# Graph Theorem for Chiral Exact Flat Bands at Charge Neutrality

Gurjyot Sethi$\dagger$,1 Bowen Xia$\dagger$,1 Dongwook Kim1 Hang Liu2,3
Xiaoyin Li1 Feng Liu1<EMAIL_ADDRESS>1Department of Materials Science &
Engineering, University of Utah, Salt Lake City, Utah 84112, USA
2Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, People’s
Republic of China
3Beijing National Laboratory for Condensed Matter Physics and Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of
China

###### Abstract

Chiral exact flat bands (FBs) at charge neutrality have attracted much recent
interest, presenting an intriguing condensed-matter system to realize exotic
many-body phenomena, as specifically shown in “magic angle” twisted bilayer
graphene for superconductivity and triangulene-based superatomic graphene for
exciton condensation. Yet, no generic physical model to realize such FBs has
been developed. Here we present a new mathematical theorem, called bipartite
double cover (BDC) theorem, and prove that the BDC of line-graph (LG) lattices
hosts at least two chiral exact flat bands of opposite chirality, i.e., yin-
yang FBs, centered-around/at charge neutrality ($E=0$) akin to the “chiral
limit” of twisted bilayer graphene. We illustrate this theorem by mapping it
exactly onto tight-binding lattice models of the BDC of LGs of hexagonal
lattice for strong topological and of triangular lattice for fragile
topological FBs, respectively. Moreover, we use orbital design principle to
realize such exotic yin-yang FBs in non-BDC lattices to instigate their real
material discovery. This work not only enables the search for exact chiral FBs
at zero energy beyond moiré heterostructures, but also opens the door to
discovering quantum semiconductors featured with FB-enabled strongly
correlated carriers.

††preprint: APS/123-QED

In recent years, the importance of topological flat bands (FBs) in realizing
exotic many-body phenomena, such as superconductivity [1, 2, 3, 4], excitonic
superfluidity [5, 6, 7, 8], and magnetism [2, 9, 10, 11], has been highlighted
through a plethora of studies [1, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 15]
following “magic angle” twisted bilayer graphene (MATBG) [1, 16, 17, 18, 19,
20, 21]. Lately, there has also been increasing interest in realizing chiral
limit of MATBG with topologically fragile exact FBs of opposite chirality at
charge neutrality [22, 23, 24, 25] using external perturbations, such as
periodic strain [23]. Another intriguing case where topological FBs emerge
near charge neutrality is a superatomic graphene lattice [5, 26, 27, 28]. It
hosts two topological strong exact FBs of opposite chirality at $E=\pm t$,
i.e., the yin-yang FBs centered symmetrically at $E=0$. These chiral exact FBs
around/at charge neutrality exhibit fascinating transport properties of
one/two-body carriers due to their overlapping Wannier functions, as
manifested in unconventional superconductivity for fragile topological FBs of
TBG [13, 29, 30] and in excitonic Bose-Einstein condensation for strong
topological yin-yang FBs of superatomic graphene [5, 6] respectively.

Beyond these specific systems, however, no generic theory or lattice design
principle for constructing exact FBs with opposite chirality around/at Fermi
level has been reported yet. On the other hand, topological FBs are also known
to exist in a special class of two-dimensional (2D) lattices called line-graph
(LG) lattices [31, 32, 33, 34, 35], where the presence of at least one FB is
guaranteed by mathematical LG theorem [36]. Compared to FBs at (around) $E=0$
in MATBG (superatomic graphene), however, FBs in LG lattices are at $E=-2t$
[37] with $t$ being the tight-binding (TB) hopping integral. Although it was
shown recently that LGs of so-called split-graph lattices contain one
additional FB at $E=0$ [35], real material with split-graph lattice has been
rarely found due to physically unrealistic hopping constraints.

In this article, we present a new graph theorem, called bipartite double cover
(BDC) theorem and prove that the BDC of LG lattices always has at least two
FBs of opposite chirality around/at the Fermi level formed by bonding and
anti-bonding pairing of the two copies of LG FB eigenfunctions, respectively.
We illustrate this theorem by mapping it exactly onto realistic physical
models of TB Hamiltonian. First, we use the BDC of Kagome lattice (the LG of
hexagonal lattice) to show that it has two strong topological FBs of opposite
chirality as the valence and conduction band edge respectively forming a
quantum semiconductor, as exemplified in the case of superatomic graphene [6].
We then show that the BDC of LG of triangular lattice has four fragile
topological exact FBs at charge neutrality, as exemplified in the case of
chiral limit of MATBG [22] but without the need of a superlattice potential or
external fields. The FB topology of both BDC-LG lattices is confirmed using
Chern number calculations and Wilson loop analyses. Finally, we elaborate on
effective orbital design principle [32] for chiral exact FBs, based on the
symmetries of BDC-LG wavefunctions to find orbitals that can be placed in the
non-LG lattices to overcome the hopping constraints in the BDC-LG lattices,
and hence to facilitate their material discovery.

Figure 1: (a) A finite cyclic graph ($G$) with 5 vertices indicated by colored
circles and 5 edges indicated by black solid lines. (b) Adjacency matrix of
$G$ ($A_{G}$) with matrix element 1 between the two circles which have an edge
between them. (c) Line-graph of $G$ ($L(G)$). (d) BDC of $L(G)$ ($B(L(G))$).

In mathematics, a graph ($G$) consists of a set of objects with some pairs of
objects coupled with each other [38], as represented by vertices (objects) and
edges (pair relation) in Fig. 1(a). Each graph is assigned with an adjacency
matrix ($A_{G}$), that has a matrix element of 1 to denote an adjacent pair of
vertices (Fig. 1(b)). An LG of $G$ ($L(G)$) is constructed to depict
adjacencies between the edges of $G$. Namely, $L(G)$ is a graph, in which each
vertex of $L(G)$ represents an edge of $G$, and the two vertices of $L(G)$ are
adjacent if and only if their corresponding edges share a common vertex in $G$
(Fig. 1(c)). $G$ is sometimes referred to as the root graph of $L(G)$. The LG
theorem states - the spectra of infinite LG ($L(X)$), i.e., the eigenvalues of
$A_{L(X)}$ with $X$ being an infinite graph, always has at least one lowest
eigenvalue at -2 with infinite multiplicity provided the number of vertices
exceeds the number of edges in the repeating unit of X [36], which works in
any dimensions, including 1D lattice [32] (also see section I. in
supplementary material (SM) [39, 40, 41, 42, 43, 44, 45]).

The striking implication of LG theorem is seen in the electronic structure of
lattices based on LGs. TB model of a lattice ($L$) is defined as the set of
atomic sites and hopping matrix elements ($t$) between the sites. Each lattice
can be associated with an infinite graph: vertex set of $X$ consisting of
exactly one vertex for each lattice point, and an edge between all the vertex
pairs for which $t\neq 0$. This leads to a one-to-one correspondence between a
TB lattice and a mathematical graph [37],

$H_{L}=tA_{X},$ (1)

where $H_{L}$ denotes the TB Hamiltonian matrix in real space. Then,
naturally, if $L$ is based on a LG, there exists at least one exact FB in the
band structure at energy $E=-2t$. The degeneracy of the FB is given by the
difference between the number of independent vertices and edges in the unit-
cell (SM Section I. [39]), which can also be explained by the index theorem or
sublattice imbalance between the root graph and LG [46, 47]. The LG FBs can be
strong or fragile topological depending on if the root graph $X$ is bipartite
or non-bipartite respectively [35, 48]. For the former, there is always a
touching point between the FB and another dispersive band which can be viewed
as a Berry flux center in analogy to Dirac point [49] and be gapped out in the
presence of spin-orbit coupling (SOC) leading to a strong topological quasi-
FB, while for the latter, fragile exact FBs are isolated from rest of the
bands [48], also at $E=-2t$.

In graph theory, a bipartite double cover (BDC) can also be constructed from a
graph. BDC is one of the covering spaces of a graph, which is known to link
discrete graphs to topological crystal structures [50]. BDC maps every vertex
$v_{i}$ of $G$ to two (double) vertices $u_{i}$ and $w_{i}$. It is called
bipartite because the two vertices $u_{i}$ and $w_{j}$ are connected by an
edge in BDC if and only if the corresponding $v_{i}$ and $v_{j}$ have an edge
between them in $G$, i.e., one can divide the vertices of BDC into two
independent subsets [51]. In Fig. 1(d), as an example, we show the BDC of a
finite LG ($L(G)$) in Fig. 1(b), denoted by $B(L(G))$. BDC can also be written
as the tensor product of graphs, $L(G)\times K_{2}$, where $K_{2}$ is simply a
two-vertex graph, consisting of a circle and a square in Fig. 1(d). Edges
exist only between a circle and a square and only if the two corresponding
black circles have an edge between them in $L(G)$ (Fig. 1(b)).

In this work we focus on the BDCs ($B(L(X))$) of infinite LGs ($L(X)$).
Following the BDC construction, the adjacency matrix is given by,

$A_{B(L(X))}=\begin{bmatrix}0&A_{L(X)}\\\ A_{L(X)}^{T}&0\\\ \end{bmatrix},$
(2)

where $A_{L(X)}$ is the adjacency matrix of $L(X)$ and the basis of
$A_{B(L(X))}$ is the set {{$u_{i}$},{$w_{i}$}} or {{circles},{squares}} so
that the only non-zero matrix elements are in the off-diagonal blocks. Note
that $A_{L(X)}$ is always a symmetric matrix, i.e., $A_{L(X)}=A_{L(X)}^{T}$
[52]. Effectively, we have two copies of $L(X)$ but formed between two vertex
subsets. Suppose one of the infinite eigenvectors of $A_{L(X)}$ corresponding
to eigenvalue -2 is $v$ satisfying, $A_{L(X)}v=A_{L(X)}^{T}v=-2v$. One can
construct a vector $V=[v\;;\;v]$ in the basis of $A_{B(L(X))}$. From Eqn. (2),
it is clear that $V$ will be an eigenvector of $A_{B(L(X)}$ satisfying
$A_{B(L(X))}V=-2V$. Similarly, one can construct a vector
$V^{\prime}=[v\;;\;-v]$ that satisfies $A_{B(L(X))}V^{\prime}=2V^{\prime}$.
Note that there are infinite possible $v$’s with eigenvalue -2 of $A_{L(X)}$,
and for each of them one symmetrically ($V$) and one anti-symmetrically
($V^{\prime}$) paired eigenvectors of $A_{B(L(X))}$ can be constructed,
formulating a new graph theorem, which we call the “BDC theorem” – The spectra
of bipartite double cover of infinite line-graphs contains at least two
eigenvalues with infinite multiplicity at -2 and 2.

One can physically understand the above theorem by carefully looking at the
structure of $A_{B(L(X))}$. The absence of matrix element between the
individual elements of $u_{i}$ or $w_{i}$ renders exact chiral symmetry to BDC
which implies that if $\lambda$ is an eigenvalue of $A_{B(L(G))}$, its
opposite $-\lambda$ should also be an eigenvalue. Alternatively, note that due
to the chiral nature of $A_{B(L(X))}$, its square matrix $A_{B(L(X))}^{2}$
would be block diagonal given by,

$A_{B(L(X))}^{2}=\begin{bmatrix}A_{L(X)}^{2}&0\\\ 0&A_{L(X)}^{2}\end{bmatrix}$
(3)

Now, if $\lambda$ is an eigenvalue of $A_{L(X)}$, $\lambda^{2}$ is an
eigenvalue of $A_{L(X)}^{2}$. From Eqn. (3), it can be easily seen that
$\lambda^{2}$ will also be an eigenvalue of $A_{B(L(X))}^{2}$. Then, using the
property of square-root matrix [47], the eigenvalue set of $A_{B(L(X))}$
should consist of $\\{\pm\lambda\\}$, implying that the spectra of
$A_{B(L(X))}$ has two copies of the spectra of $A_{L(X)}$ with one inverted
with respect to the other. Also, the eigenvectors corresponding to the two
opposite eigenvalues are formed by the bonding ([$v$ ; $v$]) and anti-bonding
([$v$ ;$-v$]) pairing of the individual $L(X)$ eigenvectors. This theorem has
important consequences on the electronic structure of a TB lattice associated
with the BDC of a LG. It guarantees a chiral symmetric band structure having
at least two FBs: one at $E=-2t$ and the other at $E=2t$.

Figure 2: (a) Line-graph of hexagonal lattice ($L(H)$) where gray lines form
the root graph ($H$) while black lines form the Kagome lattice ($L(H)$). Unit-
cell with three sites numbered 1-3 is depicted by gray dashed lines. Green
arrows denote SOC hopping directions. (b) Band structure of (a) without
(dashed blue lines) and with (solid red lines) SOC. (c) BDC construction of
(a). Unit-cell is depicted by gray dashed lines. Green arrows are the
directions of SOC hopping. (d) Band structure of (c) with $t_{0}=0$.

To gain intuition about BDC construction and confirm the existence of two FBs
in band structure, we revisit a well-known TB lattice model based on the LG of
bipartite hexagonal lattice ($H$), i.e., Kagome lattice ($L(H)$), shown in
Fig. 2(a). The energy eigen spectrum of $H_{L(H)}$ is the typical Kagome band
structure (dashed lines in Fig. 2(b)), which has an exact FB at $E=-2t$
without SOC, in accordance with the LG theorem. Considering a spinful TB model
with SOC ($\lambda$), a gap opens at the band touching point ($\Gamma$)
leading to a topologically non-trivial quasi-FB (solid lines in Fig. 2(b) at
$\lambda=0.1t$) with Chern number $C_{\uparrow(\downarrow)}=-1(+1)$. The
stable/strong topology of this FB can be confirmed by ribbon calculations
where there are clear gapless edge states (Fig. S2(a) in the SM [39]).

The BDC of Kagome lattice ($B(L(H))$) is constructed using the tensor product
of $L(H)$ with a two-vertex graph ($K_{2}$) with dark- and light-filled
circles in Fig. 2(c). To maintain the translation symmetry, $K_{2}$ is placed
along specific directions. Note that the first nearest neighbor (NN) hopping
lines in $L(H)$ are transformed into third NN hopping lines in $B(L(H))$. The
TB Hamiltonian can be directly constructed using Eqn. (1), and (2), leading to
chiral symmetric electronic band structure as shown in Fig. 2(d) having two
exact FBs at $E=-2t$ and $E=2t$. However, the two FBs are not near/at charge
neutrality as seen in the chiral limit of TBG or diatomic Kagome lattice [26].
This can be remedied by noting that the hopping between dark and light filled
atomic site in $B(L(H))$ is neglected. This is because the BDC construction
does not have an edge between the vertices of $K_{2}$ (Fig. 1(d)). Physically,
this is the shortest bond length in a lattice and usually has a hopping
integral ($t_{0}$, yellow bonds in Fig. 2(c)) larger than the third NN hopping
($t$). An important observation here is that including $t_{0}$, i.e.,
including an edge between the two vertices of $K_{2}$, will not alter the
chiral symmetry of $A_{B(L(H))}$ since this new edge introduces only a
diagonal matrix element in the off-diagonal block of $A_{B(L(H))}$. Hence, the
spinless TB Hamiltonian of $B(L(H))$ with the inclusion of $t_{0}$ can be
written as,

$H_{B^{\prime}(L(H))}=tA_{B(L(H))}+t_{0}\begin{bmatrix}0&I\\\
I&0\end{bmatrix}.$ (4)

Where $I$ is identity matrix with the dimensions of the basis set of $L(H)$.
The second term in Eqn. 4 implies that the electronic spectrum of root $L(H)$
and of its chiral copy is shifted up- and down-wards by $t_{0}$, respectively,
i.e., the band structure of $B^{\prime}(L(H))$ has two FBs, but now at
$E=-2t+t_{0}$, and $E=2t-t_{0}$.

Figure 3: (a) Band structure of Fig. 2(c) with $t_{0}=2t$ containing two exact
FBs at $E=0$. (b) Band structure of Fig. 2(c) with $t_{0}=3t$. (c) Compact
localized state (CLS) for Kagome FB in Fig. 2(b). (d) CLS for yin-yang valence
(left) and conduction (right) FB in (b) formed by bonding and anti-bonding
pairing of (c) respectively. Blue and red circles indicate the opposite phases
of the wavefunction respectively.

In Fig. 3(a) and 3(b) (dashed lines), we show the band structure obtained by
diagonalizing $H_{B^{\prime}(L(H))}$ with $t_{0}=2t$ and $t_{0}=3t$
respectively. For $t_{0}=2t$, there are two exact FBs completely degenerate at
charge neutrality ($E=0$), while for $t_{0}=3t$, the two FBs are gapped
($E=\pm t$) forming valence and conduction band edges. Thus, the chiral
symmetric band structures are guaranteed by the BDC theorem even with the
inclusion of $t_{0}$. Importantly, both FBs have a band touching point with
another dispersive band at $\Gamma$. Similar to the case of $L(H)$, they can
be gapped out via SOC. A spinful Hamiltonian can be written by adding SOC
effect containing positive (negative) hopping terms along (opposite to) the
directions indicated by green dashed arrows in Fig. 2(c). As seen from the
solid lines in Fig. 3(b), two quasi-FBs emerge with SOC having opposite
chirality given by
$C_{\uparrow(\downarrow)}^{FB_{c}}=-C_{\uparrow(\downarrow)}^{FB_{v}}=-1(+1)$,
where $FB_{v(c)}$ indicates valence (conduction) FB. They are hence called
yin-yang FBs, which have been recently shown to be an ideal platform for
exotic optoelectronic properties, such as excitonic condensation [6], and
giant circular dichroism [53]. We have confirmed the strong topology of these
FBs using ribbon calculations (Fig. S2(b) in SM [39]).

An interesting feature of strong topological FBs is their compact localized
(plaquette) states [31], manifesting a completely localized FB wavefunction
and real-space topology, as shown in Fig. 3(c) for Kagome FB. Outward hopping
from the plaquette vanishes due to opposite phases of the wavefunction on
neighboring sites that cancel each other out, leading to a destructive
interference pattern. Interestingly, each of yin-yang FBs shows also the
destructive interference patterns around a plaquette, but importantly with
bonding and anti-bonding nature for the valence and conduction FB wavefunction
respectively [26], as shown in Fig. 3(d). Here we show that this is a direct
consequence of BDC theorem involving bonding and anti-binding paring of the LG
eigenfunctions.

Figure 4: (a) Kagome-3 lattice - Line-graph of triangular lattice ($L(T)$).
Dashed red circle denotes a Wycoff position. (b) Band structure of (a). (c)
BDC of (a) ($B(L(T))$). (d) Band structure of (c) with $t_{0}=2t$. (e). Wilson
loop spectrum of the four FBs at charge neutrality in (d). (f) Same as (e) but
with two additional $s$-orbitals in $B(L(T)$) at sites corresponding to the
Wycoff position in (a).

Next, we show the BDC construction for fragile topological FBs at charge
neutrality. We consider the LG of non-bipartite triangular ($T$) lattice, so-
called Kagome-3 lattice ($L(T)$) [48, 54] as depicted in Fig. 4(a). Similar to
$L(H)$, the unit-cell of $L(T)$ also contains three sites. The difference
arises in the hopping terms. In $L(T)$, in addition to NN hopping, there are
second and part of third NN hopping terms (Fig. 4(a)) [39], leading to a TB
band structure with two exact FBs at $E=-2t$ shown in Fig. 4(b). The two FBs
are fragile topological, i.e., topological obstruction to Wannier localization
can be resolved by adding a trivial band [55, 56]. Previous studies [48, 54]
have confirmed the fragile topology of these FBs using Wilson loop analysis
[57] which we include in SM [39]. An essential point to note here is that the
topology of these FBs at $E=-2t$ is akin to the single-particle bands of MATBG
centered/at $E=0$ [54]. Following the procedure illustrated above for strong
topological FBs, one can also construct a BDC lattice of $L(T)$ as shown in
Fig. 4(c) in order to obtain fragile LG FBs centered/at $E=0$. The BDC
construction leads to doubled sites in the unit-cell with a six-band TB
Hamiltonian formulated following Eqn. (1) and (2) and the inclusion of $t_{0}$
(Eqn. (4)) [39]. The band-structure of $B(L(T))$ is shown in Fig. 4(d) for
$t_{0}=2t$. There are four exact FBs at charge neutrality, which are fragile
topological. To illustrate this, in Fig. 4(e) we calculate the Wilson loop of
each spin sector separately, since the spinful time-reversal and $S^{z}$
symmetries are conserved. The winding in the Wilson loop spectra of the four
FBs establishes their topological nature, without SOC, with two pairs each
belonging to non-trivial Euler class $|e_{2}|=1$ [58]. We next add trivial
bands to the system by introducing an additional $s$-orbital on Wycoff
position indicated by red circle in Fig. 4(a) [54]. The BDC of this extended
model becomes an eight-band model [39]. The Wilson loop spectra of six middle
bands is shown in Fig. 4(f). The non-trivial winding of FBs is now clearly
removed confirming the fragile topology of these FBs at charge neutrality.

Next, we notice that the BDC-LG lattice construction involves unusual hopping
constraints, such as the absence of second NN hopping term ($t^{\prime}=0$) in
$B(L(H))$ (Fig. 2(c)) while requiring a non-zero third NN hopping integral
($t\neq 0$). Realizing such peculiar hopping integrals in real materials could
prove significantly challenging. On the other hand, most real materials are
composed of multi atomic orbitals on each site instead of single $s$-orbital.
Therefore, we finally illustrate orbital realization of chiral exact FBs of
$B(L(H))$ following the procedure described in ref. [32] that may instigate
real material discovery of these exotic FBs. We first calculate the six
eigenstates at $\Gamma$ of the $B(L(H))$ band structure (Fig. 3(b)) and plot
them in Fig. 5(a). If one groups three sites into a “superatom” site, then the
six sites of $B(L(H))$ can be mapped onto two sites of a hexagonal lattice
with three orbitals on each site. And based on the symmetry of the
eigenstates, six eigenstates in Fig. 5(a) can be viewed as ($s,p_{x},p_{x}$)
orbitals on two hexagonal sites. Then, by choosing the two-center Slater-
Koster hopping integrals [59] as: $t_{ss\sigma}=1eV$,
$t_{sp\sigma}=0.8t_{ss\sigma}$, $t_{pp\sigma}=0.4t_{ss\sigma}$, in a spinless
TB model [39], yin-yang FBs are reproduced as shown by dashed blue lines in
Fig. 5(b). We further add the onsite SOC on $p$ orbitals and consider a
spinful model which opens a gap between dispersive band and FB (red solid
lines in Fig. 5(b). The calculated Chern numbers for two FBs are
$C_{\uparrow(\downarrow)}^{FB_{c}}=-C_{\uparrow(\downarrow)}^{FB_{v}}=-1(+1)$.
Note that the condition $t^{\prime}=0$ in single $s$-orbital model transforms
into the condition $t_{pp\pi}=0$ for $sp^{2}$-hexagonal model. Physically,
$t_{pp\pi}$ is the weakest hopping integral by symmetry. A candidate real
material should be one with $sp^{2}$ frontier orbitals on hexagonal symmetric
lattice but a large lattice parameter which reduces $t_{pp\pi}$. Indeed,
triangulene-based superatomic graphene satisfies this condition [26] and has
also been recently synthesized [60]. Similar orbital design principle can be
used for other BDC-LG constructions [39].

Figure 5: (a) Six $\Gamma$ point wavefunctions of bands denoted by blue dashed
lines in Fig. 3(b). (b) TB band structure of hexagonal lattice model with
$sp^{2}$ orbitals.

The BDC theorem, as derived and illustrated in this work, allows for the
discovery of novel exact FBs at charge neutrality leading to an intrinsic
lattice/material ($B(L(T)$) that can realize strongly correlated physics of
MATBG without the need of external fields or fine-tuning of twist angles.
Moreover, we point to the possibility of an unusual quantum semiconductor in
$B(L(H))$ with valence and conduction FBs (yin-yang FBs), which upon doping
create FB electron and hole carriers with unique transport properties due to
their non-trivial quantum metric and non-zero superfluidic weight [29]. This
can have significant implications for the discovery and design of the beyond-
Silicon electronic and optoelectronic devices. We emphasize that our BDC
design principle works for all LG lattices, both in 2D and 3D [39]. In
addition, it can be easily extended to precursor-based organic materials such
as triangulene crystals [61], and artificial lattice systems [62, 63, 64],
where the hopping parameters can be manipulated by design. One can also
consider weighted graphs with varying nearest neighbor hopping integrals [37]
to realize materials/models with flat bands of opposite chirality. Another
possible future direction could be to investigate into the properties of the
FBs in BDC lattices under external magnetic field [65, 66].

†G.S., and B.X. contributed equally to this work. We acknowledge support from
US Department of Energy (DOE)-Basic Energy Sciences (Grant No. DE-FG02-
04ER46148). All calculations were done on the CHPC at the University of Utah
and NERSC.

## References

  * Cao  _et al._ [2018] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Unconventional superconductivity in magic-angle graphene superlattices, Nature 556, 43 (2018).
  * Balents  _et al._ [2020] L. Balents, C. R. Dean, D. K. Efetov, and A. F. Young, Superconductivity and strong correlations in moiré flat bands, Nature Physics 16, 725 (2020).
  * Park  _et al._ [2021] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene, Nature 590, 249 (2021).
  * Park  _et al._ [2022] J. M. Park, Y. Cao, L.-Q. Xia, S. Sun, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Robust superconductivity in magic-angle multilayer graphene family, Nature Materials 21, 877 (2022).
  * Sethi  _et al._ [2021] G. Sethi, Y. Zhou, L. Zhu, L. Yang, and F. Liu, Flat-band-enabled triplet excitonic insulator in a diatomic kagome lattice, Physical Review Letters 126, 196403 (2021).
  * Sethi  _et al._ [2023] G. Sethi, M. Cuma, and F. Liu, Excitonic condensate in flat valence and conduction bands of opposite chirality, Physical Review Letters 130, 186401 (2023).
  * Sethi  _et al._ [2022] G. Sethi, D. Sheng, and F. Liu, Anomalous bilayer quantum hall effect, arXiv preprint arXiv:2211.04613 (2022).
  * Hu  _et al._ [2022] X. Hu, T. Hyart, D. I. Pikulin, and E. Rossi, Quantum-metric-enabled exciton condensate in double twisted bilayer graphene, Physical Review B 105, L140506 (2022).
  * Liu and Dai [2021] J. Liu and X. Dai, Orbital magnetic states in moiré graphene systems, Nature Reviews Physics 3, 367 (2021).
  * Tschirhart  _et al._ [2021] C. Tschirhart, M. Serlin, H. Polshyn, A. Shragai, Z. Xia, J. Zhu, Y. Zhang, K. Watanabe, T. Taniguchi, M. Huber, _et al._ , Imaging orbital ferromagnetism in a moiré chern insulator, Science 372, 1323 (2021).
  * Andrei  _et al._ [2021] E. Y. Andrei, D. K. Efetov, P. Jarillo-Herrero, A. H. MacDonald, K. F. Mak, T. Senthil, E. Tutuc, A. Yazdani, and A. F. Young, The marvels of moiré materials, Nature Reviews Materials 6, 201 (2021).
  * Zhang  _et al._ [2019] Y.-H. Zhang, D. Mao, Y. Cao, P. Jarillo-Herrero, and T. Senthil, Nearly flat chern bands in moiré superlattices, Physical Review B 99, 075127 (2019).
  * Törmä  _et al._ [2018] P. Törmä, L. Liang, and S. Peotta, Quantum metric and effective mass of a two-body bound state in a flat band, Physical Review B 98, 220511 (2018).
  * Kwan  _et al._ [2021] Y. H. Kwan, Y. Hu, S. H. Simon, and S. Parameswaran, Exciton band topology in spontaneous quantum anomalous hall insulators: Applications to twisted bilayer graphene, Physical Review Letters 126, 137601 (2021).
  * Song and Bernevig [2022] Z.-D. Song and B. A. Bernevig, Magic-angle twisted bilayer graphene as a topological heavy fermion problem, Physical review letters 129, 047601 (2022).
  * Kennes  _et al._ [2021] D. M. Kennes, M. Claassen, L. Xian, A. Georges, A. J. Millis, J. Hone, C. R. Dean, D. Basov, A. N. Pasupathy, and A. Rubio, Moiré heterostructures as a condensed-matter quantum simulator, 17, 155 (2021).
  * Mak and Shan [2022] K. F. Mak and J. Shan, Semiconductor moiré materials, Nature Nanotechnology 17, 686 (2022).
  * Naik  _et al._ [2022] M. H. Naik, E. C. Regan, Z. Zhang, Y.-H. Chan, Z. Li, D. Wang, Y. Yoon, C. S. Ong, W. Zhao, S. Zhao, _et al._ , Intralayer charge-transfer moiré excitons in van der waals superlattices, Nature 609, 52 (2022).
  * Wu  _et al._ [2018] F. Wu, T. Lovorn, E. Tutuc, and A. H. MacDonald, Hubbard model physics in transition metal dichalcogenide moiré bands, Physical review letters 121, 026402 (2018).
  * Bistritzer and MacDonald [2011] R. Bistritzer and A. H. MacDonald, Moiré bands in twisted double-layer graphene, Proceedings of the National Academy of Sciences 108, 12233 (2011).
  * Dos Santos  _et al._ [2012] J. L. Dos Santos, N. Peres, and A. C. Neto, Continuum model of the twisted graphene bilayer, Physical review B 86, 155449 (2012).
  * Tarnopolsky  _et al._ [2019] G. Tarnopolsky, A. J. Kruchkov, and A. Vishwanath, Origin of magic angles in twisted bilayer graphene, Physical review letters 122, 106405 (2019).
  * Wan  _et al._ [2023] X. Wan, S. Sarkar, S.-Z. Lin, and K. Sun, Topological exact flat bands in two-dimensional materials under periodic strain, Physical Review Letters 130, 216401 (2023).
  * Li  _et al._ [2022] M.-R. Li, A.-L. He, and H. Yao, Magic-angle twisted bilayer systems with quadratic band touching: Exactly flat bands with high chern number, Physical Review Research 4, 043151 (2022).
  * Gao  _et al._ [2022] Q. Gao, J. Dong, P. Ledwith, D. Parker, and E. Khalaf, Untwisting moiré physics: Almost ideal bands and fractional chern insulators in periodically strained monolayer graphene, arXiv e-prints , arXiv (2022).
  * Zhou  _et al._ [2022] Y. Zhou, G. Sethi, H. Liu, Z. Wang, and F. Liu, Excited quantum anomalous and spin hall effect: dissociation of flat-bands-enabled excitonic insulator state, Nanotechnology 33, 415001 (2022).
  * Zhou  _et al._ [2019a] Y. Zhou, G. Sethi, H. Liu, Z. Wang, and F. Liu, Excited quantum hall effect: enantiomorphic flat bands in a yin-yang kagome lattice, arXiv preprint arXiv:1908.03689 (2019a).
  * Ni  _et al._ [2020] X. Ni, Y. Zhou, G. Sethi, and F. Liu, $\pi$-orbital yin–yang kagome bands in anilato-based metal–organic frameworks, Physical Chemistry Chemical Physics 22, 25827 (2020).
  * Törmä  _et al._ [2021] P. Törmä, S. Peotta, and B. A. Bernevig, Superfluidity and quantum geometry in twisted multilayer systems, arXiv preprint arXiv:2111.00807 (2021).
  * Herzog-Arbeitman  _et al._ [2022] J. Herzog-Arbeitman, V. Peri, F. Schindler, S. D. Huber, and B. A. Bernevig, Superfluid weight bounds from symmetry and quantum geometry in flat bands, Physical review letters 128, 087002 (2022).
  * Liu  _et al._ [2014] Z. Liu, F. Liu, and Y.-S. Wu, Exotic electronic states in the world of flat bands: From theory to material, Chinese Physics B 23, 077308 (2014).
  * Liu  _et al._ [2022] H. Liu, G. Sethi, S. Meng, and F. Liu, Orbital design of flat bands in non-line-graph lattices via line-graph wave functions, Physical Review B 105, 085128 (2022).
  * Liu  _et al._ [2021] H. Liu, S. Meng, and F. Liu, Screening two-dimensional materials with topological flat bands, Physical Review Materials 5, 084203 (2021).
  * Mielke [1991] A. Mielke, Ferromagnetism in the hubbard model on line graphs and further considerations, Journal of Physics A: Mathematical and General 24, 3311 (1991).
  * Ma  _et al._ [2020] D.-S. Ma, Y. Xu, C. S. Chiu, N. Regnault, A. A. Houck, Z. Song, and B. A. Bernevig, Spin-orbit-induced topological flat bands in line and split graphs of bipartite lattices, Physical review letters 125, 266403 (2020).
  * Cvetkovic  _et al._ [2004] D. Cvetkovic, P. Rowlinson, and S. Simic,  _Spectral generalizations of line graphs: On graphs with least eigenvalue-2_ , Vol. 314 (Cambridge University Press, 2004).
  * Kollár  _et al._ [2020] A. J. Kollár, M. Fitzpatrick, P. Sarnak, and A. A. Houck, Line-graph lattices: Euclidean and non-euclidean flat bands, and implementations in circuit quantum electrodynamics, Communications in Mathematical Physics 376, 1909 (2020).
  * Brent West [2001] D. Brent West,  _Introduction to Graph Theory, vol. 2_ (Upper Saddle River, NJ: Prentice Hall, 2001).
  * [39] See supplementary material where we have included discussion regarding the relationship between incidence matrices and square-root TB models, explicit TB hamiltonians for the cases studied in the manuscript, details about the wilson loop calculations, and examples of BDC construction using 2D/3D line-graph lattices.
  * Cvetkovic  _et al._ [1980] D. Cvetkovic, M. Doob, and H. Sachs,  _Spectra of graphs. Theory and application_ (Academic Press, 1980).
  * Arkinstall  _et al._ [2017] J. Arkinstall, M. Teimourpour, L. Feng, R. El-Ganainy, and H. Schomerus, Topological tight-binding models from nontrivial square roots, Physical Review B 95, 165109 (2017).
  * Mizoguchi  _et al._ [2021] T. Mizoguchi, T. Yoshida, and Y. Hatsugai, Square-root topological semimetals, Physical Review B 103, 045136 (2021).
  * Song  _et al._ [2020] Z.-D. Song, L. Elcoro, and B. A. Bernevig, Twisted bulk-boundary correspondence of fragile topology, Science 367, 794 (2020).
  * Alexandradinata  _et al._ [2014] A. Alexandradinata, X. Dai, and B. A. Bernevig, Wilson-loop characterization of inversion-symmetric topological insulators, Physical Review B 89, 155114 (2014).
  * Zhou  _et al._ [2019b] Y. Zhou, K.-H. Jin, H. Huang, Z. Wang, and F. Liu, Weyl points created by a three-dimensional flat band, Physical Review B 99, 201105 (2019b).
  * Lieb [1989] E. H. Lieb, Two theorems on the hubbard model, Physical review letters 62, 1201 (1989).
  * Ezawa [2020] M. Ezawa, Systematic construction of square-root topological insulators and superconductors, Physical Review Research 2, 033397 (2020).
  * Chiu  _et al._ [2020] C. S. Chiu, D.-S. Ma, Z.-D. Song, B. A. Bernevig, and A. A. Houck, Fragile topology in line-graph lattices with two, three, or four gapped flat bands, Physical Review Research 2, 043414 (2020).
  * Bergman  _et al._ [2008] D. L. Bergman, C. Wu, and L. Balents, Band touching from real-space topology in frustrated hopping models, Physical Review B 78, 125104 (2008).
  * Sunada [2012] T. Sunada,  _Topological crystallography: with a view towards discrete geometric analysis_ , Vol. 6 (Springer Science & Business Media, 2012).
  * Asratian  _et al._ [1998] A. S. Asratian, T. M. Denley, and R. Häggkvist,  _Bipartite graphs and their applications_ , Vol. 131 (Cambridge university press, 1998).
  * Biggs [1993] N. Biggs,  _Algebraic graph theory_ , 67 (Cambridge university press, 1993).
  * Zhou  _et al._ [2020] Y. Zhou, G. Sethi, C. Zhang, X. Ni, and F. Liu, Giant intrinsic circular dichroism of enantiomorphic flat chern bands and flatband devices, Physical Review B 102, 125115 (2020).
  * Peri  _et al._ [2021] V. Peri, Z.-D. Song, B. A. Bernevig, and S. D. Huber, Fragile topology and flat-band superconductivity in the strong-coupling regime, Physical review letters 126, 027002 (2021).
  * Po  _et al._ [2019] H. C. Po, L. Zou, T. Senthil, and A. Vishwanath, Faithful tight-binding models and fragile topology of magic-angle bilayer graphene, Physical Review B 99, 195455 (2019).
  * Po  _et al._ [2018] H. C. Po, H. Watanabe, and A. Vishwanath, Fragile topology and wannier obstructions, Physical review letters 121, 126402 (2018).
  * Bradlyn  _et al._ [2019] B. Bradlyn, Z. Wang, J. Cano, and B. A. Bernevig, Disconnected elementary band representations, fragile topology, and wilson loops as topological indices: An example on the triangular lattice, Physical Review B 99, 045140 (2019).
  * Ahn  _et al._ [2019] J. Ahn, S. Park, and B.-J. 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, Physical Review X 9, 021013 (2019).
  * Slater and Koster [1954] J. C. Slater and G. F. Koster, Simplified lcao method for the periodic potential problem, Physical review 94, 1498 (1954).
  * Delgado  _et al._ [2023] A. Delgado, C. Dusold, J. Jiang, A. Cronin, S. G. Louie, and F. R. Fischer, Evidence for excitonic insulator ground state in triangulene kagome lattice, arXiv preprint arXiv:2301.06171 (2023).
  * Ortiz  _et al._ [2022] R. Ortiz, G. Catarina, and J. Fernández-Rossier, Theory of triangulene two-dimensional crystals, 2D Materials 10, 015015 (2022).
  * Dong  _et al._ [2021] K. Dong, T. Zhang, J. Li, Q. Wang, F. Yang, Y. Rho, D. Wang, C. P. Grigoropoulos, J. Wu, and J. Yao, Flat bands in magic-angle bilayer photonic crystals at small twists, Physical review letters 126, 223601 (2021).
  * Leykam  _et al._ [2018] D. Leykam, A. Andreanov, and S. Flach, Artificial flat band systems: from lattice models to experiments, Advances in Physics: X 3, 1473052 (2018).
  * Apaja  _et al._ [2010] V. Apaja, M. Hyrkäs, and M. Manninen, Flat bands, dirac cones, and atom dynamics in an optical lattice, Physical Review A 82, 041402 (2010).
  * Aoki  _et al._ [1996] H. Aoki, M. Ando, and H. Matsumura, Hofstadter butterflies for flat bands, Physical Review B 54, R17296 (1996).
  * Marques  _et al._ [2023] A. Marques, J. Mögerle, G. Pelegrí, S. Flannigan, R. Dias, and A. Daley, Kaleidoscopes of hofstadter butterflies and aharonov-bohm caging from 2 n-root topology in decorated square lattices, Physical Review Research 5, 023110 (2023).