondly, linear maps (2)
have unit Jacobian by definition, and thus span an $\text{SL}(2,\mathbb{R})$
subgroup of area-preserving maps; this includes an SO(2) subgroup of rotations
around the origin. Finally, to introduce an even larger subset of deformations
that will later be crucial, consider standard polar coordinates $(r,\varphi)$
defined by $x+iy=r\,e^{i\varphi}$ and ask what area-preserving diffeomorphisms
$\bm{g}:(r,\varphi)\mapsto\bm{g}(r,\varphi)$ commute with all dilations
$(r,\varphi)\mapsto(\lambda r,\varphi)$. The answer is provided by all maps of
the form

$\bm{g}(r,\varphi)=\bigg{(}\frac{r}{\sqrt{g^{\prime}(\varphi)}},\,g(\varphi)\bigg{)}$
(25)

where $g(\varphi)$ is any (orientation-preserving) circle diffeomorphism in
the sense of sec. 2.2. One readily verifies that (25) preserves the magnetic
field $\textbf{B}=B\,r\text{d}r\wedge\text{d}\varphi$: intuitively, any
angular ‘compression’ is compensated by an angle-dependent radial ‘dilation’
(and vice-versa), as in fig. 6. In fact, one can even make a stronger
statement: all deformations (25) leave the symmetric gauge potential
$\textbf{A}=\tfrac{B}{2}r^{2}\text{d}\varphi$ invariant since
$r^{2}\text{d}\varphi=\tfrac{r^{2}}{g^{\prime}(\varphi)}\text{d}g(\varphi)$.

Figure 6: The action of an edge deformation (25) on a disk, with $g(\varphi)$
of the form (21) with $k=3$, $\alpha=\cosh(1/2)$ and $\beta=\sinh(1/2)$. The
1D diffeomorphism $\varphi\mapsto g(\varphi)$ increases the angular density of
points near $\varphi=0\mod{2\pi/3}$, and decreases it near
$\varphi=\pi/3\mod{2\pi/3}$ (recall fig. 5). This local change of angular
density is compensated by a modification of radial density: regions where
angles contract become radially dilated, and vice-versa.

We shall refer to maps of the form (25) as edge deformations because they
deform the edge of any isotropic quantum Hall droplet in a non-trivial but
finite way regardless of its size (see fig. 1 above). Indeed, (25) sends any
circle $r=\text{cst}$ on a deformed curve
$r=\text{cst}/\sqrt{g^{\prime}(\varphi)}$, where $g^{\prime}(\varphi)$ is
independent of $r$. Note that edge deformations span a subgroup of
$\text{SDiff}\,\mathbb{R}^{2}$, isomorphic to the group $\text{Diff}\,S^{1}$
of circle deformations introduced in sec. 2.2. In fact, their algebra consists
of vector fields on a circle, and is thus very similar to the Virasoro algebra
of edge modes studied in [14]. Finally, note that the linear maps (2) form a
subset of the group of edge deformations, since (2) can be written as (25) in
polar coordinates, with a 1D deformation $g(\varphi)$ given by eq. (21) with
$k=2$ and

$(\alpha,\beta)=\big{(}\tfrac{1}{2}(a-ib+ic+d),\tfrac{1}{2}(a+ib+ic-d)\big{)}.$
(26)

We will exploit this coincidence at the end of sec. 4.4.

###  3.2 Divergence-free vector fields and incompressible flows

We have seen in eq. (6) that Berry phases produced by unitary transformations
can be written in terms of Lie-algebraic data as opposed to finite
transformations. In the case of 1D deformations, we exhibited this with eq.
(19), involving the 1D vector field (18) and its flow $g_{t}$. It is therefore
essential to become acquainted with the 2D version of these objects, i.e. with
the Lie algebra of the group $\text{SDiff}\,\mathbb{R}^{2}$. This will often
be useful below, not least because it allows us to introduce stream functions
that link the subject to hydrodynamics [69, sec. 4.2] and play an important
role in the quantum theory (sec. 3.3).

#### Divergence-free vector fields.

The group $\text{SDiff}\,\mathbb{R}^{2}$ consists of deformations of a plane,
so its Lie algebra consists of infinitesimal diffeomorphisms
$\textbf{x}\mapsto\textbf{x}+\bm{v}(\textbf{x})$, i.e. vector fields $\bm{v}$.
In order to preserve area, these vector fields need to be divergence-free: in
terms of Lie derivatives of the magnetic field, one has ${\cal
L}_{\bm{v}}\textbf{B}=B\,\nabla_{i}v^{i}=0$. For example, infinitesimal edge
deformations (25) are generated by vector fields of the form

$\bm{v}=v(\varphi)\partial_{\varphi}-\frac{r}{2}v^{\prime}(\varphi)\partial_{r}\qquad\text{(edge
deformations)}$ (27)

where $v(\varphi)$ is any $2\pi$-periodic function; it is immediate to verify
that (27) is indeed divergence-free. Note that the bracket of two divergence-
free vector fields is itself divergence-free, so such vector fields spans a
Lie algebra, denoted $\text{SVect}\,\mathbb{R}^{2}$ in analogy with the
notation $\text{SDiff}\,\mathbb{R}^{2}$ for the group of area-preserving
deformations. This is actually a $w_{1+\infty}$ algebra [12] whose Witt
subalgebra is that of infinitesimal edge deformations (27).

As usual for Lie groups, the map that sends Lie algebra elements on group
elements is the exponential. The latter is really a flow in the case of
diffeomorphism groups: given a vector field $\bm{v}_{t}$, say even time-
dependent, consider the time-dependent diffeomorphisms $\bm{g}_{t}$ given by
the 2D version of the logarithmic derivative (18),

$\bm{v}_{t}=\dot{\bm{g}}_{t}\circ\bm{g}_{t}^{-1}.$ (28)

This family of diffeomorphisms is, by definition, the flow of $\bm{v}_{t}$; in
the special case where $\bm{v}_{t}$ is time-independent, $\bm{g}_{t}$ is
called the exponential of $\bm{v}$. The analogy with hydrodynamics is obvious:
one can think of $\bm{v}_{t}$ as the velocity vector field of an
incompressible fluid, and $\bm{g}_{t}$ is the resulting flow so that
$\bm{g}_{t}(\textbf{x})$ is the position at time $t$ of a fluid parcel
initially at x. Notions of flow will also be important for Berry phases, owing
to the appearance of the combination (28) in eq. (10).

#### Stream functions.

On the plane, any divergence-free vector field is determined by a stream
function101010This does not hold on any surface! For example, on a torus,
translations obviously preserve area but admit no single-valued stream
function. The general statement is that all divergence-free vector fields on
$\Sigma$ admit a stream function if and only if the first cohomology group
$H^{1}(\Sigma)$ vanishes. $F$ such that $\iota_{\bm{v}}\textbf{B}=\text{d}F$,
which in Cartesian coordinates boils down to

$v^{i}=\tfrac{1}{B}\varepsilon^{ij}\partial_{j}F$ (29)

where $\varepsilon^{ij}$ is antisymmetric and $\varepsilon^{xy}\equiv+1$.
Returning to the examples of sec. 3.1, one would find that the stream
functions generating translations $\textbf{x}\mapsto\textbf{x}+\textbf{a}$ and
linear transformations (2) are respectively linear and quadratic in Cartesian
coordinates $(x,y)$, while the stream function giving rise to the edge vector
field (27) is

$F(\textbf{x})=-\frac{1}{2}Br^{2}v(\varphi)\qquad\text{(edge deformations).}$
(30)

(One can always add an arbitrary additive constant to a stream function
without affecting its vector field; we choose this constant such that $F(0)=0$
in (30).)

Stream functions highlight the abundance of area-preserving maps, and also
provide powerful methods to construct specific deformations with desired
properties. For example, any compactly supported stream function $F$ produces
a vector field $\bm{v}$ that vanishes outside of the support; the
corresponding flow is then obtained by integrating eq. (28), and consists of
diffeomorphisms that only affect the support of $F$ (see fig. 7).

Figure 7: The flow of a divergence-free vector field whose stream function is
compactly supported in the bulk of a disk (whose edge is represented by a
dashed circle). The density plot shows the stream function, whose level curves
are streamlines of the flow. It is clear that such compactly supported
functions yield deformations that only affect the bulk without touching the
edge. Conversely, one may consider stream functions that only vary in a
neighbourhood of the edge, without affecting the bulk.

As the terminology suggests, stream functions arise naturally in
incompressible hydrodynamics (see e.g. [69, sec. 4.2]). But they are also
crucial in symplectic geometry, since any area form in 2D is also a symplectic
form: the magnetic field $\textbf{B}=B\,\text{d}x\wedge\text{d}y$ says for
instance that the coordinates $x$ and $y$ are canonically conjugate, similarly
to ‘position’ and ‘momentum’ in 1D mechanics. Stream functions in that context
are more commonly called ‘Hamiltonian functions’ [56], because the
corresponding flow (28) coincides with Hamilton’s equations of motion with the
stream function seen as a Hamiltonian. However, we will steer clear of this
terminology to avoid confusion with the actual Hamiltonian of a quantum
system. The symplectic perspective is nevertheless useful thanks to its
connection with geometric quantization [44, 45, 70, 71], to which we now turn
in order to define unitary deformations of wave functions.

###  3.3 Unitary deformations are quantomorphisms

Here we carry out the 2D version of the 1D construction presented around eqs.
(11)–(14). Accordingly, consider a non-relativistic particle with electric
charge $q$ in the plane $\mathbb{R}^{2}$, whose space of states is the usual
one-body Hilbert space $L^{2}(\mathbb{R}^{2})$. We wish to act on that
particle’s wave functions with unitary area-preserving deformations. How to
proceed?

The simplest way forward is to recall the 1D definition (11), realize that its
square root term involves the Jacobian of the deformation, and adapt the
prescription to area-preserving maps (whose Jacobian is trivial):

$\big{(}\,{\cal U}[\hskip
0.50003pt\bm{g}]\Psi\big{)}(\textbf{x})\stackrel{{\scriptstyle?}}{{\equiv}}\Psi\big{(}\bm{g}(\textbf{x})\big{)}.$
(31)

This defines a unitary map for any $\bm{g}\in\text{SDiff}\,\mathbb{R}^{2}$ and
it satisfies $\,{\cal U}[\bm{f}]\circ\,{\cal U}[\hskip
0.50003pt\bm{g}]=\,{\cal U}[\hskip 0.50003pt\bm{g}\circ\bm{f}]$, so it
suffices for neutral states or in the absence of magnetic fields. But it
suffers from serious problems in the case of charged states: area-preserving
maps generally change the vector potential A (i.e.
$\bm{g}^{*}\textbf{A}\neq\textbf{A}$) even if they preserve the magnetic field
$\textbf{B}=\text{d}\textbf{A}$. This implies, on a practical level, that eq.
(31) modifies the gauge in which the vector potential is written, preventing
the computation of Berry phases which rely on the possibility of comparing
phases of wave functions. From a more conceptual standpoint, eq. (31) is
plainly ill-defined, in that it presupposes a gauge choice: changing gauges
according to $\tilde{\Psi}(\textbf{x})\equiv
e^{iq\alpha(\textbf{x})/\hbar}\Psi(\textbf{x})$ turns (31) into a different
representation $\,{\cal U}[\hskip
0.50003pt\bm{g}]\tilde{\Psi}=e^{iq(\alpha-\alpha\circ\bm{g})/\hbar}\,\tilde{\Psi}\circ\bm{g}$.
We now show how both of these issues can be fixed; the solution involves
quantomorphisms [44, 45], whose formulation in terms of fiber bundles is
reviewed in greater detail in app. A.

#### Compensating gauge transformations.

We have just pointed out that the operator (31) changes the gauge in which the
potential A is written. To correct this, one can compensate the change of
gauge by a pure gauge transformation, i.e. define

$\big{(}\,{\cal U}[\hskip
0.50003pt\bm{g}]\Psi\big{)}(\textbf{x})\stackrel{{\scriptstyle?}}{{\equiv}}e^{iq\alpha(\textbf{x})/\hbar}\,\Psi\big{(}\bm{g}(\textbf{x})\big{)}$
(32)

with a function $\alpha(\textbf{x})$ such that
$\text{d}\alpha=\textbf{A}-\bm{g}^{*}\textbf{A}$. Since the one-form
$\textbf{A}-\bm{g}^{*}\textbf{A}$ is closed, and because $\mathbb{R}^{2}$ is
simply connected, $\alpha$ exists globally111111Similarly to footnote 10, this
fails on surfaces with non-trivial first cohomology, e.g. the torus. and can
be written as an integral

$\alpha(\textbf{x})=\int_{\textbf{x}_{0}}^{\textbf{x}}(\textbf{A}-\bm{g}^{*}\textbf{A}).$
(33)

Here the integration contour is any curve going from some ‘origin’
$\textbf{x}_{0}$ to x; its choice is irrelevant since
$\textbf{A}-\bm{g}^{*}\textbf{A}$ is closed. The reference point
$\textbf{x}_{0}$ is arbitrary and may even depend on $\bm{g}$, but changing it
eventually yields the same operator up to a global phase, so the choice of
$\textbf{x}_{0}$ is ultimately unimportant as well. In the following, we pick
for simplicity the same origin $\textbf{x}_{0}=0$ for all deformations. With
this choice, for instance, eq. (32) applied to any translation
$\bm{g}(\textbf{x})=\textbf{x}+\textbf{a}$ yields

$\big{(}\,{\cal U}[\hskip 0.50003pt\bm{g}]\Psi\big{)}(\textbf{x})\equiv
e^{iqB(xa_{y}-ya_{x})/2\hbar}\,\Psi(\textbf{x}+\textbf{a})\qquad\text{(translations)}$
(34)

when $\textbf{A}=B(x\,\text{d}y-y\,\text{d}x)/2$ is written in symmetric
gauge. This is nothing but a standard (finite) magnetic translation.

At this stage we have fixed the main issue: the operators (32) no longer
modify the gauge. Furthermore, the definition (32) is nearly intrinsic: it
takes the same form in all gauge choices, up to a global phase. This last
point can be further improved by choosing, for each map $\bm{g}$, a smooth
path $\bm{\gamma}_{\bm{g}}$ linking $\textbf{x}_{0}$ to
$\bm{g}(\textbf{x}_{0})$ (for instance a straight line). One can then add a
global phase to (32) and define

$\big{(}\,{\cal U}[\hskip 0.50003pt\bm{g}]\Psi\big{)}(\textbf{x})\equiv
e^{-iq\int_{\bm{\gamma}_{\bm{g}}}\textbf{A}/\hbar}\,e^{iq\int_{\textbf{x}_{0}}^{\textbf{x}}(\textbf{A}-\bm{g}^{*}\textbf{A})/\hbar}\,\Psi\big{(}\bm{g}(\textbf{x})\big{)},$
(35)

which is our final prescription for the unitary action of area-preserving
deformations on wave functions with charge $q$. Most of the conclusions below
ultimately stem from this elementary definition. In the context of geometric
quantization, operators (35) are known as prequantum bundle automorphisms or
quantomorphisms [44, 45]. We will adopt the latter terminology, and refer
again to app. A for a number of technical details, including the construction
of quantomorphisms on surfaces more general than the plane where integrals
such as (33) may be globally ill-defined. In particular, we show there that
(35) is essentially the unique well-defined unitary action of deformations on
quantum wave functions.

Recall from sec. 2.2 that 1D diffeomorphisms provide geometry-dependent
Hamiltonians (14), involving a deformed metric and potential. In a similar
way, the intuitive meaning of quantomorphisms (35) becomes clearer upon acting
with some $\,{\cal U}[\hskip 0.50003pt\bm{g}]$ on the Hamiltonian

$H=\frac{1}{2M}(\textbf{p}-q\textbf{A})^{2}+V(\textbf{x})$ (36)

for an electron of mass $M$ in some potential $V$. A brute force computation
then yields

$\,{\cal U}[\hskip 0.50003pt\bm{g}]H\,{\cal U}[\hskip
0.50003pt\bm{g}]^{\dagger}=\frac{1}{2M}\,\big{(}p_{j}-qA_{j}(\textbf{x})\big{)}G^{jk}(\textbf{x})\big{(}p_{k}-qA_{k}(\textbf{x})\big{)}+V\big{(}\bm{g}(\textbf{x})\big{)}$
(37)

where $G^{jk}(\textbf{x})$ is the (inverse) metric induced by the deformation
$\bm{g}$:

$G_{ij}(\textbf{x})\equiv\frac{\partial g^{k}}{\partial x^{i}}\frac{\partial
g^{k}}{\partial x^{j}}\qquad\Rightarrow\qquad
G^{ij}(\textbf{x})=\left(\frac{\partial(g^{-1})^{i}}{\partial
y^{k}}\frac{\partial(g^{-1})^{j}}{\partial
y^{k}}\right)_{\textbf{y}=\bm{g}(\textbf{x})}.$ (38)

The operator (37) is thus the deformed Hamiltonian produced by
$\bm{g}\in\text{SDiff}\,\mathbb{R}^{2}$, again involving a modified metric
($\delta^{ij}\to G^{ij}$) and a deformed potential ($V\to V\circ\bm{g}$). Note
that the vector potential is unchanged in (37), as guaranteed by the
compensating gauge transformation in (32)–(35). We stress, in particular, that
the transformation (38) in the case of linear maps (2) reduces to the metric
redefinitions considered in [19, eq. (10)] and [20, eq. (3.2)]: the
corresponding linear response will eventually coincide with the Hall
viscosity.

#### Aharonov-Bohm extension.

A natural question at this point is whether the assignment (35) furnishes a
representation of the group $\text{SDiff}\,\mathbb{R}^{2}$, since this is a
necessary condition for the arguments of sec. 2.1 to hold. To answer this,
compose two operators of the form (35) and find that they satisfy eq. (8) with
a non-zero central extension

$\frac{\hbar}{q}\,\mathsf{C}(\bm{g},\bm{f})=\int_{\bm{\gamma}_{\bm{g}\circ\bm{f}}}\textbf{A}-\int_{\bm{g}(\bm{\gamma}_{\bm{f}})}\textbf{A}-\int_{\bm{\gamma}_{\bm{g}}}\textbf{A}.$
(39)

One may view this as an AB phase, since it is the holonomy of A along the
(curved) triangle
$\bm{\gamma}_{\bm{g}\circ\bm{f}}-\bm{g}(\bm{\gamma}_{\bm{f}})-\bm{\gamma}_{\bm{g}}$,
i.e. the flux of the magnetic field through the surface enclosed by that
triangle: see fig. 8. In particular, it is manifestly gauge-invariant. Its
presence states that the operators (35) furnish a projective representation of
the group of area-preserving maps, implying that their Berry phases have
inhomogeneous term as in eq. (10); this will be important in sec. 4.2. Note
that the cocycle (39) is non-trivial, i.e. it cannot be absorbed by a
redefinition of the operators $\,{\cal U}[\hskip 0.50003pt\bm{g}]$. One can
prove this by noting that (39) reduces to the (obviously non-trivial)
Heisenberg central extension of $\mathbb{R}^{2}$ in the case of magnetic
translations (34), as we now confirm by investigating infinitesimal
transformations.

$\textbf{x}_{0}$$\bm{g}\big{(}\bm{f}(\textbf{x}_{0})\big{)}$$\bm{g}(\textbf{x}_{0})$$\bm{\gamma}_{\bm{g}\circ\bm{f}}$$-\bm{g}(\bm{\gamma}_{\bm{f}})$$-\bm{\gamma}_{\bm{g}}$
Figure 8: The central extension (39) measures the area of a triangle built out
of the paths $\bm{\gamma}_{\bm{g}\circ\bm{f}}$,
$-\bm{g}(\bm{\gamma}_{\bm{f}})$ and $-\bm{\gamma}_{\bm{g}}$, i.e. the flux of
the magnetic field through that triangle. This is a special case of a standard
geometric construction in group cohomology (see e.g. [72, prop. 4.2] or [73]
and references therein). In particular, the cocycle identity (9) that ensures
associativity is automatically satisfied, since it amounts to the statement
that the area of a curved quadrilateral can be decomposed in two different
ways as the sum of the areas of two triangles.

#### Infinitesimal quantomorphisms.

We will soon need to implement infinitesimal diffeomorphisms (i.e. divergence-
free vector fields) in quantum mechanics. Accordingly, define Hermitian
operators $\mathfrak{u}[\bm{v}]\equiv-i\partial_{\epsilon}\big{|}_{0}\,{\cal
U}[e^{\epsilon\bm{v}}]$ as in eqs. (7) and (12), where $\bm{v}$ is any
divergence-free vector field. A direct computation starting from the main
definition (35) then yields the differential operator [74, 75]

$\mathfrak{u}[\bm{v}]=-i\nabla_{\bm{v}}-\frac{q}{\hbar}F_{\bm{v}}\qquad\text{with}\qquad
F_{\bm{v}}(\textbf{x})\equiv\int_{\textbf{x}_{0}}^{\textbf{x}}\iota_{\bm{v}}\textbf{B}.$
(40)

Here the covariant derivative
$\nabla_{\bm{v}}=v^{j}(\partial_{j}-i\tfrac{q}{\hbar}A_{j})$ is a local
combination of mechanical momenta $\textbf{p}-q\textbf{A}$, generating
parallel transport of charged wave functions. As for $F_{\bm{v}}$, it is a
stream function that originates from the compensating gauge transformation
introduced in eq. (32). Note that $F_{\bm{v}}$ is now fixed uniquely by the
condition $F_{\bm{v}}(\textbf{x}_{0})=0$, so changing the reference point
$\textbf{x}_{0}$ shifts the stream function $F_{\bm{v}}$ by an additive
constant. This reflects the unavoidable global phase ambiguity in the
definition (35) of $\,{\cal U}[\hskip 0.50003pt\bm{g}]$, which in turns allows
the representation to be projective.

Let us illustrate eq. (40) with the generators of deformations listed around
eq. (30), taking $\textbf{x}_{0}=0$ for simplicity. As a first example,
translations are generated by constant vector fields $\bm{v}=\partial_{i}$,
with linear stream functions
$F_{\partial_{i}}(\textbf{x})=B\varepsilon_{ij}x^{j}$. The corresponding
Hermitian operator (40),

$\hbar\mathfrak{u}[\partial_{j}]=p_{j}-qA_{j}(\textbf{x})-qB\varepsilon_{jk}x^{k}\qquad\text{(translations)},$
(41)

is nothing but the gauge-invariant generator of magnetic translations, and
could also have been obtained by differentiating eq. (34) with respect to the
components of the translation vector a. Note that this implies the usual
Heisenberg commutator of magnetic translations,
$\big{[}\hbar\mathfrak{u}[\partial_{x}],\hbar\mathfrak{u}[\partial_{y}]\big{]}=-i\hbar
qB$, confirming that the cocycle (39) is non-trivial. This is actually a
special case of the commutator of arbitrary operators (40),

$\big{[}\mathfrak{u}[\bm{v}],\mathfrak{u}[\bm{w}]\big{]}=i\mathfrak{u}\big{[}[\bm{v},\bm{w}]\big{]}-\tfrac{iq}{\hbar}\textbf{B}(\bm{v},\bm{w})\big{|}_{\textbf{x}_{0}},$
(42)

which exhibits a ‘Lichnerowicz’ central charge due to the magnetic field (see
e.g. [76]).

A less elementary example that will be essential below is provided by edge
deformations (25) whose vector fields (27) and stream functions (30) are
uniquely labelled by a 1D vector field $v(\varphi)\partial_{\varphi}$. Written
in an arbitrary gauge, the corresponding Hermitian operators (40) are

$\mathfrak{u}[\bm{v}]=-iv(\varphi)\partial_{\varphi}+i\frac{r}{2}v^{\prime}(\varphi)\partial_{r}-\frac{q}{\hbar}\,\langle\textbf{A},\bm{v}\rangle+\frac{qB}{2\hbar}r^{2}v(\varphi)\qquad\text{(edge
deformations)}$ (43)

where $\langle\textbf{A},\bm{v}\rangle\equiv\iota_{\bm{v}}\textbf{A}\equiv
v^{i}A_{i}$ is the pairing of the one-form A with the vector field $\bm{v}$,
and the stream function (30) is apparent on the right-hand side. This
simplifies neatly in symmetric gauge: when
$\textbf{A}=\tfrac{1}{2}Br^{2}\text{d}\varphi$, the term
$\langle\textbf{A},\bm{v}\rangle$ cancels the stream function and eq. (43)
becomes

$\mathfrak{u}[\bm{v}]=-iv(\varphi)\partial_{\varphi}+i\frac{r}{2}v^{\prime}(\varphi)\partial_{r}\qquad\begin{pmatrix}\text{edge
deformations}\\\ \text{in symmetric gauge}\end{pmatrix}.$ (44)

At the level of the group $\text{SDiff}\,\mathbb{R}^{2}$, the simplification
means that the action (35) of edge deformations in symmetric gauge reduces to
the naive ansatz (31). This will greatly simplify the identification of the
Hall viscosity in our setup.

##  4 Adiabatic deformations of planar droplets

This section presents our main result: explicit Berry phases due to adiabatic
area-preserving deformations acting unitarily on a many-body droplet of
electrons in the plane. As a preliminary, we first display such phases for
electrically neutral states. Adding a magnetic field produces the extension
(39), and the associated one-body phases can be derived following eq. (10).
They turn out to consist of two terms, both separately gauge-invariant (see
eq. (49)): the first involves the current density, and the second is an
Aharonov-Bohm (AB) phase sensitive to the wave function’s probability density.
Their many-body generalization thus involves the current and density of the
full droplet (see eq. (50)).

We eventually apply this to edge deformations (25) and observe that the AB
phase is super-extensive in the thermodynamic limit ($\propto N^{2}$ for $N\gg
1$ electrons). By contrast, the contribution of the current is extensive at
very strong magnetic fields, i.e. for genuine quantum Hall (QH) droplets. The
corresponding finite Berry curvature per unit area measures the jump of the
current at the edge and is reminiscent of the Hall viscosity, with which it
coincides up to an overall factor for both integer and fractional QH states.
The Hall viscosity as such is recovered by restricting attention to linear
deformations of the plane that only affect the metric without touching the
potential.

###  4.1 Invitation: Berry phases of neutral states

To start, let us compute Berry phases due to deformations of neutral planar
wave functions. This preliminary will exhibit a key aspect of the charged case
as well: geometric phases measure current. As in sec. 3, we only consider
area-preserving maps, whose action on wave functions is now given by eq. (31)
since $q=0$. This induces deformations of the metric and the one-body
potential, as in eq. (37), albeit without vector potential A. The
corresponding infinitesimal operators are given by the $q=0$ version of (40):

$\mathfrak{u}[\bm{v}]=\frac{1}{\hbar}v^{j}p_{j}=-iv^{j}\partial_{j}$ (45)

for any divergence-free vector field $\bm{v}$. The commutators of these
operators reproduce the Lie bracket of vector fields, since eq. (42) holds
with $q=0$; there is no central extension.

#### Berry phases from deformations.

Let $\Psi$ be a normalized eigenfunction of some one-body Hamiltonian (36)
with $q=0$, and assume its energy is isolated and non-degenerate; this is
typically the case if $\Psi$ has sufficiently low energy and the one-body
potential $V(\textbf{x})$ is bounded from below, with few enough symmetries.
Let $\bm{g}_{t}$ be a closed curve of area-preserving maps, and assume these
deformations act adiabatically through eq. (31) on the system initially
prepared in the state $\Psi$. What is the ensuing Berry phase?

The answer is provided by the same derivation as in the 1D case of sec. 2.2,
and can similarly be obtained in two equivalent ways. The first is to use the
middle formula of eq. (6), write the scalar product as a planar integral, rely
on the unitary action (31), and evaluate the time derivative by brute force.
The second relies on the Lie-algebraic expression on the far right-hand side
of eq. (6), using the infinitesimal operator (45) and replacing $\bm{v}$ by
the time derivative (28) of the transformations $\bm{g}_{t}$. Regardless of
one’s approach, the result is

${\cal B}_{\Psi}[\hskip
0.50003pt\bm{g}_{t}]=-\frac{M}{\hbar}\oint\text{d}t\,\int\text{d}^{2}\textbf{x}\,\big{<}\bm{j},\dot{\bm{g}}_{t}\circ\bm{g}_{t}^{-1}\big{>}=-\frac{M}{\hbar}\oint\text{d}t\,\int\text{d}^{2}\textbf{x}\,\langle\bm{j},\bm{v}_{t}\rangle$
(46)

where $\bm{j}=\tfrac{\hbar}{2Mi}(\Psi^{*}\text{d}\Psi-\Psi\,\text{d}\Psi^{*})$
is the probability current of $\Psi$, seen as a one-form so that
$\langle\bm{j},\bm{v}\rangle\equiv\iota_{\bm{v}}\bm{j}=v^{i}j_{i}$ denotes the
pairing of a one-form with a vector field, as in (43).

Similarly to the 1D case of sec. 2.2, we view the phase (46) as a functional
of the path $\bm{g}_{t}$ that depends parametrically on the current $\bm{j}$.
In particular, states that carry no current have vanishing Berry phases. The
AB effect will change this conclusion for charged wave functions in a magnetic
field.

#### Many-body phases.

The generalization of eq. (46) to fermionic or bosonic Fock spaces is
immediate. Indeed, suppose
$\Psi(\textbf{x}_{1},\textbf{x}_{2},...,\textbf{x}_{N})$ is the ground state
wave function for $N$ particles governed by some many-body Hamiltonian,
possibly including interactions. One can then act on $\Psi$ with an $N$-fold
tensor product of time-dependent unitary deformations (35) and use the
adiabatic theorem [77, 65] to obtain the resulting Berry phases just as we did
in sec. 2.1. In the case at hand, these phases are given by (46) up to the
replacement of the one-body probability current $\bm{j}$ by the many-body
numerical current density $\bm{J}$:

${\cal B}[\hskip
0.50003pt\bm{g}_{t}]=-\frac{M}{\hbar}\oint\text{d}t\,\text{d}^{2}\textbf{x}\,\langle\bm{J},\dot{\bm{g}}\circ\bm{g}^{-1}\rangle.$
(47)

One has e.g. $\bm{J}=\sum_{i=1}^{N}\bm{j}_{i}$ for free electrons, where the
sum runs over occupied one-body states; but we emphasize that the phase
formula (47) holds even when the constituent particles interact. In any case,
the absence of current immediately implies ${\cal B}=0$.

###  4.2 Berry phases of charged states

Let us now ask how eqs. (46)–(47) generalize to charged droplets. As in sec.
4.1, we start by deriving Berry phases for one-body wave functions; the many-
body generalization will then be straightforward. This time, however, we first
exploit the intuition provided by eq. (46) to avoid computations and ‘guess’
the form of deformational Berry phases for charged states. This is then
confirmed by a detailed proof based on the unitary action (35).

#### Inferring Berry phases.

The neutral result (46) exhibits the dependence of Berry phases on current. A
similar phenomenon may be anticipated for charged states, except that the
current in (46) must be replaced by its gauge-invariant version

$\bm{j}=\tfrac{\hbar}{2Mi}\big{(}\Psi^{*}\,\nabla\Psi-(\nabla\Psi)^{*}\,\Psi\big{)}=\tfrac{\hbar}{2Mi}\big{(}\Psi^{*}(\text{d}-iq\textbf{A}/\hbar)\Psi-\Psi(\text{d}+iq\textbf{A}/\hbar)\Psi^{*}\big{)}.$
(48)

The integral (46) with a current given by (48) is thus gauge-invariant, but it
cannot be the complete Berry phase of a charged wave function acted upon by
adiabatic deformations. Indeed, the AB effect [51] yields phases that even
affect states carrying no current (think for instance of Gaussian wave
functions subjected to time-dependent translations). One therefore expects an
additional AB contribution to the earlier phase (46), resulting in

$\Bigg{.}{\cal B}_{\Psi}[\hskip
0.50003pt\bm{g}_{t}]=-\frac{M}{\hbar}\oint\text{d}t\,\text{d}^{2}\textbf{x}\,\langle\bm{j},\dot{\bm{g}}_{t}\circ\bm{g}_{t}^{-1}\rangle+\frac{q}{\hbar}\int\text{d}^{2}\textbf{x}\,|\Psi(\textbf{x})|^{2}\oint_{\bm{g}^{-1}_{t}(\textbf{x})}\textbf{A}$
(49)

where the current $\bm{j}$ in (48) is gauge-invariant, as are the density
$|\Psi|^{2}$ and the holonomy of A in the second term. This satisfies both the
requirement that Berry phases measure current and the existence of the AB
effect. The presence of a holonomy along the inverse path $\bm{g}_{t}^{-1}$ in
(49) is a technical detail that stems from the definition (35): had all
$\bm{g}$’s on the right-hand side of that formula been replaced by
$\bm{g}^{-1}$’s, one would have obtained a left group action instead of a
right one, and the AB holonomy in (49) would have read
$\oint_{\bm{g}_{t}(\textbf{x})}\textbf{A}$.

As before, the many-body generalization of (49) is straightforward since the
action of deformations on $N$-body wave functions is an $N$-fold tensor
product of one-particle formulas (35). The current $\bm{j}$ in (49) is thus
replaced by its many-body analogue $\bm{J}$ and the probability density
$|\Psi|^{2}$ is replaced by the many-body numerical density
$\rho(\textbf{x})$, resulting in the many-body Berry phase

$\Bigg{.}{\cal B}[\hskip
0.50003pt\bm{g}_{t}]=-\frac{M}{\hbar}\oint\text{d}t\,\text{d}^{2}\textbf{x}\,\langle\bm{J},\bm{v}_{t}\rangle+\frac{q}{\hbar}\int\text{d}^{2}\textbf{x}\,\rho(\textbf{x})\oint_{\bm{g}^{-1}_{t}(\textbf{x})}\textbf{A}.$
(50)

For free electrons,
$\rho(\textbf{x})=\sum_{i=1}^{N}|\Psi_{i}(\textbf{x})|^{2}$ is a sum over
occupied one-body states, but we stress again that eq. (50) holds even for
interacting states. The key point is that (50) is only sensitive to two
universal properties of any charged droplet: its current and density.

#### Deriving Berry phases.

Having guessed eq. (49), let us prove that it follows (in the sense of sec.
2.1) from the action (35) of deformations on wave functions. The computation
highlights the geometric structure of quantomorphisms, but does not affect
later applications; the hasty reader may therefore skip it and go straight to
sec. 4.3.

We have seen that eq. (35) furnishes a projective representation of area-
preserving maps, so the ensuing Berry phase is given by (10) and involves a
term due to the extension (39):

${\cal B}_{\Psi}[\hskip
0.50003pt\bm{g}_{t}]=\oint\text{d}t\,\langle\Psi|\big{(}i\nabla_{\bm{v}_{t}}+\frac{q}{\hbar}F_{\bm{v}_{t}}\big{)}|\Psi\rangle-\oint\text{d}t\,\partial_{\tau}\mathsf{C}(\bm{g}_{\tau},\bm{g}_{t}^{-1})\big{|}_{\tau=t}$
(51)

where $\bm{v}_{t}$ is the velocity vector field (28) and we used the
expression (40) of the Lie algebra operator $\mathfrak{u}[\bm{v}_{t}]$. The
covariant derivative then gives rise to the gauge-invariant current (48),
which yields the first piece of the phase on the right-hand side of (49):

${\cal B}_{\Psi}[\hskip
0.50003pt\bm{g}_{t}]=-\frac{M}{\hbar}\oint\text{d}t\,\text{d}^{2}\textbf{x}\,\langle\bm{j},\bm{v}_{t}\rangle+\frac{q}{\hbar}\oint\text{d}t\,\text{d}^{2}\textbf{x}\,|\Psi(\textbf{x})|^{2}\,F_{\bm{v}_{t}}(\textbf{x})-\oint\text{d}t\,\partial_{\tau}\mathsf{C}(\bm{g}_{\tau},\bm{g}_{t}^{-1})\big{|}_{\tau=t}.$
(52)

It only remains to evaluate the contribution of the stream function and that
of the central extension. We begin with the former and rely on its definition
in (40) to write

$\oint\text{d}t\,F_{\bm{v}_{t}}(\textbf{x})=\oint\text{d}t\,\int_{\textbf{x}_{0}}^{\textbf{x}}\iota_{\dot{\bm{g}}_{t}\circ\bm{g}_{t}^{-1}}\textbf{B}=\oint\text{d}t\,\int_{\textbf{x}_{0}}^{\textbf{x}}\iota_{\dot{\bm{g}}_{t}\circ\bm{g}_{t}^{-1}}\big{(}(\bm{g}_{t}^{-1})^{*}\textbf{B}\big{)},$
(53)

where the second equality was obtained thanks to the fact that $\bm{g}_{t}$’s
preserve area. Since the integral from $\textbf{x}_{0}$ to x is taken along
any path connecting them, let $\bm{\gamma}(s)$ be such a path with
$\bm{\gamma}(0)=\textbf{x}_{0}$ and $\bm{\gamma}(1)=\textbf{x}$, and unpack
the integral on the far right-hand side of (53) as

$\oint\text{d}t\,F_{\bm{v}_{t}}(\textbf{x})=-\oint_{0}^{T}\\!\\!\text{d}t\int_{0}^{1}\\!\\!\text{d}s\,\textbf{B}_{\bm{g}^{-1}_{t}(\bm{\gamma}(s))}\Big{(}\frac{\partial}{\partial
t}\bm{g}^{-1}_{t}\big{(}\bm{\gamma}(s)\big{)},\frac{\partial}{\partial
s}\bm{g}^{-1}_{t}\big{(}\bm{\gamma}(s)\big{)}\Big{)}.$ (54)

This is a surface integral of the magnetic field B, with two boundaries (see
fig. 9): one is the loop $\bm{g}^{-1}_{t}(\textbf{x})$, the other is
$\bm{g}^{-1}_{t}(\textbf{x}_{0})$. By Stokes’s theorem, eq. (54) can thus be
recast as

$\oint\text{d}t\,F_{\bm{v}_{t}}(\textbf{x})=\oint_{\bm{g}_{t}^{-1}(\textbf{x})}\textbf{A}-\oint_{\bm{g}_{t}^{-1}(\textbf{x}_{0})}\textbf{A},$
(55)

which is nearly what we need to match eq. (52) with the expected result (49):
the only difference between the two now resides in the holonomy of A along
$\bm{g}_{t}^{-1}(\textbf{x}_{0})$ in the very last term of eq. (55). This is
where the cocycle (39) finally comes to the rescue: we saw in fig. 8 that it
measures the area of a (curved) triangle. In the case at hand, eq. (52)
involves

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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{60.43852pt}{-1.25055pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\textbf{x}_{0}$}}
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}$ (56)

where the second equality was obtained by acting on the left triangle with the
area-preserving map $\bm{g}_{\tau}^{-1}$. One can then differentiate
$\mathsf{C}(\bm{g}_{\tau},\bm{g}_{t}^{-1})$ to find
$\partial_{\tau}\mathsf{C}(\bm{g}_{\tau},\bm{g}_{t}^{-1})=-\tfrac{q}{\hbar}\textbf{A}_{\bm{g}_{t}^{-1}(\textbf{x}_{0})}(\partial_{t}\bm{g}_{t}^{-1}(\textbf{x}_{0}))$
up to a total time derivative, since $\tau$ only appears in the argument of
the vertex at $\bm{g}_{\tau}^{-1}(\textbf{x}_{0})$ (see app. B for details).
Integrating over $t$ then cancels the holonomy along
$\bm{g}_{t}^{-1}(\textbf{x}_{0})$ in (55), eventually reproducing the AB term
in eq. (49).

$\bm{g}_{t}^{-1}(\textbf{x})$x$\bm{g}_{t}^{-1}(\bm{\gamma}(s))$$\bm{g}_{t}^{-1}(\textbf{x}_{0})$$\textbf{x}_{0}$
Figure 9: In (54), the magnetic field $\textbf{B}=\text{d}\textbf{A}$ is
integrated over a surface whose points are $\bm{g}_{t}^{-1}(\bm{\gamma}(s))$,
where $t\in[0,T]$, $s\in[0,1]$ and $\bm{\gamma}(s)$ connects some origin
$\textbf{x}_{0}$ to the point x. Each $\bm{g}_{t}$ is a deformation and
$\bm{g}_{T}=\bm{g}_{0}$; for definiteness, the cartoon on the left assumes
$\bm{g}_{0}=e$ the identity, but this is not required in the proof of eq.
(50). The time parameter $t$ increases along the black curves, so the two
black loops are the images of $\bm{g}_{t}^{-1}(\textbf{x}_{0})$ and
$\bm{g}_{t}^{-1}(\textbf{x})$ for $t\in[0,T]$. The auxiliary parameter $s$
increases along red and green lines, which are thus paths of the form
$\bm{g}_{t}^{-1}(\bm{\gamma}(s))$ at fixed $t$. The simplification of (54) in
two holonomies (55) follows from the cancellation between integrals over red
and green curves when these curves overlap, which only leaves out the two
yellow surfaces with boundaries $\bm{g}_{t}^{-1}(\textbf{x}_{0})$ and
$\bm{g}_{t}^{-1}(\textbf{x})$.

###  4.3 Examples of adiabatic deformations

Eq. (50) is general and holds for any sequence of deformations, so we can
apply it to specific families of area-preserving maps (recall the examples of
sec. 3.1). We first study translations and rotations, then turn to edge
deformations (25).

#### Translations and rotations.

Consider a sequence of time-dependent translations
$\textbf{x}\mapsto\textbf{x}+\textbf{a}_{t}$ (with
$\textbf{a}_{T}=\textbf{a}_{0}$). Their only effect on the Hamiltonian (37) is
to shift the location of the potential, since the kinetic term is translation-
invariant. The same can be confirmed in terms of Berry phases: the velocity
vector field (28) of overall translations is a total derivative
$\bm{v}_{t}=\dot{\textbf{a}}_{t}$, so the current integral in the Berry phase
(50) vanishes and only the AB contribution remains. The latter actually
involves the same holonomy $\oint_{\textbf{a}_{t}}\textbf{A}$ at all positions
(since translations do not depend on x), and one eventually finds

${\cal B}[\hskip
0.50003pt\bm{g}_{t}]=-\frac{q}{\hbar}\,N\,\oint_{\textbf{a}_{t}}\textbf{A}\qquad\text{(translations)}$
(57)

where $N=\int\text{d}^{2}\textbf{x}\,\rho(\textbf{x})$ is the total number of
particles in the droplet. This is manifestly just an overall AB phase, as was
to be expected (with a minus sign stemming from our choice of right group
actions in eqs. (3) and (35)). One can think of it as the response of a
charged droplet to adiabatic changes of the location of the potential well.

Another elementary example is provided by rotations around the origin: in that
case, both the current and the density generally contribute to the phase (50).
Using polar coordinates $(r,\varphi)$, adiabatic rotations read
$\varphi\mapsto\varphi+\theta_{t}$ with $\theta_{T}=\theta_{0}+2\pi n$ for
some integer $n$. The AB holonomy of (50) thus reads
$\oint\textbf{A}=-Br^{2}\pi n$ and the velocity vector field (28) is purely
angular:
$\dot{\bm{g}}_{t}\circ\bm{g}_{t}^{-1}=\dot{\theta}_{t}\,\partial_{\varphi}$.
Thus the current one-form

$\bm{J}=J_{\varphi}(r,\varphi)\text{d}\varphi+J_{r}(r,\varphi)\text{d}r$ (58)

only contributes to the phase (50) through its angular component
$J_{\varphi}$, and the complete phase (50) for adiabatic rotations reads

${\cal B}[\hskip 0.50003pt\bm{g}_{t}]=-2\pi
n\frac{M}{\hbar}\int\text{d}^{2}\textbf{x}\Big{[}J_{\varphi}(r,\varphi)+\frac{\omega_{c}}{2}r^{2}\rho(r,\varphi)\Big{]}\qquad\text{(rotations)}$
(59)

where $\omega_{c}\equiv qB/M$ is the cyclotron frequency. This generalizes the
1D result (20) and similarly measures the average current, save for an
additional density contribution that now makes the Berry phase super-extensive
in the thermodynamic limit. Note that (59) vanishes modulo $2\pi$ in the case
of isotropic states. Indeed, for one-body wave functions, its integrand

$J_{\varphi}+\frac{\omega_{c}}{2}r^{2}\rho=\frac{\hbar}{2Mi}(\Psi^{*}\partial_{\varphi}\Psi-\Psi\partial_{\varphi}\Psi^{*})$
(60)

is just $\hbar/M$ times the angular momentum of $\Psi$. Since angular momentum
is an integer, the phase (59) is indeed an integer multiple of $2\pi$. The
same is true of isotropic states whose total angular momentum is the sum of
angular momenta of their constituents. Following the remark below eq. (7),
this means that the parameter space for deformations of isotropic states is
(at most) a quotient space $\text{SDiff}\,\mathbb{R}^{2}/S^{1}$.

#### Edge deformations.

The maps (25) are ‘rotations with extra wiggles’, so their Berry phases are
expected to generalize eq. (59). Indeed, working once more in symmetric gauge
$\textbf{A}=\tfrac{1}{2}Br^{2}\text{d}\varphi$, the AB holonomy of eq. (50) is

$\oint_{\bm{g}_{t}^{-1}(\textbf{x})}\textbf{A}=-\frac{Br^{2}}{2}\oint\text{d}t\,\dot{g}_{t}\big{(}g_{t}^{-1}(\varphi)\big{)}$
(61)

where the $g_{t}(\varphi)$’s are 1D diffeomorphisms that determine adiabatic
edge deformations (25). The appearance of a 1D Berry phase analogous to eq.
(16) is thus manifest. As for the fluid velocity (28) of edge deformations, it
is a vector field of the form (27):

$\dot{\bm{g}}_{t}\circ\bm{g}_{t}^{-1}\equiv\bm{v}_{t}=\dot{g}\big{(}g^{-1}(\varphi)\big{)}\partial_{\varphi}-\frac{r}{2}\big{(}\dot{g}\circ
g^{-1}\big{)}^{\prime}(\varphi)\partial_{r}\qquad\text{(edge deformations)}.$
(62)

It only remains to pair this with the current (58), yielding the Berry phase

${\cal B}[\hskip
0.50003pt\bm{g}_{t}]=-\frac{M}{\hbar}\oint\text{d}t\,\text{d}\varphi\int_{0}^{\infty}r\,\text{d}r\bigg{[}J_{\varphi}(r,\varphi)+\frac{r}{2}\partial_{\varphi}J_{r}(r,\varphi)+\frac{\omega_{c}}{2}r^{2}\rho(r,\varphi)\bigg{]}\,\dot{g}_{t}\big{(}g_{t}^{-1}(\varphi)\big{)}.$
(63)

Remarkably, this is just a phase (16) produced by 1D deformations
$g_{t}(\varphi)$, now involving an effective 1D current

$J_{\text{eff}}(\varphi)\equiv\int_{0}^{\infty}r\,\text{d}r\bigg{[}J_{\varphi}(r,\varphi)+\frac{r}{2}\partial_{\varphi}J_{r}(r,\varphi)+\frac{\omega_{c}}{2}r^{2}\rho(r,\varphi)\bigg{]}.$
(64)

As in eq. (59), the contribution involving $r^{2}\times\rho$ in (64) is super-
extensive in the thermodynamic limit. Furthermore, a simplification occurs
again in isotropic states, for which (64) is $\varphi$-independent and reduces
to $\hbar/M$ times angular momentum.

###  4.4 Deformations of Hall droplets and Hall viscosity

Eq. (50) holds for any charged many-body ground state in the plane, so we now
apply it to QH droplets with finite area consisting of $N$ spin-polarized
electrons. We begin by recalling basic aspects of planar quantum mechanics in
the limit of strong magnetic fields, then study the Berry phases (50) due to
edge deformations in the same regime. This leads to the observation that the
current contribution to the Berry phase is extensive in the thermodynamic
limit. From there we derive an analogue of the Hall viscosity thanks to the
effective current (64), and eventually show how Hall viscosity itself is
recovered in our formalism.

#### Landau levels and QH droplets.

Consider the Hamiltonian (36) with a strong magnetic field
$\textbf{B}=\text{d}\textbf{A}$ and a weak confining potential
$V(\textbf{x})$. For simplicity, we assume that $V(\textbf{x})$ is isotropic
and harmonic, so $V(\textbf{x})=\kappa\,r^{2}/2$ for some ‘stiffness’
$\kappa\ll M\omega_{c}^{2}$.121212We will eventually assume the stronger
relation $\kappa\ll M\omega_{c}^{2}/N$ for $N\gg 1$ electrons: see eqs.
(68)–(69). Then the energy spectrum can be organized in well-defined Landau
levels, with non-degenerate energies thanks to the presence of
$V(\textbf{x})$. Indeed, in symmetric gauge
$\textbf{A}=\tfrac{1}{2}Br^{2}\text{d}\varphi$, an orthonormal basis of energy
eigenstates with angular momentum $\hbar(m-n)$ in the $n^{\text{th}}$ level is
given by wave functions that involve generalized Laguerre polynomials
$L_{n}^{a}(x)\equiv\tfrac{1}{n!}x^{-a}(\partial_{x}-1)^{n}x^{n+a}$,

$\Psi_{n,m}(\textbf{x})=\frac{1}{\sqrt{2\pi\ell^{2}}}\,\sqrt{\frac{n!}{m!}}\,z^{m-n}\,L_{n}^{m-n}(|z|^{2})\,e^{-|z|^{2}/2}.$
(65)

Here $\ell^{2}\equiv\hbar/\sqrt{q^{2}B^{2}+4M\kappa}$ is the (squared)
magnetic length corrected by harmonic effects and
$z\equiv(x+iy)/\sqrt{2\ell^{2}}$ is a dimensionless complex coordinate. At
very strong magnetic fields, the energy of each such state is
$\hbar\omega_{c}n+\kappa\ell_{B}^{2}m$ up to an irrelevant additive constant,
where $\ell_{B}\equiv\sqrt{\hbar/qB}$ is the standard magnetic length. The
corresponding (gauge-invariant) probability current (48) is purely angular:

$\bm{j}_{n,m}=\frac{\hbar}{M}\,\frac{1}{2\pi\ell^{2}}\,\frac{|z|^{2m-2n}}{m!}\,n!\big{[}L_{n}^{m-n}(|z|^{2})\big{]}^{2}\,e^{-|z|^{2}}\Big{(}m-n-\frac{\ell^{2}}{\ell_{B}^{2}}|z|^{2}\Big{)}\,\text{d}\varphi.$
(66)

Both the density $|\Psi_{n,m}(\textbf{x})|^{2}$ and the current only depend on
the radius $|z|$, and they are localized around $|z|=\sqrt{m}$ (see the left
panel of fig. 10 for the lowest level, $n=0$). Qualitatively similar
conclusions hold in weak anharmonic but isotropic traps [78].

Now let an isotropic QH droplet consist of $N\gg 1$ occupied one-body states
(65) with the lowest possible energy. These states belong to some subset
$\\{0,1,...,\nu-1\\}$ of the available Landau levels, where $\nu$ is some
integer filling fraction. In the thermodynamic limit, each level contains
$\sim N/\nu$ occupied states. Then the many-body density $\rho(r)$ is nearly
constant and equal to $\nu/(2\pi\ell^{2})$ in the bulk, before falling sharply
to zero near the edge located at $r_{\text{edge}}=\sqrt{2\ell^{2}N/\nu}$ (see
the right panel of fig. 10). The ground state thus forms a disk-shaped droplet
with roughly uniform density and area $2\pi\ell^{2}N/\nu$. As for the purely
angular current $\bm{J}=J_{\varphi}(r)\text{d}\varphi$, its bulk behaviour is
determined by the Hall law $J_{\varphi}(r)=\tfrac{\nu}{2\pi\hbar}r\partial
V/\partial r$ in terms of the filling fraction $\nu$, followed by a robust
jump localized at the edge [52, 53].

Figure 10: Left: Probability densities (blue) and current densities (yellow)
of states in the lowest Landau level, given by eqs. (65)–(66) with $n=0$ and
$m=1,9,25,49$. Right: Many-body density (blue) and many-body current (yellow)
of isotropic droplets, obtained by summing over the densities and currents of
states (65) with $n=0$ and $m$ ranging from $0$ to $N-1=64$. The density is
constant for $|z|\lesssim\sqrt{N}$ and drops to zero at the edge
$|z|\sim\sqrt{N}$, while the current grows slowly following the gradient of
the potential in the bulk, then ‘jumps’ near the edge. Similar plots hold upon
including higher Landau levels, save for additional oscillations in both one-
body and many-body observables (see e.g. [79]).

#### Adiabatic edge deformations.

Knowing currents and densities, the Berry phases (50) follow for any sequence
of adiabatic deformations. For instance, bulk effects are exhibited by
deformations that only affect the interior of the droplet while fixing its
boundary, as in fig. 7 above. Conversely, the jump of the current at the edge
contributes to Berry phases produced by deformations $\bm{g}_{t}$ that move
the boundary without affecting the bulk. Let us now focus on edge deformations
(25), whose Berry phases are given in full generality by eq. (63). As we saw
around eq. (60), the resulting 1D current (64) is really just the total
angular momentum (since the droplet is isotropic). The one-body phase (63) for
a state (65) thus reads ${\cal B}_{n,m}[\hskip
0.50003pt\bm{g}_{t}]=-(m-n)\oint\frac{\text{d}t\,\text{d}\varphi}{2\pi}\,\dot{g}_{t}\big{(}g_{t}^{-1}(\varphi)\big{)}$,
and its many-body generalization is

${\cal B}[\hskip
0.50003pt\bm{g}_{t}]=-L\oint\frac{\text{d}t\,\text{d}\varphi}{2\pi}\,\dot{g}_{t}\big{(}g_{t}^{-1}(\varphi)\big{)}\qquad\text{with}\qquad
L\sim\frac{N^{2}}{2\nu}.$ (67)

Here $L$ is the droplet’s total angular momentum when the occupied Landau
levels are $n=0,1,...,\nu-1$, each containing $\sim N/\nu$ states in the
thermodynamic limit. The Berry phase is thus super-extensive, as observed
below eq. (64). Note that this also holds for fractional QH states; for
example, the angular momentum of the Laughlin wave function [80] at filling
$\nu$ is $L=N(N-1)/(2\nu)$, in accordance with (67).

A weakness of (67) is to completely miss the fact that the total Berry phase
(50) contains two pieces: a trivial AB phase, and an additional current term.
It is instructive to compute the current contribution separately, which we do
now. Accordingly, consider the ‘truncated’ phase (46), with a gauge-invariant
current (48). Applying this to a one-body state (65) yields a phase (67) whose
coefficient $L$ is replaced by the integrated current

$\frac{2\pi M}{\hbar}J_{\text{int}}\equiv\frac{2\pi
M}{\hbar}\int_{0}^{\infty}r\,\text{d}r\,j_{n,m}(r)\stackrel{{\scriptstyle\text{\eqref{jemm}}}}{{=}}\big{(}1-\tfrac{\ell^{2}}{\ell_{B}^{2}}\big{)}m-\big{(}1+\tfrac{\ell^{2}}{\ell_{B}^{2}}\big{)}n-\tfrac{\ell^{2}}{\ell_{B}^{2}}.$
(68)

This still depends on $m$, so the corresponding sum over $N\gg 1$ occupied
states is super-extensive. However, the $m$ dependence is only due to the
confining potential and disappears in the limit of strong magnetic fields.
Assuming indeed that $\kappa\ll M\omega_{c}^{2}/N$, one may expand
$\ell^{2}/\ell_{B}^{2}\sim 1-2M\kappa/q^{2}B^{2}$ and find
$J_{\text{int}}=-\tfrac{\hbar}{2\pi M}(2n+1)$, which now only depends on the
Landau level $n$ [53, eq. (18)]. The part of the many-body phase (67) due only
to the current then becomes extensive:

${\cal B}_{\text{current}}[\hskip
0.50003pt\bm{g}_{t}]\sim\sum_{n=0}^{\nu-1}(2n+1)\sum_{m=0}^{N/\nu}\oint\frac{\text{d}t\,\text{d}\varphi}{2\pi}\,\dot{g}_{t}\big{(}g_{t}^{-1}(\varphi)\big{)}\sim
N\nu\,\oint\frac{\text{d}t\,\text{d}\varphi}{2\pi}\,\dot{g}_{t}\big{(}g_{t}^{-1}(\varphi)\big{)}.$
(69)

We stress that the final coefficient $N\nu$ appearing here is really an
integral (64) of the (angular component of the) many-body current:

$\frac{N\nu}{2\pi}=-\frac{M}{\hbar}\int_{0}^{\infty}r\,\text{d}r\,J_{\varphi}(r)\Big{|}_{\text{$\nu$
levels with $N/\nu\gg 1$ states, infinite $\textbf{B}$}}.$ (70)

Eq. (69) is thus an extensive Berry phase that is independent of the confining
potential, and that stems entirely from the many-body current at the edge. In
fact, the same contribution would have been obtained for any deformation of
the droplet that acts non-trivially on its boundary, regardless of the
detailed form of edge deformations (25). The coefficient (70) is universal in
that sense; we now show that it is analogous to the Hall viscosity.

#### Hall viscosity revisited.

The Berry phases (67) and (69) can obviously be applied to explicit edge
deformations, as done in sec. 2.3. In particular, the Berry curvature (24)
holds for SL(2,$\mathbb{R}$) transformations (21), except that the one-body
parameter $s$ is now replaced by the angular momentum $m-n$ in (67), or by the
coefficient $-2n-1$ stemming from the strong magnetic field limit of eq. (68).
The Berry curvature for a single state (65) thus reads

${\cal
F}_{n,m}=\tfrac{4}{k}\,\sinh(2\lambda)\,\text{d}\lambda\wedge\text{d}\theta\times\begin{cases}n-m&\text{for
full phase \eqref{bbexx}},\\\ 2n+1&\text{for pure current phase
\eqref{betod}}.\end{cases}$ (71)

For $k=2$, this is the curvature associated with linear deformations (2) whose
Jacobian matrix is constant in Cartesian coordinates. It is then customary
[19, 20] to rewrite (71) in terms of a complex parameter
$\tau=\tau_{1}+i\tau_{2}$ (with $\tau_{2}>0$). The latter can be read off from
the transformation of the Euclidean metric under linear maps (2), namely [19,
eq. (10)]

$(a\,\text{d}x+b\,\text{d}y)^{2}+(c\,\text{d}x+d\,\text{d}y)^{2}\equiv\frac{1}{\tau_{2}}\big{(}\text{d}x^{2}+2\tau_{1}\,\text{d}x\,\text{d}y+|\tau|^{2}\,\text{d}y^{2}\big{)},$
(72)

which yields $\tau=(ab+cd+i)/(a^{2}+c^{2})$. The correspondence (26) between
$(a,b,c,d)$ and the complex parameters
$(\alpha,\beta)=(e^{i(\chi+\theta)}\cosh\lambda,e^{i(\chi-\theta)}\sinh\lambda)$
then gives

$\tau=\frac{-\sin(2\theta)\sinh(2\lambda)+i}{\cosh(2\lambda)+\cos(2\theta)\,\sinh(2\lambda)},$
(73)

finally allowing us to rewrite the one-body Berry curvature (71) with $k=2$ as

${\cal
F}_{n,m}=-\tfrac{1}{2}\,\frac{\text{d}\tau_{1}\wedge\text{d}\tau_{2}}{\tau_{2}^{2}}\times\begin{cases}n-m&\text{for
full phase \eqref{bbexx}},\\\ 2n+1&\text{for pure current phase
\eqref{betod}}.\end{cases}$ (74)

Note that the ‘pure current’ result with its coefficient $2n+1=2(n+1/2)$
reproduces [20, eq. (3.21)] up to an overall factor 2. It is thus very close,
but not quite identical, to the Berry curvature normally associated with the
Hall viscosity of a state in the $n^{\text{th}}$ Landau level.

This mild discrepancy is not a mere matter of conventions. To see its origin,
suppose we had limited the discussion to linear deformations (2) from the
outset. We would then have found that their unitary action (35) on wave
functions reduces to eq. (31) in symmetric gauge, since the latter is
preserved by edge deformations (25). The corresponding Lie algebra operators
(44) can be expressed as quadratic combinations of ladder operators
$a^{\dagger}=\bar{z}/2-\partial$ and $b=\bar{z}/2+\partial$, which
respectively raise the Landau level and decrease angular momentum within a
level. Indeed, in terms of the coordinates $(\lambda,\theta,\chi)$ defined
above eq. (23) and expanding the group element (21) at $k=2$ up to first order
in $\lambda$ and $\chi+\theta$, one has

$\begin{split}\mathfrak{u}[\bm{v}]=&-i\lambda e^{-i\chi+i\theta}a^{2}+i\lambda
e^{i\chi-i\theta}(a^{\dagger})^{2}-(\chi+\theta)(a^{\dagger}a+aa^{\dagger})\\\
&-i\lambda e^{i\chi-i\theta}b^{2}+i\lambda
e^{-i\chi+i\theta}(b^{\dagger})^{2}+(\chi+\theta)(b^{\dagger}b+bb^{\dagger})\end{split}$
(75)

for infinitesimal linear deformations. The factorization between $a$ and $b$
pieces is manifest, respectively corresponding to deformations of the metric
and the (slowly varying) confining potential in the Hamiltonian (36). The
resulting Berry phase (49) consists of two parts, respectively involving
expectation values of $\tfrac{1}{2}(a^{\dagger}a+aa^{\dagger})$ and
$\tfrac{1}{2}(b^{\dagger}b+bb^{\dagger})$. In fact, these two parts can be
read off from the first line in the curvature (74) upon writing
$n-m=(n+1/2)-(m+1/2)$:

${\cal F}_{n,m,\text{
full}}=\underbrace{-(n+1/2)\tfrac{1}{2}\,\frac{\text{d}\tau_{1}\wedge\text{d}\tau_{2}}{\tau_{2}^{2}}}_{\displaystyle{\cal
F}_{a}}+\underbrace{(m+1/2)\tfrac{1}{2}\,\frac{\text{d}\tau_{1}\wedge\text{d}\tau_{2}}{\tau_{2}^{2}}}_{\displaystyle{\cal
F}_{b}}.$ (76)

Here the term involving $n+1/2$ coincides precisely with [20, eq. (3.21)];
there is no factor 2 mismatch, and it originates indeed from metric
deformations produced by the $a$ factors in (75), as in [20, eqs.
(3.4)–(3.9)]. Standard formulas for the Hall viscosity then follow in the case
of a droplet with $\nu$ filled levels: the sum over occupied states is the
same as in (69) and yields an extensive total Berry curvature ${\cal
F}_{a}=-\tfrac{1}{4}\,N\nu\,\text{d}\tau_{1}\wedge\text{d}\tau_{2}/\tau_{2}^{2}$.
In the regime of linear response where $\tau$ is close to $i$, the coefficient
of the area form $\text{d}\tau_{1}\wedge\text{d}\tau_{2}/\tau_{2}^{2}$ is
interpreted as an odd viscosity $\eta_{H}$ times the droplet’s area
$2\pi\ell_{B}^{2}N/\nu$, with the standard value

$\eta_{H}=\frac{\hbar}{4}\,N\nu\times\frac{\nu}{2\pi\ell_{B}^{2}N}=\frac{\hbar}{4}\,\frac{\nu^{2}}{2\pi\ell_{B}^{2}}\qquad\text{(integer
$\nu$)}$ (77)

for integer QH states [19, 20]. We stress that this was obtained without ever
referring to a torus. In particular, the complex number (73) may label any
point on the hyperbolic upper half-plane, and should not be interpreted as a
modular parameter [81].

At this point, it is natural to wonder if the separation between $a$ and $b$
pieces in (70) has anything to do with the distinct contributions of current
and density in the complete Berry phase (50). Indeed, we saw in eq. (74) that
the Berry curvature due only to the current coincides with ${\cal F}_{a}$ in
(76) up to an overall factor 2. This stems from the integrated current (68):
at strong magnetic fields, the coefficient $n-m$ in the full curvature (74)
splits as

$n-m=\underbrace{(2n+1)}_{\displaystyle\text{current}}-\underbrace{(m+n+1)}_{\displaystyle\text{AB
phase}},$ (78)

which is crucially not the same splitting as in (76). In particular, the fact
that the AB phase itself depends on the level $n$ ultimatly gives rise to a
current contribution whose Berry curvature is twice the curvature of the Hall
viscosity for integer QH states. One should not be concerned about this
discrepancy: there is no inherent reason for the current portion of the Berry
phase (50) to be related to viscosity. The fact that the corresponding Berry
curvatures match up to normalization is simply a result of the unique
invariant area form on a hyperbolic plane (recall footnote 1). Moreover, it is
important to note that the distinction between $a$ and $b$ contributions in
the operator (75) is specific to linear deformations (2), and does not align
with the separation of (50) into current and AB contributions. Identifying the
part of the Berry phase (50) arising solely from metric deformations would
pose a significant challenge, extending beyond the scope of the
quantomorphisms studied here.

#### Fractional QH states.

It is tempting to speculate that this factor 2 is a robust property of QH
droplets. Indeed, we have just shown that it holds for integer QH states at
strong magnetic fields, allowing us to view the Hall viscosity (77) as a
quantized integral

$\eta_{H}=-\lim_{\begin{subarray}{c}\nu\text{ levels,}\\\
N\to\infty,\;\textbf{B}\to\infty\end{subarray}}\frac{\nu}{2\pi\ell_{B}^{2}N}\times\frac{\pi
M}{2}\int_{0}^{\infty}r\,\text{d}r\,J_{\varphi}(r)\qquad\text{(integer
$\nu$)}$ (79)

that measures the net jump of the current’s ‘stream function’ across the edge
of a droplet. A similar integral can be devised for fractional QH states, up
to a key change in normalization that spoils the aforementioned factor 2. Let
us exhibit this in the case of a Laughlin wave function for $N$ electrons at
filling $\nu$: then the full Berry phase for edge deformations is given by
(67) with a total angular momentum $L=N(N-1)/(2\nu)$. The latter may be seen
as an integral of current and density through the ‘effective current’ (64),
namely

$\frac{N(N-1)}{2\nu}=\frac{M}{\hbar}\int\text{d}^{2}\textbf{x}\,J_{\varphi}(r)+\frac{1}{\ell_{B}^{2}}\int\text{d}^{2}\textbf{x}\,r^{2}\,\rho(r)$
(80)

where nothing depends on $\varphi$ since the Laughlin wave function is
isotropic. Here the current integral is the coefficient of the ‘pure current’
Berry phase meant to mimic the Hall viscosity, and its value can be obtained
by plugging in eq. (80) the sum rule [82, eq. (B.2)]

$\int\text{d}^{2}\textbf{x}\,r^{2}\rho(r)=\ell_{B}^{2}N\frac{N-1+2\nu}{\nu}$
(81)

for the density of an isotropic Coulomb gas. Much more generally, the current
integral is readily found in second quantization for any isotropic QH droplet:

$\int_{0}^{\infty}r\,\text{d}r\,J_{\varphi}(r)=\sum_{m=0}^{\infty}\,\langle
a^{\dagger}_{m}a_{m}\rangle\int_{0}^{\infty}r\,\text{d}r\,j_{0,m}(r)\stackrel{{\scriptstyle\text{\eqref{jet}}}}{{\sim}}-\frac{\hbar}{2\pi
M}\sum_{m=0}^{\infty}\langle a^{\dagger}_{m}a_{m}\rangle=-\frac{\hbar N}{2\pi
M},$ (82)

where $a_{m}^{\dagger}$ is the Fock space creation operator for the
$m^{\text{th}}$ isotropic orbital in the lowest Landau level, and
$\langle...\rangle$ denotes expectation values in the droplet’s ground state.
One thus concludes that the ‘pure current’ Berry curvature associated with
linear quantomorphisms of any isotropic QH droplet is

${\cal
F}_{\text{current}}=-\frac{N}{2}\frac{\text{d}\tau_{1}\wedge\text{d}\tau_{2}}{\tau_{2}^{2}},$
(83)

where we use the same hyperbolic coordinates as in eq. (74). This crucially
differs from the Berry curvature associated with metric deformations of
Laughlin states on a torus, calculated in [27, 23] thanks to the plasma
analogy and given by

${\cal
F}_{\text{metric}}=-\frac{N}{4\nu}\frac{\text{d}\tau_{1}\wedge\text{d}\tau_{2}}{\tau_{2}^{2}},\quad\text{with
viscosity}\quad\eta_{H}=\frac{\hbar}{4}\frac{1}{2\pi\ell_{B}^{2}}\qquad\text{(integer
$1/\nu$)}.$ (84)

Relating this value of viscosity to an integral of the current requires that
the right-hand side of (79) be divided by $\nu$; eq. (79) is thus not valid
for fractional QH states, as announced. Put differently, the coefficient of
the pure current curvature (83) divided by twice the droplet’s area predicts a
Hall viscosity $\hbar\nu/(8\pi\ell_{B}^{2})$, which differs from (84) by a
factor $\nu$.131313Amusingly, the incorrect value
$\hbar\nu/(8\pi\ell_{B}^{2})$ is the one that was initially stated in [22],
before the correction to $\eta_{H}=\hbar/(8\pi\ell_{B}^{2})$ in [23]. It is
unclear if this is just a coincidence or if there is a deeper reason for the
matching. A similar mismatch occurs, for instance, in the Moore-Read state at
filling $\nu=1/2$, where [27] predicts a Hall viscosity
$\eta_{H}=\tfrac{3}{2}\hbar/(8\pi\ell_{B}^{2})$ while our universal current
Berry curvature (83) predicts a response coefficient
$\tfrac{1}{2}\hbar/(8\pi\ell_{B}^{2})$. As previously mentioned, this mismatch
is not contradictory; instead, it is a distinguishing characteristic. Hall
viscosity is a specific response to pure metric deformations, whereas our
current Berry curvature (83) encompasses both metric and potential
perturbations, and fundamentally does not have any direct connection to
viscosity.

##  5 Conclusion and outlook

This work was devoted to the Berry phases produced by adiabatic deformations
of many-body quantum systems. Specifically, we implemented deformations
through unitary operators in the Hilbert space, allowing us to interpret the
resulting changes of the Hamiltonian (4) as actual deformations of the metric
and potential (recall eqs. (14) and (37)).

In all cases, the response involves the current of the state being acted upon.
We applied this to 1D quantum wires with potentially observable consequences,
reproducing in an elementary way the Berry phases (16) studied in [50] in
conformal field theory. But a much more prominent application of our approach
was that of 2D planar droplets in a strong magnetic field. The formalism
developed for that situation in sec. 3 was based on unitary area-preserving
maps generalizing magnetic translations, known as ‘quantomorphisms’ in
geometric quantization [44, 45]. In particular, we introduced the ‘edge
deformations’ (25) that generalize linear maps (2) and deform the boundary of
thermodynamically large isotropic droplets. We then turned to our main result,
namely an analytical derivation of explicit Berry phases (50) produced by
adiabatic quantomorphisms. As emphasized there, such phases consist of two
separately gauge-invariant terms—a current piece and an Aharonov-Bohm (AB)
piece. This is true for any many-body ground state, regardless of the presence
of interactions, and thus applies to both integer and fractional quantum Hall
(QH) states.

While our formulas hold for all area-preserving deformations, we focused
repeatedly on edge deformations, eventually showing that their Berry phases
include, besides trivial AB phases, a contribution reminiscent of the Hall
viscosity [19, 20]. The latter actually emerges as a subleading contribution
to the phase, measuring the response to linear deformations produced by
mechanical momenta that change the metric, as opposed to magnetic translations
that do not (recall the splitting (75)). For integer QH states in the limit of
extremely strong magnetic fields, the same viscosity was obtained, up to a
factor 2, from a bulk integral (79) of a boundary current. We also considered
fractional QH states, for which the factor 2 had to replaced by other
coefficients with otherwise identical conclusions. As explained in the
manuscript, such mismatches should not be seen as paradoxes compared with
earlier works on the Hall viscosity, since the splitting between current terms
and AB terms in the Berry phase (50) generally has nothing to do with the
splitting of quantomorphisms in metric and potential deformations. What is
remarkable instead is that the Berry phase (50) contains an extensive gauge-
invariant piece that just happens to be proportional to the Hall viscosity.

These results open the door to a number of follow-up questions. First, a
detailed study of linear response is in order, both for non-interacting QH
states and for their fractional peers, to clarify the relation between the
integral viscosity formula (79) and the more standard link between viscosity
and stresses [83, §2]. One could also imagine going beyond the leading order
of adiabatic linear response by evaluating the (functional) quantum metric
associated with adiabatic planar deformations [84]. Indeed, it is the metric
of parameter space, rather than the Berry curvature, that eventually justifies
the use of hyperbolic geometry in the context of the Hall viscosity [20].

A thornier issue is that the deformations defined here acted both on the
metric and on the potential (see eq. (37)), calling for a modification that
would allow metric and potential to transform independently. For example,
densities and currents of QH droplets in arbitrary (disordered) potentials
were studied in [85, 86], providing formulas that can presumably be used to
deduce Berry phases due to potential deformations alone. Combining this with
compensating quantomorphisms would likely yield deformations of the metric
alone, as was achieved independently in [27, 36] for fractional QH states
viewed as conformal blocks. The link between these approaches seems tenuous
and requires clarification, not to mention their relation with earlier works
on infinitesimal deformations in the lowest Landau level [12, 13, 14, 15, 16].
We intend to investigate this in the near future; see also [78].

Finally, it goes without saying that our predictions call for experiments.
Even in the simplest 1D quantum wires treated here as toy models, observing
the phases (16) or the corresponding adiabatic response would be fascinating.
This may require a reformulation of our formalism in terms of lattice models,
along the lines of what was done e.g. in conformal field theories with a
Floquet drive [87, 88]. Even more importantly, implementing 2D planar
deformations in QH simulators could provide the striking measure of the Hall
viscosity that is still lacking at the moment, despite recent breakthroughs in
graphene [38]. This could again be done either on a lattice or in the
continuum [42, 43]. Our formalism thus paves the way for a number of new
geometric observations in mesoscopic systems, with conceptual implications for
topological phases in general.

## Acknowledgements

We are grateful to Laurent Charles for many enlightening discussions on
quantomorphisms; to Hansueli Jud and Clément Tauber for introducing us to the
Hall viscosity; to Giandomenico Palumbo, Nicolas Regnault, Shinsei Ryu and
Jean-Marie Stéphan for insightful comments on the initial preprint; and
finally to Mathieu Beauvillain, Nathan Goldman, Bastien Lapierre, Per Moosavi
and Marios Petropoulos for collaboration on related subjects. Many thanks also
to the organizers of the workshop ‘Mathematical Aspects of Quantum Phases of
Matter’ (held in Bȩdlewo, Poland in July 2021), where a preliminary version of
the results of this paper was presented. The work of B.O. is supported by the
European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie grant agreement No. 846244. B.O and B.E. are also
supported by the ANR grant TopO No. ANR-17-CE30-0013-01.

##  Appendix A Quantomorphisms on closed surfaces

This appendix accompanies sec. 3.3 and provides a self-contained review of the
action of (certain) area-preserving diffeomorphisms on closed, oriented
surfaces such as spheres or tori. In the context of geometric quantization
(see [44, 45, 70, 71]), the action that we consider coincides with
automorphisms of prequantum line bundles, also known as prequantum operators
or quantomorphisms [89]. A summary of the main points is as follows:

  * •

The Hilbert space of a charged particle coupled to a background U(1) gauge
field A is a space of sections for a Hermitian line bundle $L$ on some base
manifold $\Sigma$, with compatible connection $\nabla$. This geometric data
$(L,\nabla)$ is fully captured by (and can be reconstructed from) the
knowledge of all holonomies of A.

  * •

Not all area-preserving diffeomorphisms can be lifted to unitary
quantomorphisms acting on the Hilbert space: only holonomy-preserving ones
can. The corresponding transformation of wave functions is unique up to a
global phase, and can be constructed explicitly using parallel transport. When
the base space $\Sigma$ is the plane, this action reduces to eq. (35) since
all area-preserving maps also preserve holonomies. This is not so on generic
surfaces (e.g. the torus), in which case eq. (35) only holds locally.

  * •

The Lie algebra of the group of holonomy-preserving diffeomorphisms consists
of all Hamiltonian vector fields, that is, vector fields $\bm{v}$ for which
the one-form $\iota_{\bm{v}}\textbf{B}$ is exact, where
$\textbf{B}=\text{d}\textbf{A}$ is the curvature of the connection. The
unitary action of any such vector field $\bm{v}$ is given by the Hermitian
operator (40):

$\mathfrak{u}[\bm{v}]=-i\nabla_{\bm{v}}-F_{\bm{v}}\qquad\text{with}\qquad\iota_{\bm{v}}\textbf{B}=\text{d}F_{\bm{v}},$
(A.1)

where $\nabla_{\bm{v}}$ is the covariant derivative along ${\bm{v}}$ and
$F_{\bm{v}}$ is a stream function (a ‘Hamiltonian’) for $\bm{v}$ in the sense
of sec. 3.2.

The plan of the appendix is as follows. We begin by recalling the fiber bundle
approach to quantum mechanics, where charged wave functions on $\Sigma$ are
viewed as sections of a complex line bundle equipped with a connection. We
then ask how diffeomorphisms of $\Sigma$ can be implemented unitarily, and
show that only Hamiltonian area-preserving maps are allowed, leading to the
quantomorphisms defined locally in eq. (35). Finally, we turn to infinitesimal
quantomorphisms and recover the earlier expression (40), now derived thanks to
bundle geometry. We adopt units such that $q/\hbar=1$ for simplicity.

We stress that the material of this appendix is not original, and is for the
most part a simple translation of the lecture notes [89] from the language of
principal U$(1)$ bundles to that of the associated line bundles. We hope to
have made it accessible to physicists: the only required background is some
familiarity with line bundles, as reviewed e.g. in [61, chaps. 9–10]. We also
refer to [90] for a pedagogical introduction to geometric quantization that
overlaps with our presentation here.

###  A.1 Line bundles, connections and holonomies

Let us briefly review the geometric treatment of quantum mechanics, in which
wave functions are sections of a Hermitian line bundle [44, 91, 92].
Background electromagnetic fields are then incorporated thanks to a
connection, generally with some non-zero curvature ($=$ magnetic field). This
approach is very general, but we focus on the case relevant for the quantum
Hall effect, namely a charged particle moving on some 2D surface $\Sigma$. The
latter is assumed to be connected, closed (i.e. compact and without boundary),
and oriented. We endow it with an area form ${\bm{\omega}}$, i.e. a nowhere-
vanishing two-form; this is typically the Riemannian area form
${\bm{\omega}}=\sqrt{g}\,\text{d}x\wedge\text{d}y$ associated with a
Riemannian metric $g$, but the metric will play no role in the following.

#### Line bundles and sections.

How to describe wave functions on $\Sigma$? The starting point is to think of
$\Sigma$ as the base space of a complex line bundle $\pi:L\to\Sigma$. Wave
functions are sections of this bundle, i.e. maps $\Psi:\Sigma\to
L:\textbf{x}\mapsto\Psi(\textbf{x})$ such that
$\pi(\Psi(\textbf{x}))=\textbf{x}$ for any point x in $\Sigma$. To make sense
of the fact that wave functions must be square-integrable, we assume that $L$
is endowed with a Hermitian structure. Scalar products of sections thus read

$\langle\Phi|\Psi\rangle=\int_{\Sigma}h(\Phi,\Psi)\,{\bm{\omega}}$ (A.2)

where $h$ is a Hermitian metric, i.e. a collection of Hermitian inner products
$h_{\textbf{x}}\big{(}\Phi(\textbf{x}),\Psi(\textbf{x})\big{)}$ on each fiber
at $\textbf{x}\in\Sigma$.

The line bundle of sec. 3.3 was a trivial direct product
$L=\Sigma\times\mathbb{C}$ with $\Sigma=\mathbb{R}^{2}$. Sections were thus
functions $\textbf{x}\mapsto\Psi(\textbf{x})$ and the Hermitian structure
(A.2) was given by
$h_{\textbf{x}}\big{(}\Phi(\textbf{x}),\Psi(\textbf{x})\big{)}=\Phi^{*}(\textbf{x})\Psi(\textbf{x})$.
In general, however, the line bundle is non-trivial and cannot be written
globally as a product manifold. Relatedly, sections cannot be written as mere
functions on $\Sigma$. One therefore has to choose some covering of $\Sigma$
by a (finite) collection of open sets $U_{a}$ labelled by some index
$a=1,2,...$. These sets can always be taken small enough that their preimage
$\pi^{-1}(U_{a})\cong U_{a}\times\mathbb{C}$ be a trivial bundle
$\pi:\pi^{-1}(U_{a})\to U_{a}$; we then refer to the open covering
$\\{U_{a}\\}$ as trivializing. This can be used to locally write any section
as a function. Indeed, choose for each $U_{a}$ a section $\sigma_{a}:U_{a}\to
L$ that vanishes nowhere on $U_{a}$; we refer to this as a choice of frame on
the line bundle. Given a frame, write any global section as (no implicit
summation over $a$!)

$\Psi(\textbf{x})=\psi_{a}(\textbf{x})\,\sigma_{a}(\textbf{x})\qquad\forall\,\textbf{x}\in
U_{a},$ (A.3)

with $\psi_{a}$ some complex function on $U_{a}$. The frame is said to be
unitary if
$h_{\textbf{x}}\big{(}\sigma_{a}(\textbf{x}),\sigma_{a}(\textbf{x})\big{)}=1$
for all $\textbf{x}\in U_{a}$. Note that the choice of frame is not unique:
for any unitary frame $\\{\sigma_{a}\\}$, one can use local gauge
transformations on the $U_{a}$’s to define a new unitary frame

$\tilde{\sigma}_{a}(\textbf{x})\equiv
e^{-i\alpha_{a}(\textbf{x})}\sigma_{a}(\textbf{x}).$ (A.4)

The corresponding local wave function $\psi_{a}$ transforms into
$\tilde{\psi}_{a}(\textbf{x})=e^{i\alpha_{a}(\textbf{x})}\psi(\textbf{x})$.
Also note that, given a unitary frame, wave functions on non-empty overlaps
$U_{a}\cap U_{b}$ are related by similar gauge transformations: one has
$\psi_{a}(\textbf{x})=e^{i\alpha_{ab}(\textbf{x})}\psi_{b}(\textbf{x})$.

#### Background field as a connection.

We now wish to couple our quantum system on $\Sigma$ to some background U(1)
gauge field. This is achieved by endowing the line bundle $L\to\Sigma$ with a
connection $\nabla$ that defines a covariant derivative of sections, i.e. a
notion of parallel transport along $\Sigma$. The latter is readily written in
a familiar form upon using some local trivialization $\\{U_{a}\\}$ and some
unitary frame $\\{\sigma_{a}\\}$: one can then specify the connection on each
$U_{a}$ by a local one-form $\textbf{A}_{a}$ such that

$\nabla_{\bm{v}}\,\sigma_{a}\equiv-i\langle\textbf{A}_{a},\bm{v}\rangle\,\sigma_{a},$
(A.5)

for any vector field $\bm{v}$ on $U_{a}$, with
$\langle\textbf{A}_{a},\bm{v}\rangle\equiv\iota_{\bm{v}}\textbf{A}_{a}=v^{j}A_{a,j}$
as in eq. (43). Thus the covariant derivative of an arbitrary section $\Psi$,
written locally as (A.3), reads

$\nabla_{\bm{v}}\Psi=\Big{(}v^{j}\partial_{j}\psi_{a}-i\langle\textbf{A}_{a},\bm{v}\rangle\,\psi_{a}\Big{)}\sigma_{a}.$
(A.6)

The term in parentheses on the right-hand side is the familiar form of a
covariant derivative, expressed in terms of components $v^{j}$ and involving
the local gauge field $\textbf{A}_{a}$.

As usual, we require the connection to be unitary, i.e. that the connection
one-forms $\textbf{A}_{a}$ be real for any unitary frame. (In fact, if
$\textbf{A}_{a}$ is real in a unitary frame, then it is real in any other
unitary frame.) Parallel transport along any curve in $\Sigma$ is thus an
isometry: it preserves the Hermitian product (A.2). In particular, parallel
transport along a closed, oriented loop $\bm{\gamma}$ in $\Sigma$ only affects
a wave function by an overall phase, namely the holonomy of the connection
around $\bm{\gamma}$. From the perspective of sec. 4, this holonomy is nothing
but an Aharonov-Bohm phase. It turns out that the full geometric data of the
line bundle $L$ with connection $\nabla$ is encoded in such holonomies: two
Hermitian bundles with connection are isomorphic if and only if all their
holonomies coincide (see e.g. [93]).

Note that changing the local frame changes connection one-forms: the local
gauge transformations (A.4) affect the gauge fields $\textbf{A}_{a}$ defined
in (A.5) by changing them into
$\tilde{\textbf{A}}_{a}=\textbf{A}_{a}+\text{d}\alpha_{a}$. On overlaps where
$\psi_{a}(\textbf{x})=e^{i\alpha_{ab}(\textbf{x})}\psi_{b}(\textbf{x})$, one
similarly finds $\textbf{A}_{a}=\textbf{A}_{b}+\text{d}\alpha_{ab}$.

#### Magnetic field as a curvature.

Suppose one is given a Hermitian line bundle with some unitary connection
$\nabla$. Then the corresponding curvature two-form B is defined by the
commutator of covariant derivatives,

$\nabla_{\bm{v}}\nabla_{\bm{w}}-\nabla_{\bm{w}}\nabla_{\bm{v}}-\nabla_{[\bm{v},\bm{w}]}=-i\textbf{B}(\bm{v},\bm{w}).$
(A.7)

In terms of a local trivialization with some local frame, this yields the
usual magnetic field $\textbf{B}=\text{d}\textbf{A}_{a}$ on $U_{a}$. We shall
assume that B is uniform: given the area form ${\bm{\omega}}$ on $\Sigma$, we
let $\textbf{B}=B{\bm{\omega}}$ so that
$\int_{\Sigma}\textbf{B}=B\times\text{area}(\Sigma)$, and assume $B>0$ for
definiteness.

There is little more to be said about uniform magnetic fields when $\Sigma$ is
the plane, as was the case in the main text. However, subtleties occur on
closed manifolds, where the two-form B is not exact (as it would otherwise
violate Stokes’s theorem) and is subject to Dirac’s quantization condition

$\displaystyle\frac{1}{2\pi}\int_{\Sigma}\textbf{B}\in\mathbb{Z}.$ (A.8)

This integer is the Chern number of the line bundle. As usual, the bundle is
non-trivial if the Chern number is non-zero.

###  A.2 Lifting diffeomorphisms to line bundles

Having recalled the bundle picture of quantum mechanics, we now use it to
study deformations of the base manifold. Namely, given a diffeomorphism
$\bm{g}:\Sigma\to\Sigma$, we wish to define a corresponding unitary operator
acting on the Hilbert space of square-integrable sections $\Psi:\Sigma\to L$.
The most naive solution is to send $\Psi$ on its pullback
$\bm{g}^{*}\Psi=\Psi\circ\bm{g}$, possibly adding a prefactor as in eq. (11)
to account for the fact that a wave function is a half-density.141414We will
eventually see in eq. (A.12) that $\bm{g}$ needs to preserve area, so no
square root appears in practice. But this does not quite make sense:
$(\bm{g}^{*}\Psi)(\textbf{x})$ lies in the fiber $L_{\bm{g}(\textbf{x})}$
instead of $L_{\textbf{x}}$, so $\bm{g}^{*}\Psi$ is no longer a section of the
bundle $\pi:L\to\Sigma$ and fails to be a bona fide wave function. On the
other hand, $\bm{g}^{*}\Psi$ is a section of the pullback bundle
$\pi^{\prime}:\bm{g}^{*}L\to\Sigma$, so we need a way to identify these two
bundles. This is achieved thanks to a bundle isomorphism, i.e. a
diffeomorphism $\bm{G}:L\to\bm{g}^{*}L$ such that the following diagram
commutes:

${L}$${\bm{g}^{*}L}$${\Sigma}$${\Sigma}$$\scriptstyle{\pi}$$\scriptstyle{\bm{G}}$$\scriptstyle{\pi^{\prime}}$id

and whose restriction to each fiber is a linear map. Morally, the isomorphism
is just a collection of (invertible) linear maps
$\bm{G}_{\textbf{x}}:L_{\textbf{x}}\to L_{\bm{g}(\textbf{x})}$, depending
smoothly on $\textbf{x}\in\Sigma$. Provided such a map exists, we may assume
without loss of generality that each $\bm{G}_{\textbf{x}}$ is unitary. (If
$\bm{G}_{\textbf{x}}$ is not norm-preserving, it can be rescaled fiberwise
$\bm{G}_{\textbf{x}}\to\lambda_{\textbf{x}}\bm{G}_{\textbf{x}}$ with some
smooth rescaling factor $\lambda:\Sigma\to\mathbb{R}^{+}$.) Alternatively, one
can interpret $\bm{G}$ as a bundle map covering $\bm{g}$, in the sense that
the following diagram commutes:

${L}$${L}$${\Sigma}$${\Sigma}$$\scriptstyle{\pi}$$\scriptstyle{\bm{G}}$$\scriptstyle{\pi}$$\scriptstyle{\bm{g}}$

The bundle map $\bm{G}$ thus sends $\bm{g}^{*}\Psi$ back to $L$, yielding the
following unitary action of $\bm{g}$ on sections of $L$:

$\,{\cal U}[\hskip 0.50003pt\bm{g}]\Psi\equiv\bm{G}^{-1}\circ\Psi\circ\bm{g}.$
(A.9)

This is the elementary starting point that will eventually yield
quantomorphisms.

In practice, the prescription (A.9) is subject to a number of conditions.
First, it requires a bundle isomorphism between $L$ and $\bm{g}^{*}L$. The
latter exists if and only if $\bm{g}^{*}L$ and $L$ have the same first Chern
class $c_{1}[L]=c_{1}[\hskip 0.50003pt\bm{g}^{*}L]\in
H^{2}(\Sigma,\mathbb{Z})$, which here boils down to $\bm{g}$ being
orientation-preserving.151515For compact oriented $\Sigma$, $H^{1}(\Sigma)$ is
torsion-free and the real first Chern class $[\textbf{B}]\in
H^{2}(\Sigma,\mathbb{R})$ classifies line bundles. Then $\bm{g}^{*}\textbf{B}$
and B are cohomologous iff
$\int_{\Sigma}\bm{g}^{*}\textbf{B}=\int_{\Sigma}\textbf{B}$, i.e. if $\bm{g}$
preserves orientation. A second, much stronger requirement is that $\bm{G}$
must not affect the coupling of wave functions to the background gauge field.
This is to say that the operator (A.9) needs to leave the connection
invariant:

$\,{\cal U}[\hskip 0.50003pt\bm{g}]^{{\dagger}}\,\nabla_{\bm{v}}\,{\cal
U}[\hskip 0.50003pt\bm{g}]=\nabla_{\bm{g}_{*}\bm{v}}$ (A.10)

where $\bm{v}$ is any vector field on $\Sigma$ and $\bm{g}_{*}\bm{v}$ is its
pushforward by $\bm{g}$. Thus the image of the connection $\nabla$ under the
bundle isomorphism $\bm{G}:L\to\bm{g}^{*}L$ must be the pullback connection
$\bm{g}^{*}\nabla$. Equivalently, the bundle map $\bm{G}:L\to L$ covering
$\bm{g}$ must preserve the connection $\nabla$. Unitary bundle isomorphisms
that satisfy these criteria are known as prequantum bundle automorphisms or
quantomorphisms [44, 45]. As emphasized in sec. 3.3, magnetic translations of
electronic wave functions provide a well known example of such maps in
physics.

The key constraint (A.10) limits sharply the set of deformations of $\Sigma$
that admit a well-defined unitary action on charged wave functions. Indeed,
$(L,\nabla)$ and $(\bm{g}^{*}L,\bm{g}^{*}\nabla)$ need to be isomorphic as
Hermitian line bundles with connection, so all their holonomies must coincide.
As a result, a diffeomorphism $\bm{g}:\Sigma\to\Sigma$ admits a unitary,
connection-preserving lift $\bm{G}:L\to L$ if and only if it is holonomy-
preserving: for any cycle $\bm{\gamma}$ in $\Sigma$,

$\text{hol}(\bm{\gamma})=\text{hol}(\bm{g}\circ\bm{\gamma}).$ (A.11)

Note that holonomy-preserving maps necessarily preserve area: if the cycle
$\bm{\gamma}$ is the boundary of some domain $S$, then
$\text{hol}(\bm{\gamma})=B\times\text{area}(S)$ and eq. (A.11) entails the
local condition

$\bm{g}^{*}\textbf{B}=\textbf{B}.$ (A.12)

But there is also a global aspect, since eq. (A.11) even holds for loops that