**footnotetext: These authors contributed equally to the development of this
work

# Emergent QED3 from half-filled flat Chern bands

Xue-Yang Song∗ Hart Goldman∗ Liang Fu  Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA 02139

###### Abstract

In recent years, two-dimensional Dirac materials patterned with a superlattice
structure have emerged as a rich platform for exploring correlated and
topological quantum matter. In this work, we propose that by subjecting Dirac
electrons to a periodic magnetic field with triangular lattice symmetry it is
possible to realize a _quantum critical phase_ of $N_{f}=3$ Dirac fermion
species strongly coupled to an emergent gauge field, or 2+1-D quantum
electrodynamics (QED3). We demonstrate explicitly that the QED3 phase
naturally arises from a Dirac composite fermion (CF) picture, where the
periodic magnetic field manifests as a periodic CF potential and transforms
the CF Fermi surface into gapless Fermi points. We further show that by
breaking the particle-hole symmetry of the TI surface – either by doping or by
introducing a periodic electrostatic potential with zero mean – our quantum
critical phase gives way to a sequence of fractional Chern insulator phases.
Our theory illustrates the rich menagerie of quantum phases possible around
half filling of a flat Chern band.

_Introduction._ The unprecedented tunability of two-dimensional (2d) van der
Waals materials has led to new opportunities to explore novel quantum phases
of matter. In particular, a plethora of correlated and topological electron
states has been found by engineering 2d systems with a superlattice structure.
Major examples include moiré potentials established by stacking 2d materials
with a twist angle between layers or a lattice mismatch [1, 2, 3, 4, 5, 6],
spatially varying strains [7, 8], and buckling in graphene sheets [9]. The
superlattice patterning of these materials commonly provides a spatial
modulation at the length scale of $10$nm, which can give rise to flat bands
hosting strong interaction effects.

In the presence of Coulomb interactions, partially filled flat Chern bands at
odd-denominator filling fractions can support fractional Chern insulators
(FCIs) [10, 11, 12, 13, 14, 15, 16]. Like ordinary fractional quantum Hall
(FQH) phases, FCIs exhibit a fractionally quantized Hall conductivity,
topological order, and fractional excitations, but enriched with the symmetry
of the periodic lattice. The search for FCIs in 2d materials has garnered much
recent experimental attention [17, 18].

Less explored is the case of flat Chern bands at half filling. Ordinary Landau
levels (LLs) in a uniform magnetic field are experimentally observed to host a
range of correlated phases at half filling, depending on the LL index, $n$.
These include the composite Fermi liquid ($n=0$) [19, 20], a non-Abelian FQH
state ($n=1$) [21, 22, 23, 24], and charge ordered states (higher $n$) [25,
26, 27, 28, 29, 30]. We are therefore driven to ask: Can topological flat
bands beyond Landau levels support new types of quantum phases at half
filling? If so, in what material platforms can they be found?

In this paper, we demonstrate the emergence of an exotic _quantum critical
phase_ in half-filled flat Chern bands, and we propose a realistic material
platform for its physical realization. This phase can be described in terms of
$N_{f}=3$ species of emergent Dirac fermions strongly interacting through a
dynamical U$(1)$ gauge field, or $2+1$-D quantum electrodynamics (QED3). As a
strongly interacting conformal field theory, many of the universal properties
of QED3 are unknown outside of large-$N_{f}$ limits, and numerical methods
[31, 32, 33, 34, 35, 36, 37, 38] have thus far made limited progress, making
the possibility of its physical realization all the more tantalizing.

We show that the quantum critical QED3 state can be realized from a system of
two-dimensional Dirac electrons subject to a periodic magnetic field with
triangular lattice symmetry and two flux quanta per unit cell. In the limit of
a uniform magnetic field and at charge neutrality, the $n=0$ Landau level is
half filled, and, in the presence of Coulomb interactions, the ground state is
a composite Fermi liquid. When a periodic modulation in the magnetic field is
introduced, the composite fermions (CFs) experience a periodic electric
potential, leading to the formation of CF bands. By solving the CF band
structure at mean field level, we find that massless Dirac cones appear at
three $M$-points of the Brillouin zone. For the case of two flux quanta per
unit cell, these emergent Dirac cones cross the Fermi level, i.e. the CF Fermi
surface is transformed into three CF Fermi points by periodic field
modulation. The low energy physics of the resulting quantum critical state is
thus governed by $N_{f}=3$ QED3.

Crucial to our construction is the particle-hole symmetry of the physical
electrons at half filling of the Chern band, which protects the emergent Dirac
fermions of our QED3 phase from becoming gapped. We also require the
electronic Chern band to be sufficiently flat: A large enough bandwidth will
instead favor an ordinary Fermi liquid state at half filling. These two
conditions are simultaneously satisfied in two-dimensional Dirac electron
systems at charge neutrality, such as topological insulator (TI) surface
states, subject to a periodic magnetic field.

We further show that when the electronic particle-hole symmetry is broken with
a periodic electrostatic potential, the QED3 quantum critical phase gives way
to FCI phases with topological orders of $\nu=\pm 1/3$ Laughlin states,
consistent with earlier studies on FCI phase transitions [39]. We also find by
doping away from half filling a Jain sequence of further FCI phases. Dirac
materials in a periodic magnetic field are thus a simple and robust setup
realizing both QED3 and FCI phases from a half-filled flat Chern band.

Figure 1: (a) The schematic setup: Topological insulator surface under a
periodic magnetic field, forming a triangular vortex lattice in our case,
which could arise from proximity to type-II superconductors. The low-energy
theory consists of $3$ Dirac fermions interacting with photons (QED3). (b) The
Landau level dispersion for free Dirac electrons under a periodic magnetic
field where the zeroth Landau level is exactly flat.

_Setup._ We begin by considering the TI surface in a uniform magnetic field.
The surface of the TI supports a single “Dirac electron,” $\Psi$, with action

$\displaystyle
S_{\Psi}=\int_{t,\bm{x}}\,i\bar{\Psi}(\partial_{\mu}-iA_{\mu})\gamma^{\mu}\Psi\,,$
(1)

where $A_{\mu}$ is the background electromagnetic (EM) field,
$\gamma^{\mu}=(\sigma^{z},i\sigma^{x},i\sigma^{y})$ are the Dirac gamma
matrices, and we take the Dirac electrons’ velocity to be $v=1$. Throughout
the manuscript, we will use boldface to denote spatial vectors, as well as the
notation $\int_{t,\bm{x}}\equiv\int dt\,d^{2}\bm{x}$. We also work in units of
$\hbar=c=k_{B}=1$ unless otherwise noted. In the presence of a uniform
magnetic field, ${B_{0}=\bm{\nabla}\times\bm{A}_{(0)}}$,
$\bm{A}_{(0)}=\frac{B_{0}}{2}(y,-x)$, time-reversal symmetry ($\mathcal{T}$)
is broken while particle-hole symmetry ($\mathcal{PH}$) [40] remains intact
(see Supplemental Material). Consequently, the Dirac electrons form positive
and negative-energy Landau levels. At charge neutrality, the particle-hole
symmetry $\mathcal{PH}$ (which exchanges empty and filled states) guarantees
that the $n=0$ Landau level at zero energy is exactly at half filling for any
$B_{0}$.

It is well-known that in the presence of Coulomb interaction, the half-filled
$n=0$ Landau level is a strongly correlated metallic phase known as a
composite Fermi liquid (CFL) [19, 20, 41]. The existence of the CFL can be
explained using flux attachment [42, 43, 44, 45], in which each electron is
transmuted into a composite fermion via adiabatic attachment of two flux
quanta. Attaching flux screens the external magnetic field completely,
allowing the composite fermions to form a Fermi surface coupled to a
fluctuating U$(1)$ gauge field. Although the traditional flux attachment
procedure breaks $\mathcal{PH}$ – one must decide whether to attach flux to
electrons or holes – Son proposed a new CFL theory [46] where $\mathcal{PH}$
is manifest,

$\displaystyle
S_{\psi}=\int_{t,\bm{x}}\left[i\bar{\psi}(\partial_{\mu}-ia_{\mu})\gamma^{\mu}\psi+\frac{1}{4\pi}Ada\right]\,.$
(2)

Here the composite fermions, $\psi$, are Dirac fermions coupled to a
fluctuating U$(1)$ gauge field, $a_{\mu}$, and we have defined the emergent
magnetic field, $b_{*}=\bm{\nabla}\times\bm{a}$ . From the point of view of
the Dirac CFL, the physical electrons in the $n=0$ Landau level are double-
vortices of the emergent gauge field [47, 48, 49, 50, 51, 52, 53, 54],

$\displaystyle\rho_{e}=\Psi^{\dagger}\Psi\leftrightarrow\frac{1}{4\pi}\,b_{*}\,,\qquad
J^{i}_{e}=\bar{\Psi}\gamma^{i}\Psi\leftrightarrow\frac{1}{4\pi}\,\varepsilon^{ij}\,e_{j}\,,$
(3)

where $e_{i}=\partial_{i}a_{t}-\partial_{t}a_{i}$. Similarly, like in the well
known boson-vortex duality [55, 56, 57], the Dirac CFs are vortices of the
physical magnetic field: $a_{t}$ acts as a Lagrange multiplier fixing the CF
density, $\rho_{\psi}=\psi^{\dagger}\psi$,

$\displaystyle\rho_{\psi}=\psi^{\dagger}\psi=-\frac{1}{2}\frac{B}{2\pi}\,,$
(4)

meaning that a single CF corresponds to two flux quanta. The CFL thus has a
circular Fermi surface fixed by the external magnetic field, with Fermi wave
vector, ${k_{F}=\sqrt{|4\pi\,\rho_{\psi}|}=\sqrt{|B|}}$.

Importantly, the $\mathcal{PH}$ symmetry of the half-filled LL problem appears
as a time-reversal symmetry of the Dirac CFs [46], which feel no magnetic
field. For clarity, we will denote this symmetry as
$\mathcal{T}_{\mathrm{CF}}$ when discussing the action of physical
$\mathcal{PH}$ symmetry on Dirac CFs. The detailed symmetry action on the CFs
can be found in the Supplemental Material.

Figure 2: The evolution of the CF Fermi surface at unit filling of the
periodic potential. As the strength of the periodic modulation, $|B_{1}|$ is
increased, the CF bands fold at the Brillouin zone edge, and the original
circular Fermi surface is deformed. When $|B_{1}|/|B_{0}|\sim O(1)$, three
isolated Dirac cones emerge at the Fermi energy.

_Emergent $N_{f}=3$ QED3 from periodic magnetic field._ We proceed to consider
the case where the TI surface is subject to a periodic magnetic field. One way
such a setup can be established is by placing TI film on top of a type-II
superconductor, where an external magnetic field, $H$, induces an Abrikosov
vortex lattice, which in turn periodically modulates the magnetic field, $B$,
felt by the TI. However, in such a setup the flux per unit cell is fixed to
the superconducting flux quantum, $h/2e$, rather than the desired case of
$2h/e$. Alternatively, a periodic $B$-field can be induced by an array of
micromagnets [58, 59]. As shown by a recent study [60], in Dirac electron
systems under a periodic $B$-field, there remains a perfectly flat Chern band
at zero energy, which leads to competing FCI and Wigner crystal states at odd-
denominator filling fractions. Our work studies the case of half filling,
which corresponds to charge neutrality. It is important to note that the
particle-hole symmetry $\mathcal{PH}$ remains exact even when the magnetic
field is nonuniform.

We now consider a magnetic field with both uniform and spatially oscillating
components,

$\displaystyle B(\bm{x})$
$\displaystyle=B_{0}+2B_{1}\sum_{n=1}^{6}\cos\left(\bm{Q}_{n}\cdot\bm{x}\right)\,.$
(5)

Here the oscillatory component, which is a two-dimensional periodic function,
defines a triangular lattice with lattice constant $\mathfrak{a}$.
$\bm{Q}_{n}=\frac{4\pi}{\sqrt{3}\,\mathfrak{a}}(\sin(\frac{\pi(n-1)}{3}),\cos(\frac{\pi(n-1)}{3}))$,
$n=1,\dots,6$, are the reciprocal lattice vectors. The CF filling per
triangular lattice unit cell is equal to half the number of flux quanta per
unit cell, by Eq. (4).

In the Dirac CF variables, a slowly varying magnetic field leads to a slowly
varying CF density, $\rho_{\psi}(\bm{x})=-B(\bm{x})/4\pi$. Rather than
implement this identity as a constraint, it is convenient to instead enforce
this relation on average by introducing a scalar potential for the CFs of the
same symmetry as the magnetic field $B(\bm{x})$,

$\displaystyle S_{V}$
$\displaystyle=-\int_{t,\bm{x}}V_{\mathrm{CF}}(\bm{x})\,\psi^{\dagger}\psi(t,\bm{x})\,,$
$\displaystyle V_{\mathrm{CF}}(\bm{x})$
$\displaystyle=\mu_{0}+2V_{1}\sum_{n=1}^{6}\cos\left(\bm{Q}_{n}\cdot\bm{x}\right)\,.$
(6)

The coefficient, $V_{1}\propto\ell_{B_{0}}B_{1}/4\pi$,
$\ell_{B_{0}}=1/\sqrt{B_{0}}$, are self-consistently determined by enforcing
Eq. (4) for the mean density, $\langle\rho_{\psi}\rangle$ (see Supplemental
Material) 111This procedure is valid by the equivalence of thermodynamic
ensembles: We have passed from the canonical to the grand canonical
formulation of the CF theory..

The presence of the periodic potential $V_{\mathrm{CF}}(\bm{x})$ leads to band
folding. When the periodic potential in Eq. (6) is made sufficiently strong,
$B_{1}/B_{0}\sim\mathcal{O}(1)$, the mean field Dirac CF band structure
exhibits three Dirac cones connecting the first and second bands at $E>0$ (see
Fig. 2). These Dirac cones are degenerate and located at the $M$-points of the
Brillouin zone (BZ), consistent with the $C_{6}$ lattice rotation symmetry and
the CF time-reversal symmetry, $\mathcal{T}_{\mathrm{CF}}$. This CF band
structure is depicted in the top panel of Fig. 3, which is obtained by solving
the mean field CF Hamiltonian numerically (see Supplemental Material). Of
particular interest to us is the case of unit filling of the triangular
lattice, which we denote $f=1$. In this case, there are two flux quanta –
hence a single CF – at each triangular lattice unit cell. Consequently, the
Fermi level of the CFs is exactly at the Dirac points, i.e. the original CF
Fermi surface is transformed into three Fermi points by the periodic
modulation of magnetic field. Reintroducing gauge fluctuations, the theory
then finds itself in an exotic _quantum critical phase_ governed by QED3 with
$N_{f}=3$ fermion species.

Figure 3: The dispersion of the CFs under a periodic potential, Eq. (6), with
$V_{1}=2/(\sqrt{3}\mathfrak{a})$. (Top) When the electronic $\mathcal{PH}$
symmetry is enforced, three Dirac cones appear in the CF spectrum at the $M$
points of the BZ. At filling, $f=1$, the chemical potential (dashed line)
crosses each of the Dirac points. (Right) There are $3$ Dirac cones along the
plotted trajectory at the Fermi energy. (Bottom) Introducing a commensurate,
periodic scalar potential for the electrons (magnetic field for the CFs)
breaks $\mathcal{PH}$ and causes a gap to open.

We now explicitly calculate the CF dispersion near the BZ $M$-points to
leading order in the periodic potential [61]. Without loss of generality, we
focus on the points, $\pm\bm{M}_{1}=\pm(0,2\pi/(\sqrt{3}\mathfrak{a}))$. The
eigenspinors of the mean field CF Dirac Hamiltonian about these points are

$\displaystyle\xi_{+}(\bm{M}_{1}+\delta\bm{k})=\frac{1}{\sqrt{2}}\,\left(1,-e^{-i\frac{\sqrt{3}\mathfrak{a}}{2\pi}\delta
k_{x}}\right)\,,$ (7)
$\displaystyle\xi_{-}(-\bm{M}_{1}+\delta\bm{k})=\frac{1}{\sqrt{2}}\,\left(1,e^{i\frac{\sqrt{3}\mathfrak{a}}{2\pi}\delta
k_{x}}\right)\,,$ (8)

for small deviations of $\delta\bm{k}=(\delta k_{x},\delta k_{y})$ from
$\pm\bm{M}_{1}$. The periodic potential, Eq. (6), leads to scattering between
the states, $\xi_{\pm}$, as their momenta differ by $\bm{Q}_{1}$. In momentum
space, the term which connects these points is
$V_{1}\,\psi^{\dagger}(\bm{k})\,\psi(\bm{k}+\bm{Q}_{1})$. Since the
unperturbed eigenspinors of Dirac CFs at $\pm\bm{M}_{1}$ form a Kramers pair,
the scattering matrix element between them vanishes. In the vicinity of
$M_{1}$-point, the scattering matrix element depends linearly on
$\delta\bm{k}$ as

$\displaystyle
V_{1}\,\xi_{+}^{\dagger}(\delta\bm{k})\,\xi_{-}(\delta\bm{k})=i\frac{\sqrt{3}\,V_{1}\mathfrak{a}}{2\pi}\,\delta
k_{x}\,.$ (9)

The CF dispersion near the $M_{1}$-point is reconstructed due to scattering
off of the periodic potential. If we let
$\chi_{1}(\delta\bm{k})=c_{+}(\delta\bm{k})v_{+}(\delta\bm{k})+c_{-}(\delta\bm{k})v_{-}(\delta\bm{k})$,
$v_{\pm}=(\xi_{+}\pm\xi_{-})/\sqrt{2}$, be a generic spinor near the
$M_{1}$-point, we find a Dirac Hamiltonian,

$\displaystyle\mathcal{H}_{1}=\delta
k_{y}\,\chi_{1}^{\dagger}\tau^{x}\chi_{1}+\frac{\sqrt{3}V_{1}\mathfrak{a}}{2\pi}\delta
k_{x}\,\chi_{1}^{\dagger}\tau^{y}\chi_{1}+\dots\,,$ (10)

where $\tau^{x,y,z}$ are the Pauli matrices in the $v_{\pm}$ basis. The
corresponding results for the Dirac cones near the $M_{2}$ and $M_{3}$-points
can be obtained using this result by acting with $C_{3}$ rotations. Notice
that the band velocity is anisotropic and thus changes by a $C_{3}$ rotation
between each $M$-point.

Our mean field result can be substantiated by general considerations based on
symmetry. Because the mean field Hamiltonian involves a single flavor of Dirac
CF protected by $\mathcal{T}_{\mathrm{CF}}$ symmetry with
$\mathcal{T}^{2}_{\mathrm{CF}}=-\mathcal{I}$, the CF energy spectrum should be
gapless, and the total number of degenerate Dirac cones must be odd (as in the
ordinary case of TI surface states in the presence of time reversal symmetry).
By Kramers’ theorem, the existence of a Dirac point at momentum $\bm{k}$
implies the presence of a degenerate state at $-\bm{k}$. This leads to fermion
doubling unless the Dirac cone appears _only_ at BZ points that are left
invariant under $\mathcal{T}_{\mathrm{CF}}$, i.e. points satisfying
$\bm{k}=-\bm{k}\mod\bm{Q}_{n}$. For the triangular lattice, such points of the
BZ are the $\Gamma$-point and the three $M$-points. In particular, the $C_{3}$
symmetry relating the $M$-points implies that Dirac cones at the $M$-points
must all be degenerate. It is thus natural for the Dirac CF system discussed
here to form three Dirac cones at the $M$-points.

Introducing gauge fluctuations to the mean field result, Eq. (10), one obtains
at $f=1$ three Dirac fermions coupled to the U$(1)$ gauge field, $a_{\mu}$.
The band velocities of these Dirac fermions are anisotropic, each differing by
a $C_{3}$ rotation. However, the large-$N_{f}$ QED3 fixed point is stable to
both velocity anisotropy and differences in the velocity of each Dirac species
[62]. We therefore reasonably expect that the long wavelength theory has
emergent Lorentz invariance, and each $\chi_{I}$ fermion has the same velocity
$v\equiv 1$. Hence, $N_{f}=3$ QED3 arises as an effective theory at length
scales much greater than the period of the oscillatory magnetic field
$\mathfrak{a}$,

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle=\int_{t,\bm{x}}\left[\sum_{I=1}^{3}i\bar{\chi}_{I}(\partial_{\mu}-ia_{\mu})\gamma^{\mu}\chi_{I}+\frac{1}{4\pi}A^{\prime}da\right]\,,$
(11)

where again $\gamma^{\mu}=(\tau^{z},i\tau^{x},i\tau^{y})$, and we define
${A^{\prime}\equiv A_{\mathrm{total}}-A}$ to be a background probe field on
top of the original background field, $A_{\mu}$, that produces $B(\bm{x})$.
Notice that what started as a discrete $C_{3}$ symmetry in the UV has been
enhanced to an emergent SU$(3)$ flavor symmetry, $\chi_{I}\rightarrow
U_{IJ}\chi_{J}$, at low energies 222There is a possibility the theory in Eq.
(11) may be unstable to dynamical mass generation, leading to a trivial state
with finite correlation length. See the discussion in Ref. [63] for a review..
See Supplemental Material for a discussion of how discrete symmetries such as
parity and time-reversal act in the effective $N_{f}=3$ QED3 theory.

Furthermore, our mean field arguments do not depend on the detailed choice of
density-density interaction potential, $V_{\mathrm{int}}(\rho_{e})$, for the
Dirac electrons, which simply maps to a flux-flux interaction,
$V_{\mathrm{int}}(b_{*})$, in the dual theory. In particular, we note that
instantaneous Coulomb interactions are expected to be marginally irrelevant at
the QED3 fixed point [64], while a Lorentz invariant generalization of Coulomb
interactions is exactly marginal and leads to a line of fixed points with
self-dual properties [65, 66, 51, 53].

_QED 3 as a FCI plateau transition._ In ordinary quantum Hall systems,
incompressible FQH phases are achieved when the filling is tuned away from the
CFL state at $\nu=1/2$, with Hall conductivity set by the filling. In striking
contrast, here we show that incompressible states with fractional Hall
conductivity, known as fractional Chern insulator (FCI) states, can be induced
_fixed at half-filling_ for the electronic Chern band, by introducing a
periodic chemical potential.

Our QED3 state is protected by the electronic $\mathcal{PH}$ symmetry. When
$\mathcal{PH}$ is broken, this critical state can become unstable and
transition into a new state. By introducing a periodic electrostatic potential
to break $\mathcal{PH}$, we find the QED3 state gives way to FCI phases, while
maintaining half filling of the electronic Chern band. We consider a potential
that is commensurate with the vortex lattice,

$\displaystyle
V_{e}(\bm{x})=2\mu_{1}\sum_{n=1}^{6}\cos(\bm{Q}_{n}\cdot\bm{x})\,,$ (12)

which has zero spatial mean. It generates the singlet mass operator,
$m\Delta_{\mathrm{singlet}}=m\sum_{I}\bar{\chi}_{I}\chi_{I}$, with
${\operatorname{sgn}(m)=\operatorname{sgn}(\mu_{1})}$.

From the fermion-vortex duality, Eq. (3), we see that $V_{e}(\bm{x})$ sources
a periodic magnetic field felt by the CFs, $\overline{b_{*}}(\bm{x})$, which
we choose to write in terms of a vector potential,
${\overline{a}_{i}(\bm{x})=\varepsilon_{ij}\partial_{j}\overline{\phi}(\bm{x})}$,
where $\overline{\phi}(\bm{x})$ is a bounded, real-valued function with the
same periodicity as $\overline{a}$. We find numerically in the lower panel of
Fig. 3 that the combination of periodic potential and periodic gauge field
opens a gap in the mean field CF dispersion. In other words, the emergent
Dirac fermion in our QED3 state now acquires a finite mass.

The origin of gap opening can also be understood by a perturbative analysis.
We proceed analogously to the arguments leading to Eq. (10). For momenta near
$\bm{M}_{1}$, the relevant scattering process induced by the periodic gauge
field has wave vectors $\bm{Q}_{1}$, $\bm{Q}_{4}=-\bm{Q}_{1}$. Denoting the
amplitude of $\overline{\phi}(\bm{x})$ by $\Phi$ and using the matrix element
$\xi_{+}(0)\gamma^{t}\gamma^{x}\xi_{-}(0)=i$, we find that the mean field
Hamiltonian contains a Dirac mass term,

$\displaystyle\mathcal{H}_{1}^{\prime}$
$\displaystyle=\overline{a}_{x}(\bm{Q}_{1})\,\psi^{\dagger}(\bm{M}_{1})\gamma^{x}\psi(-\bm{M}_{1})+\mathrm{h.c.}$
(13)
$\displaystyle\propto\frac{\pi\Phi}{\mathfrak{a}}\,\chi_{1}^{\dagger}\,\tau^{z}\chi_{1}\,,$
(14)

where $\chi_{1}$ was defined above Eq. (10) and
$\operatorname{sgn}(\Phi)=\operatorname{sgn}(\mu_{1})$. The analogous
calculation can be readily performed for the other two species, yielding a
mass, $m\sum_{I}\bar{\chi}_{I}\chi_{I}$,
${\operatorname{sgn}(m)=\operatorname{sgn}(\mu_{1})}$.

When the $\chi_{I}$ fermions become massive, one obtains a CF Chern insulator
with Chern number $3\operatorname{sgn}(m)/2$. Integrating out the fermions and
gauge fluctuations, we see that the periodic scalar potential in Eq. (12)
tunes an FCI transition across which the Hall conductivity changes by

$\displaystyle\Delta\sigma_{xy}$ $\displaystyle=\frac{1}{3}\frac{e^{2}}{h}\,,$
(15)

with the two topological orders on each side of the transition corresponding
to the $\nu=\pm 1/3$ Laughlin states (see Supplemental Material).

(a)  (b)

Figure 4: Two ways to access FCI phases. (Left) By tuning a periodic scalar
potential with strength $\mu_{1}$ while remaining fixed at half filling, the
emergent Dirac fermions become massive, leading to FCI states. Note that in
this figure the Hall conductivity is that which would be measured on a single
TI surface. (Right) By doping away from $N_{f}=3$ QED3 with a uniform electron
density, $\rho_{e}$, and magnetic field $B^{\prime}=B-B_{0}$. In this case,
the emergent $N_{f}=3$ QED3 phase gives way to a sequence of incompressible
FCI states. The magenta segment on the $B$-axis denotes the CFL phase. The
green dot corresponds to the case of two flux quanta per unit cell, where the
quantum critical QED3 phase appears.

It is also of interest to consider scalar potentials that break the $C_{6}$
lattice symmetry by having different amplitudes, $\mu_{1}^{(n)}$, for each
$\bm{Q}_{n}$ vector. Such deformations generically introduce an octet of mass
operators of the form,
${\Delta^{b}_{\mathrm{octet}}=\sum_{I,J}\,\bar{\chi}_{I}\,t^{b}_{IJ}\,\chi_{J}}$,
where $t^{b}$, $b=1,\dots,8$, are the generators of $\mathrm{SU}(3)$. These
operators can tune the theory to (1) the FCI states described above, as well
as (2) a Chern insulator state with integer Hall response occurring when one
of the $\chi_{I}$ fermions receives a mass with opposite sign from the other
two, and finally (3) a $\mathcal{PH}$-preserving state where one of the
emergent Dirac fermions is gapless and the remaining two are gapped with
opposite sign.

_Doping QED 3: FCI Jain sequence._ We now consider finite doping from charge
neutrality — which breaks the electronic $\mathcal{PH}$ — and varying $B_{0}$
away from the case of two flux per unit cell. We find that our emergent QED3
theory gives way to a family of FCI states, in analogy with how the CFL is a
parent state for the celebrated Jain sequence of observed Abelian FQH phases.
At finite doping $\rho_{e}$, the $\chi_{I}$ fermions feel a uniform emergent
magnetic field, $b_{*}$, and form their own Landau levels (including the
zeroth Landau level). When an integer number of these Landau levels are
completely filled, an incompressible FCI appears. This occurs when the
$\chi_{I}$ fermions are at filling,

$\displaystyle\nu_{\chi}=2\pi\sum_{I}\frac{\langle\chi_{I}^{\dagger}\chi_{I}\rangle}{b_{*}}=3\left(q-\frac{1}{2}\right)\,,\qquad
q\in\mathbb{Z}\,.$ (16)

Here the $1/2$ term comes from the zeroth Landau level of the emergent Dirac
fermions in our QED3 state, and the factor of $3$ accounts for the three
flavors. From Eq. (11), we observe that the emergent Dirac fermion density is
set by the deviation from $B_{0}$, denoted
$B^{\prime}=\varepsilon_{ij}\partial_{i}A^{\prime}_{j}=-(4\pi)\sum_{I}\chi^{\dagger}_{I}\chi_{I}$,
and that the background charge density leads to an emergent magnetic field,
$\rho_{e}=b_{*}/4\pi$. Combining these relations with Eq. (16), we see by the
Streda formula that the resulting incompressible FCI phase has Hall
conductivity,

$\displaystyle\sigma_{xy}=2\pi\frac{d\rho_{e}}{dB^{\prime}}=-\frac{1}{12q-6}\frac{e^{2}}{h}.$
(17)

Here $\sigma_{xy}$ is the Hall conductivity associated with a single TI
surface, of which the contribution from the half-filled Chern band is
$\sigma^{0}_{xy}=\sigma_{xy}+1/2$ in units of $e^{2}/h$ (because Chern bands
at negative energies contribute $-1/2$). Comparing
$\sigma^{0}_{xy}=(3q-2)/(6q-3)$ with the Jain sequence,
$\sigma^{\mathrm{Jain}}_{xy}=p/(2p+1)$, we find that each of these fractions
lies on the principal Jain sequence, and has the same topological order as
their associated Jain states. Importantly, only the Jain sequence states
satisfying $p=3q-2$ appear on the FCI sequence in Eq. (17). One therefore
expects to measure a Landau fan of FCI states in a Chern band near half
filling, with slopes given by Eq. (17), as shown in Fig. 4.

_Universal properties of the emergent QED 3 state._ Among the most basic
universal properties of the $N_{f}=3$ QED3 theory in Eq. (11) are the scaling
dimensions of the mass operators, denoted $[\Delta_{\mathrm{singlet}}]$ and
$[\Delta_{\mathrm{octet}}]$, which determine the divergence of the correlation
length as the transition is crossed. For example, tuning the FCI transition
with a periodic scalar potential will lead to a diverging correlation length
exponent, $\xi\sim(\mu_{1})^{-\nu}$, with
${\nu=1/(3-[\Delta_{\mathrm{singlet}}])}$. The values of
$[\Delta_{\mathrm{singlet}}]$ and $[\Delta_{\mathrm{octet}}]$ can be
calculated in the large-$N_{f}$ expansion [67, 68, 62, 69], but the validity
of extending these results to small $N_{f}$ is unclear. Were one to establish
an experimental or numerical realization of our setup, it could be possible to
measure these exponents, either directly from the correlation functions of the
mass operators in numerics, or by measuring the scaling of the DC
conductivity, $\sigma_{xx}\sim f(\omega/T,\mu_{1}/T^{\nu z})$ with $z=1$ [70].

The DC conductivity at quantum phase transitions in 2d is another essential
piece of universal data. Because the EM current of the theory in Eq. (11) is
the emergent electric field, $J_{i}=\varepsilon_{ij}e_{j}/4\pi$,
$e_{i}=\partial_{i}a_{t}-\partial_{t}a_{i}$, one obtains for a rotationally
invariant system,

$\displaystyle\sigma\left(\omega\over
T\right)=\frac{i\omega}{(4\pi)^{2}}\,\Pi^{-1}\left(\omega\over T\right)\,,$
(18)

where $\Pi_{xx}=\Pi_{yy}\equiv\Pi$ is the polarization of the emergent gauge
field. Both the DC ($\omega/T\rightarrow 0$) and optical ($T/\omega\rightarrow
0$) conductivities are universal numbers of $\mathcal{O}(e^{2}/h)$, but they
do not necessarily take the same value [71]. They can again be calculated in
the large-$N_{f}$ limit, which gives [72]

$\displaystyle\sigma\left({\omega\over T}\rightarrow
0\right)\rightarrow\infty\,,\,\sigma\left({T\over\omega}\rightarrow
0\right)=\frac{2}{\pi N_{f}}\frac{e^{2}}{h}\,,$ (19)

suggesting that in the clean limit the theory may be a perfect conductor with
diverging DC conductivity and vanishing density of states. Note that to obtain
this result, one computes _the resistivity_ in the large-$N_{f}$ limit, using
${\Pi(\omega,\bm{q}=0)=iN_{f}\omega(\frac{1}{16}+\mathcal{O}(N_{f}^{-1}))}$
(for $T=0$), and then inverts the result to obtain the conductivity. This is
why the leading contribution to the conductivity goes as $1/N_{f}$.

Similarly, the electronic compressibility is given by the correlator of the
emergent magnetic field, since $\rho_{e}=b_{*}/4\pi$. Because of the scale
invariance of $N_{f}=3$ QED3, the static compressibility should vanish
linearly as $T\rightarrow 0$, $\kappa(T)\sim T$. This is in contrast to the
CFL phase, which has finite compressibility as $T\rightarrow 0$.

We finally comment that truly realistic systems have quenched disorder, which
will cause QED3 theories to run to a new fixed point governed by an interplay
of disorder and interactions [73, 74, 72, 75, 64]. In the $N_{f}=3$ theory,
$\mathcal{PH}$-symmetric disorder (a random magnetic field component for the
electrons) is exactly marginal, leading to a line of fixed points with varying
dynamical critical exponent, $z>1$. For generic disorder breaking
$\mathcal{PH}$, the theory runs to strong disorder and gives way to diffusion
($z=2$), leading to a quantum critical point governed by a gauged non-linear
sigma model (NLSM) [76, 77] which at mean field level resembles Pruisken’s
theory of the IQH transition [78, 79]. Determining the universal properties of
NLSMs of this type remains an important open problem.

_Discussion._ In this work, we have demonstrated that half-filled Landau
levels of Dirac electrons in spatially periodic magnetic fields can give rise
to exotic quantum critical states – namely, $N_{f}=3$ QED3 – without any fine
tuning. This critical state is protected by the particle-hole symmetry of
Dirac electron systems (such as TI surface or graphene) at charge neutrality.
More generally, an emergent particle-hole symmetry may be found in other Chern
band systems at half filling, as it is the case in half-filled Landau levels
at high magnetic fields. Thus, the exotic quantum critical physics described
in this work may be a more general feature of half-filled Chern bands.
Speaking more broadly, half-filled Chern bands provide ample opportunities for
realizing novel quantum states of matter and call for much further study.

_Acknowledgements._ We thank P.A. Lee, T. Senthil, A. Vishwanath, and J. Wang
for discussions. HG is especially grateful to R. Sohal for enlightening
conversations. This work is funded by the Air Force Office of Scientific
Research (AFOSR) under award FA9550-22-1-0432. HG and XYS were supported by
the Gordon and Betty Moore Foundation EPiQS Initiative through Grant No.
GBMF8684 at the Massachusetts Institute of Technology. LF is partly supported
by the David and Lucile Packard Foundation.

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## Supplemental Material

###  .1 Mean field CF band structure

Here we describe the numerical calculation of the mean-field CF band structure
in the presence of a a periodic modulation, Eq. (5). After mapping to the
Dirac CF picture, one has a Dirac cone lying at the $\Gamma$ point below the
Fermi energy. As described in the main text, the periodic magnetic field can
be viewed as a periodic CF chemical potential. Thus, at leading order, the
periodic magnetic field scatters the CFs with a momentum difference of the
reciprocal super-lattice vectors, $\bm{Q}_{n}$. In momentum space, the single
particle Hamiltonian is

$\displaystyle\mathcal{H}_{CF}=\int
d^{2}\bm{p}\,\bar{\psi}(\bm{p})\,(\gamma^{i}p_{i})\,\psi(\bm{p})+V_{1}\sum_{n=1}^{6}\psi^{\dagger}(\bm{p})\psi(\bm{p}+\bm{Q}_{n}).$
(20)

Figure 5: The dispersion of the CFs under a periodic potential, Eq. (6), with
$V_{1}=2/(\sqrt{3}\mathfrak{a})$ showing more bands and Dirac fermions at
$M,K$ points. Physical settings are identical to main text Fig 3.

For a particular $\bm{p}$ in the first Brillouin zone, the scattering process
involves states with momenta
$\bm{p},\bm{p}+\bm{Q}_{n},\bm{p}+2\bm{Q}_{n}\cdots$ and we take the momentum
cutoff to $\bm{p}+3\bm{Q}_{n}$ for the numerical diagonalization. Fig 6(a)
shows a dispersion for small $B_{1}$ which for CFs deforms the circular Fermi
surface into particle/hole pockets.

When solving for the mean field of CFs under an internal magnetic field
$b^{*}$ from chemical potential modulation on physical electrons, we take

$\displaystyle\overline{a}_{i}=\epsilon^{ij}\partial_{j}\phi(\bm{x}),\phi(\bm{x})=\phi_{0}\sum_{n=1}^{6}\cos(\bm{Q}_{n}\cdot\bm{x}).$
(21)

The vector potential $\overline{a}$ thus has nontrivial Fourier component at
reciprocal lattice vectors
$\overline{a}_{i}(Q_{k})=i\epsilon^{ij}Q_{k,j}\phi_{0}$ where $k=1\cdots
6,i,j=x,y$. This connects CFs states with momentum difference of $Q_{k}$, i.e.
resulting in additional terms

$\displaystyle\mathcal{H}_{a}=\sum_{n=1}^{6}\overline{a}_{i}(Q_{k})\overline{\psi}_{\bm{p}}\gamma^{i}\psi_{\bm{p}+\bm{Q}_{n}}\,.$
(22)

Figure 6: (a) The dispersion of CFs under exact diagonalization when $B_{1}$
is small compared with $B_{0}$. The dashed line indicates Fermi energy that
crosses bands, creating CF particle and hole pockets plotted in fig 2(a). (b)
An illustration of the integration in eq (30) near a particular intersection
point between circular Fermi surface and the Brillouin zone. The red shade
denotes momenta $\bm{p}$’s which scatter to $p-Q_{1}$ and give nonzero
contribution to the integral.

###  .2 Periodic potential of CFs from periodic magnetic field of physical
electrons

In order to perform our mean field analysis, it was necessary to pass to a CF
description with fixed chemical potential, rather than fixed density (i.e.
pass from the canonical to the grand canonical ensemble). Here we describe how
to calculate the profile of the needed CF chemical potential,
$V_{\mathrm{CF}}(\bm{x})$ in Eq. (6), induced by a periodic modulation in the
magnetic field. From dimensional analysis, it is natural to guess the leading
behavior,

$\displaystyle|V_{1}|\sim\frac{|B_{1}|}{\sqrt{|B_{0}|}}+\dots\,,$ (23)

where $V_{1}$ is defined in Eq. (6) as the amplitude of the oscillating part
of the CF chemical potential,

$\displaystyle
V_{\mathrm{CF}}(\bm{x})=\mu_{0}+V_{1}\sum_{n=1,\dots,6}e^{i\bm{Q}_{n}\cdot\bm{x}}\,.$
(24)

Indeed, Eq. (23) makes physical sense – the quantity has dimensions of energy
and vanishes as $B_{1}\rightarrow 0$ (the periodic potential turns off) or
$B_{0}\rightarrow\infty$ (the periodic modulation is small on the scale of the
Fermi wave vector).

The proportionality coefficient in Eq. (23) can be determined by enforcing
self-consistently,

$\displaystyle\langle\psi^{\dagger}\psi(\bm{x})\rangle_{V_{\mathrm{CF}}(\bm{x})}=\frac{B(\bm{x})}{4\pi}\,,$
(25)

working power-by-power in $B_{1}/B_{0}$. Wick rotating to imaginary time,
$t=-i\tau$, we single out the $\bm{Q}_{n}$ Fourier components of the density
one-point function,

$\displaystyle\langle\psi^{\dagger}\psi(\bm{x})\rangle$
$\displaystyle=\frac{1}{2\pi}\int
d^{2}\bm{x}\,\operatorname{Tr}[G_{\psi^{\dagger}\psi}(\bm{x},\bm{x},\tau\rightarrow
0)]e^{i\bm{Q}_{n}\cdot\bm{x}}=\frac{B(\bm{Q}_{n})}{4\pi}\,,\qquad
G_{\psi^{\dagger}\psi}(\bm{x},\bm{x},i\omega)=(i\omega+i\gamma_{0}\gamma^{i}\partial_{i}+V_{\mathrm{CF}}(\bm{x}))^{-1}.$
(26)

In the regime where $|V_{1}|\ll|\mu_{0}|$, i.e. $B_{1}\ll B_{0}$, we can
expand Eq. (26) with a small parameter $\frac{V_{1}}{\mu_{0}}$,

$\displaystyle G_{\psi^{\dagger}\psi}(\bm{x},i\omega)=G_{0}+\delta
G=G_{0}-G_{0}(V_{\mathrm{CF}}(\bm{x})-\mu_{0})G_{0}+\dots\,,\qquad
G_{0}=(i\omega+i\gamma_{0}\gamma^{i}\partial_{i}+\mu_{0})^{-1}\,.$ (27)

Keeping only the leading (one-loop) contribution, we obtain for the
oscillating part, $\delta G$,

$\displaystyle\delta G=-\int
d^{2}\bm{p}d\omega\operatorname{Tr}[G_{0}(\bm{p},i\omega)V_{1}G_{0}(\bm{p}+\bm{Q}_{n},i\omega)]=\frac{B_{1}(\bm{Q}_{n})}{4\pi}.$
(28)

We find it convenient to express the Green’s function in the eigen-energy
basis for each momentum $\bm{p}$, rewriting
$G_{0}(\bm{p},i\omega)=P(\bm{p})(i\omega+|p|+\mu_{0})^{-1}+(1-P(\bm{p}))(i\omega-|p|+\mu_{0})^{-1}$
in terms of the projection operator, $P(\bm{p})$, that project the Dirac
spinor onto the positive energy band. The frequency integral has the general
form,

$\displaystyle\int
d\omega\,(i\omega+\epsilon_{1})^{-1}(i\omega+\epsilon_{2})^{-1}=1-\Theta(-\epsilon_{1}\epsilon_{2})\frac{1}{|\epsilon_{1}-\epsilon_{2}|}\,.$
(29)

Here $\Theta(X)$ is the Heaviside step function, which tells us that the
negative energy states (well below the chemical potential) do not contribute.
Furthermore, for states with positive energy, we observe that only when
$\bm{p}$ and $p+\bm{Q}_{n}$ are above and below the Fermi surface, will the
integral in Eq. (28) be nonzero. The dominant contribution comes when the
energies at momenta $\bm{p},\bm{p}+\bm{Q}_{n}$ are close. For the specific
$\bm{Q}_{n}$ for the hexagonal Brillouin zone, one concludes that the integral
is most singular at the momenta where the original circular Fermi surface (for
integer filling, i.e. with the same area as the first Brillouin zone)
intersects the hexagonal BZ. Linearizing the deviation from one intersection
point as $\delta\bm{p}$, we have for the integral in Eq. (28) (see Fig. 6(b))

$\displaystyle\int d(\delta p)d(\delta
p_{\parallel})\frac{\xi_{\bm{p},\bm{p}+\bm{Q}_{n}}}{v_{\perp}\delta
p+v_{\parallel}\delta p_{\parallel}}V_{1}=\frac{B_{1}(\bm{Q}_{n})}{4\pi},$
(30)

where $\xi_{\bm{p},\bm{p}+\bm{Q}_{n}}=\operatorname{Tr}[P(p)P(p+\bm{Q}_{n})]$
and $\delta p_{\parallel}$ is the momentum projected along $\bm{Q}_{n}$ (not
to be confused with the momentum along the Fermi surface). The above integral
is regular and can be calculated numerically. However, since the regime of
interest corresponds to the case $B_{1}/B_{0},V_{1}/V_{0}\sim\mathcal{O}(1)$,
we do not expect the final coefficient, $C\sim\mathcal{O}(1)$ to be physically
meaningful. We therefore confirm the expectation from dimensional analysis and
estimate,

$\displaystyle C\sqrt{B_{0}}\,V_{1}=\frac{B_{1}(\bm{Q}_{n})}{4\pi}\,$ (31)

where the factor $\sqrt{B_{0}}$ comes in from the $|\bm{Q}_{n}|$ dependence on
the unit cell size and hence $B_{0}$, given that unit cell size $S$ satisfies
$B_{0}S=4\pi$. We have $|\bm{Q}_{n}|\propto\sqrt{B_{0}}$ and the expression
(30) scales as $|\bm{Q}_{n}|\sim\sqrt{B_{0}}$.

###  .3 Position of the emergent CF Dirac cones

We show that the emergent Dirac cones for CFs are pinned at certain high-
symmetry points - including $M_{i},K(K^{\prime})$ at BZ edge and $\Gamma$ at
BZ center.

Two threads of arguments are presented: From the anomaly of the CFs, i.e.
parity anomaly in $\mathcal{PH}\rtimes U(1)_{\psi}$, at mean-field level for a
fixed energy, the spectrum has to be either gapless or possess odd number of
Dirac cones to match the anomaly. For the Dirac cone case, note that
$\mathcal{PH}$ sends momenta $k\rightarrow-k$ due to its anti-unitary action.
Hence generically the Dirac cones come in pairs, except at high symmetry point
$M_{i}$’s since $\pm M_{i}$ differ by a reciprocal lattice vector and one
could have odd number of Dirac cones pinned at $M_{i}$’s. Another possibility
is a single Dirac cone at $\Gamma$, which is $\mathcal{PH}$ invariant. For the
filling $1$ for CFs focused in the main text, we have the case for Dirac cones
emerging at $M_{i}$’s.

Another argument follows from symmetry of the CF action: $\mathcal{PH}$ leaves
$M_{i},\Gamma$’s invariant and squares to $-1$. Hence there is a Kramer’s
degeneracy at $M_{i},\Gamma$’s.

To see the appearance of Dirac cones at $K,\tilde{K}=-K$ points, which in our
case occurs between the second and third Bloch bands in energy (along with the
Dirac cone at $\Gamma$ to cancel the parity anomaly), we consider two relevant
symmetries $C_{3},\mathcal{PH}\cdot\mathcal{P}$, with the action on the CFs

$\displaystyle C_{3}:\psi(r)\rightarrow
e^{-i\sigma^{z}\frac{\pi}{3}}\psi(C_{3}(r)),$
$\displaystyle\mathcal{PH}\cdot\mathcal{P}:\psi\rightarrow\sigma^{z}\psi.$
(32)

We show that acting on the eigenstates with momenta $K$ and its $C_{3}$
equivalents $K^{\prime},K^{\prime\prime}$, the two symmetries do not commute,
which implies degeneracy. The Bloch wavefunctions with positive energy for the
CFL action eq (2) are taken to be

$\displaystyle\psi_{K}:(1,e^{i\phi})^{T},\psi_{K^{\prime}}:C_{3}(\psi_{K})=(e^{-2i\phi},e^{i3\phi})^{T},$
$\displaystyle\psi_{K^{\prime\prime}}:-C_{3}^{-1}(\psi_{K})=(e^{-4i\phi},e^{i5\phi})^{T}.$
(33)

Under such basis, the transformation matrix for $C_{3},\mathcal{PH\cdot P}$
reads

$\displaystyle C_{3}:\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right),$ (37) $\displaystyle\mathcal{PH\cdot
P}:\left(\begin{array}[]{ccc}0&e^{2i\phi}&0\\\ e^{2i\phi}&0&0\\\
0&0&-e^{2i\phi}\end{array}\right).$ (41)

which do not commute. Hence the degeneracy at $K,\tilde{K}=-K$ are enforced.

###  .4 Symmetry action on the emergent Dirac cones

Here we specify how different symmetries act on the emergent Dirac fermions,
$\chi_{I}$, $I=1,2,3$, of the emergent QED3 theory. We start with the lattice
translations and $C_{3}$ rotations,

$\displaystyle T_{i}:\chi_{I}$ $\displaystyle\rightarrow
e^{i\bm{M}_{I}\cdot\bm{x}}\,\chi_{I},$ $\displaystyle C_{3}:\chi_{I}$
$\displaystyle\rightarrow
e^{i\gamma^{t}\frac{\pi}{3}}\,\chi_{(I+1)\operatorname{mod}3},$ (42)

Here $\bm{M}_{I}$ are the reciprocal lattice vectors corresponding to the
three $M$-points. As described in the main text, at long wavelengths the
$C_{3}$ rotation symmetry is enhanced to an emergent SU$(3)$ flavor symmetry.

The discrete symmetries of most interest to us are the anti-unitary
(electronic) particle-hole and parity symmetries, which we denote
$\mathcal{CT}$ (the product of charge conjugation and time-reversal) and
$\mathcal{CP}$,

$\displaystyle\mathcal{CT}:\chi_{I}$
$\displaystyle\rightarrow-i\tau^{y}\chi_{I}(-t,\bm{x})\,,\qquad(a_{0},a_{i})\rightarrow(a_{0},-a_{i})$
$\displaystyle\mathcal{CP}:\chi_{I}$
$\displaystyle\rightarrow\tau^{x}\chi_{s(I)}(t,-x,y)\,,\qquad(a_{0},a_{i})\rightarrow(a_{0},-a_{x},a_{y})\,,$
(43)

where $s(I)$ permutes the Dirac fermion species as $s(1)=1,s(I=2,3)=5-I$. The
emergent $N_{f}=3$ QED3 theory is invariant under these symmetries. Both are
broken by the introduction of a periodic scalar potential, i.e. the Dirac mass
operator.

###  .5 Topological orders of the FCI states

Here we consider the topological orders of the fractional Chern insulator
states considered in the main text. This requires introducing an auxiliary
gauge field, $b_{\mu}$, to the $N_{f}=3$ QED3 theory such that all Chern-
Simons terms are properly quantized and the theory is gauge invariant,

$\displaystyle
S=\int_{t,\bm{x}}\sum_{I=1}^{3}i\bar{\chi}_{I}(\partial_{\mu}-ia_{\mu})\gamma^{\mu}\chi_{I}-\frac{1}{8\pi}ada-\frac{2}{4\pi}bdb+\frac{1}{2\pi}bd(a+A^{\prime})-\frac{1}{8\pi}A^{\prime}dA^{\prime}\,.$
(44)

Note here that $a_{\mu}$ is a $\mathrm{spin}_{c}$ connection, while $b_{\mu}$
is an ordinary U$(1)$ connection. See e.g. Refs. [49, 80] for a more detailed
discussion of the difference between $\mathrm{spin}_{c}$ and U$(1)$
connections. For our needs, the only important consequence of this distinction
is that anyons associated with $a_{\mu}$ will have their statistics shifted by
$\pi$, since spinc connections couple to fermions. Note also that we continue
to consider the theory on the surface of a TI, the bulk of which gives rise to
the final $A^{\prime}dA^{\prime}/8\pi$ term.

One can easily recover the action discussed in the main text, Eq. (11), by
integrating out $b_{\mu}$, meaning that the local physics of the two theories
is equivalent. However, unlike Eq. (11), this theory is gauge invariant on any
manifold. This is crucial for correctly diagnosing the topological order on
integrating out the $\chi_{I}$ fields deep in a FCI phase.

We begin by considering the effect of a singlet mass term,
$\mathcal{L}_{\mathrm{mass}}=-m\sum_{I}\bar{\chi}_{I}\chi_{I}$, which
generated by the periodic scalar potential in Eq. (12). This deformation
causes the $\chi_{I}$ fermions to form Chern insulators, each with Chern
number $\operatorname{sgn}(m)/2$. The resulting state is a FCI described by
the topological quantum field theory (TQFT) with Lagrangian,

$\displaystyle\mathcal{L}_{\mathrm{FCI}}$
$\displaystyle=\frac{3\operatorname{sgn}(m)-1}{2}\frac{1}{4\pi}ada-\frac{2}{4\pi}bdb+\frac{1}{2\pi}bd(a+A^{\prime})-\frac{1}{8\pi}A^{\prime}dA^{\prime}\,.$
(45)

The Hall conductivity of this state can be calculated by integrating out both
gauge fields,

$\displaystyle\mathcal{L}_{\mathrm{response}}$
$\displaystyle=\left(\frac{1}{2}-\frac{\operatorname{sgn}(m)}{6}\right)\frac{1}{4\pi}A^{\prime}dA^{\prime}-\frac{1}{8\pi}A^{\prime}dA^{\prime}\,,$
(46)

where we have separated out the bulk TI contribution. Indeed, the first term
has coefficient $1/3$ ($m>0$) or $2/3$ ($m<0$), suggesting that these states
are have the topological order of the $\nu=\pm 1/3$ Laughlin states. We can
furthermore calculate the ground state degeneracy on the torus, which is
$|\det K|=3$ (where $K$ is the $(3\operatorname{sgn}(m)/2-1/2,-2,1)$
$K$-matrix corresponding to Eq. (45), in agreement with this expectation.

The equivalence can be established more concretely at the level of the TQFT.
We start with the case $m>0$. In this case, the first term in Eq. (45) is a
trivial TQFT, $ada/4\pi$. Indeed, in this case we can integrate out $a$ in a
gauge invariant way to obtain,

$\displaystyle\mathcal{L}_{m>0}=-\frac{3}{4\pi}bdb+\frac{1}{2\pi}bdA^{\prime}-\frac{1}{8\pi}A^{\prime}dA^{\prime}\,.$
(47)

This is of course the usual Laughlin U$(1)_{-3}$ state, with the same
charge-$1/3$ anyonic quasiparticles as the usual Laughlin FQH state. Notice
that the fluctuating gauge field here is an ordinary U$(1)$ connection, as the
spinc connection has been eliminated.

We now consider the case of $m<0$. Here we cannot immediately integrate out
$a_{\mu}$, but we can nevertheless argue its equivalence to a Laughlin state.
We start by using a trick to simplify the TQFT (which in this case
incorporates the same physics as the more rigorous notion of level-rank
duality; see Refs. [53, 80] for an introduction accessible to condensed matter
physicists). We start by introducing a background composite fermion world-
line, $j^{\mu}_{f}$, coupling to $a_{\mu}$. Because $j^{\mu}_{f}$ describes a
fermion, we can use flux attachment to rewrite it in terms of a bosonic world-
line variable, $J^{\mu}$, with a single flux attached,

$\displaystyle j_{f}^{\mu}a_{\mu}\rightarrow
J^{\mu}\,\beta_{\mu}+\frac{1}{4\pi}(\beta+a)d(\beta+a)\,,$ (48)

where $\beta_{\mu}$ is a new U$(1)$ gauge field implementing the flux
attachment. Its role is to explicitly implement the aforementioned shift of
the anyon statistics by $\pi$ associated with the fact that $a_{\mu}$ is a
spinc connection. Therefore, rather than working with a TQFT where one probes
with fermionic lines, we can equivalently work with a different representation
of the TQFT where all of the probes are bosonic. We therefore arrive at an
equivalent TQFT,

$\displaystyle\mathcal{L}_{m<0}$ $\displaystyle=\frac{1}{4\pi}\beta
d\beta+\frac{1}{2\pi}\beta
da-\frac{1}{4\pi}ada-\frac{2}{4\pi}bdb+\frac{1}{2\pi}bd(a+A^{\prime})-\frac{1}{8\pi}A^{\prime}dA^{\prime}\,,$
(49)

where we have dropped the dependence on the probe, $J^{\mu}$. Now again the
theory involves a trivial TQFT for $a_{\mu}$, and we can integrate it out in
the same way as before to obtain,

$\displaystyle\mathcal{L}_{m<0}$ $\displaystyle=\frac{2}{4\pi}\beta
d\beta+\frac{1}{2\pi}bd(\beta+A^{\prime})-\frac{1}{4\pi}bdb-\frac{1}{8\pi}A^{\prime}dA^{\prime}\,,$
(50)

which, incidentally, is the TQFT we would have found had we invoked the level-
rank duality between U$(1)_{-2}$ (with a spinc connection) and U$(1)_{2}$
(with an ordinary U$(1)$ connection). We can integrate out $b_{\mu}$ to obtain
the properly quantized, single-component TQFT,

$\displaystyle\mathcal{L}_{m<0}$ $\displaystyle=\frac{3}{4\pi}\beta
d\beta+\frac{1}{2\pi}\beta dA^{\prime}+\frac{1}{8\pi}A^{\prime}dA^{\prime}\,.$
(51)

From here, one can easily confirm that this theory has the correct Hall
conductivity and ground state degeneracy. We therefore observe that the $m<0$
state is equivalent to a U$(1)_{3}$ TQFT, which is the particle-hole conjugate
of the U$(1)_{-3}$ state found for $m>0$.

In the case of the FCI sequence accessed by filling $q\in\mathbb{Z}$ Landau
levels of the $\chi_{I}$ fermions, the TQFT one obtains is

$\displaystyle\mathcal{L}_{\textrm{FCI Jain}}$
$\displaystyle=\frac{3q-2}{4\pi}ada-\frac{2}{4\pi}bdb+\frac{1}{2\pi}bd(a+A^{\prime})-\frac{1}{8\pi}A^{\prime}dA^{\prime}\,.$
(52)

Notice that the $q=0$ and $q=1$ states are the same as the states we found by
turning on mass operators with $m<0$ and $m>0$ respectively. This general
sequence of states have $K$-matrices that each correspond to a state on the
principal Jain sequence. Indeed, each Jain state can be described by a
$2\times 2$ $K$-matrix with label $(k,-2,1)$, $k\in\mathbb{Z}$, where the
gauge field with level $k$ is a spinc connection (i.e. couples to composite
fermions). The anyon content of each of these states can be found discussed in
numerous references, e.g. Refs. [45, 81].