††thanks: All the authors have contributed equally to this work.††thanks: All
the authors have contributed equally to this work.††thanks: All the authors
have contributed equally to this work.

# Gapping fragile topological bands by interactions

Ari M. Turner  Physics Department, Technion, Haifa 320003, Israel  Erez Berg
Department of Condensed Matter Physics, Weizmann Institute of Science Rehovot
7610001, Israel  Ady Stern  Department of Condensed Matter Physics, Weizmann
Institute of Science Rehovot 7610001, Israel

###### Abstract

We consider the stability of fragile topological bands protected by space-time
inversion symmetry in the presence of strong electron-electron interactions.
At the single-particle level, the topological nature of the bands prevents the
opening of a gap between them. In contrast, we show that when the fragile
bands are half filled, interactions can open a gap in the many-body spectrum
without breaking any symmetry or mixing degrees of freedom from remote bands.
Furthermore, the resulting ground state is not topologically ordered. Thus, a
fragile topological band structure does not present an obstruction to forming
a “featureless insulator” ground state. Our construction relies on the
formation of fermionic bound states of two electrons and one hole, known as
“trions”. The trions form a band whose coupling to the electronic band enables
the gap opening. This result may be relevant to the gapped state indicated by
recent experiments in magic angle twisted bilayer graphene at charge
neutrality.

_Introduction.–_ Topology has been recognized as a key player in condensed
matter physics over the last few decades. Much work has been devoted to
classifying the topological properties of single-particle band structures
Schnyder  _et al._ (2008); Kitaev (2009); Hasan and Kane (2010); Qi and Zhang
(2011); Fu (2011); Kruthoff  _et al._ (2017); Bradlyn  _et al._ (2017); Po
_et al._ (2017). However, it is always important to ask whether the
topological structure of the single-particle states survives in the presence
of strong electronic correlations Fidkowski and Kitaev (2010, 2011); Turner
_et al._ (2011); Fidkowski  _et al._ (2013); Kimchi  _et al._ (2013); Morimoto
_et al._ (2015); Queiroz  _et al._ (2016); Lee  _et al._ (2018).

In this work, we examine the stability of “fragile topological bands” Po  _et
al._ (2018a); Zou  _et al._ (2018); Song  _et al._ (2019); Po  _et al._ (2019)
in the presence of electron-electron interactions. A set of bands possesses
fragile topology if the subspace of their states cannot be spanned by a
complete basis of localized Wannier functions that preserves all the
symmetries of the system, while a localized Wannier description exists upon
adding topologically trivial bands Po  _et al._ (2018b); Ahn  _et al._ (2019);
Bouhon  _et al._ (2019); Bradlyn  _et al._ (2019). Fragile topology arises in
the nearly-flat bands in magic angle twisted bilayer graphene (MATBG) Lopes
dos Santos  _et al._ (2007); Bistritzer and MacDonald (2011); Cao  _et al._
(2018a, b).

Physically, fragile topology implies that without interactions, the set of
bands cannot be separated from one another by an energy gap in the entire
Brillouin zone (BZ) unless a symmetry is broken. Here, we find that in the
presence of interactions, the system can become gapped even in the absence of
any symmetry breaking or mixing with remote bands. Unlike interaction-induced
gapping that arises elsewhere Wang  _et al._ (2013); Bonderson  _et al._
(2013); Burnell  _et al._ (2014); Metlitski  _et al._ (2015); Mross  _et al._
(2016), here the gapped ground state is not required to have topological
order.

We provide a proof of principle for such a “featureless insulator” by
introducing an interaction that uses states from the fragile bands to create a
band of charged excitations composed of two electrons and one hole, and then
coupling these excitations to the electronic band to open a gap in the many-
body spectrum. The gap opens when the electron density in the system
corresponds to an integer number of electrons per site, as required by the
generalized Lieb-Schulz-Mattis theorem Lieb  _et al._ (1961); Oshikawa (2000);
Hastings (2004). Our mechanism is illustrated in two models that possess
fragile topological bands protected by space-time inversion symmetry
$C_{2}\mathcal{T}_{+}$. The first model is defined on a square lattice and the
second on a honeycomb lattice. The second model has the same symmetries and
Wannier obstructions as MATBG Zou  _et al._ (2018).

Figure 1: (a,b) Winding of the vector $(h_{1}(\bm{k}),h_{2}(\bm{k}))$ of the
effective low-energy Hamiltonian [Eq. (3)] around the Dirac points. (a) For
$\Delta_{x}>0$ and $\Delta_{xz}=0$ [Eq. (2)], there are two Dirac points, both
with winding number $+1$. (b) For $\Delta_{x}=-\Delta_{xz}$, the two Dirac
points merge into a single quadratic band touching at $\bm{k}=0$ with a
winding number of $+2$. (c) Spectrum of the three-band Hamiltonian (4). When a
third band is added, the quadratic band touching can be gapped, separating the
lowest band from the other two. In our construction, the additional band
consists of trion bound states. (d) Bloch wavefunction of the lowest band in
panel (c). The wavefunction can be chosen to be real. The in-plane components
represent the amplitudes of the two original electronic bands, and the out of
plane component is the amplitude of the third (trion) band.

Model for fragile topological bands.– The first model starts from two pairs of
$C_{2}$–symmetric, spinless Chern bands of opposite Chern numbers 111We thank
B. A. Bernevig, R. Queiroz, and Z. Song for explaining this model to us.. The
model has four orbitals on each site of a square lattice. The Hamiltonian is

$H_{\rm{CB}}({\bm{k}})=d_{z}({\bm{k}})\sigma_{z}+d_{x}({\bm{k}})\sigma_{x}\tau_{z}+d_{y}({\bm{k}})\sigma_{y}.$
(1)

Here, the Pauli matrices $\sigma,\tau$ act on the four-dimensional Hilbert
space of each site, $d_{z}=m+t_{1}\sin^{2}(k_{x}/2)+t_{1}\sin^{2}(k_{y}/2)$,
$d_{x}=t_{2}\sin(k_{x})$, $d_{y}=t_{2}\sin(k_{y})$, and $\bm{k}$ is measured
in units of the inverse lattice spacing. For every $\bm{k}$, each band of
$H_{\rm{CB}}$ contains two degenerate states, labeled by $\tau_{z}=\pm 1$. For
positive $t_{1}$ and $-t_{1}<m<0$, the Chern numbers of the bands are
$C=\pm\tau_{z}$.

The Hamiltonian (1) possesses an anti-unitary symmetry
$\mathcal{T}_{+}=K\tau_{x}\times(\bm{k}\rightarrow-\bm{k})$ (where $K$ is
complex conjugation, and $\mathcal{T}_{+}^{2}=1$), as well as a unitary
inversion symmetry $C_{2}=\sigma_{z}\times(\bm{k}\rightarrow-\bm{k})$, where
$(\bm{k}\rightarrow-\bm{k})$ signifies a change in the sign of the momentum,
and a mirror symmetry $M_{y}=\tau_{x}\times(k_{x}\rightarrow-k_{x})$.

The crucial symmetry is the product $C_{2}{\mathcal{T}}_{+}$, which leaves the
momentum unchanged, and in momentum space may be written as
$C_{2}\mathcal{T}_{+}=K\tau_{x}\sigma_{z}$. We focus on the pair of low-energy
bands, denoting their two degenerate Bloch states by
$|\psi_{\pm}(\bm{k})\rangle$ ($\pm$ denotes the eigenvalue of $\tau_{z}$).
These states may be chosen such that
$|\psi_{-}(\bm{k})\rangle=C_{2}\mathcal{T}_{+}|\psi_{+}(\bm{k})\rangle$.
Explicitly, we choose
${\left|\psi_{+}(\bm{k})\right\rangle}=\left(\cos\frac{\theta_{\bm{k}}}{2},\sin\frac{\theta_{\bm{k}}}{2}e^{i\phi_{\bm{k}}}\right)^{T}\otimes{\left|\tau^{z}=1\right\rangle}$,
where $\cos\theta_{\bm{k}}=-d_{z}({\bm{k}})/|\bm{d}(\bm{k})|$ and
$e^{i\phi_{\bm{k}}}=-(d_{x}+id_{y})/|d_{x}+id_{y}|$.

We add to (1) two terms,

$\delta H=\Delta_{x}\tau_{x}+\Delta_{xz}\tau_{x}\sigma_{z}.$ (2)

The essential features of $H_{\rm{CB}}+\delta H$ can be understood at the
level of first-order degenerate perturbation theory in $\delta H$. We define
Pauli matrices $\eta_{x,y,z}$ that act in the subspace spanned by
${\left|\psi_{\pm}\right\rangle}$, such that
$\eta^{z}{\left|\psi_{\pm}\right\rangle}=\pm{\left|\psi_{\pm}\right\rangle}$
and $\eta^{x}{\left|\psi_{\pm}\right\rangle}={\left|\psi_{\mp}\right\rangle}$.
The symmetry $C_{2}\mathcal{T}_{+}$ acts in this subspace as $\eta_{x}K$.
Hence, the effective Hamiltonian projected to the two lowest-energy bands is

$H_{\rm{eff}}(\bm{k})=h_{0}(\bm{k})+h_{1}(\bm{k})\eta_{x}+h_{2}(\bm{k})\eta_{y},$
(3)

where $h_{0}={\left\langle\psi_{+}\right|}(H_{\rm{CB}}+\delta
H){\left|\psi_{+}\right\rangle}$,
$h_{1}-ih_{2}={\left\langle\psi_{+}\right|}\delta
H{\left|\psi_{-}\right\rangle}$. A third term proportional to $\eta_{z}$ is
ruled out by $C_{2}\mathcal{T}_{+}$.

The eigenstates of Eq. (3) are determined by the direction of the planar
vector field $(h_{1}(\bm{k}),h_{2}(\bm{k}))$. The two bands touch when
$h_{1}(\bm{k})=h_{2}(\bm{k})=0$. Each band touching can be characterized by
the winding of the phase $\alpha_{\bm{k}}\equiv\arg(h_{1}+ih_{2})$ as $\bm{k}$
is taken around the band touching point. A map of $\alpha_{\bm{k}}$ vs.
$\bm{k}$ is shown in Fig. 1a for $\Delta_{x}>0$ and $\Delta_{xz}=0$, for which
there are two Dirac points in the BZ. The two Dirac points have the same
winding number, $+1$. Consequently, even when the Dirac points coincide at
$\bm{k}=0$ (for $\Delta_{x}=-\Delta_{xz}$), they do not annihilate each other
to form a gap; rather, they form a quadratic band touching with a winding
number of $+2$ (see Fig. 1b). The fact that the total winding number of all
band touchings is non-zero indicates that the two bands possess fragile
topology Po  _et al._ (2018b).

A nonzero net winding number sounds impossible at first, since the total
winding number around all the singular points of $\alpha_{\bm{k}}$ in the BZ
must vanish due to its periodicity. However, there are additional singular
points that are not associated with band touchings. Such points occur at
momenta for which the basis wave functions
${\left|\psi_{\pm}(\bm{k})\right\rangle}$ are singular. The existence of such
singularities is a consequence of ${\left|\psi_{\pm}(\bm{k})\right\rangle}$
spanning Chern bands. For our choice of gauge,
${\left|\psi_{\pm}\right\rangle}$ are singular at $\bm{k}=(\pi,0)$, $(0,\pi)$,
$(\pi,\pi)$, and $\alpha_{\bm{k}}$ winds around these points (see Fig. 1a,b).

Interaction-induced insulating states.– Physically, the fragile topological
character of the lowest-energy bands implies that as long as the chemical
potential lies within these bands, the system must be a (semi-)metal.
Interactions between electrons can open a gap at a filling of one electron per
unit cell, either by spontaneously breaking the $C_{2}\mathcal{T}_{+}$ Xie and
MacDonald (2020) or translational Kang and Vafek (2020) symmetry, or by mixing
remote bands Xie and MacDonald (2020).

Here, we ask whether a gap can form even without breaking any symmetry or
involving degrees of freedom beyond those of the two fragile bands.
Furthermore, can such an insulating state have a unique ground state on a
torus (a “featureless insulator”), or does it have to possess topological
order?

Explicitly, we consider the Hamiltonian (3) for the two lowest bands at a
filling of one electron per site, and add arbitrary finite-range interactions.
We ask whether there is an insulating ground state that preserves
$C_{2}\mathcal{T}_{+}$, translation, and charge conservation symmetries. We
allow modifications of the band structure that preserves these symmetries.
Using this freedom, we shift the two Dirac points to the $\Gamma$-point by
choosing $\Delta_{xz}=-\Delta_{x}$. Furthermore, $h_{0}({\bm{k}})$ is chosen
such that there is no $\bm{k}\neq 0$ whose energy $\varepsilon_{\bm{k}}$
equals $\varepsilon_{\bm{k}=0}$.

Our affirmative answer to the question above relies on the following strategy.
We note that the band touching at $\Gamma$ is protected only as long as there
are no additional bands to mix with. Consequently, a gap may open if the
interactions effectively generate another band with low-energy states near
$\Gamma$, consisting of degrees of freedom from other parts of the BZ. Such a
band can be formed when the interactions produce “trions” Kheng  _et al._
(1993); Finkelstein  _et al._ (1995); Huard  _et al._ (2000), namely bound
states of two electrons and a hole whose momenta add up to the $\Gamma$ point.
Trion bound states, known to occur in certain semiconductors, are fermionic,
charge-$e$ states. By hybridizing the trions with electronic states near
$\bm{k}=0$, we show that a gap in the spectrum can open without breaking any
symmetry and without affecting the structure of the filled band away from the
$\Gamma$ point.

Opening a gap by hybridization with an additional band.– We now show that the
presence of a third trivial band near $\Gamma$ allows for the desired energy
gap. The third band is taken to have a parabolic dispersion,
$\varepsilon_{\rm{t}}(\bm{k})=-\varepsilon_{\rm{t},0}+\frac{k^{2}}{2m_{\rm{t}}}$,
where $\varepsilon_{t,0}>0$. The subscript ‘t’ stands for the third band, but
also anticipates the trions. We perform a unitary transformation on the two
fragile bands, $U=e^{i\pi\eta_{y}/4}e^{i\pi\eta_{x}/4}$. Then,
$C_{2}\mathcal{T}$ changes to $U(\eta_{x}K)U^{\dagger}=iK$, and the
Hamiltonian is real. Near $\bm{k}=0$, it is

$H_{\rm{3-bands}}(\bm{k})=\left(\begin{array}[]{ccc}\frac{k_{y}^{2}-k_{x}^{2}}{2m}&-\frac{k_{x}k_{y}}{m}&g_{1}(\bm{k})\\\
-\frac{k_{x}k_{y}}{m}&\frac{k_{x}^{2}-k_{y}^{2}}{2m}&g_{2}(\bm{k})\\\
g_{1}(\bm{k})&g_{2}(\bm{k})&\frac{k^{2}}{2m_{\rm{t}}}-\varepsilon_{\rm{t},0}\end{array}\right).$
(4)

Here, the upper left $2\times 2$ block gives the uncoupled quadratic band
touching at $\bm{k}=0$, with $m\propto 1/\Delta_{x}$. Choosing
$(g_{1},g_{2})\propto(k_{x},k_{y})$ hybridizes the third band with the lowest
of the two original bands, separating them by a gap SI . The resulting
spectrum is shown in Fig. 1c. This mechanism of opening a gap is related to
the ones discussed in Refs. Wu  _et al._ (2019); Ahn  _et al._ (2019); Bouhon
_et al._ (2020); Kang and Vafek (2020).

The eigenstates of (4) may be chosen to be real. As such, they correspond to
unit vectors in three dimensions. For $g_{1}=g_{2}=0$, the vectors that
correspond to each of the two original bands lie in the $x-y$ plane, and form
a vortex structure around $\Gamma$. The limit ${\bm{k}}\rightarrow 0$ is then
singular. This enforces a band touching between the two original bands. The
hybridization with the third band allows the unit vector that corresponds to
the wave function to acquire a $z$–component, and turns the vortex into a
meron (Fig. 1d). Importantly, the hybridization changes the lowest-energy band
only at small momenta, and leaves both the energies and the wave functions
largely unaffected far from $\Gamma$ 222Such a gapping mechanism is impossible
near a single Dirac point of the original bands; in that case, the lowest
energy band has a Berry phase of $\pi$ going around the Dirac point, which
enforces a band touching between that band and the bands above it..

Formation of a trion band.– Returning to our original problem with two
isolated bands [Eq. (3)], we now show that interactions can drive the system
into an insulating phase without breaking any symmetry. This is done through
the formation of a low-momentum band of trions. The trions provide the
additional fermionic degree of freedom near $\Gamma$ needed to gap out the
quadratic band touching of the two original bands. Importantly, within our
construction, the trions and the electrons near the quadratic band touching
are independent degrees of freedom, since the trions are made of states whose
momenta are far from the quadratic band touching.

To create the trions, imagine deforming the single-particle dispersion to
include two local minima in the conduction band at $\bm{k}_{1,2}$ away from
$\bm{k}=0$, and a local maximum in the valence band at
$\bm{k}_{\rm{h}}=\bm{k}_{1}+\bm{k}_{2}$ (see Fig. 2a). The dispersions close
to $\bm{k}_{1},\bm{k}_{2},\bm{k}_{\rm{h}}$ are quadratic, and are taken to be
$\varepsilon_{i,\bm{k}}=\frac{(\bm{k}-\bm{k}_{i})^{2}}{2m_{\rm{e}}}+\varepsilon_{\rm{e},0}$
(where $i=1,2$) and
$\varepsilon_{\mathrm{h},\bm{k}}=-\frac{(\bm{k}-\bm{k}_{\rm{h}})^{2}}{2m_{\rm{h}}}-\varepsilon_{\rm{h},0}$.
Here, $m_{\rm{e,h}}$ are the effective masses of the electron and hole
valleys, and $\varepsilon_{\rm{e},0},\varepsilon_{\rm{h},0}>0$ are their
energy offsets relative to the Fermi energy (Fig. 2b).

Figure 2: (a) To facilitate the formation of trion bound states near
$\bm{k}=0$, we first deform the single-particle dispersion to contain two
electron valleys centered at $\bm{k}_{1,2}$, and a hole valley at
$\bm{k}_{\rm{h}}=\bm{k}_{1}+\bm{k}_{2}$. (b) Schematic cut of
$\varepsilon_{\bm{k}}$ on a trajectory
$\Gamma\rightarrow\bm{k}_{1}\rightarrow\bm{k}_{\rm{h}}\rightarrow\bm{k}_{2}$ .
The trions form due to interactions between the two electron valleys and the
hole valley. The minimum of the trion dispersion is at $\bm{k}=0$ (dashed blue
line).

We choose the interaction in the vicinity of the three valleys to couple the
electrons only to the holes, $V_{\rm{eff}}=-u_{0}\sum_{i=1,2}\int
d^{2}r\,\psi^{\dagger}_{{\mathrm{e}},i}\psi^{\dagger}_{\rm{h}}\psi^{\vphantom{\dagger}}_{\rm{h}}\psi^{\vphantom{\dagger}}_{{\mathrm{e}},i}$,
where $\psi_{{\mathrm{e}},i}^{\dagger}(\bm{r})$ creates an electron in a
wavepacket centered at $\bm{r}$ whose momenta are localized near $\bm{k}_{i}$,
$\psi_{\rm{h}}^{\dagger}(\bm{r})$ similarly creates a hole in the valence band
near $\bm{k}_{\rm{h}}$, and $u_{0}>0$ is a coupling constant.

This interaction creates bound states, excitons and trions. The excitons are
composed of one electron and one hole, carrying a non-zero momentum near
$\bm{k}_{i}+\bm{k}_{\rm{h}}$, with binding energy $\varepsilon_{\rm{b}}$ 333In
$d=2$, any attractive interaction between electrons and holes yields an
exciton bound state.. The bottom of the excitation band is at energy
$\varepsilon_{\rm{ex}}=\varepsilon_{\rm{e},0}+\varepsilon_{\rm{h},0}-\varepsilon_{\rm{b}}$;
to avoid the excitons from condensing and breaking translational symmetry, we
require $\varepsilon_{\rm{ex}}>0$. The trion is composed of two electrons and
one hole, and carries a momentum near zero. Setting, for simplicity,
$m_{\rm{h}}\gg m_{\rm{e}}$, the bottom of the trion band is at
$-\varepsilon_{\rm{t},0}=2\varepsilon_{\rm{e},0}+\varepsilon_{\rm{h},0}-2\varepsilon_{\rm{b}}$.
The combination of the two requirements $\varepsilon_{\rm{ex}}>0$ and
$\varepsilon_{\rm{t},0}<0$ gives
$\varepsilon_{\rm{e},0}+\frac{1}{2}\varepsilon_{\rm{h},0}<\varepsilon_{\rm{b}}<\varepsilon_{\rm{h},0}+\varepsilon_{\rm{e},0}$.
Close to $\bm{k}=0$, the trion energy is
$\varepsilon_{\rm{t}}(\bm{k})=-\varepsilon_{\rm{t},0}+\frac{k^{2}}{2m_{\rm{t}}}$.
Due to the Galilean invariance of our model near the electron and hole
valleys, $m_{\rm{t}}=2m_{\rm{e}}+m_{\rm{h}}$. At large enough momenta, the
trion band merges into the two-electron, one-hole continuum, and ceases to be
a well-defined bound state. Since $\varepsilon_{\rm{t},0}>0$, the many-body
ground state includes a non-zero density of trions in a pocket centered at
$\bm{k}=0$.

The final step in our construction couples the trions to electrons in the
lower original band near $\bm{k}=0$, in the way described in Eq. (4). The
trion creation operator in real space is

$\displaystyle t^{\dagger}(\bm{r}_{\rm{t}})$ $\displaystyle=$
$\displaystyle\int
d^{2}{r_{\rm{e},1}}d^{2}{r_{\rm{e},2}}\,F^{\star}(\bm{r}_{\rm{e},1}-\bm{r}_{\rm{t}},{\bm{r}_{\rm{e},2}-\bm{r}_{\rm{t}}})$
(5) $\displaystyle\times$
$\displaystyle\psi^{\dagger}_{\rm{h}}({\bm{r}_{\rm{t}}})\psi^{\dagger}_{\rm{e},1}(\bm{r}_{\rm{e},1}-\bm{r}_{\rm{t}})\psi^{\dagger}_{\rm{e},2}({\bm{r}_{\rm{e},2}-\bm{r}_{\rm{t}}}),$

where the trion wavefunction $F(\bm{r},\bm{r}^{\prime})$ decays at long
distances. We introduce an interaction $V_{\rm{t}-\Gamma}$ that hybridizes the
trions with the electrons near the $\Gamma$ point:

$V_{\rm{t}-\Gamma}=u\sum_{j=1,2}\int
d^{2}r\,t^{\dagger}(\bm{r})\frac{1}{i}\partial^{\vphantom{\dagger}}_{r_{j}}\psi^{\vphantom{\dagger}}_{j}(\bm{r})+\rm{h.c.}$
(6)

with a coupling constant $u$. The operators $\psi^{\dagger}_{j={1,2}}(\bm{r})$
create electrons near $\Gamma$ (the subscript labels the two eigenstates of
$\eta_{z}$), and $r_{j=1,2}$ are the two spatial components of $\bm{r}$. This
interaction has the same effect as the $g_{1,2}$ terms in Eq. (4). Note that
$V_{\rm{t}-\Gamma}$ is a two-particle short-range interaction, as can be seen
by inserting Eq. (5) into Eq. (6).

Figure 3: Gapping fragile topological bands with $C_{3}$ symmetry. (a) BZ of
the four-band model on the honeycomb lattice SI . The lower two bands touch at
Dirac points (red) at $K$ and $K^{\prime}$, whose winding numbers are $+1$.
(b) The two fragile bands can be deformed into a configuration with quadratic
touchings at $K$ and $K^{\prime}$ whose winding numbers are $-2$ (blue), and
quadratic touchings at the $M$ points with winding $+2$. (c) To gap out the
quadratic touching at $K$, we form three trion bands $t_{1,2,3}$ in a $C_{3}$
symmetric fashion around $K$. The symmetric combination
$\frac{1}{\sqrt{3}}(t_{1}+t_{2}+t_{3})$ can hybridize with the lower band at
$K$ without breaking $C_{3}$, opening a gap in the spectrum (see Fig. 1c). The
quadratic touching at $K^{\prime}$ may be gapped in a similar way. The
quadratic touching at $M$ can be gapped by introducing a pair of electron
valleys and a hole valley in a mirror symmetric way, forming a trion band, and
hybridizing it with the electrons at $M$.

The interaction $V_{\rm{t}-\Gamma}$ achieves our goal: it opens a gap in the
many-body spectrum, leading to a featureless insulating ground state without
involving degrees of freedom external to the two fragile topological bands.

Several comments are in order. First, our treatment of the trions as
essentially free fermions is justified in the limit where their size is much
smaller than the spacing between them. In this limit, the interaction energy
between trions can be neglected, because it scales to zero faster than the gap
SI . Second, the resulting ground state is not a Slater determinant in terms
of the original electrons. However, the wavefunction deviates significantly
from a Slater determinant only in the vicinity of
$\Gamma,\bm{k}_{1},\bm{k}_{2},\bm{k}_{\rm{h}}$. The filled states near
$\Gamma$ are depicted in Fig. 1d, with the $z$ component proportional to the
trion amplitude. Importantly, the spectrum does not contain any fractionalized
excitations, and hence the system is not topologically ordered. Finally, our
construction relies on merging the two Dirac points into a quadratic band
touching. This is impossible when the Dirac points are pinned to a particular
point in the BZ, e.g., due to $C_{3}$ symmetry (as in MATBG). This case is
treated next.

_Gapping in the presence of $C_{3}$ symmetry.–_ To address this case, we use a
$C_{3}$ symmetric model that gives a pair of fragile topological bands on a
honeycomb lattice, described in Ref. Zou  _et al._ (2018) (see SI ). As
before, the fragile topology is protected by $C_{2}\mathcal{T}_{+}$ and
translational symmetries. Fig. 3a shows the BZ, indicating the positions of
the two Dirac points at $K$ and $K^{\prime}$ and their winding numbers, which
are both $+1$. Importantly, the Dirac points cannot be moved away from $K$ and
$K^{\prime}$ without breaking the $C_{3}$.

To demonstrate that this does does not prohibit gapping by interactions, we
deform the single-particle Hamiltonian, turning $K,K^{\prime}$ into quadratic
touching points at the cost of adding three additional quadratic touchings
half-way between $K$ and $K^{\prime}$, at the $M$ points (Fig. 3b). The
deformation from Fig. 3a to Fig. 3b can be performed by first nucleating at
$M$ two Dirac points with winding numbers $-1$ and two with winding numbers
$+1$. The $-1$ Dirac points are pulled apart and moved towards the $K$ and
$K^{\prime}$ points, leaving at $M$ quadratic band touching with winding $+2$.
The $K$ and $K^{\prime}$ points absorb three Dirac points with winding $-1$
each, consistently with $C_{3}$ symmetry, turning into quadratic band
touchings with winding numbers of $-2$ ani . With all gap-closures being
quadratic, the interaction-driven gapping mechanism described above can be
applied, as explained in Fig. 2.

_Discussion.–_ We end with a brief discussion of the possible implications of
our results for magic-angle twisted bilayer graphene. The narrow bands in this
system have a fragile topological character of the type discussed here Po  _et
al._ (2018b); Zou  _et al._ (2018). This system has an additional spin and
valley multiplicity. Our mechanism for opening a gap without symmetry breaking
is, in principle, applicable for each of the spin and valley flavors
separately 444This multiplicity allows for an additional gapping mechanism by
spontaneously breaking the valley charge conservation Bultinck  _et al._
(2020)..

Whether there is a gap at charge neutrality in MATBG is presently unresolved,
with some experiments observing semi-metallic behavior Cao  _et al._ (2018a)
and others observing a gap Lu  _et al._ (2019). Our work implies that a gap
may occur even in the absence of any broken symmetry. The mechanism we
outlined here serves as a proof of principle. Given the strongly interacting
nature of MATBG, it is possible that other, less fine-tuned mechanisms may
lead to the same gapped, symmetry-preserving state. For instance, a variant of
our mechanism can be used to construct a featureless insulating state on the
honeycomb lattice at half filling SI ; the same phase can be constructed via a
different mechanism that does not involve trions Kimchi  _et al._ (2013); Lee
_et al._ (2018).

Our mechanism for a gap opening invokes the formation of an additional band of
trions. Since the trion has the same quantum numbers as an electron, the trion
band should be observable in angle-resolved photoemission or scanning
tunneling microscopy experiments.

To conclude, our work elucidates an aspect in which “fragile topology” lives
up to its name: while the topological character prevents the bands from being
separated by a gap at the single-particle level, no such fundamental
obstruction exists once interactions are included.

###### Acknowledgements.

We thank B. Andrei Bernevig, Eslam Khalaf, Raquel Queiroz, Robert-Jan Slager,
Zhida Song, and Oskar Vafek for useful discussions. E.B. and A.S. acknowledge
support from the Israeli Science Foundation Quantum Science and Technology
grant no. 2074/19, and from the CRC 183 of the Deutsche Forschungsgemeinschaft
(Project C02). This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme [grant agreements No. 788715 and 817799, Projects LEGOTOP (AS) and
HQMAT (EB)]. A.M.T. acknowledges support from the Israeli Science Foundation
grant no. 1939/18.

## References

  * Schnyder  _et al._ (2008) Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, and Andreas W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions,” Phys. Rev. B 78, 195125 (2008).
  * Kitaev (2009) Alexei Kitaev, “Periodic table for topological insulators and superconductors,” in  _AIP conference proceedings_ , Vol. 1134 (American Institute of Physics, 2009) pp. 22–30.
  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
  * Qi and Zhang (2011) Xiao-Liang Qi and Shou-Cheng Zhang, “Topological insulators and superconductors,” Rev. Mod. Phys. 83, 1057–1110 (2011).
  * Fu (2011) Liang Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011).
  * Kruthoff  _et al._ (2017) Jorrit Kruthoff, Jan de Boer, Jasper van Wezel, Charles L. Kane, and Robert-Jan Slager, “Topological classification of crystalline insulators through band structure combinatorics,” Phys. Rev. X 7, 041069 (2017).
  * Bradlyn  _et al._ (2017) Barry Bradlyn, L Elcoro, Jennifer Cano, MG Vergniory, Zhijun Wang, C Felser, MI Aroyo, and B Andrei Bernevig, “Topological quantum chemistry,” Nature 547, 298–305 (2017).
  * Po  _et al._ (2017) Hoi Chun Po, Ashvin Vishwanath, and Haruki Watanabe, “Symmetry-based indicators of band topology in the 230 space groups,” Nature communications 8, 1–9 (2017).
  * Fidkowski and Kitaev (2010) Lukasz Fidkowski and Alexei Kitaev, “Effects of interactions on the topological classification of free fermion systems,” Phys. Rev. B 81, 134509 (2010).
  * Fidkowski and Kitaev (2011) Lukasz Fidkowski and Alexei Kitaev, “Topological phases of fermions in one dimension,” Phys. Rev. B 83, 075103 (2011).
  * Turner  _et al._ (2011) Ari M. Turner, Frank Pollmann, and Erez Berg, “Topological phases of one-dimensional fermions: An entanglement point of view,” Phys. Rev. B 83, 075102 (2011).
  * Fidkowski  _et al._ (2013) Lukasz Fidkowski, Xie Chen, and Ashvin Vishwanath, “Non-abelian topological order on the surface of a 3d topological superconductor from an exactly solved model,” Phys. Rev. X 3, 041016 (2013).
  * Kimchi  _et al._ (2013) Itamar Kimchi, SA Parameswaran, Ari M Turner, Fa Wang, and Ashvin Vishwanath, “Featureless and nonfractionalized mott insulators on the honeycomb lattice at 1/2 site filling,” Proceedings of the National Academy of Sciences 110, 16378–16383 (2013).
  * Morimoto  _et al._ (2015) Takahiro Morimoto, Akira Furusaki, and Christopher Mudry, “Breakdown of the topological classification $\mathbb{Z}$ for gapped phases of noninteracting fermions by quartic interactions,” Phys. Rev. B 92, 125104 (2015).
  * Queiroz  _et al._ (2016) Raquel Queiroz, Eslam Khalaf, and Ady Stern, “Dimensional hierarchy of fermionic interacting topological phases,” Phys. Rev. Lett. 117, 206405 (2016).
  * Lee  _et al._ (2018) Jong Yeon Lee, Ari M. Turner, and Ashvin Vishwanath, “Building symmetry-enriched topological phases from a bipartite lattice construction and anyon condensation,” Phys. Rev. B 98, 214416 (2018).
  * Po  _et al._ (2018a) Hoi Chun Po, Liujun Zou, Ashvin Vishwanath, and T. Senthil, “Origin of mott insulating behavior and superconductivity in twisted bilayer graphene,” Phys. Rev. X 8, 031089 (2018a).
  * Zou  _et al._ (2018) Liujun Zou, Hoi Chun Po, Ashvin Vishwanath, and T. Senthil, “Band structure of twisted bilayer graphene: Emergent symmetries, commensurate approximants, and Wannier obstructions,” Phys. Rev. B 98, 085435 (2018).
  * Song  _et al._ (2019) Zhida Song, Zhijun Wang, Wujun Shi, Gang Li, Chen Fang, and B. Andrei Bernevig, “All magic angles in twisted bilayer graphene are topological,” Phys. Rev. Lett. 123, 036401 (2019).
  * Po  _et al._ (2019) Hoi Chun Po, Liujun Zou, T. Senthil, and Ashvin Vishwanath, “Faithful tight-binding models and fragile topology of magic-angle bilayer graphene,” Phys. Rev. B 99, 195455 (2019).
  * Po  _et al._ (2018b) Hoi Chun Po, Haruki Watanabe, and Ashvin Vishwanath, “Fragile topology and wannier obstructions,” Phys. Rev. Lett. 121, 126402 (2018b).
  * Ahn  _et al._ (2019) Junyeong Ahn, Sungjoon Park, and Bohm-Jung Yang, “Failure of nielsen-ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: Application to twisted bilayer graphene at magic angle,” Phys. Rev. X 9, 021013 (2019).
  * Bouhon  _et al._ (2019) Adrien Bouhon, Annica M. Black-Schaffer, and Robert-Jan Slager, “Wilson loop approach to fragile topology of split elementary band representations and topological crystalline insulators with time-reversal symmetry,” Phys. Rev. B 100, 195135 (2019).
  * Bradlyn  _et al._ (2019) Barry Bradlyn, Zhijun Wang, Jennifer Cano, and B. Andrei Bernevig, “Disconnected elementary band representations, fragile topology, and Wilson loops as topological indices: An example on the triangular lattice,” Phys. Rev. B 99, 045140 (2019).
  * Lopes dos Santos  _et al._ (2007) J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H. Castro Neto, “Graphene bilayer with a twist: Electronic structure,” Phys. Rev. Lett. 99, 256802 (2007).
  * Bistritzer and MacDonald (2011) Rafi Bistritzer and Allan H MacDonald, “Moiré bands in twisted double-layer graphene,” Proceedings of the National Academy of Sciences 108, 12233–12237 (2011).
  * Cao  _et al._ (2018a) Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang, Spencer L Tomarken, Jason Y Luo, Javier D Sanchez-Yamagishi, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, _et al._ , “Correlated insulator behaviour at half-filling in magic-angle graphene superlattices,” Nature 556, 80 (2018a).
  * Cao  _et al._ (2018b) Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, and Pablo Jarillo-Herrero, “Unconventional superconductivity in magic-angle graphene superlattices,” Nature 556, 43–50 (2018b).
  * Wang  _et al._ (2013) Chong Wang, Andrew C. Potter, and T. Senthil, “Gapped symmetry preserving surface state for the electron topological insulator,” Phys. Rev. B 88, 115137 (2013).
  * Bonderson  _et al._ (2013) Parsa Bonderson, Chetan Nayak, and Xiao-Liang Qi, “A time-reversal invariant topological phase at the surface of a 3d topological insulator,” Journal of Statistical Mechanics: Theory and Experiment 2013, P09016 (2013).
  * Burnell  _et al._ (2014) F. J. Burnell, Xie Chen, Lukasz Fidkowski, and Ashvin Vishwanath, “Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order,” Phys. Rev. B 90, 245122 (2014).
  * Metlitski  _et al._ (2015) Max A. Metlitski, C. L. Kane, and Matthew P. A. Fisher, “Symmetry-respecting topologically ordered surface phase of three-dimensional electron topological insulators,” Phys. Rev. B 92, 125111 (2015).
  * Mross  _et al._ (2016) David F. Mross, Andrew Essin, Jason Alicea, and Ady Stern, “Anomalous quasiparticle symmetries and non-abelian defects on symmetrically gapped surfaces of weak topological insulators,” Phys. Rev. Lett. 116, 036803 (2016).
  * Lieb  _et al._ (1961) Elliott Lieb, Theodore Schultz, and Daniel Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics 16, 407–466 (1961).
  * Oshikawa (2000) Masaki Oshikawa, “Commensurability, excitation gap, and topology in quantum many-particle systems on a periodic lattice,” Phys. Rev. Lett. 84, 1535–1538 (2000).
  * Hastings (2004) M. B. Hastings, “Lieb-Schultz-Mattis in higher dimensions,” Phys. Rev. B 69, 104431 (2004).
  * Note (1) We thank B. A. Bernevig, R. Queiroz, and Z. Song for explaining this model to us.
  * Xie and MacDonald (2020) Ming Xie and A. H. MacDonald, “Nature of the correlated insulator states in twisted bilayer graphene,” Phys. Rev. Lett. 124, 097601 (2020).
  * Kang and Vafek (2020) Jian Kang and Oskar Vafek, “Non-abelian dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic-angle twisted bilayer graphene,” Phys. Rev. B 102, 035161 (2020).
  * Kheng  _et al._ (1993) K. Kheng, R. T. Cox, Merle Y. d’ Aubigné, Franck Bassani, K. Saminadayar, and S. Tatarenko, “Observation of negatively charged excitons ${\mathit{x}}^{\mathrm{-}}$ in semiconductor quantum wells,” Phys. Rev. Lett. 71, 1752–1755 (1993).
  * Finkelstein  _et al._ (1995) Gleb Finkelstein, Hadas Shtrikman, and Israel Bar-Joseph, “Optical spectroscopy of a two-dimensional electron gas near the metal-insulator transition,” Phys. Rev. Lett. 74, 976–979 (1995).
  * Huard  _et al._ (2000) V. Huard, R. T. Cox, K. Saminadayar, A. Arnoult, and S. Tatarenko, “Bound states in optical absorption of semiconductor quantum wells containing a two-dimensional electron gas,” Phys. Rev. Lett. 84, 187–190 (2000).
  * (43) See Supplementary Information, that includes Ref. Haldane (1988) for a color representation of $\alpha_{\bm{k}}$, a derivation of the spectrum of $H_{\rm{3-band}}$, a description of the honeycomb model with fragile topological bands, and a construction of a featureless insulator on the honeycomb lattice.
  * Wu  _et al._ (2019) QuanSheng Wu, Alexey A Soluyanov, and Tomáš Bzdušek, “Non-abelian band topology in noninteracting metals,” Science 365, 1273–1277 (2019).
  * Bouhon  _et al._ (2020) Adrien Bouhon, QuanSheng Wu, Robert-Jan Slager, Hongming Weng, Oleg V Yazyev, and Tomáš Bzdušek, “Non-abelian reciprocal braiding of weyl points and its manifestation in zrte,” Nature Physics 16, 1137–1143 (2020).
  * Note (2) Such a gapping mechanism is impossible near a single Dirac point of the original bands; in that case, the lowest energy band has a Berry phase of $\pi$ going around the Dirac point, which enforces a band touching between that band and the bands above it.
  * Note (3) In $d=2$, any attractive interaction between electrons and holes yields an exciton bound state.
  * (48) For an animation of this process, see: https://youtu.be/78USiNqM8I0.
  * Note (4) This multiplicity allows for an additional gapping mechanism by spontaneously breaking the valley charge conservation Bultinck  _et al._ (2020).
  * Lu  _et al._ (2019) Xiaobo Lu, Petr Stepanov, Wei Yang, Ming Xie, Mohammed Ali Aamir, Ipsita Das, Carles Urgell, Kenji Watanabe, Takashi Taniguchi, Guangyu Zhang, _et al._ , “Superconductors, orbital magnets and correlated states in magic-angle bilayer graphene,” Nature 574, 653–657 (2019).
  * Haldane (1988) F. D. M. Haldane, “Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the Parity Anomaly,” Phys. Rev. Lett. 61, 2015–2018 (1988).
  * Bultinck  _et al._ (2020) Nick Bultinck, Eslam Khalaf, Shang Liu, Shubhayu Chatterjee, Ashvin Vishwanath, and Michael P. Zaletel, “Ground state and hidden symmetry of magic-angle graphene at even integer filling,” Phys. Rev. X 10, 031034 (2020).

## Supplementary Information

###  .1 Colormap representation of the winding around the Dirac points

For the reader’s convenience, we display the angle
$\alpha_{\bm{k}}\equiv\arg(h_{1}(\bm{k})+ih_{1}(\bm{k}))$ as a colormap in
Fig. S1. The color scale ranges from $-\pi$ to $\pi$. Note that the lines of
discontinuities in $\alpha_{\bm{k}}$ by $2\pi$, that emanate from the band
touchings, are not physical. The winding numbers of the different band
touchings are shown in the figure. The angle $\alpha_{\bm{k}}$ depends on our
choice of phase for the wavefunctions of the two fragile bands, but the
relative winding numbers of the Dirac cones do not depend on this gauge
choice.

###  .2 Diagonalization of the three-band Hamiltonian

Here, we diagonalize the three-band Hamiltonian, Eq. (4) in the main text. We
choose $g_{1}(\bm{k})=\Delta\,k_{x}$, $g_{2}(\bm{k})=\Delta\,k_{y}$, where
$\Delta$ represents the strength of the hybridization between the additional
third band and the original two bands. The diagonalization is simplified by
exploiting the transformation law of the Hamiltonian under rotation. In
particular, $H_{\rm{3-band}}$ satisfies

$H_{\rm{3-band}}\left(R_{\phi}\bm{k}\right)=R^{T}_{\phi}H_{\rm{3-band}}(\bm{k})R_{\phi},$
(S1)

where $R_{\phi}$ is the following orthogonal matrix:

$R_{\phi}=\left(\begin{array}[]{ccc}\cos(\phi)&\sin(\phi)&0\\\
-\sin(\phi)&\cos(\phi)&0\\\ 0&0&1\end{array}\right).$ (S2)

Thus, to find the spectrum, it is sufficient to consider a momentum $\bm{k}$
along the $x$ axis. The spectrum at a different values of $\bm{k}$ can then be
found by rotating ${\bm{k}}$ onto the $x$ axis by Eq. (S1), and using the fact
that the spectrum is invariant under an orthogonal transformation.

From these considerations, we find that the three eigenvalues of
$H_{\rm{3-band}}$ are given by

$\displaystyle\varepsilon_{1}(\bm{k})$ $\displaystyle=\frac{k^{2}}{2m},$ (S3)
$\displaystyle\varepsilon_{2,3}(\bm{k})$
$\displaystyle=\frac{1}{2}\left[\varepsilon_{\rm{t}}(k)-\frac{k^{2}}{2m}\right]\pm\sqrt{\Delta^{2}k^{2}+\frac{1}{4}\left[\varepsilon_{\rm{t}}(k)+\frac{k^{2}}{2m}\right]^{2}},$

where $\varepsilon_{\rm{t}}(k)\equiv-\varepsilon_{\rm{t},0}+k^{2}/(2m)$. The
spectrum is shown in Fig. S2. As can be seen in the figure, the lowest band is
separated from the other two bands by a gap if $\varepsilon_{\rm{t},0}>0$.

Figure S1: The angle $\alpha_{\bm{k}}$ for the square lattice model as a
function of $\bm{k}$, represented by color. The same angle is represented in
Fig. 1 of the main text by arrows. (a) $\Delta_{x}>0$, $\Delta_{xz}=0$; in
this case, there are two Dirac points, both with a winding of $+1$ (indicated
in red). (b) $\Delta_{x}=-\Delta_{xz}$. In this case, the Dirac points merge
into a single quadratic band touching at $\bm{k}=0$ with winding number $+2$.

Our choice of $g_{1,2}(\bm{k})$, for which the problem is rotationally
symmetric, is made mostly for convenience. It is not essential in order to
demonstrate the possibility of separating the lowest of the three bands of
$H_{\rm{3-band}}$ from the other two by a gap. We note, however, that in the
$C_{3}$ symmetric case discussed in the main text, for the quadratic band
touchings at $K$ and $K^{\prime}$, the Hamiltonian near these points is
required to be $C_{3}$ symmetric. The form we have chosen satisfies this
automatically, as $C_{3}$ is a subgroup of the continuous rotational symmetry
of $H_{\rm{3-band}}$. (One can check that within the $C_{3}$ symmetric model,
described in Sec. .4 below, the states at the quadratic band touchings
transform according to $R_{\phi}$ above when written in a real basis.)

Interestingly, even after the third band hybridizes with the original two, the
quadratic band touching at $\bm{k}=0$ persists. This is a general feature of
our three-band problem. As long as $C_{2}\mathcal{T}_{+}$ symmetry is
preserved, and hence the Hamiltonian can be chosen to be real, three
eigenstates of the Hamiltonian are specified by an orthogonal matrix that
depends on $\bm{k}$. Since $\pi_{1}[O(3)]=\mathbb{Z}_{2}$, the eigenstates
along a closed path that encircles the origin are characterized by a
$\mathbb{Z}_{2}$ topological invariant. The presence of a quadratic band
touching at the origin corresponds to this invariant being non-trivial.
Therefore, a quadratic band touching between two of the three bands within the
closed path cannot disappear unless $C_{2}\mathcal{T}_{+}$ is broken or the
band touching crosses the path. This feature generalizes to any number $N\geq
2$ of bands with $C_{2}\mathcal{T}_{+}$ symmetry, such that the eigenstates
can be chosen to be real. Note, however, that this does not preclude a many-
body gapped state with an integer filling per site, since the band touching
can occur either below or above the Fermi level.

###  .3 Treatment of interactions between trions

Here, we present a more systematic discussion of the effective interacting
Hamiltonian that describes the low-energy degrees of freedom of the system.
Crucially, we show that the interactions between the trion bound states can be
neglected in the limit where the trions are dilute in the ground state,
justifying the treatment of the trions as essentially non-interacting
particles.

The degrees of freedom near the chemical potential include the gapless
electrons near $\Gamma$ and the trions. Within our construction, there are no
other low-energy degrees of freedom. In particular, the electrons and holes
near $\bm{k}_{1,e},$ $\bm{k}_{2,e}$ and $\bm{k}_{h}$, as well as any bosonic
excitons formed by combining an electron and a hole, remain gapped, as
discussed in the manuscript.

The low-energy effective Hamiltonian is written as

$H_{{\rm eff}}=H_{\Gamma}+H_{t}+H_{\Gamma t}+H_{int}.$ (S4)

Here, $H_{\Gamma}$ is the bare Hamiltonian of electrons near $\Gamma$. As in
the main text, the basis is chosen such that the states near $\Gamma$ are
spanned by $|u_{1,2}(\bm{k})\rangle$, which are smooth near $\Gamma$, and
$C_{2}\mathcal{T}_{+}$ symmetry acts simply as complex conjugation. The
corresponding creation operators are $c_{\bm{k},1}^{\dagger}$,
$c_{\bm{k},2}^{\dagger}$. In terms of these operators, $H_{\Gamma}$ is written
as

$H_{\Gamma}=\sum_{|\bm{k}|\leq
k_{0}}c_{\bm{k}}^{\dagger}h^{\vphantom{\dagger}}_{\bm{k}}c^{\vphantom{\dagger}}_{\bm{k}},$
(S5)

where $c_{\bm{k}}^{\dagger}=(c_{\bm{k},1}^{\dagger},c_{\bm{k},2}^{\dagger})$
and

$h_{\bm{k}}=\frac{1}{2m}\left(\begin{array}[]{cc}k_{y}^{2}-k_{x}^{2}&k_{x}k_{y}\\\
k_{x}k_{y}&k_{x}^{2}-k_{y}^{2}\end{array}\right),$ (S6)

forming the quadratic band touching. The momentum summation in (S5) is
restricted to a disk of radius $k_{0}$ around the $\Gamma$ point, such that
$k_{0}$ serves as the momentum cutoff of our low-energy theory. Without the
coupling to the trions, $H_{\Gamma}$ has a gapless spectrum.

Next, we discuss the trions. A single trion excitation is created by an
operator $t_{\bm{k}}^{\dagger}$, which is composed entirely of electron and
hole states outside the disk of radius $k_{0}$ of $\Gamma$, and hence can be
treated as an independent fermionic degree of freedom. Since the energy of the
trions is assumed to be minimal at $\bm{k}=0$, their effective Hamiltonian can
be written as

$H_{t}=\sum_{|\bm{k}|\leq
k_{0}}\left(\frac{k^{2}}{2m_{t}}-\varepsilon_{t,0}\right)t_{\bm{k}}^{\dagger}t_{\bm{k}}.$
(S7)

The parameters are chosen such that the characteristic momentum of the trions,
$k_{t}\equiv\sqrt{m_{t}\varepsilon_{t,0}}$, is much smaller than $k_{0}$. To
simplfy our discussion, we assume that $m\sim m_{t}$.

The $H_{\Gamma t}$ term hybridizes the electrons and holes with the trions,
and is chosen to be

$H_{\Gamma t}=\sum_{|\bm{k}|\leq
k_{0}}\Delta\,t_{\bm{k}}^{\dagger}\left(k_{x}c_{1\bm{k}}+k_{y}c_{2\bm{k}}\right)+h.c.,$
(S8)

where $\Delta$ is real. We choose $\Delta\sim k_{t}/m_{t}\ll k_{0}/m_{t}$,
such the hybridization between electrons and trions outside of the disk of
radius $k_{0}$ is negligible, ensuring the consistency of our low-energy
treatment. In terms of the original electrons and holes, $H_{\Gamma t}$ is a
local two-particle interaction between an electron near $\Gamma$ and electrons
near $\bm{k}_{1,e},\bm{k}_{2,e},$ or $\bm{k}_{h}$, as discussed below Eq. (6)
of the manuscript.

The trions necessarily interact with each other, as can be seen by considering
the Fourier transformed operator $t^{\dagger}(\bm{r})$ that creates a trion
centered at point $\bm{r}$ in real space (Eq. 5 in the manuscript). The trion
has a characteristic size $a_{t}\sim 1/\delta k_{t}$, where $\delta k_{t}$ is
the spread in momentum space around $\bm{k}_{1,e},\bm{k}_{2,e},\bm{k}_{h}$ of
the states that participate in the trion excitation. $\delta k_{t}$ is
determined by the trion binding energy (and hence by the strength of the
interaction between electrons and holes that binds them together to form the
trion). Two trions within $a_{t}$ of each other interact due to their spatial
overlap.

To account for this effect, we include an inter-trion effective interaction

$H_{int}=\frac{1}{L^{2}}\sum_{\bm{k}_{1},\dots,\bm{k}_{4}}V_{\bm{k}_{1},\dots,\bm{k}_{4}}t_{\bm{k}_{4}}^{\dagger}t_{\bm{k}_{3}}^{\dagger}t_{\bm{k}_{2}}t_{\bm{k}_{1}}\delta_{\bm{k}_{1}+\bm{k}_{2}-\bm{k}_{3}-\bm{k}_{4}},$
(S9)

where $L$ is the linear dimension of the system. The details of
$V_{\bm{k}_{1},\dots,\bm{k}_{4}}$ do not matter, as long as it is real (thus
preserving $C_{2}\mathcal{T}_{+}$ symmetry) and smoothly decays over a
distance $\delta k_{t}$ in momentum space (i.e., its range is $a_{t}\sim
1/\delta k_{t}$ in real space). $V_{\bm{k}_{1},\dots,\bm{k}_{4}}$ can be
computed, in principle, by calculating the scattering amplitude of a pair of
trion excitations from momenta $\left(\bm{k}_{1},\bm{k}_{2}\right)$ to
$\left(\bm{k}_{3},\bm{k}_{4}\right)$. Three-trion interactions and higher
order many-body terms are also allowed, but become successively unimportant in
the dilute trion limit, as we shall now discuss.

The effective Hamiltonian (S4) is tractable in the limit $k_{t}a_{t}\ll 1$.
Physically, in this limit, the typical distance between trions present in the
ground state, which is of the order of $1/k_{t}$, is much larger than the size
of a trion, $a_{t}$. In this limit, the effects of the inter-trion interaction
are expected to be small. Indeed, expanding
$V_{\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},\bm{k}_{4}}$ at small momenta and using
the property
$V_{\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},\bm{k}_{4}}=-V_{\bm{k}_{1},\bm{k}_{2},\bm{k}_{4},\bm{k}_{3}}=-V_{\bm{k}_{2},\bm{k}_{1},\bm{k}_{3},\bm{k}_{4}}$
(which follows from the Pauli principle) and the rotational symmetry of the
effective Hamiltonian, we obtain

$V_{\bm{k}_{1},\bm{k}_{2},\bm{k}_{3},\bm{k}_{4}}=V_{0}a_{t}^{2}(\bm{k}_{1}-\bm{k}_{2})\cdot(\bm{k}_{3}-\bm{k}_{4})+O(k^{4}).$
(S10)

The absence of a constant term in the limit $k\to 0$ is characteristic of
spinless fermions, which cannot occupy the same point in space.

Since the typical momentum of trions in the ground state is $k_{t}$, the
interaction matrix elements are suppressed by a factor $(k_{t}a_{t})^{2}\ll
1$. This justifies a perturbative treatment of $H_{int}$. To zeroth order,
$H_{{\rm eff}}$ is non-interacting; it is identical to the three-band
Hamiltonian (Eq. 4 of the main text), where the trions are treated as a new
‘flavor’ of free fermions. The resulting zeroth-order spectrum is gapped, with
a gap $E_{g}\sim|\Delta|k_{t}$. Turning on infinitesimal interactions
renormalizes the magnitude of the gap, but does not close it immediately.

To verify the consistency of the perturbative analysis, we estimate the first-
order interaction correction to the energy of a trion excitation, $\delta E$,
and compare it to the gap. Using perturbation theory, we obtain $\delta E\sim
V_{0}\left(a_{t}k_{t}\right)^{2}n_{t}\sim
m_{t}V_{0}\left(a_{t}k_{t}\right)^{2}\cdot\varepsilon_{t,0}$, where $n_{t}\sim
k_{t}^{2}$ is the density of trions in the ground state. With our choice
$\Delta\sim\varepsilon_{t,0}/k_{t}$, we find that the zeroth order gap
satisfies $E_{g}\sim\varepsilon_{t,0}\gg\delta E$. Hence, we have obtained a
fully gapped state.

Finally, the effective Hamiltonian (S4) is manifestly $C_{2}\mathcal{T}_{+}$
symmetric. Our perturbative solution of $H_{{\rm eff}}$ in the limit
$k_{t}a_{t}\ll 1$ does not break $C_{2}\mathcal{T}_{+}$ (or any other
symmetry) spontaneously, as can be seen from the fact that the resulting
ground state in the limit $k_{t}a_{t}\rightarrow 0$ is a
$C_{2}\mathcal{T}_{+}$ symmetric Slater determinant. Infinitesimal
interactions cannot cause spontaneous symmetry breaking in a gapped state, and
hence $C_{2}\mathcal{T}_{+}$ is preserved for a finite range of $k_{t}a_{t}$.

###  .4 Model for fragile topological bands with $C_{3}$ symmetry

We briefly review a model for fragile topological bands on a honeycomb
lattice, satisfying $C_{3}$, $C_{2}\mathcal{T}_{+}$, and $M_{y}$ symmetries
Zou  _et al._ (2018); Song  _et al._ (2019). The model contains two orbitals
of spinless electrons on each sublattice of the honeycomb lattice, such that
there are four bands in total. As before, the construction starts from two
degenerate pairs of Chern bands with opposite chiralities, each forming a
Haldane model Haldane (1988). We then add a term that couples the lower two
Chern bands, separating them everywhere in the BZ except at the $K$ and
$K^{\prime}$ points.

Figure S2: Spectrum of $H_{\rm{3-band}}$, given in Eq. (4) of the main text,
with $g_{1}(\bm{k})=\Delta\,k_{x}$ and $g_{2}(\bm{k})=\Delta\,k_{y}$. With
this choice, the spectrum is rotationally symmetric, depending only on
$|\bm{k}|$. We have used $m_{\rm{t}}=5m$. In panel (a),
$\varepsilon_{\rm{t},0}=2m\Delta^{2}$, whereas in (b),
$\varepsilon_{\rm{t},0}=-2m\Delta^{2}$.

The Hamiltonian is given by

$\displaystyle H$
$\displaystyle=-2t_{1}\left[f_{x}(\bm{k})\sigma^{x}+f_{y}(\bm{k})\sigma^{y}\right]$
$\displaystyle-2t_{2}\,g(\bm{k})\tau^{z}\sigma^{z}-2t_{3}\left[f_{x}(\bm{k})\sigma^{x}+f_{y}(\bm{k})\sigma^{y}\right]\tau^{x},$
(S11)

where $\sigma^{x,y,z}$ act on the sublattice degrees of freedom,
$\tau^{x,y,z}$ act on the two orbitals, $t_{1}$ is the amplitude for nearest-
neighbor hopping that preserves the orbital index, $t_{2}$ is a next-nearest
neighbor, orbital-dependent hopping amplitude, and $t_{3}$ is a nearest-
neighbor orbital-flipping hopping term. The functions $f_{x,y}$ and $g$ are
defined as:

$\displaystyle f_{x}(\bm{k})$
$\displaystyle=\cos\left(\frac{\sqrt{3}}{2}k_{x}\right)\cos\left(\frac{k_{y}}{2}\right)+\cos(k_{y}),$
$\displaystyle f_{y}(\bm{k})$
$\displaystyle=\cos\left(\frac{\sqrt{3}}{2}k_{x}\right)\sin\left(\frac{k_{y}}{2}\right)-\sin(k_{y}),$
$\displaystyle g(\bm{k})$
$\displaystyle=\sin\left(\frac{\sqrt{3}}{2}k_{x}\right)\left[\cos\left(\frac{\sqrt{3}}{2}k_{x}\right)-\cos\left(\frac{3}{2}k_{y}\right)\right].$
(S12)

Figure S3: (a) Dirac points in the BZ of a graphene-like honeycomb lattice.
The two Dirac points at $K$ and $K^{\prime}$ have opposite winding numbers.
(b) After a deformation of the single-particle Hamiltonian, the $K$ and
$K^{\prime}$ points carry quadratic band touchings with opposite winding
numbers. The deformation can be done without breaking any lattice symmetry.

The symmetries of the model are represented as:

$\displaystyle C_{2}\mathcal{T}_{+}$ $\displaystyle=\sigma^{x}\tau^{x}K,$
$\displaystyle C_{3}$ $\displaystyle=(\bm{k}\rightarrow R_{3}^{z}\bm{k}),$
$\displaystyle M_{y}$ $\displaystyle=\tau^{x}\times(k_{x}\rightarrow-k_{x}),$
(S13)

where $R_{3}^{z}$ denotes a rotation matrix by $2\pi/3$ around the $z$ axis.

Similarly to the square lattice model presented in the main text, the
honeycomb model (S11) yields two pairs of bands that possess fragile topology.
Each pair of bands touch contains two Dirac points at the $K$ and $K^{\prime}$
points in the BZ. The fragile topological character of the bands can be
deduced by computing the winding numbers of the two Dirac points, as in the
square lattice model. Alternatively, as explained in Ref. Zou  _et al._
(2018), due to the mirror symmetry $M_{y}$ of the model, there is a simple way
to determine the relative chirality of the two Dirac points. This is done by
examining the $M_{y}$ eigenvalues of the two eigenstates of the lower bands.
Their product can be checked to be $-1$, confirming that the winding numbers
of the Dirac points are the same Zou  _et al._ (2018).

###  .5 Featureless insulator on the honeycomb lattice

Here, we show that the method we applied in the main text for gapping the
Dirac points of fragile topological bands with $C_{3}$ symmetry can be used to
show that a system of electrons on a honeycomb lattice at half filling can
form a featureless insulator ground state. As in the fragile case, the
featureless insulator, that does not break any symmetry and has no topological
order, is impossible in the absence of interactions Kimchi  _et al._ (2013).
The existence of a featureless insulator on the honeycomb lattice was derived
using a different approach in Ref. Lee  _et al._ (2018).

As in the case of fragile topological bands, our mechanism proceeds by
deforming the single-particle Hamiltonian to form quadratic band touchings,
that can then become gapped by hybridizing with a band of trion excitations.
We start from a graphene-like single-particle Hamiltonian, in which the
spectrum possesses two Dirac points at $K$ and $K^{\prime}$ (see Fig. S3a). In
contrast to the fragile case, the two Dirac points carry opposite winding
numbers. On the honeycomb lattice, the mirror symmetry ${M}_{y}$ that takes
$k_{x}\to-k_{x}$ flips the winding number. This can be seen, e.g., from the
fact that the two Bloch states at the $M$ point have the same ${M}_{y}$
eigenvalue Zou  _et al._ (2018).

This property of mirror symmetry allows us to smoothly deform this band
structure to create quadratic band touchings. To achieve this, we nucleate a
pair of new Dirac points with opposite winding numbers at each $M$ point, and
shift one of the Dirac points toward $K$ and the other toward $K^{\prime}$.
When this is done in a $C_{3}$ symmetric way, three Dirac points with winding
number $-1$ merge with the original Dirac point of winding $+1$ at the $K$
point, leaving a quadratic band touching of winding $-2$ (see Fig. S3b). By
mirror symmetry, this procedure leaves the $K^{\prime}$ point with winding
$+2$. Each quadratic band touching can now be gapped following a similar
mechanism to that described in the main text, by forming a trion band from
states away from the $K$ and $K^{\prime}$ points and hybridizing them with
electrons near those points.