Isolated flat bands in 2D lattices based on a path-exchange symmetry

Jun-Hyung Bae1, Tigran Sedrakyan2, Saurabh Maiti1,3

1 Department of Physics, Concordia University, Montreal, QC H4B 1R6, Canada

2 Department of Physics, University of Massachusetts, Amherst, MA 01003, USA

3 Centre for Research in Molecular Modelling, Concordia University, Montreal,
QC H4B 1R6, Canada

## Abstract

The increased ability to engineer two-dimensional (2D) systems, either using
materials, photonic lattices, or cold atoms, has led to the search for 2D
structures with interesting properties. One such property is the presence of
flat bands. Typically, the presence of these requires long-ranged hoppings,
fine-tuning of nearest neighbor hoppings, or breaking time-reversal symmetry
by using a staggered flux distribution in the unit cell. We provide a
prescription based on carrying out projections from a parent system to
generate different flat band systems. We identify the conditions for
maintaining the flatness and identify a path-exchange symmetry in such systems
that cause the flat band to be degenerate with the other dispersive ones.
Breaking this symmetry leads to lifting the degeneracy while still preserving
the flatness of the band. This technique does not require changing the
topology nor breaking time-reversal symmetry as was suggested earlier in the
literature. The prescription also eliminates the need for any fine-tuning.
Moreover, it is shown that the subsequent projected systems inherit the
precise fine-tuning conditions that were discussed in the literature for
similar systems, in order to have and isolate a flat band. As examples, we
demonstrate the use of our prescription to arrive at the flat band conditions
for popular systems like the Kagomé, the Lieb, and the Dice lattices. Finally,
we are also able to show that a flat band exists in a recently proposed chiral
spin-liquid state of the Kagomé lattice only if it is associated with a gauge
field that produces a flux modulation of the Chern-Simons type.

###### Contents

  1. 1 Introduction
  2. 2 Kagomé: naïve attempts to lift the flat band degeneracy
    1. 2.1 Onsite perturbations and strain
    2. 2.2 Breaking TRS
  3. 3 Kagomé: Flat band preserving parameterization
    1. 3.1 Physical meaning behind the $r$-parameter
    2. 3.2 Model with $|r|>1$
  4. 4 A prescription to generate flat band systems
    1. 4.1 Flat bands beyond the bipartite condition
  5. 5 Isolating the flat band
  6. 6 Application of the prescription to other lattices
    1. 6.1 Lieb lattice and its projections
    2. 6.2 Dice lattice and its projections
  7. 7 Flat band with Chern-Simons flux distribution
  8. 8 Conclusion
  9. A Useful relations between matrix elements $\tilde{\alpha}_{i}$
  10. B Properties of non-square matrices
  11. C Identifying the path-exchange symmetry

##  1 Introduction

The term ‘flat band systems’ has recently attracted a lot of attention. There
are at least two contexts in which this term is used. The first, and probably
the more popular, is in the context of Moiré bands[1, 2] in twisted, layered
Vanderwaals systems (epitomized by twisted bi-layer Graphene[3, 4, 5, 6, 7, 8,
9]) where the multiple band-foldings due to enlarging of the unit-cell results
in bands which can have significant regions in the Brillouin zone (BZ) where
they disperse very weakly. This leads to an enhanced density of states, and if
the chemical potential is around this region, many of the physics of itinerant
electrons manifest themselves as a strongly correlated problem as the relevant
dimensionless parameter $\nu_{F}U$ (where $\nu_{F}$ is the density of states
at the Fermi surface and $U$ is some scale of interaction in the problem)
could be made large even for small $U$. Another aspect driving the system
towards strong correlation physics is the fact that the competition from the
kinetic energy (which is characterized by the dispersiveness of a band) falls
off due to the reduction of the bandwidth.

The second context in which the term ‘flat band’ is used is in the technically
strict sense where systems have perfectly flat dispersionless bands. Some
systems that are popularly discussed are the Kagomé lattice[10], the Lieb
lattice[11], and the Dice lattice[12]. While there aren’t any natural systems
with these specific lattice structures, some of them can be realized in cross-
sections of crystals,[13] while some could be artificially engineered[14, 15,
16, 17, 18]. It should be noted that these systems have a perfectly flat band
within the nearest neighbor (nn) approximation. Beyond the nn, the flatness is
disturbed, of course, and the flat band acquires a bandwidth that is generally
still much smaller than that of the ‘flat bands’ in the Vanderwaals systems.
In this work, we reserve the term ‘flat’ for this second context in the rest
of this article.

Investigating flat bands is of fundamental interest[19] for a variety of
reasons: it offers novel perspectives on topology[20, 21]; if they are
topological, then it is expected that exotic physics of the fractional Quantum
Hall effect could be observed in zero-field and at high temperatures[22]; a
universal low-energy behavior that is different from the Fermi liquid can be
expected as in the theory of the half-filled flat Landau level[23, 24, 25];
spin-liquid and chiral spin-liquid behaviors are associated with the presence
of a flat energy manifold of excitations and serves as a platform to explore
the role of Chern-Simons gauge field [26, 27, 28, 29, 30]; the presence of a
flat band serves as a possible resolution to the fermion-doubling problem in
lattice-based field theories;[31] to name a few. Perhaps the most intriguing
yet achievable application of isolated flat band systems would be exploring
the physics of the Sachdev-Ye-Kitaev (SYK) model [32, 33, 34]: e.g.,
introducing disorder leads to maximal chaos exhibiting black-hole like
behavior (finite entropy at zero temperature) for which there are already
various proposals for implementation [35, 36, 37, 38, 39, 40]. However, many
of these interesting effects only manifest themselves if the flat band is
isolated (gapped) from the rest of the system. It is thus desirable to have a
design prescription that achieves precisely this.

This desire has certainly been recognized by many. Investigation into the
existence of the flat band itself revealed that the flat band would be
degenerate with dispersive bands, with the degeneracy being protected by
topology [41]. It was suggested in Ref. [42] that breaking Time Reversal
Symmetry (TRS) was crucial to isolate the flat bands from the dispersive ones
by considering staggered fluxes through the unit cell. In a sequence of works
[43, 44, 45], it was demonstrated that breaking TRS was not necessary, but one
would need to fine-tune the system using compact localized states for
destructive interference of the electronic states, which would lead to a
dispersionless band. There are general considerations from permutation
symmetries in graph theory[46] and latent symmetries (associated with
destructive interference across certain paths)[47] that can also explain the
formation of flat bands on general grounds in general lattices. Certain
special symmetry properties of the Hamiltonian (antiunitary-Parity-Time) can
also lead to flat bands[48]. Reference [49] presented an interesting
parameterization of the Kagomé lattice that also successfully isolated the
flat band without the consideration of any special symmetries except what the
authors identified as inversion [we shall show later (Sec. 5) that it does not
have to do much with inversion]. Some works explore the possibility of having
a flat band on general grounds, but they either require precise long-ranged
hoppings[50] (which is not so desirable in material systems nor in photonic
lattices nor ultra-cold atoms) or non-hermitian matrices[51].

In this work, we add to the existing body of literature and show that it is
possible to have isolated flat bands without breaking TRS, without using long-
ranged hopping, without losing hermiticity, and without fine-tuning a system.
At this point, we note that very recently in Refs [52, 53], the authors
presented a comprehensive analysis on the design of flat bands achieving the
same goals of avoiding long-range hopping and fine-tuning. It provides an
elegant and robust generalization of our results below, along with the
topological classification of these bands. We encourage the reader to refer to
this work for a more detailed analysis. While the fundamental requirements for
both our works remain similar, we provide an alternative perspective through a
focus on providing precise prescriptions to isolate a degenerate flat band and
investigate the role of gauge fields in achieving the same. We start from a
sufficient condition for the flatband to exist and arrive at a sufficient
condition to isolate (gap out) the flatband. The distinguishing feature of our
approach is that while the previous works focus on the properties of the
Hamiltonian with a flat band, our method involves arriving at flat band
systems by performing projections from a ‘parent system.’ In fact, we show
that talking about symmetries of the parent system allows for a simpler
interpretation of the flat band in the projected systems. To emphasize this
point, we first begin with the Kagomé lattice and show that naive attempts to
isolate the flat band (via different onsite energies and applying strain)
destroy the flat band. However, in addition to already existing prescriptions,
we were able to identify another parameterization that preserves the flat band
for all ranges of the parameter. But this technique (and the others discussed
in the literature) often requires a very specific relationship between various
hopping parameters. We then present our main result where we introduce a
parent system with certain special properties that guarantees a flat band. And
upon performing projections (to be detailed later in the text), we show that
the projected systems automatically inherit the various conditions presented
earlier for the existence of and isolation of the flat band.

We find this condition by first exploring bipartite systems with different
system sizes and using the fact that such a system has the number of flat
bands equal to the difference in the size of the subsystems[12], thus
establishing a sufficient (but not necessary) condition to have flat bands. We
then perform a Hilbert-space projection to project out the smaller subsystem,
and we show that the larger subsystem will necessarily have flat bands. We
explicitly demonstrate that our earlier parameterization of the Kagomé lattice
and also some cases discussed earlier in the literature[49] are a special case
of this projection prescription.

We also identify a symmetry associated with the bipartite system (and not the
projected systems), which, when broken, isolates the flat band in the main
system and its projected subsystems. We refer to this as a path-exchange
symmetry with respect to a property which is the set ratios of the paths (in
this case hoppings) from different components of one subsystem to the other.
If this set of ratios is the same for two atoms of the larger subsystem, then
the whole system is said to have an exchangeable path between the subsystems.
We show that the number of exchangeable paths directly controls the degree of
degeneracy of the flat band with other dispersive bands. This more fundamental
symmetry can map to spatial symmetries like mirror and inversion under special
conditions and does not necessarily translate to something simple in the
projected system and probably explains why there exist so many different
attempts to understand the origin of the flat band in various systems. We then
relax the bipartite condition in the subsystem that is being projected out and
show that our main results still hold.

We also apply our prescription to the Lieb and Dice lattices and demonstrate
the existence of other lattice structures where flat bands are present and can
be isolated by breaking the path-exchange symmetry. In none of these cases
where we isolate the flat band do we break time-reversal symmetry as was
required in the staggered flux technique of Ref. [42]. Finally, as an
application of our prescription, we are able to present scenarios where the
hexagonal lattice (such as Graphene) or the chequerboard-like square lattice
could also have a flat band. Since our prescription includes carrying out
projections from a nn model, we believe that this prescription should be
amenable to realization in photonic lattices and circuit QED systems[54, 55,
56].

As a closing example, we consider the case of a chiral spin-liquid state of
fermionized bosons in Kagomé lattice subject to a Chern-Simon’s (CS) field. It
was proposed that this state has a flux distribution that would preserve the
flat band as opposed to the situation with the regular flux which grows with
the area[27]. We apply our results in the presence of a regular Maxwellian
flux and show that our projection procedure exactly reproduces the proposed CS
flux distribution. The presence of the flat band in such a system was seen as
a consequence of the characteristic flux distribution within the unit cell.
From our work we now understand that the flat band is actually guaranteed from
a more fundamental level.

The rest of the text is organized as follows. In Section 2 we introduce the
Kagomé lattice and show that the common ideas to modify the lattice ends up
disturbing the flat band. In Section 3, we introduce our parameterization that
preserves the flat band and discuss the physical meaning of the
parameterization. In Section 4 we introduce the projection prescription in
terms of bi-partite systems to generate flat bands. In Section 5 we identify
the path-exchange symmetry in our bi-partite systems, which upon being broken,
isolates the flat band. We then show that the bipartite condition is not a
strict requirement. In Section 6, we demonstrate all of the above ideas in the
Lieb and Dice lattices. In Section 7, we demonstrate why the propsoed Chern-
Simons type flux distribution on a Kagomé lattice guarantees a flat band,
whereas the usual Maxwellian flux distribution does not. Finally, we summarize
our results in Section 8. The Appendix includes some details and proofs that
did not find their place in the main text.

##  2 Kagomé: naïve attempts to lift the flat band degeneracy

We start by considering the Hamiltonian for the Kagomé lattice within a tight-
binding model with only nn hoppings:

$\displaystyle H_{\rm Kg}$ $\displaystyle=$
$\displaystyle-t\begin{pmatrix}0&1+e^{-i\vec{k}\cdot\vec{R}_{1}}&1+e^{-i\vec{k}\cdot\vec{R}_{2}}\\\
1+e^{i\vec{k}\cdot\vec{R}_{1}}&0&1+e^{-i\vec{k}\cdot\vec{R}_{3}}\\\
1+e^{i\vec{k}\cdot\vec{R}_{2}}&1+e^{i\vec{k}\cdot\vec{R}_{3}}&0\end{pmatrix},$

where $t$ is the nn hopping matrix element, $\vec{R}_{1}=(1,0)$,
$\vec{R}_{2}=(\frac{1}{2},\frac{\sqrt{3}}{2})$ are the translation vectors of
the lattice, $\vec{R}_{3}=\vec{R}_{2}-\vec{R}_{1}$, and $\vec{k}$ is the Bloch
momentum in the first Brillouin zone (fBZ) and is made dimensionless by
absorbing the lattice constant $a$. This Hamiltonian is written in the basis
$\hat{\Psi}_{\vec{k}}=(\hat{c}_{\vec{k},A},\hat{c}_{\vec{k},B},\hat{c}_{\vec{k},C})^{T}$,
where $A,B,C$ are the three atoms within the unit cell [see Fig. 1(a)]. The
spectrum contains a flat band as shown in Fig. 1(b).

(a)

(b)

Figure 1:  (a) Kagomé lattice with three atoms $A,B,C$ in the unit cell and
the translation vectors $\vec{R}_{1}$ and $\vec{R}_{2}$. The brown
parallelogram is a unit cell (b) The energy spectrum for the Kagome lattice.
Note the presence of the flat band that is degenerate with the dispersing
middle band at the $\Gamma$-point (0,0).

In order to lift the degeneracy at the $\Gamma$-point, one could envision
multiple ways to perturb the system: break the similarity of $A,B,C$ (sub-
lattice symmetry), break or lower some translational or point-group symmetry,
or break Time-reversal symmetry (TRS). We shall briefly demonstrate below that
while the standard ways to apply these perturbations may lift the degeneracy
at the $\Gamma$-point, they also destroy the flatness of the band. Further,
the degeneracy point sometimes just gets moved to other points in the fBZ.

###  2.1 Onsite perturbations and strain

Consider first making the three atoms different by subjecting them to
different on-site potentials. To model this, one could add the following term
to the Hamiltonian: $\delta H_{\rm site}=-t~{}{\rm Diag}(0,\Delta,-\Delta)$.
For $\Delta\ll 1$, the $\Gamma$-point eigenvalues are
$t\left(2\pm\frac{\Delta}{\sqrt{3}}\right)+\mathcal{O}(\Delta^{2})$ and
$-4t+\mathcal{O}(\Delta^{2})$. However, as seen in Fig. 2(a), numerical
diagonalization shows that this condition just splits the quadratic
$\Gamma$-point degeneracy to two Dirac points. So the degeneracy of the bands
is not really lifted. This also destroys the flat band.

(a)

(b)

Figure 2:  (a) Spectrum for the Kagomé system with different onsite energies
($\Delta=0.5$). The $\Gamma$-point degeneracy is lifted, but it splits into
two Dirac points at different $k$ values. The flatness of the flat band is
also lost. (b) Spectrum for Kagomé lattice under a uniaxial strain along the
diagonal that intersects the $B-C$ bond in Fig. 1a. Once again, the flat band
is lost. Here $\delta t=0.5$.

One can also consider modeling the effect of the strain. For simplicity,
consider a uniaxial strain applied to the system. This would have two effects
on the system: alter the translation vectors and alter the hoppings. The
change in translation vectors will only act as a change in “gauge” in the
$k$-space [57, 58, 59, 60] and hence not really alter the qualitative aspects
of the spectrum. The change in hoppings alters the symmetry properties of the
$3\times 3$ Bloch Hamiltonian. For the orientation shown in Fig. 1(a),
applying a strain along the diagonal that intersects the $B-C$ bonds would
result in the following Hamiltonian:

$H_{\rm Kg,strain}=-\begin{pmatrix}0&(t-\delta
t)(1+e^{-i\vec{k}\cdot\vec{R}_{1}})&(t-\delta
t)(1+e^{-i\vec{k}\cdot\vec{R}_{2}})\\\ (t-\delta
t)(1+e^{i\vec{k}\cdot\vec{R}_{1}})&0&(t+\delta
t)(1+e^{-i\vec{k}\cdot\vec{R}_{3}})\\\ (t-\delta
t)(1+e^{i\vec{k}\cdot\vec{R}_{2}})&(t+\delta
t)(1+e^{i\vec{k}\cdot\vec{R}_{3})}&0\end{pmatrix}$ (2)

The full spectrum plotted in Fig. 2(b) shows that the flat band is lost.
Moreover, the quadratic touching point shifts away from the $\Gamma$-point as
it splits into two Dirac points. This splitting of the $\Gamma$-point
degeneracy into Dirac points is the same phenomenon that was discussed in
Refs. [61, 62], although not in the context of strain.

###  2.2 Breaking TRS

In another attempt to lift the degeneracy of the flat band, we may also
consider breaking TRS by applying an out-of-plane magnetic field to the
system. Since we are working within a spin-less model, the only effect of the
magnetic field would be the orbital effect. We address this by constructing a
Hofstadter model for the Kagome lattice (see, e.g. [63]. In Fig. 3 we show, as
a representative case, the spectrum for a flux per unit cell of $\pi\phi_{0}$
(where $\phi_{0}=\frac{h}{e}$ is the single electron flux quantum) and that
the flat band becomes dispersive. There are, however, ways to break TRS and
still preserve the flat band. This requires using staggered flux as introduced
in Ref. [42] or a Chern-Simons flux (which is focused through one part of the
unit cell) like in Ref. [27], and we shall not dwell on this point as it is
beyond the scope of this work.

(a)

(b)

Figure 3:  (a) Energy bands of the Kagomé lattice with $\pi\phi_{0}$ flux
distributed uniformly in the unit cell. (b) A zoomed-in version of the top
bands showing the dispersive nature. This remains true for any other uniform
flux linked to the unit cell.

##  3 Kagomé: Flat band preserving parameterization

While it seems that any perturbation we apply to the system lifts the flatness
of the band, one may wonder if there could be a parameterization that
preserves the flatness of the band. To tackle this question, let us consider
the Kagomé Hamiltonian in the generic form

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle-t\left(\begin{array}[]{ccc}0&\alpha_{1}&\alpha_{2}\\\
\alpha_{1}^{*}&0&\alpha_{3}\\\
\alpha^{*}_{2}&\alpha^{*}_{3}&0\end{array}\right).$ (6)

The eigenvalues are the roots of the equation:

$\displaystyle-(E/t)^{3}+(E/t)\left(|\alpha_{1}|^{2}+|\alpha_{2}|^{2}+|\alpha_{3}|^{2}\right)-~{}~{}~{}~{}$
$\displaystyle 2{\rm Re}\left[\alpha_{1}^{*}\alpha_{2}\alpha^{*}_{3}\right]=0$
. (7)

To have a flat band at $E/t=f$ (independent of $\vec{k}$), we necessarily need
the $\vec{k}$-dependence of $\alpha_{i}$ to be such that for all $\vec{k}\in$
fBZ

$|\alpha_{1}|^{2}+|\alpha_{2}|^{2}+|\alpha_{3}|^{2}=\frac{2{\rm
Re}\left[\alpha_{1}^{*}\alpha_{2}\alpha^{*}_{3}\right]}{f}+f^{2}.$ (8)

In fact, if such a flat band were to exist, then Eq. (8) would have to be
satisfied for some real value of $f$. Plugging this into the characteristic
equation leads us to

$-(E/t)^{3}+(E/t)\left(\frac{A}{f}+f^{2}\right)-A=0,$ (9)

where $A\equiv 2{\rm Re}\left[\alpha_{1}^{*}\alpha_{2}\alpha^{*}_{3}\right]$.
The eigenvalues would then be

$\displaystyle E_{0}$ $\displaystyle=$ $\displaystyle tf;$ $\displaystyle
E_{+}$ $\displaystyle=$
$\displaystyle\frac{tf}{2}\left(-1+\sqrt{1+4A/f^{3}}\right);$ $\displaystyle
E_{-}$ $\displaystyle=$
$\displaystyle\frac{tf}{2}\left(-1-\sqrt{1+4A/f^{3}}\right).$ (10)

However, if there exists no such $f$, then the above expressions are no longer
the solution. One could then ask what are the conditions for which $f$ could
exist or equivalently, under what conditions would Eq. (8) be satisfied. Note
that this equation presents one constraint on the parameters of this equation
which are $\alpha_{i}$ ($i\in\\{1,2,3\\}$). At this stage, it might seem like
one should always be able to satisfy this. However, note that the
$\alpha_{i}$’s are, in turn, functions of $k_{x},k_{y}$ and $f$. The $k_{i}$’s
are subject to the condition that they must be bound to the fBZ. But given
that the $k_{i}$’s appear as arguments of periodic functions, this bound is
not really an additional constraint. But, the $\alpha_{i}$’s themselves are
not independent. In fact, the Kagomé Hamiltonian’s structure ensures that
$(\alpha_{1}-1)^{*}(\alpha_{2}-1)=\alpha_{3}-1$, which are two more
constraints on the parameters. Thus, we have a situation with 3 constraints
and 3 parameters. Since they are not linear, they may or may not be satisfied
in general.

For completeness, we could ask if any of the dispersive bands ($E_{\pm}$)
could intersect the flat band $E_{0}$. Setting them equal to each other
immediately establishes that for real values of $A$ and $f$, only $E_{+}$
could intersect with $E_{0}$ and this would happen at those $\vec{k}$-points
where

$2{\rm Re}\left[\alpha_{1}^{*}\alpha_{2}\alpha^{*}_{3}\right]\equiv A=2f^{3}.$
(11)

In fact, for the case of the Kagomé lattice, we note that
$\alpha_{i}=1+e^{-i\vec{k}\cdot\vec{R}_{i}}$ and Eq. (8) is satisfied for
$f=2$ and for all $\vec{k}$, establishing the condition for the flat band.
Further, Eq. (11) is also satisfied only at the $\Gamma$-point indicating that
the flat band would be degenerate with the dispersive band at the
$\Gamma$-point.

One may now wonder if there are other scenarios, other than the Kagomé
lattice, where Eq. (8) could be satisfied. The answer is affirmative and a
family of scenarios is presented below. Consider the modification where
$\alpha_{i}$ is changed from $1+e^{-i\vec{k}\cdot\vec{R}_{i}}$ to
$(1+r)+(1-r)e^{-i\vec{k}\cdot\vec{R}_{i}}$, with $|r|<1$. The physical meaning
of the $r$-parameter will be discussed in a subsequent section. In fact, one
could factor out $1+r$ and absorb it into the hopping element $t$ yielding the
following transformation $t\rightarrow\tilde{t}=t(1+r)$, and

$\alpha_{i}\rightarrow\tilde{\alpha}_{i}=1+\frac{1-r}{1+r}e^{-i\vec{k}\cdot\vec{R}_{i}}=1+e^{-i\vec{k}\cdot\vec{R}_{i}-h},$

where $h$ is defined through the equation $r=\tanh\frac{h}{2}$. This allows us
to express

$\tilde{\alpha}_{i}=2\cos\left(\frac{\vec{k}\cdot\vec{R}_{i}}{2}-\frac{ih}{2}\right)e^{-i\frac{\vec{k}\cdot\vec{R}_{i}}{2}-\frac{h}{2}}.$

From these definitions, we observe that the flat band condition of Eq. (8)

$\displaystyle\sum_{i=3}|\tilde{\alpha}_{i}|^{2}$ $\displaystyle=$
$\displaystyle\frac{2{\rm
Re}[\tilde{\alpha}_{1}^{*}\tilde{\alpha}_{2}\tilde{\alpha}_{3}^{*}]}{f}+f^{2}$
(12)

can now be satisfied with $f=2e^{-\frac{h}{2}}\cosh\frac{h}{2}$ (see Appendix
A for details). The spectrum is given by Eq. (3) but with
$\alpha_{i}\rightarrow\tilde{\alpha}_{i}$ and $t\rightarrow\tilde{t}$. The
condition for band-touchings also remains the same as Eq. (11) but with
$\alpha_{i}\rightarrow\tilde{\alpha}_{i}$. A direct evaluation shows that if
the condition is satisfied for $\alpha_{i}$, then it is automatically
satisfied for $\tilde{\alpha}_{i}$. This means that the flat band remains flat
at $E=\tilde{t}f=2\tilde{t}e^{-\frac{h}{2}}\cosh\frac{h}{2}=2t$ . The
introduction of the parameter $r$ neither lifts nor moves the degeneracy at
the $\Gamma$-point. In fact, observe that when $r\rightarrow 0$,
$\tilde{\alpha}_{i}\rightarrow\alpha_{i}$, and the formulae smoothly connect
to the original Kagome lattice. Thus, we have a family of flat band systems
characterized by the parameter $r$.

The spectra for the modified lattices are shown in Fig. 4 for various values
of $r$. As observed above, for any value of $r$, the flat band is preserved.
Although the Dirac point at the $K$ and $K^{\prime}$ points of the fBZ are
gapped out, the $r$ parameter preserves the degeneracy at the $\Gamma$-point
consistent with the analysis above.

(a) $r=0.1$

(b) $r=0.3$

(c) $r=0.5$

(d) $r=0.7$

(e) $r=0.9$

Figure 4:  Evolution of the spectrum for the Kagomé lattice with the parameter
$r$. The flat band is located at $E=\tilde{t}f=2t$. As $r$ increases, the
Dirac points gap out, but the flat band maintains a quadratic band touching
with the dispersive band. As $r\rightarrow 1$, the dispersive band merges with
the flat band. The spectrum remains the same under $r\rightarrow-r$.

###  3.1 Physical meaning behind the $r$-parameter

The introduction of the $r$ parameter allowed for the same parameterization of
the flat band condition as the original Kagome lattice but with a modified
(complex) phase. Further, this parameterization also allows us to interpret
$t(1+r)$ as hopping within the unit cell and $t(1-r)$ and hopping outside the
unit cell (because this is the term in the Bloch Hamiltonian associated with
the translation phase factor). For $r>0$, it would correspond to bringing the
3 atoms closer together (without altering the lattice constant) towards one
corner of the unit cell, as shown in Fig. 5. For $r<0$ the deformation takes
the atoms toward the other corner of the unit cell. This deformation is
analogous to the ‘breathing’ anisotropy considered in frustrated Kagomé
lattices[64]. Further, the case with $r>1$ corresponds to negative
$t_{inter}$, which is equivalent to having a $\pi$ phase attached to the
hopping element. This case will be discussed in more detail in the next
subsection.

Observe from Fig. 5 that when $r=1$, the intercell hopping $t(1-r)=0$
(molecular limit) and we do not hop to the neighboring unit cells and thus we
get a non-dispersive 3-level system. Because of the non-dispersive nature, the
bands are trivially flat, and two of them are degenerate because we are still
satisfying Eq. (11) for touching of the flat band with one other band. For
$r<0$, we effectively swap intercell and intracell hoppings. This just amounts
to mirroring the unit cell about a diagonal and thus does not change anything
in the spectrum.

(a) $r>0$

(b) $r<0$

Figure 5:  Modified Kagomé lattice. For $r>0$, the basic modification is such
that it brings the atoms together towards one corner of the unit cell. For
$r<0$ the atoms are moved closer toward the other corner of the unit cell. The
original Kagomé lattice is restored at $r=0$.

###  3.2 Model with $|r|>1$

When $|r|>1$ one of either the inter-cell hopping or the intra-cell hopping
parameters goes negative. From a solid-state point of view, this is clearly
unphysical, however, we still have a well-defined spectrum and eigenfunctions.
One could imagine the bonds with negative hoping to be associated with a phase
of $\pi$. In Fig. 6 (a) we show the resulting flux distribution. This is
clearly a $2\pi$-flux per unit cell, but the flux is modulated within the unit
cell such that the flux is concentrated in the hexagonal region and one of the
two triangular regions in the unit cell. This could be viewed as the flux
through the closed structure in the unit cell (the triangle ABC) having zero
flux, and the flux is only “in between” these closed structures. In the limit
$r\rightarrow\infty$ (recall that the energy of the flat band is independent
of $r$) the lattice returns to the Kagomé form factor, albeit with the
modified flux. Although there is flux distribution within the unit cell, TRS
is still preserved because the phase is $\pi$ which is the same as $-\pi$. In
fact, this is exactly one of the cases arrived at in Ref. [42] but is
naturally included in our parameterization.

The spectrum for $|r|>1$ is shown in Fig. 6 (b). As discussed in an earlier
subsection, at $r=1$, the dispersing band merges with the original flat band.
However, as $r$ becomes greater than 1, the dispersing band moves above the
flat band. This is an interesting example where one band completely passes
through another one. Note also that the $\Gamma$-point remains degenerate.

(a)

(b)

Figure 6:  (a) Modified Kagomé lattice for $r>1$. The negative hopping could
be seen as the bond having a $\pi$-phase that results in characteristic
$2\pi$-flux distribution per unit cell as shown. (b) The energy spectrum for
$r=1.2$. The flat band is preserved and so is the degeneracy at the
$\Gamma$-point. However, note that the dispersive band is now above the flat
band.

Another application of models with $|r|>1$ would be in scenarios where a
coupled system of oscillators is mapped to the tight-binding model. This is
realizable in photonic-crystal systems and even in cold atom systems. Thus, we
have demonstrated that there exists a parameterization in terms of the change
of the basis of the unit-cell, mathematically realized by introducing the
parameter $r$, that preserves the flat band. At this stage, this is just one
additional means to discuss the condition for flat bands. However, we shall
now describe a prescription to generate flat bands on general grounds, which
includes all the cases discussed above (and in the literature).

##  4 A prescription to generate flat band systems

In the previous sections, we presented a detailed analysis of the Kagomé
system and its modifications that would preserve the flat band. The approach
was rather direct where we searched for the parameters that would keep a band
dispersionless ($\vec{k}$-independent). This approach, however, is not
generalizable to investigate other systems as they would have to be dealt with
on a case-by-case basis. However, the presence of the flat band in the Kagomé
lattice could be deduced in a rather interesting manner. Consider a bipartite
system as shown in Fig. 7 with 2 and 3 atoms per unit cell and hoppings only
between the respective subsystems: consisting of X, Y atoms and A, B, and C
atoms.

(a)

(b)

Figure 7:  (a) Bipartite system with 2 and 3 atoms per unit cell. (b) The
energy spectrum for the 5 atoms per unit cell structure is particle-hole
symmetric. This is guaranteed by the bipartite nature of the system.

The Hamiltonian is given by

$\displaystyle H_{5}$ $\displaystyle=$
$\displaystyle-t\begin{pmatrix}0&0&1&1&1\\\
0&0&1&e^{-i\vec{k}\cdot\vec{R}_{1}}&e^{-i\vec{k}\cdot\vec{R}_{2}}\\\
1&1&0&0&0\\\ 1&e^{i\vec{k}\cdot\vec{R}_{1}}&0&0&0\\\
1&e^{i\vec{k}\cdot\vec{R}_{2}}&0&0&0\\\ \end{pmatrix},$ (13)

where the basis is
$\hat{\Psi}_{\vec{k}}=(\hat{c}_{\vec{k},X},\hat{c}_{\vec{k},Y},\hat{c}_{\vec{k},A},\hat{c}_{\vec{k},B},\hat{c}_{\vec{k},C})^{T}$.
There is a well known property of a bipartite Hamiltonian,
$H_{(n+m)\times(n+m)}$ (of subsystem sizes $n$ and $m$), one can always
construct the matrix $\mathcal{C}\equiv$ Diag($1_{n\times n},-1_{m\times m}$)
such that $\\{\mathcal{C},H\\}=0$ (i.e. it anti-commutes with the
Hamiltonian). This implies that for every state with energy $E$, there must
exist another orthogonal state at energy $-E$. This is commonly known as a
particle-hole symmetric spectrum. Fig. 7(b) shows the numerically evaluated
spectrum for $H_{5}$. It is easy to argue from this property that if $m+n$ is
odd, we need to have a state at $E=0$ and thereby guaranteeing a flat
band[12]. However, we wish to show that there is a more fundamental reason for
having flat bands which goes beyond this standard argument.

Starting from a bipartite system, consider projecting out one subsystem, by
using the Löwdin’s method[65], at some energy scale of interest $E_{0}$. This
is a method that projects out one subspace from the system and the scale
$E_{0}$ plays the role of chemical potential in most cases. Let us suppose
that we would like to project out the subsystems A, B, and C and express the
Hamiltonian purely in terms of the states of the other subsystems X, Y. To
apply the Löwdin’s method, first view the Hamiltonian $H_{5}$ as blocks

$\begin{pmatrix}[H_{GG}]_{2\times 2}&[H_{GK}]_{2\times 3}\\\ [H_{KG}]_{3\times
2}&[H_{KK}]_{3\times 3}\end{pmatrix}.$

Then, the effective Hamiltonian projected onto the $G$ space (consisting of
X,Y) would be

$H_{\rm eff,G}(E_{0})=H_{GG}+H_{GK}[E_{0}-H_{KK}]^{-1}H_{KG}.$ (14)

Because of the bipartite nature with $K\rightarrow$ Kagomé and $G\rightarrow$
Graphene, $H_{GG}=0$ and $H_{KK}=0$. This yields

$H_{\rm eff,G}(E_{0})=\frac{H_{GK}H_{KG}}{E_{0}}.$ (15)

Similarly, the effective Hamiltonian projected onto the $K$ space is

$H_{\rm eff,K}(E_{0})=\frac{H_{KG}H_{GK}}{E_{0}}.$ (16)

Usually, Löwdin’s method is used in the perturbative sense at some energy
scale that separates out the states far away from that energy scale. However,
due to the bipartite nature, we can perform an exact projection, as outlined
above. There are two observations of interest for the Hamiltonians $H_{\rm
eff,G}$ and $H_{\rm eff,K}$:

  1. 1.

We can recognize

$H_{\rm
eff,G}=\frac{t^{2}}{E_{0}}\begin{pmatrix}3&1+e^{i\vec{k}\cdot\vec{R}_{1}}+e^{i\vec{k}\cdot\vec{R}_{2}}\\\
1+e^{-i\vec{k}\cdot\vec{R}_{1}}+e^{-i\vec{k}\cdot\vec{R}_{2}}&3\end{pmatrix}$
(17)

as the Hamiltonian for Graphene and

$H_{\rm
eff,K}=\frac{t^{2}}{E_{0}}\begin{pmatrix}2&1+e^{-i\vec{k}\cdot\vec{R}_{1}}&1+e^{-i\vec{k}\cdot\vec{R}_{2}}\\\
1+e^{iR_{1}}&2&1+e^{-i\vec{k}\cdot\vec{R}_{3}}\\\
1+e^{i\vec{k}\cdot\vec{R}_{2}}&1+e^{i\vec{k}\cdot\vec{R}_{3}}&2\end{pmatrix}$
(18)

as the Hamiltonian for the Kagomé lattice. This could have been expected since
the second order hops connects each subsystem to itself.

  2. 2.

Note that $H_{KG}=H_{GK}^{\dagger}$. A non-square matrix $M$ has the property
that $M^{\dagger}M$ and $MM^{\dagger}$, which are of different ranks, share
the same eigenvalues, with additional zeroes making up for the difference in
ranks (see Appendix B). Thus, our two subsystems will be such that $H_{\rm
eff,K}\sim H_{KG}H_{GK}$ will have the same eigenvalues as $H_{\rm eff,G}\sim
H_{GK}H_{KG}$ but an additional $0$ due to the rank mismatch. This zero is
$\vec{k}$ independent and hence results in a flatband. This is the more
fundamental reason behind the formation of flat bands and most other
conditions, if not all, are derivable from this construction. We demonstrate a
few other cases later in this article.

This explains why the Kagomé lattice spectrum has a flat band and also why the
rest of the spectrum is exactly the same as Graphene.

From point (2) above, we can conclude a general rule that if one constructs a
bipartite system, projecting out the smaller subsystem will result in a flat
band in the new effective system. The generality of the prescription outlined
above implies that the detailed structure (the dimensions or even the matrix
elements) of $H_{GK}$ does not matter. In fact, our $r$ parameterization is a
special case of this general rule. To demonstrate this point, consider first,
the $H_{5}$ Hamiltonian for the original Kagomé and Graphene lattices. Here

$H_{GK}=-t\begin{pmatrix}1&1&1\\\
1&e^{-i\vec{k}\cdot\vec{R}_{1}}&e^{-i\vec{k}\cdot\vec{R}_{2}}\end{pmatrix}.$

This can be generalized to

$H_{GK}=-\begin{pmatrix}t_{G_{1}K_{1}}&t_{G_{1}K_{2}}&t_{G_{1}K_{3}}\\\
t_{G_{2}K_{1}}&t_{G_{2}K_{2}}e^{-i\vec{k}\cdot\vec{R}_{1}}&t_{G_{2}K_{3}}e^{-i\vec{k}\cdot\vec{R}_{2}}\end{pmatrix}$

and the flat band would still persist owing to the size mismatch of the two
subsystems. The $r$ parameterization (and all of the ensuing discussion)
corresponds to the choice of $t_{G_{1}K_{i}}=\sqrt{t(1+r)}$ and
$t_{G_{2}K_{i}}=\sqrt{t(1-r)}$, for $i\in\\{1,2,3\\}$, and hence preserves the
flat band. We emphasize that the prescription we provide (based on bi-
partiteness) is a sufficient condition to get a flat band but not a necessary
one.

###  4.1 Flat bands beyond the bipartite condition

A bipartite system with different system sizes having a flat band is a rather
straightforward result. However, we now wish to show that this condition is
not necessary. We can start from the bipartite system and then allow for
hoppings or energies in the subsystem to be projected out. This would still
guarantee the presence of the flat band (the particle-hole symmetry will no
longer persist, of course) in the parent system and also in the projected
subsystem (with the larger size). The existence of the flat band is controlled
solely by the existence of the non-square coupling matrix between a subsystem
that does not talk to itself and another subsystem with a smaller size.

To see this, let us first note that the effective Hamiltonian now goes from

$H_{KG}H_{GK}\rightarrow H_{KG}[E_{0}-H_{GG}]^{-1}H_{GK},$

where $H_{GG}$ is an arbitrary matrix. The matrix $E_{0}-H_{GG}$ can be
diagonalized as $M\Lambda M^{\dagger}$, where $\Lambda$ is the diagonal matrix
of the eigenvalues of $H_{GG}$ and $M$ is the matrix of eigenvectors of
$H_{GG}$. Thus, $[E_{0}-H_{GG}]^{-1}=M[E_{0}-\Lambda]^{-1}M^{\dagger}$. This
allows us to write

$H_{KG}[E_{0}-H_{GG}]^{-1}H_{GK}=\underbrace{H_{KG}M}_{\tilde{H}_{KG}}[E_{0}-\Lambda]^{-1}\underbrace{M^{\dagger}H_{GK}}_{\tilde{H}_{GK}}.$

Due to Hermiticity we have $H_{GK}=H^{\dagger}_{KG}$ and this implies
$\tilde{H}_{GK}=\tilde{H}^{\dagger}_{KG}$. Thus, we arrive at the form
$[\tilde{H}_{KG}]_{ac}D_{cd}[\tilde{H}_{GK}]_{db}$, where
$D_{ab}=d_{a}\delta_{ab}$ is a diagonal matrix. Since,
$\tilde{H}_{KG}=\tilde{H}^{\dagger}_{GK}$, the above product can be written as
$[\bar{H}_{KG}]_{ac}[\bar{H}_{GK}]_{cb}$, where
$[\bar{H}_{KG}]_{ab}=[\tilde{H}_{KG}]_{ab}\sqrt{d_{a}}$. Since this maintains
$\bar{H}_{KG}=\bar{H}^{\dagger}_{GK}$, we can map this non-bipartite system to
a strict bipartite system and apply all the same arguments: the size mismatch
of the two systems would result in a zero eigenvalue and hence a flat band. In
this sense, we only need the bipartiteness to choose the subsystems and then
allow the smaller subsystem to have any hoppings and the resulting system
would still have a flat band. We demonstrate this in Fig. 8 where we include
the off-diagonal elements in $H_{GG}$ as
$t^{\prime}(1+0.1e^{i\vec{k}\cdot\vec{R}_{1}}+0.5e^{i\vec{k}\cdot\vec{R}_{2}})$
and its c.c with $t^{\prime}=0.5E_{0}$ (this simply accounts some hoppings in
the GG subspace). The particle-hole symmetry is lost, but the flat band still
exists.

(a)

(b)

Figure 8:  (a) The spectrum of the $H_{5}$ system still has a flat band with
$H_{GG}\neq 0$ (but $H_{KK}=0$), which is in violation of the bipartite
condition. (b) The flat band in the projected subsystem

Lastly, we show that deviation from the stated condition above destroys the
flat band. Consider the simple case of adding different onsite energies to the
sites of the subsystem of interest. Consider

$H_{KK}=\begin{pmatrix}E_{a}&0&0\\\ 0&0&0\\\ 0&0&0\end{pmatrix}.$

The spectrum for such a case is shown in Fig. 9(a) and the ensuing projected
system also shown in (b) also loses the flatness. As another example for
demonstrating that violating the stated condition destroys the flatness,
consider the strained Hamiltonian in Eq. (2) which does not have a flat band.
This is so because arriving at this form from the $H_{5}$ formulation requires
choosing

$H_{KK}=\begin{pmatrix}0&c_{1}&c_{2}\\\ c_{1}^{*}&0&c_{3}\\\
c_{2}^{*}&c_{3}^{*}&0\end{pmatrix}$ (19)

with $c_{1}=-\frac{t\delta
t}{E_{0}}(1+e^{-i\vec{k}\cdot\vec{R}_{1}}),c_{2}=-\frac{t\delta
t}{E_{0}}(1+e^{-i\vec{k}\cdot\vec{R}_{2}}),c_{3}=\frac{t\delta
t}{E_{0}}(1+e^{-i\vec{k}\cdot\vec{R}_{3}})$. This violates our stated
condition that $H_{KK}$ still needs to be of the ‘bipartite’ nature. Indeed
after Löwdin’s projection we get

$H_{\rm eff}=H_{\rm eff,K}+H_{KK},$ (20)

where $H_{\rm eff,K}$ is the same as in Eq. (18). This is nothing but the
strained Hamiltonian of Eq (2), with an overall scale factor of
$\frac{t}{E_{0}}$. The spectrum is shown in Fig. 9(c). Physically including
$c_{i}$ amounts to including nnn terms in the $H_{5}$ system.

(a) $H_{5}$ with onsite energy

(b) Projected Kagomé with on-site energy

(c) Projected Kagomé with non-bipartite terms

Figure 9:  (a) The energy spectrum for the $H_{5}$ system with an on-site
energy $E_{a}=t$. (b) The spectrum after projection from the $H_{5}$ system
onto the ABC subsystem. (c) The spectrum after projection from the $H_{5}$
system with the non-bipartite terms added to the ABC subsystem. Here, $\delta
t=0.5t$. In both cases, $(b)$ and $(c)$, the loss of our stated flat band
condition indeed results in the loss of the flat band.

##  5 Isolating the flat band

Having formulated a technique to generate a family of flat band systems, we
note that in the cases we looked at, the flat band always appeared degenerate
with a dispersive band. However, if we let $t_{G_{i}K_{j}}$ to be different
from each other, we preserve the flat band (since any change within the matrix
$H_{GK}$ is allowed) as well as isolate it from the dispersive band. As an
example, consider the case where we displace one of the $a,b$, or $c$ atoms
such that it is closer to $x$ and further from $y$ in Fig. 10(b) such that
with
$t_{G_{1}K_{1}}=t_{G_{1}K_{2}}=t_{G_{2}K_{1}}=t_{G_{2}K_{2}}=t,t_{G_{2}K_{3}}=t+\delta
t$ and $t_{G_{1}K_{3}}=t-\delta t$. After projecting the system onto the
Kagomé form, an explicit calculation shows that the gap at the $\Gamma$-point
is $\frac{\delta t^{2}}{3t}$ (for $\delta t\ll t$).

#### The path-exchange symmetry

The above change falls under case shown in Fig. 10(d). This differs from the
cases in Fig. 10(a), (b), and (c) in the following way. Consider the sets of
paths taking us from the larger subsystem ($ABC$) to the smaller ($XY$):
$\\{AX,AY\\},\\{BX,BY\\},\\{CX,CY\\}$. Consider the set
$\\{r^{AB}_{X},r^{AB}_{Y}\\}$ created from the ratios
$r^{AB}_{X}\equiv\frac{AX}{BX}$ and $r^{AB}_{Y}\equiv\frac{AY}{BY}$. The
elements of this set correspond to the ratio of paths-to-other-subsystem
between two atoms of the larger subsystem. Special situations arise when all
the entries of the set are identical (i.e., can be reduced to a unit set).
When we find a pair AB where the set of ratios could be reduced to a unit set,
then we say we have a path-exchange in the system (physically, the presence of
a unit set between pairs of atoms means that the paths to hop across
subsystems are exchangeable between these two atoms). We show in Appendix C
that for a bipartite system (of size $m,n$ with $m>n$), the presence of $r$
such path-exchanges leads to a $2(r-[m-n])+1$-fold degeneracy of the flat band
with dispersive bands.

Returning to Fig. 10, note that in (a) our sets are
$\\{\frac{AX}{BX},\frac{AY}{BY}\\}\rightarrow\\{1,1\\}$ for AB;
$\\{\frac{AX}{CX},\frac{AY}{CY}\\}\rightarrow\\{1,1\\}$ for AC; and
$\\{\frac{BX}{CX},\frac{BY}{CY}\\}\rightarrow\\{1,1\\}$ for BC. Since both B
and C can be exchanged with A, we say that there are two path-exchanges. This
leads to a triple degeneracy in the $H_{5}$ system. In (b) the ratios for the
same pairs are $\\{\alpha,\alpha\\},\\{\alpha,\alpha\\}$, and
$\\{\alpha,\alpha\\}$, where $\alpha\neq 1$. Thus, we still have two path-
exchanges and the degeneracy remains the same. In (c) the ratios are
$\\{1,1\\},\\{1,1\\},\text{and }\\{1,1\\}$ (despite the different hoppings)
and hence the degeneracy is still 3. And finally in (d) the ratios are
$\\{1,1\\},\\{1,\beta\\},\text{and }\\{1,\beta\\}$. Since we reduce the number
of path-exchanges in the system by 1, this lowers the degeneracy to 1
(basically lifting the degeneracy) as shown in Fig. 10(e) and (f).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10:  Hoppings in the unit cell of the $H_{5}$ system. The pink bonds
are different from the grey bonds. the situations in (a), (b), and (c) do not
lift the flat band degeneracy but that in (d) does. This is because, in the
former three, the paths for individual atoms for going from the ABC subsystem
to XY subsystem are exchangeable. Whereas in (d) that condition is broken. (e)
Isolated flat band in the spectrum of the exchange-broken $H_{5}$ system and
(f) the same for the projected Kagomé subsystem with the exchange-breaking
perturbation being $\delta t=0.5t$. The flat band is preserved and isolated
from the dispersing bands.

#### Note

In general, with hopping parameters $t_{G_{i}K_{j}}$, the projected
Hamiltonian in the Kagomé subspace is

$\displaystyle H_{\rm
eff,K}=\frac{1}{E_{0}}\times~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\tiny{\left(\begin{array}[]{ccc}t_{G_{1}K_{1}}^{2}+t_{G_{2}K_{1}}^{2}&t_{G_{1}K_{1}}\,t_{G_{1}K_{2}}+t_{G_{2}K_{1}}\,t_{G_{2}K_{2}}\,e^{-i\vec{k}\cdot\vec{R}_{1}}&t_{G_{1}K_{1}}\,t_{G_{1}K_{3}}+t_{G_{2}K_{1}}\,t_{G_{2}K_{3}}\,e^{-i\vec{k}\cdot\vec{R}_{2}}\\\
c.c.&t_{G_{1}K_{2}}^{2}+t_{G_{2}K_{2}}^{2}&t_{G_{1}K_{2}}\,t_{G_{1}K_{3}}+t_{G_{2}K_{2}}\,t_{G_{2}K_{3}}\,e^{-i\vec{k}\cdot(\vec{R}_{2}-\vec{R}_{1})}\\\
c.c.&c.c.&t_{G_{1}K_{3}}^{2}+t_{G_{2}K_{3}}^{2}\end{array}\right)}$ (24)

This form is the same as in Ref. [49] where the authors decomposed their
Kagomé Hamiltonian into contributions from ‘upper’ triangles and ‘lower
triangles’. In their interpretation, in order to isolate the flat band they
hypothesized breaking inversion (in the bonds) in the unit cell. This is not
technically correct. It is not sufficient to break inversion at the level of
the hoppings between the nearest neighbor atoms: the changes in hoppings need
to be correlated in a definite manner which was only evident because of the
author’s chosen parameterization. One example of inversion broken deformation
of the Kagomé unit cell is the $r$-parameterization discussed in Sec. 2. This
parameterization breaks inversion but does not lift the flat band degeneracy.
To understand why, consider the $H_{5}$ Hamiltonian with the parameterization
above where $t_{G_{i}K_{j}}=t$ except $t_{G_{2}K_{3}}=t+\delta t$ and
$t_{G_{1}K_{3}}=t-\delta t$. This results in the following projected
Hamiltonian in the Kagomé subspace:

$H_{\rm
eff,K}(\Delta)=\frac{t^{2}}{E_{0}}\begin{pmatrix}2&1+e^{-i\vec{k}\cdot\vec{R}_{1}}&(1-\Delta)+(1+\Delta)e^{-i\vec{k}\cdot\vec{R}_{2}}\\\
1+e^{i\vec{k}\cdot\vec{R}_{1}}&2&(1-\Delta)+(1+\Delta)e^{-i\vec{k}\cdot(\vec{R}_{2}-\vec{R}_{1})}\\\
(1+\Delta)e^{i\vec{k}\cdot\vec{R}_{2}}&(1+\Delta)e^{i\vec{k}\cdot(\vec{R}_{2}-\vec{R}_{1})}&2+\Delta^{2}\end{pmatrix}$
(26)

where $\Delta=\frac{\delta t}{t}$. Notice that in the Kagomé subspace, this
isn’t just an arbitrary breaking of the mirror in the Kagomé lattice as the
$\Delta$ has to enter the onsite energy in precisely the stated manner to
preserve the flatness of the band. However, in the $H_{5}$ system, it is
sufficient to break the path-exchange symmetry in the bonds, in any manner
possible with no other conditions, and we arrive at the appropriate
Hamiltonian in the projected basis with all the necessary conditions built-in.

In this regard, we emphasize that while the statement in Ref. [49] that their
stated perturbation that isolates the flat band also breaks inversion is
correct, we wish to state that breaking inversion does not necessarily gap out
the flat band and that inversion may not have anything to do with the problem
at hand. This is evident from the $r$-parameterization presented earlier which
also breaks inversion but preserves the degeneracy of the flat band.

We note in passing that this path-exchange symmetry is a fundamental symmetry
of bipartite graph. Certain spatial symmetries such as mirror, rotations,
inversions can be a special case of this symmetry. It is thus easy to
associate the spatial symmetries to the cause of degeneracies. This is not
incorrect, but they are not fundamental. In the examples we present in this
article, most of the path-exchange symmetries can be broken by breaking a
mirror symmetry or a Cn symmetry in the physical lattice.

##  6 Application of the prescription to other lattices

Thus far we derived the well-known lattices like the Kagomé and Graphene
systems as projections from a parent $H_{5}$ system and indicated that
isolating the flat band requires breaking the path-exchange symmetry in their
parent system. We now demonstrate that the other systems that are discussed in
the literature (like Lieb and Dice) are actually the parent systems to other
flat-band lattices and that the flat band could be isolated by breaking the
path-exchange symmetry of the parent system. We shall only show the results
for the strict bipartite case to keep the discussion simple, but as we have
shown, this is not needed.

###  6.1 Lieb lattice and its projections

The Lieb lattice shown in Fig. 11(a) is another realizable example of the
prescription. The system is bipartite within the nn approximation and the
Hamiltonian is given by

$\displaystyle H_{\rm Lb}$ $\displaystyle=$
$\displaystyle-t\begin{pmatrix}0&1+e^{-i\vec{k}\cdot\vec{R}_{1}}&1+e^{-i\vec{k}\cdot\vec{R}_{2}}\\\
1+e^{i\vec{k}\cdot\vec{R}_{1}}&0&0\\\
1+e^{i\vec{k}\cdot\vec{R}_{2}}&0&0\end{pmatrix},$

where all the terms have an analogous meaning as in Eq. (2), but in this case
$\vec{R}_{2}=(0,1)$. Here, however, the Lieb lattice is the analog of the
$H_{5}$ Hamiltonian, prior to projection. Hence, the spectrum is particle-hole
symmetric as shown in Fig. 11(b).

(a)

(b)

Figure 11:  (a) Lieb lattice with three atoms $A,B,C$ in the unit cell and the
translation vectors $\vec{R}_{1}$ and $\vec{R}_{2}$. (b) The energy spectrum
for Lieb lattice. Note the presence of the flat band at the center of the
particle-hole symmetric spectrum.

Carrying out the projections using the Löwdin’s method, the two subsystems are

$H_{\rm
eff,Sq}(E_{0})=\frac{2t^{2}}{E_{0}}\left[2+\cos(\vec{k}\cdot\vec{R}_{1})+\cos(\vec{k}\cdot\vec{R}_{2})\right],$
(28)

which is the regular square lattice; and the other Hamiltonian is

$H_{\rm
eff,xSq}(E_{0})=\frac{t^{2}}{E_{0}}\begin{pmatrix}2+2\cos(\vec{k}\cdot\vec{R}_{1})&1+e^{i\vec{k}\cdot\vec{R}_{1}}+e^{-i\vec{k}\cdot\vec{R}_{2}}+e^{i\vec{k}\cdot(\vec{R}_{1}-\vec{R}_{2})}\\\
1+e^{-i\vec{k}\cdot\vec{R}_{1}}+e^{i\vec{k}\cdot\vec{R}_{2}}+e^{-i\vec{k}\cdot(\vec{R}_{1}-\vec{R}_{2})}&2+2\cos(\vec{k}\cdot\vec{R}_{2})\end{pmatrix},$
(29)

which is also a square lattice with 2-atoms per unit cell, which we may refer
to as the extended square lattice. The lattices and spectrum of these two
effective systems are shown in Fig. 12. It is worth noting that without the
projection technique, one would need to know the exact ratios of the nn and
nnn hoppings to ensure the presence of the flat band in the two band system.
However, this technique ensures that the projected systems already have the
selected ratios to have the flat band.

(a)

(b)

(c)

(d)

Figure 12:  (a) and (b) The two subsystems (square and extended square
lattices) from projecting out the Lieb lattice. (c) and (d) The energy
spectrum for the respective subsystems. Note the presence of the flat band in
the extended square lattice.

Further, like we identified $H_{GK}$ in $H_{5}$, one can identify the matrix
$H_{SX}$ with

$H_{SX}=\begin{pmatrix}1+e^{-i\vec{k}\cdot\vec{R}_{1}}&1+e^{-i\vec{k}\cdot\vec{R}_{2}}\end{pmatrix},$
(30)

which can be generalized to

$H_{SX}=\begin{pmatrix}t_{AB}+\tilde{t}_{AB}e^{-i\vec{k}\cdot\vec{R}_{1}}&t_{AC}+\tilde{t}_{AC}e^{-i\vec{k}\cdot\vec{R}_{2}}\end{pmatrix}.$
(31)

We can now break the path-exchange symmetry in the Lieb lattice [see Fig.
13(b)] by selecting the parameterization $t_{AC}=(t-\delta t)$ and
$\tilde{t}_{AC}=(t+\delta t)$. We show in Fig. 13(d) the spectrum of the
exchange-broken Lieb lattice projected onto the extended square lattice system
which shows the isolated flat band.

(a)

(b)

(c)

(d)

(e)

Figure 13:  Hoppings in the unit cell of the Lieb lattice with A being one
system and BC being the other. In situations (a) and (b), where the path-
exchange between the subsystems is preserved, the flat band degeneracy is not
lifted degeneracy, but it is in (c) where the condition is broken. (d)
Isolated flat band in the spectrum of the exchange-broken Lieb lattice and (e)
the same but projected onto the extended square lattice. Here, the exchange-
breaking perturbation is $\delta t=0.5t$.

###  6.2 Dice lattice and its projections

(a)

(b)

Figure 14:  (a) Dice lattice with three atoms A, B, C in the unit cell and the
translation vectors $\vec{R}_{1}$ and $\vec{R}_{2}$. (b) The respective energy
spectrum with the flat band as well as particle-hole symmetry.

The Dice lattice [shown in the Fig. 14(a)] has the following Hamiltonian:

$H_{\rm
Dc}=-t\begin{pmatrix}0&1+e^{-i\vec{k}\cdot\vec{R}_{1}}+e^{-i\vec{k}\cdot\vec{R}_{2}}&1+e^{-i\vec{k}\cdot\vec{R}_{1}}+e^{i\vec{k}\cdot(\vec{R}_{2}-\vec{R}_{1})}\\\
1+e^{i\vec{k}\cdot\vec{R}_{1}}+e^{i\vec{k}\cdot\vec{R}_{2}}&0&0\\\
1+e^{i\vec{k}\cdot\vec{R}_{1}}+e^{-i\vec{k}\cdot(\vec{R}_{2}-\vec{R}_{1})}&0&0\end{pmatrix}$
(32)

where $\vec{R}_{1}=(1,0)$ and
$\vec{R}_{2}=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$. The two subsystems
are the triangular lattice

$H_{\rm
eff,T}(E_{0})=\frac{4t^{2}}{E_{0}}\left[\frac{3}{2}+\cos(\vec{k}\cdot\vec{R}_{1})+\cos(\vec{k}\cdot\vec{R}_{2})+\cos(\vec{k}\cdot(\vec{R}_{1}-\vec{R}_{2}))\right]$
(33)

and something which is a Graphene lattice with nnn and nnnn hoppings (which we
may call the extended Graphene lattice):

$H_{\rm
eff,xG}(E_{0})=\frac{t^{2}}{E_{0}}\begin{pmatrix}|h(\vec{k})|^{2}&e^{-i\vec{k}\cdot\vec{R}_{1}}h^{2}(\vec{k})\\\
e^{i\vec{k}\cdot\vec{R}_{1}}h^{*2}(\vec{k})&|h(\vec{k})|^{2}\end{pmatrix},$
(34)

where
$h(\vec{k})=1+e^{i\vec{k}\cdot\vec{R}_{1}}+e^{i\vec{k}\cdot\vec{R}_{2}}$.

(a)

(b)

(c)

(d)

Figure 15: (a) and (b) The two projected subsystems of the Dice lattice. (c)
and (d) Their respective energy dispersion. Note the flat band in the extended
Graphene.

We can now define the following matrix

$H_{TX}=\begin{pmatrix}t_{AB}^{1}+t_{AB}^{2}e^{-i\vec{k}\cdot\vec{R}_{1}}+t_{AB}^{3}e^{-i\vec{k}\cdot\vec{R}_{2}}~{}~{}&~{}~{}t_{AC}^{1}+t_{AC}^{2}e^{-i\vec{k}\cdot\vec{R}_{1}}+t_{AC}^{3}e^{i\vec{k}\cdot(\vec{R}_{2}-\vec{R}_{1})}\end{pmatrix}.$
(35)

We can now break the path-exchange symmetry in the Dice lattice [see Fig.
16(b)] by selecting the parameterization $t_{AC}^{1}=(t-\delta
t),t_{AC}^{2}=(t-\delta t)$ and $t_{AC}^{3}=(t+\delta t)$. We show in Fig.
16(d) the spectrum of the exchange-broken Dice lattice projected onto the
extended Graphene lattice which shows the isolated flat band.

(a)

(b)

(c)

(d)

(e)

Figure 16:  Hoppings in the unit cell of the Dice lattice. A is one subsystem
and BC is the other. In the situations (a) and (b) the exchange-symmetry is
preserved and so is the flat band degeneracy, whereas in (c) it is broken and
the flat band degeneracy is lifted. (d) Isolated flat band in the spectrum of
the exchange-broken Dice lattice and (e) the same but projected onto the
extended Graphene lattice. Here, the exchange-breaking perturbation is $\delta
t=t$.

##  7 Flat band with Chern-Simons flux distribution

In this section we demonstrate that the properties we outlined above also
apply to systems with reduced translation symmetry such as in situations where
the lattice is subject to a flux $\phi=2\pi p/q$ per unit cell. Such a system
is usually modelled as a Hofstadter problem by attaching phases on the bonds
such that when the pattern is extended to the entire lattice one finds a unit
cell that is enlarged $q$-fold.

First, let us note a feature of the projection of the subsystems. Although we
introduced the projection in the ${\bf k}$-space, the same principles apply in
the real space. This can be seen by performing a unitary transformation from
the Bloch basis to tight-binding basis. It is advantageous to work in the real
space as one would not have to worry about the unit-cell enlargement due to
the flux attachment to the bonds. For definiteness, consider again the
bipartite $H_{5}$ lattice with atoms $\\{X_{i},Y_{i}\\}$ from one subsystem
and $\\{A_{i},B_{i},C_{i}\\}$ in another subsystem, where $i$ marks the unit
cell. By a direct calculation it can be shown that if one were to project out
the XY subsystem then the effective hopping that is induced in the ABC
subsystem, say $t_{PQ}$ (where $P,Q\in\\{A,B,C\\}$), would be given by the sum
of products of hoppings taking $P\rightarrow\\{X,Y\\}\rightarrow Q$. In the
$H_{5}$ case every sum only consists of one term such that
$t_{PQ}=t_{PX}t_{XQ}$ or $t_{PY}t_{YQ}$. This means that the phase associated
with the effective bond $PQ$ would be the sum of phases along the path from
$P$ to $Q$ in the original system.

Figure 17:  The hoppings inside the unit cell of the $H_{5}$ lattice, but with
phases $\phi_{1,2,3}$ attached as shown. Upon projecting out the XY subsystem,
the effective hoppings (dashed lines) acquire phases $\phi_{AB,BC,CA}$.

Second, it is clear that since we started out with a bipartite system, we
would end up with a projected system with a flat band. However, it is
interesting to observe the situation depicted in Fig. 17 where the parent
system has phases $\phi_{1}$, $\phi_{2}$ and $\phi_{3}$ attached to the
hoppings between the subsystems. Upon projection, using the result presented
above, we find that $\phi_{AB}=\phi_{1}+\phi_{2}$,
$\phi_{BC}=\phi_{3}-\phi_{2}$, and $\phi_{CA}=-\phi_{3}-\phi_{1}$. This
results in $\phi_{AB}+\phi_{BC}+\phi_{CA}=\oint\vec{A}\cdot d\vec{l}=0$, i.e.
the flux enclosed in the triangular regions is zero. This is in fact the
scenario that is predicted to happen for in a Kagomé chiral spin-liquid which
is modelled as fermions subject to a Chern-Simons (CS) field. Thus, the flux
distribution that is implemented by the projection technique naturally
accounts for the CS nature of the flux distribution and is probably a good
tool to use to model systems with CS fields. It follows from this that in a
Maxwell-like flux distribution (where the flux is proportional to the area
enclosed, see Fig. 18), a flat band is not guaranteed. This is the reason why
lattices with Maxwell-like fluxes have dispersive bands (in general), whereas
those with CS-type fluxes can have flat bands. This comparison has been
discussed in Fig. 4 of Ref. [27]. In that work, it was also pointed out that
the flat band was gapped from the other dispersive bands. This is now easy to
understand ( e.g., see Fig. 19) that the path-exchange symmetry is broken due
to the phase attachments, which isolates the flat band.

(a) Maxwell type flux distribution

(b) Chern-Simons type flux distribution

Figure 18:  Under Maxwell type field, the flux is proportional to area whereas
in the CS case, the flux is concentrated in one region of the unit cell, with
the triangular parts having zero flux.

As an explicit example let us consider the case with $\phi=\pi$. Here the unit
cell is doubled ($q=2$). The Hamiltonian is given by

$H^{\phi=\pi}=\begin{pmatrix}0&H_{GK}^{\pi}\\\ H_{KG}^{\pi}&0\end{pmatrix},$
(36)

where

$H_{KG}^{\pi}=-t\begin{pmatrix}1&0&0&e^{-i\vec{k}\cdot\vec{R}_{2}}\\\
0&e^{-i\pi}&e^{-i\vec{k}\cdot\vec{R}_{2}}&0\\\ 1&0&1&0\\\ 0&1&0&1\\\
1&0&e^{i\vec{k}\cdot(\vec{R}_{1}-\vec{R}_{2})}&0\\\
0&e^{-i\pi}&0&e^{i\vec{k}\cdot(\vec{R}_{1}-\vec{R}_{2})}\end{pmatrix}$ (37)

and $R_{1}=(1,0)$ and $R_{2}=(\frac{1}{2},\frac{\sqrt{3}}{2})$. The phase
attachments are shown in Fig. 19 and the band structure for the $H_{5}$ system
and the projected systems are shown in Fig. 20, which clearly show the
isolated flat band. This feature remains for all values of $\phi\in(0,2\pi)$.

Figure 19:  Chern-Simons flux attachment prescription for $\phi=\pi$. All the
flux is concentrated on the middle hexagon and the flux through the triangular
regions is zero. The unit cell is doubled. The purple lines indicate the
hoppings in the projected subsystem and each parallelogram formed by these
lines is the original unit cell. The pink lines represent the hoppings of the
parent system. The arrows indicate the phases, which are all zero except the
three in purple and pink each with ($\pi,-\pi,2\pi$).

(a)

(b)

Figure 20:  (a) Energy spectrum of $H_{5}$ with CS field with flux $\phi=\pi$
(b) Energy spectrum of $H_{\rm eff,K}$. In both cases notice the flat band at
$E=0$. We are only showing the bands closest to the flat band.

##  8 Conclusion

Although there exist many works in the literature on designing systems with
flat bands, they suffer from one or many of the following shortcomings: the
need for long-ranged hoppings; the need for fine-tuning of parameters even
with nn approximation; the need for staggered fluxes. While these are all
valid techniques, they could be seen as shortcomings because implementing
long-range hoppings is often a design challenge. So is fine-tuning parameters
of the Hamiltonian. Breaking a discrete symmetry such as the time-reversal
symmetry is certainly viable, however, generating staggered flux may not
always be a straightforward task. In this work, we proposed a straightforward
design method based on the bipartiteness of a system (which is not strictly
necessary) and Löwdin’s projection to generate flat band systems. In this
prescription, we were able to identify that if there exists a path-exchange
symmetry in the bipartite system, with different sizes of the subsystems, the
flat band would appear degenerate with other dispersive bands. And by breaking
the symmetry, it is possible to isolate the flat band. We then showed that
projecting out various subsystems from this parent bipartite system still
leaves behind a subsystem with an isolated flat band. We demonstrated the
verity of all the above points of the prescription by applying it to the
$H_{5}$ lattice (from which we can project out the hexagonal lattice and the
Kagomé lattice: the latter having the spectrum of the former plus a flat
band), Lieb and Dice lattices (from which we can project out the square and
triangular lattices, respectively, and special lattices that share their
respective spectra plus a flat band). We also showed that relaxing the
bipartite condition in the subsystem that is being projected out still
maintains the flat band. In this sense, we only need to couple a non-hopping
(or weakly hopping) system to a background system, which is smaller in size,
to get a flat band.

Apart from this main result, we even developed a special parameterization of
the Kagomé lattice to show that it is possible to devise flat band conditions,
different from what is already in the literature and showed that this
construction is automatically implemented if arrived at from the parent
bipartite system. We even showed that starting from the parent bipartite
system, we can reproduce the conditions for the existence and isolation of
flat bands that were discussed previously in the literature (about staggered
$\pi$ fluxes[42] and inversion broken Kagomé[49]). Finally, we demonstrated
that the existence of a flat band in the chiral spin-liquid state of the
Kagomé lattice that originates due to coupling to a Chern Simons like field is
also consistent with the rules we identify in this work. This also suggests
that the Chern Simons type flux in lattices could be understood in terms of
Maxwellian flux coupled to bipartite systems.

We also believe that the prescription suggested here should be practically
implementable in photonic-crystal lattices or even in cavity QED techniques
such as in Refs. [54, 55]. Isolating the various subsystems can be done, e.g.
simply by introducing appropriate onsite energies to split the subsystems in
energy, and then tuning the energy scale of any probe to be near the
respective onsite energies of interest. This will a subject for a future work.

There are several interesting avenues to pursue using this construction. It
may be possible to study the role of electronic correlations in Kagomé and
Graphene by studying them in the parent $H_{5}$ system. Given that we have a
systematic way to isolate flat bands, it can serve as a test bench to
investigate fractional Quantum Hall state formation in the absence of a
magnetic field. Further, it was stated in Ref. [41] that the band touching was
protected by topology in the Kagomé system. However, we have seen that the
band touching is protected by a path exchange symmetry in a parent bipartite
system. This could suggest a connection between topology and exchange-
symmetries in a higher-dimensional Hilbert space. It will also be interesting
to explore the link between the path-exchange symmetry and the Shubnikov
groups as elaborated on in Ref. [52].

## Acknowledgments

The authors would like to acknowledge useful exchanges with A. Andreanov.

#### Funding information

This work was funded by the Natural Sciences and Engineering Research Council
of Canada (NSERC) Grant No. RGPIN-2019-05486 (S.M.).

## APPENDIX

##  Appendix A Useful relations between matrix elements $\tilde{\alpha}_{i}$

There are a couple of identities involving the matrix elements of the
Hamiltonian that can be derived straightforwardly. For the parameter $|r|<1$,
we observe that

$|\tilde{\alpha}_{i}|^{2}=2e^{-h}[\cosh h+\cos(\vec{k}\cdot\vec{R}_{i})],$

and the quantity $A=2{\rm
Re}[\tilde{\alpha}_{1}^{*}\tilde{\alpha}_{2}\tilde{\alpha}_{3}^{*}]$ evaluates
to

$A=2[1+(e^{-h}+e^{-2h})\sum_{i}\cos(\vec{k}\cdot\vec{R}_{i})+e^{-3h}].$

Using these we arrive at the relation

$\sum_{i}|\tilde{\alpha}_{i}|^{2}=2e^{-h}[3\cosh
h+\sum_{i}\cos(\vec{k}\cdot\vec{R}_{i})].$

Observing that both $\sum_{i}|\tilde{\alpha}_{i}|^{2}$ and $A$ have
$\sum_{i}\cos(\vec{k}\cdot\vec{R}_{i})$ in them as the only
$\vec{k}$-dependent term, it is possible to search for $f$ that satisfies the
flat band condition in Eq. (8) by equating their coefficients, which leads to
$f=2e^{-\frac{h}{2}}\cosh\frac{h}{2}$.

If $|r|>1$, then the identification $\frac{1-r}{1+r}=e^{-h}$ can still be made
if $h$ is extended to the complex plane. In particular, we note that as
$|r|>1$, $h$ acquires a step jump in the imaginary part from $0$ to $\pm
i\pi$. The sign is ambiguous but this does not affect our analysis and we
shall stick to the choice of $i\pi$. That is, $h\rightarrow h_{c}$ such that
$e^{-{\rm Re}[h]}=\frac{r-1}{r+1}$. This extension to the complex plane also
results in $[e^{-h_{c}}]^{*}=e^{-h_{c}}$, thus acting like a real number. This
ensures that all the above identities above are still valid with $h\rightarrow
h_{c}$. In that case, the resulting parameter $f$ is given by
$f=2e^{-\frac{h_{c}}{2}}\cosh\frac{h_{c}}{2}=2e^{-\frac{{\rm
Re}h_{c}}{2}}\sinh\frac{{\rm Re}h_{c}}{2}$, where $\frac{1-r}{1+r}=e^{-{\rm
Re}h_{c}}$.

##  Appendix B Properties of non-square matrices

Consider a non-square matrix $M_{n\times m}$ with $m>n$ for definiteness. Two
square matrices could be constructed from this: $[MM^{\dagger}]_{n\times n}$
and $[M^{\dagger}M]_{m\times m}$. Now construct the square matrix
$\tilde{M}_{m\times m}$ by padding $m-n$ rows of zero to $M$. Then we have

$\tilde{M}\tilde{M}^{\dagger}=\begin{pmatrix}[MM^{\dagger}]_{n\times
n}&0_{n\times(n-m)}\\\ 0_{n\times(n-m)}&0_{(n-m)\times(n-m)}\end{pmatrix}$

and

$\tilde{M}^{\dagger}\tilde{M}=M^{\dagger}M.$

From properties of matrices we have
Det[$\tilde{M}\tilde{M}^{\dagger}$]=Det[$\tilde{M}^{\dagger}\tilde{M}$] which
is the product of their eigenvalues. Since $\tilde{M}\tilde{M}^{\dagger}$
explicitly has $m-n$ $0$’s as eigenvalues, and that this construction is
possible for any $M$, it follows that $\tilde{M}^{\dagger}\tilde{M}$ and hence
$M^{\dagger}M$ must have (i) the same number of $0$ eigenvalues; (ii) the same
non-zero eigenvalues.

In other words, what we have argued is that for a non-square matrix
$M_{n\times m}$, $MM^{\dagger}$ and $M^{\dagger}M$ have the same eigenvalues
with one of them having $|n-m|$ zeros as additional eigenvalues.

##  Appendix C Identifying the path-exchange symmetry

Consider a generic flat band system generated based on a bipartite system
whose subsystems have size $n\times n$ and $m\times m$ with $n<m$. In such a
system, there are $m-n$ flat bands at $E=0$. This means that there are $m-n$
additional redundant equations (rows) in the system for any ${\bf k}$. If the
flat bands are additionally degenerate with any of the other dispersive bands
at a particular ${\bf k}$-point, then at that ${\bf k}$-point there must by
additional equations (rows) that are redundant. A redundant equation (row) can
be reduced to a scaled version of the other linearly independent equations
(rows). For generic scenarios in nn bipartite systems, it is sufficient to
just look at the scaling property. In a bipartite system of the form

$\hat{H}=\begin{pmatrix}0&\hat{h}^{\dagger}_{n\times m}\\\ \hat{h}_{m\times
n}&0\end{pmatrix}$ (38)

all the information is contained in the block $\hat{h}_{m\times n}$, which has
$m$ equations and $n$ unknowns. If such a system has $E=0$ solutions, then
only $n$ of these are linearly independent. If one identifies $r$ rows that
could be scaled (we call them $r$ zero rows), then linear algebra dictates
that there would be $p=2(r-[m-n])+1$ degeneracies in the system. That is, for
every new row, we introduce two degeneracies. The factor of 2 arises from the
fact that each time we reduce a row in $\hat{h}$, we reduce one in
$\hat{h}^{\dagger}$ (this won’t be the case for non-bipartite cases as will be
discussed shortly). To extract the physical meaning behind this, note that
each redundant row, if not trivially zero, is a scaled version of the linearly
independent rows. Since a given row in the $\hat{h}$ block contains the
hopping from one atom, say $A$, of larger subsystem to all the other atoms of
the smaller subsystem, the redundancy of another row would imply that another
atom, say $B$, from the larger subsystem has a scaled set of hoppings of $A$.
That is, the set $\\{AX_{i}\\}$ and set $\\{BX_{i}\\}$ (where $\\{X_{i}\\}$ is
the set of atoms in the smaller subsystem) are related via
$\\{AX_{i}\\}=\alpha\\{BX_{i}\\}$. If $AX_{i}$ and $BX_{i}$ are not trivial,
then we can construct the set of ratios $\\{\frac{AX_{i}}{BX_{i}}\\}$. A
redundancy of a row requires the set $\\{\frac{AX_{i}}{BX_{i}}\\}$ to be a
unit set and we say that the paths from $A$ and from $B$ are exchangeable. In
simpler terms, for a bipartite system, for $r$ path-exchanges we would have
$2(r-[m-n])+1$ degeneracies. Thus, breaking/reducing the number of path-
exchanges would lift/reduce the degeneracies. In this sense $r$ simply counts
the number of atoms in the larger subsystem that could be exchanged. If we now
consider the system

$\hat{H}=\begin{pmatrix}\hat{s}_{n\times n}&\hat{h}^{\dagger}_{n\times m}\\\
\hat{h}_{m\times n}&0\end{pmatrix},$ (39)

where $\hat{s}$ is non-trivial, then for $r$ path-exchanges, we would have
$p=(r-[m-n])+1$ degeneracies. We lose the factor of 2 due to presence of
$\hat{s}$.

As an example, consider the bipartite $H_{5}$ lattice with $m=3,n=2$. Here the
formula that applies is $p=2(r-1)+1=2r-1$. The number of reducible rows ($r$)
can be 1 or 2. Thus, we can have a degeneracy $p$ of 1 (non degenerate) or 3
(triply degenerate) at $E=0$ or none. So, if the $\hat{h}$ block looks like

$\begin{pmatrix}t_{1}&t_{2}\\\ t_{3}&t_{4}e^{i\vec{k}\cdot\vec{R}_{1}}\\\
t_{5}&t_{6}e^{i\vec{k}\cdot\vec{R}_{2}}\end{pmatrix},$ (40)

the case with two path-exchanges would look like

$\frac{t_{3}}{t_{1}}=\frac{t_{4}e^{i\vec{k}\cdot\vec{R}_{1}}}{t_{2}},~{}\text{and}~{}~{}\frac{t_{6}e^{i\vec{k}\cdot\vec{R}_{2}}}{t_{2}}=\frac{t_{5}}{t_{1}},$
(41)

then $p=2(2-[m-n])+1=3$, a triple degeneracy. Because the hoppings are real,
this condition is only satisfied at the $\Gamma$-point (${\bf k}=0$). Breaking
any of the path-exchange reduces the degeneracy by 2 and hence removing the
degeneracy in this case.

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