# Flat-band-based multifractality in the all-bands-flat diamond chain

Aamna Ahmed  Department of Physics, Indian Institute of Science Education and
Research, Bhopal, Madhya Pradesh 462066, India
Ajith Ramachandran  Department of Physics, Christ College, Irinjalakuda,
Kerala 680125, India
Ivan M. Khaymovich Nordita, Stockholm University and KTH Royal Institute of
Technology Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden
Auditya Sharma  Department of Physics, Indian Institute of Science Education
and Research, Bhopal, Madhya Pradesh 462066, India

###### Abstract

We study the effect of quasiperiodic Aubry-André disorder on the energy
spectrum and eigenstates of a one-dimensional all-bands-flat (ABF) diamond
chain. The ABF diamond chain possesses three dispersionless flat bands with
all the eigenstates compactly localized on two unit cells in the zero disorder
limit. The fate of the compact localized states in the presence of the
disorder depends on the symmetry of the applied potential. We consider two
cases here: a symmetric one, where the same disorder is applied to the top and
bottom sites of a unit cell and an antisymmetric one, where the disorder
applied to the top and bottom sites are of equal magnitude but with opposite
signs. Remarkably, the symmetrically perturbed lattice preserves compact
localization, although the degeneracy is lifted. When the lattice is perturbed
antisymmetrically, not only is the degeneracy is lifted but compact
localization is also destroyed. Fascinatingly, all eigenstates exhibit a
multifractal nature below a critical strength of the applied potential. A
central band of eigenstates continue to display an extended yet non-ergodic
behaviour for arbitrarily large strengths of the potential. All other
eigenstates exhibit the familiar Anderson localization above the critical
potential strength. We show how the antisymmetric disordered model can be
mapped to a $\frac{\pi}{4}$ rotated square lattice with nearest and selective
next-nearest neighbour hopping and a staggered magnetic field - such models
have been shown to exhibit multifractality. Surprisingly, the antisymmetric
disorder (with an even number of unit cells) preserves chiral symmetry - we
show this by explicitly writing down the chiral operator.

††preprint: APS/123-QED

##  I INTRODUCTION

Highly degenerate dispersionless or flat band (FB) [1, 2, 3, 4, 5] systems,
which support compact localized eigenstates (CLS) [6, 7] have been of great
interest over the last decade. The localization properties and associated
repressed transport have been discussed in the context of engineered lattices
in one, two and three dimensions, such as diamond [8, 9, 10], cross-stitch
[11, 12, 13], dice [14, 15, 16, 17], honeycomb [18, 19], kagome [20, 21, 22]
and pyrochlore lattices [23, 24]. The compact localized states have been
experimentally found to exist in a range of setups such as Hubbard model
systems [25, 26], photonic systems [27, 28], exciton-polariton condensates
[20] and Bose-Einstein condensates [29].

In the most familiar type of localization, namely Anderson localization [30,
31], which is induced by on-site disorder, the ‘spread’ of a state dies down
exponentially with a well-defined notion of a characteristic localization
length [32, 33]. Compact localization, in contrast, is much stronger with the
span restricted strictly to a few unit cells, with zero probability amplitude
elsewhere. Paradoxically the interplay of both these types of strong
localization results in a drop in localization. When a tiny amount of uniform
disorder is turned on in a compactly localized all-bands-flat (ABF) diamond
chain, the eigenstates exhibit an extremely weak flat-band-based localization
[34]. The disorder, in fact, facilitates the hybridization of the large-scale
degenerate network of compact localized eigenstates, and we see weak
localization that is on the cusp of delocalization. In this work, we show that
when the ABF chain is subjected instead to a quasiperiodic Aubry-André ($AA$)
disorder [35, 36], the eigenstates are in fact extended but non-ergodic thus
exhibiting multifractality, a delicate phenomenon that has attracted a wave of
interest in recent times [37, 38, 39, 40, 41, 42, 43, 44, 43, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55]. The familiar $1$-D Aubry-André Harper (AAH)
tight-binding model [56, 57, 48], which is endowed with self-duality [35, 58]
exhibits multifractality [31, 59] only at the metal-insulator transition that
occurs at a critical potential strength. Here, we report a robust flat-band-
based multifractality(FBM) that is seen in an extensive region of the phase
diagram.

Moreover, we find that the symmetry of the applied external potential is
crucial [60, 61, 62, 63, 64, 65, 66, 67, 64, 68, 69, 70, 71]. A symmetric
disorder, where the same disorder is applied to the top and bottom sites of a
unit cell, causes a complete lift of degeneracy; however, remarkably we find
that the CLS remain robust [72]. Although all the eigenstates are compactly
localized over two unit cells, these cannot be obtained by translating the
coefficients by an integer number of unit cells since the disorder breaks the
translation symmetry. In contrast, when we apply the $AA$ potential in an
antisymmetric manner, where the disorder applied to the top and bottom sites
of a unit cell are of equal magnitude but opposite sign, we find that both the
degeneracy and compact localization are destroyed. A tiny disorder hybridizes
the different compact localized states, resulting in flat-band-based
multifractality. When the strength of the disorder is higher than a critical
value (where inter-band hybridization becomes possible), we recover
conventional Anderson localization for all the eigenstates except those in a
central band. Another of our striking findings is that when the lattice is
perturbed anti-symmetrically, the chiral symmetry of the Hamiltonian is left
intact despite the presence of disorder. We show this with an explicit
construction of the chiral symmetry operator that anticommutes with the
Hamiltonian independent of the strength of the disorder.

The paper is organized as follows. In Section II, the ABF diamond chain, as
well as the Aubry-André potential, are described. In Section III, we discuss
the effects of applying the $AA$ potential in the symmetric configuration that
leaves the compact localization robust. In Section IV, we discuss the
antisymmetric application of disorder, which supports multifractal states. In
Section V, we present an analytic treatment in support of the numerical
results discussed in Section IV. We then summarize our results in Section VI.
The details of the lattice transformations used in Section V and numerical
analysis of a variety of complementary quantities in support of the main
findings are presented in the Appendix.

##  II Model

We consider the ABF diamond lattice [34], where each unit cell $n$ consists of
three sites labelled by $u_{n}$ (up), $c_{n}$ (center) and $d_{n}$ (down)
respectively. The corresponding single particle states are conveniently
represented in Dirac notation as $\ket{u_{n}}$, $\ket{c_{n}}$ and
$\ket{d_{n}}$ respectively. The Hamiltonian is given by:

$\displaystyle\hat{H}=-J\sum_{n=1}^{N/3}\left(-\ket{{u}_{n}}\bra{{c}_{n}}+\ket{{d}_{n}}\bra{{c}_{n}}+\ket{{c}_{n}}\bra{{u}_{n+1}}+\ket{{c}_{n}}\bra{{d}_{n+1}}+\text{
H.c.
}\right)+\sum_{n=1}^{N/3}\left(\zeta_{n}^{u}\ket{{u}_{n}}\bra{{u}_{n}}+\zeta_{n}^{c}\ket{{c}_{n}}\bra{{c}_{n}}+\zeta_{n}^{d}\ket{{d}_{n}}\bra{{d}_{n}}\right).$
(1)

Since each unit cell has $3$ sites, the total number of sites denoted by $N$
should be a multiple of $3$. All the energy terms are represented in hopping
amplitude $J$ units, assuming $J=1$ for simplicity. For each site of the
$n^{\text{th}}$ unit cell, we include on-site energy using independent Aubry-
André potentials

$\zeta_{n}^{\alpha}=\lambda_{\alpha}\cos(2\pi nb+\theta_{p}),$ (2)

where the strength of the potential is $\lambda_{\alpha}$ and $\alpha$ can
take the values $u$, $c$ and $d$. The quasi-periodicity parameter $b$ must be
an irrational number which we set to be the golden mean $(\sqrt{5}-1)/2$.
$\theta_{p}$ is an arbitrary global phase chosen randomly from a uniform
distribution in the range $[0,2\pi]$. Also, periodic boundary conditions have
been assumed [73].

The ABF diamond chain is a particular case of the Hamiltonian for which the
on-site energies $\zeta_{n}$ are zero. This system possesses three flat bands
at energies $\pm 2$ and $0$ and no dispersive bands. Consequently, the system
possesses only compact localized eigenstates. Hence, the system is highly
degenerate and also a good insulator. The Hamiltonian in the zero-disorder
limit possesses chiral symmetry [74, 75, 76], which is represented by a
$k$-independent operator $\Gamma_{0}=\text{diag}(1,1,-1)$. This operator is
unitary since $\Gamma_{0}^{2}=\mathbb{I}$ and
$\Gamma_{0}\mathcal{H}^{\dagger}(k)\Gamma_{0}^{-1}=-\mathcal{H}(k)$. Since the
chiral operator $\Gamma_{0}$ anti-commutes with the Hamiltonian, for each
eigenvalue $E$ with eigenvector $\ket{\phi_{E}}$ the negative $-E$ is also an
eigenvalue with eigenvector $\Gamma_{0}\ket{\phi_{E}}$.

In upcoming sections, we introduce the quasiperiodic Aubry-André potential on
the ABF diamond chain, and investigate how it affects the spectrum and the
compact localized states. Broadly there are two natural ways in which the on-
site energies on the up and down sites may be correlated: a symmetric
configuration in which $\zeta_{n}^{u}=\zeta_{n}^{d}$ and an antisymmetric
configuration in which $\zeta_{n}^{u}=-\zeta_{n}^{d}$. Furthermore, we discuss
two sub-cases within the symmetric setup: one in which the Aubry-André
potential is applied on the $u$ and $d$ sites, and the second one in which the
Aubry-André potential is applied on the $c$ sites while keeping the $u$ and
$d$ sites at zero potential.

##  III SYMMETRIC APPLICATION OF AUBRY-ANDRÉ POTENTIAL

(a) (b)

(c) (d) (e)

Figure 1: The spectrum of the ABF diamond lattice in the symmetric case with
(a) increasing strength of the quasiperiodic potential $\lambda$ and (b)
uniform uncorrelated random disorder with increasing strength $\Delta$ on the
$u$ and $d$ sites. Schematic representations of the diamond chain: compact
localized states from the (c) lower band ($E=-2.49$), (d) middle band
($E=-0.025$) and (e) upper band ($E=2.51$) at $\lambda=2$ (the amplitudes are
obtained numerically). The system size is $N=126$.

###  III.1 AA potential on the $u$ and $d$ sites

First, we consider the symmetric configuration:

$\zeta_{n}^{u}=\zeta_{n}^{d}\qquad\text{and}\qquad\zeta_{n}^{c}=0.$ (3)

The introduction of the $AA$ potential i.e. $\zeta_{n}^{u}=\lambda\cos(2\pi
nb+\theta_{p})$ lifts the degeneracy of all the flat bands, as can be seen
from Fig. 1(a). The eigenstates of the Hamiltonian are found to reside on two
unit cells even in the presence of the $AA$ potential (see Fig. 1(c)–1(e)).
Remarkably, the compactness of the eigenstates is preserved at higher
strengths of the potential, even after all the bands mix and the system
exhibits a single-band energy spectrum. Also, although the spectrum appears to
be symmetric with respect to $E=0$, a closer look reveals that this is not
quite true. There is no requirement that every energy $E$ comes with its
negative counterpart $-E$ since the disorder breaks the chiral symmetry.

Interestingly, as shown in Fig. 1(b), when the applied potential is drawn from
a uniform uncorrelated random distribution $[-\Delta,\Delta]$, the results are
similar (to those with quasiperiodic disorder of the same strength $\lambda$).
We conclude that the presence of the compact localized states is a consequence
of the symmetric application of the disorder and not the details of the
applied potential. There exists a useful transformation to a new lattice (see
Section A.1), which in the disorder-free limit, takes the system to a set of
completely uncoupled sites. From Fig. 8(a), we observe that the transformation
results in a lattice made of three-site unit cells but with an absence of
inter-cell hopping, indicating the preservation of the CLSs (see Section A.2).

(a) (b)

(c) (d) (e)

Figure 2: The spectrum of the ABF diamond lattice with (a) $AA$ potential with
increasing quasiperiodic strength $\lambda$ and (b) uniform uncorrelated
random disorder with increasing strength $\Delta$ on the central $c$ sites.
Schematic representations of the diamond chain with the $AA$ potential only on
the $c$ sites. CLSs for the (c) lower band ($E=-2.12$), (d) middle band
($E=0$) (e) upper band ($E=1.89$) at $\lambda=5$. The system size is $N=126$.

###  III.2 AA potential on the $c$ sites

Next, we consider the case when disorder is introduced only on the $c$ sites
i.e.

$\zeta_{n}^{u}=\zeta_{n}^{d}=0\quad\text{and}\quad\zeta_{n}^{c}\neq 0.$ (4)

This is a special type of symmetric configuration. The energy spectrum with
increasing strength of potential has been plotted for both the $AA$ potential
i.e. $\zeta_{n}^{c}=\lambda\cos(2\pi nb+\theta_{p})$ (see Fig. 2(a)) and for
the uniform disorder case (see Fig. 2(b)) i.e. $\zeta_{n}^{c}$ drawn uniformly
from $[-\Delta,\Delta]$. We observe that in both the cases, the degeneracies
of the eigenstates are broken for the upper and lower bands, while the
flatband at $E=0$ remains robust even at higher disorder strengths. Further,
the associated eigenstates preserve compact localization in both cases. We
have numerically verified the same, as shown in the case of $AA$ potential
with $\lambda=5$ in Figs. 2(c)–2(e). With the help of the transformation
discussed in Section A.1, we show that the lattice is made of three-site unit
cells but with an absence of inter-cell hopping, indicating the preservation
of the CLSs (see Fig. 8(b)). Additionally, each unit cell has a decoupled site
which strengthens the observation of a robust flatband even in the presence of
the disorder.

##  IV Antisymmetric APPLICATION OF AUBRY-ANDRÉ POTENTIAL

As a further step, we consider the impact of the application of the $AA$
potential in an antisymmetric manner, defined by

$\zeta_{n}^{u}=-\zeta_{n}^{d}=\lambda\cos(2\pi
nb+\theta_{p})\quad\text{and}\quad\zeta_{n}^{c}=0.$ (5)

Here we observe that the introduction of the tinest of disorder results in the
loss of compact localization of the energy eigenstates. Hence we explore the
localization characteristics as a function of the disorder strength $\lambda$
with the aid of several measures: inverse participation ratio (IPR),
multifractal dimensions and level spacings.

(a) (b) (c)

Figure 3: $I_{2}$ and $D_{2}$ in the anti-symmetric case. (a) The spectrum as
a function of increasing strength $\lambda$, where the colour denotes the
value of $I_{2}$. The system size is $N=6000$. (b) $I_{2}$ averaged over the
eigenstates with increasing strength of $AA$ potential $\lambda$ for various
system sizes averaged over $50$ values of $\theta_{p}$. Inset of (b) shows the
scaling of $\left\langle I_{2}\right\rangle$ with system size for
$\lambda=0.01$ and $10$. Here the fitting for the multifractal states at
$\lambda=0.01$ ($\left\langle I_{2}\right\rangle_{0.01}\sim N^{-0.63}$) is
shown with red color and the fitting for the states at $\lambda=10$
($\left\langle I_{2}\right\rangle_{10}\sim N^{-0.54}$) is shown with black
color. (c) The spectrum as a function of increasing strength $\lambda$, where
the colour denotes the value of the fractal dimension $D_{2}$, defined in Eq.
(7), for all the single particle eigenstates. The system size is $N=6000$. The
red solid line in panels (a) and (c) given by $\lambda=4/|E|$ shows the
transition between multifractal and localized states, conjectured in [77] from
the analogy to the extended Harper problem (see Sec V for more details). In
(b) $I_{2}$ averaged over the eigenstates drawn from the inner and outer
regions separated by the fractal mobility edge are shown separately. Solid
lines with filled symbols correspond to states in the inner region and dashed
lines with open symbols to states in the outer region. We observe that the
open symbols corresponding to different system sizes overlap indicating that
these states are localized.

The localization characteristics of the eigenstates can be understood with the
help of the inverse participation ratio (IPR), which is defined as:

${\color[rgb]{0,0,0}I_{2}}=\sum_{n=1}^{\frac{N}{3}}\sum_{\alpha=u,c,d}\left|\psi_{k}(\alpha_{n})\right|^{4}$
(6)

where the $k^{\text{th}}$ normalized single-particle eigenstate
$\left|\psi_{k}\right\rangle=$
$\sum_{n,\alpha}\psi_{k}(\alpha_{n})|\alpha_{n}\rangle$ is written in terms of
the Wannier basis $|\alpha_{n}\rangle$, representing the eigenstate of a
single particle localized at the site $\alpha$ $(\alpha=u,c,d)$ in the
$n^{th}$ unit cell of the lattice. For a completely localized eigenstate
${\color[rgb]{0,0,0}I_{2}}=O(1)$ (when the state is localized on a few sites),
while for a perfectly delocalized eigenstate
${\color[rgb]{0,0,0}I_{2}}=O(1)/N$. Figure 3(a) shows the disorder-averaged
IPR as a function of the strength of the antisymmetric $AA$ potential for the
entire spectrum. The detuned spectrum is confined within the limit
$-2\lambda\leq E\leq 2\lambda$, with the bandwidth showing a roughly linear
relation with $\lambda$. The introduction of the potential lifts the
degeneracy of all the bands and also modifies the localization properties of
the eigenstates. We observe that all the eigenstates for $\lambda\lesssim 1.5$
as well as those associated with the central band for any strength of the
potential are extended while the remaining eigenstates show localization at
higher strengths of the potential.

It turns out that the central band of extended eigenstates are well-described
by a fractal mobility edge $|E|<4/\lambda$ conjectured in a recent preprint
[77]. Figure 3(b) shows the IPR averaged over the eigenstates separated by the
fractal mobility edge $\lambda=4/|E|$ [77]. The IPR of states constituting the
inner section shows system-size dependence and is $\simeq 10^{-3}$, which is a
signature of the extended nature of the eigenstates. However for the states
comprising the outer section, IPR is independent of the system size and is
close to unity, which is an indication of Anderson localization.

###  IV.1 Multifractal analysis

We also analyze a quantity called the fractal dimension $D_{q}$ [31, 78, 79]
which is defined as:

$D_{q}=\frac{S_{q}}{\ln(N)},$ (7)

where $N$ is the dimension of the Hilbert space and $S_{q}$ are the
participation entropies obtained from the $k^{\text{th}}$ eigenstate
$\left|\psi_{k}\right\rangle=$
$\sum_{n,\alpha}\psi_{k}(\alpha_{n})|\alpha_{n}\rangle$ using the relation
$S_{q}=\frac{1}{1-q}\ln I_{q}$ where

$I_{q}=\sum_{n=1}^{\frac{N}{3}}\sum_{\alpha=u,c,d}\left|\psi_{k}(\alpha_{n})\right|^{2q}$
(8)

are the $q^{\text{th}}$ order moments.

(a)

(b)

Figure 4: In the antisymmetric case (a) fractal dimension $D_{q}$ vs $q$ for
various system sizes. (b) Fractal dimension $D_{q}$ vs
$\frac{1}{\text{ln}(N)}$ for various $q$ values and system sizes ranging from
$N=60$ to $N=6000$. Here quasiperiodic strength $\lambda=0.01$ and $D_{q}$ is
averaged over all the eigenstates for at least $50$ values of $\theta_{p}$.

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 5:  In the antisymmetric case, (a)–(c) level spacing $s_{k}^{e-o}$
(red) and $s_{k}^{o-e}$ (blue) at quasiperiodic strength $\lambda=0.01,2$ and
$10$ respectively with averaging done over $50$ values of $\theta_{p}$.
(d)–(f) Single-particle energy spectra $E_{k}$ for $\lambda=0.01,2$, and $10$
respectively. Inset shows the broken degeneracy for the states in the central
band, and (g)–(i) are corresponding level-spacing $s_{k}$. Here index is the
serial number of energy (gap) levels divided by the total number of gaps, and
system size is $N=6000$. For $\lambda=2$ and $10$, the indices corresponding
to the fractal mobility edges $|E|=4/\lambda$, are determined from (e)–(f). We
have plotted them in the corresponding Figs. (b)–(c) and Figs. (h)–(i) using
the vertical red lines.

While considering the ensemble average, the fractal dimension can be defined
in two forms [80, 81]; the first one uses arithmetically ensemble-averaged
moments $\left\langle I_{q}\right\rangle$ after which the logarithm is taken:

$\tilde{D}_{q}=\frac{1}{1-q}\frac{1}{\ln N}\ln\left\langle
I_{q}\right\rangle.$ (9)

In the second approach, the averaging is done in a geometric fashion, i.e.,
after taking the logarithm:

$D_{q}=\frac{1}{1-q}\frac{1}{\ln N}\left\langle\ln I_{q}\right\rangle.$ (10)

Also, $\tilde{D}_{q}$ is a lower bound to $D_{q}$ from Jensen’s inequality
i.e. $\tilde{D}_{q}\leq D_{q}$, since logarithm is a concave function. For
$q=2$, one obtains $S_{2}=-\ln(I_{2})$. Figure 3(c) shows $D_{2}$ as a
function of the strength of the antisymmetric $AA$ potential for the entire
spectrum. We observe the presence of the fractal mobility edge conjectured
recently [77] using an analogy to the extended Harper problem (see Sec V). The
fractal mobility edge ($\lambda\leq 4/|E|$) is shown by the red curves in Fig
3(c), separating the multifractal wave functions (extended but non-ergodic)
with $0<D_{2}<1$, from the Anderson localized eigenstates (red colour
signifies $D_{2}\approx 0$). Here we also study the scaling of IPR
($\left\langle I_{2}\right\rangle$) with system size (see inset of Fig. 3(b)).
The IPR scales as  $N^{-\tilde{D}_{2}}$ [82]. We observe from the inset of
Fig. 3(b) that for $\lambda=0.01$, IPR scales as $N^{-0.63}$, which reaffirms
the multifractal nature of the eigenstates at low potential strengths.

The fractal dimension in the limit $N\rightarrow\infty$ is given by [31]:

$D_{q}^{\infty}=\text{lim}_{N\rightarrow\infty}D_{q}.$ (11)

For a perfectly delocalized state $D_{q}^{\infty}=1$ while for a localized
state $S_{q}$ is a constant, as observed for Anderson localization and results
in vanishing $D_{q}^{\infty}$ for all $q>0$. For intermediate cases,
$0<D_{q}^{\infty}<1$ , which is a sign that the state is extended but non-
ergodic. Further, the eigenstates are multifractal if $D_{q}^{\infty}$ depends
non-trivially on $q>0$ while for a constant $0<D_{q}^{\infty}<1$, the states
are fractal. The $q$-dependence of the fractal dimension $D_{q}$ is shown in
Fig. 4(a). We observe that for all $q>0$, $0<D_{q}<1$ with a non-trivial
dependence on the moment $q$. This indicates that all the eigenstates in the
low-$\lambda$ region exhibit multifractal nature.

For large enough $N$ the IPR is given by the expression

$\displaystyle I_{q}=c_{q}N^{-D_{q}^{\infty}}\ ,$ (12)

with a certain $c_{q}$, weakly-dependent on $N$. This leads to the finite-size
$D_{q}$ being linear in $1/\ln N$ (see Fig. 4(b)) and allows one to extract
$D_{q}^{\infty}$ via a linear extrapolation in $1/\ln N$ [81]. We observe that
in the limit of $N\rightarrow\infty$, the fractal dimension tends to a value
significantly lower than unity. It can be concluded that the multifractality
seen here is robust against increasing system sizes. Moreover, we have also
verified that the system size dependence of $D_{q}$ seen here is very similar
to what is displayed by the AAH model at the critical point.

Another useful method to distinguish between localized, multifractal and
delocalized phases is to carry out an analysis of the even-odd (odd-even)
spacings of the energy eigenvalues $E_{k}$ (arranged in ascending order) [49].
They are defined as $s_{k}^{e-o}=E_{2k}-E_{2k-1}$ and
$s_{k}^{o-e}=E_{2k+1}-E_{2k}$ for even-odd and odd-even cases respectively.
For a localized state, the gap vanishes as both the spacings exhibit the same
form. In the multifractal case, the distributions of both spacings are
strongly scattered. We observe from Fig. 5(a) that at $\lambda=0.01$, both the
spacings are scattered for the entire energy spectrum, indicating that all the
eigenstates are multifractal in the low potential regime. From Fig. 5(b), we
observe that while there is strong scattering corresponding to the states at
the centre ($|E|<4/\lambda$), the gap begins to vanish as one move toward the
edges. At higher potential strengths, for example $\lambda=100$ (see Fig.
5(c)), except for the level spacing at the centre, the gap completely
disappears, indicating localization.

It is well known that multifractal states are characterized by a broad
distribution in energy gaps [83, 84]. To study this aspect, we show plots of
$E_{k}$ and $s_{k}=E_{k+1}-E_{k}$ for various $\lambda$, in Fig. 5(d)–5(i). We
observe that at low potential strength, i.e. $\lambda=0.01$, the energy
spectrum has a large number of subbands (see inset of Fig. 5(d)), and
fluctuations are observed in the spacing of these gaps (see Fig. 5(g)). We
also observe that the level spacing distribution of all the eigenstates at
$\lambda=0.01$ follows an inverse power law [56], indicating that all the
eigenstates are multifractal in the low potential regime. From Fig. 3(c), the
transition point from the fully extended regime to the mixed one with the
mobility edge, $|E|=4/\lambda$, is is observed to be around $\lambda\simeq
1.5$. The magnitude of the gap between the energy levels becomes larger as one
moves toward the edges (Fig. 5(e)), which is accompanied by a decrease in the
magnitude of fluctuations in those gaps (see Fig. 5(h)). At higher potential
strengths, for example $\lambda=10$ (see Fig. 5(f)), the spectrum is
completely pure point-like, and all the eigenstates except those in the
central band show reduced fluctuations, indicating localization (see Fig.
5(i)). We conclude that the presence of the $AA$ potential in the
antisymmetric case transforms the CLSs into multifractal states at low
potential strengths below a critical value (i.e. below $\lambda\simeq 1.5$,
which is about the gap between the flat bands in the zero disorder limit). At
higher potential strengths where the bands hybridize, we observe that all the
eigenstates localize except those in the central part of the spectrum, which
display multifractal nature.

###  IV.2 Chiral Symmetry

The addition of diagonal disorder in the ABF diamond lattice breaks
translational invariance. One would also expect diagonal disorder to break the
chiral symmetry [85, 86, 87] as observed in the symmetric case. However,
remarkably in the antisymmetric case, when $N$ is even, we observe pairs of
eigenvalues ${\pm E}$, despite the on-site disorder. We infer that the chiral
symmetry of the lattice is not broken. Indeed this is confirmed explicitly by
the identification of the chiral operator $\Gamma$ which is required to be a
local operator [74]:

$\Gamma=\gamma_{1}\oplus-\gamma_{2}\oplus\gamma_{3}\cdots=\bigoplus_{i=1}^{n}(-1)^{i-1}\gamma_{i},$
(13)

where each of the matrices $\gamma_{i}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&1\end{pmatrix}_{i}$ acts on the $i^{\text{th}}$ unit cell and $n$ is the
total number of unit cells. We can verify that
$\Gamma^{-1}H^{\dagger}\Gamma=-H$, and that $\Gamma$ is involutory since
$\Gamma\Gamma^{\dagger}=\mathbb{I}$. When periodic boundary conditions are
imposed, the chiral symmetry is valid only for an even number of unit cells.

Before we conclude, we remark that multifractal states have been reported to
exist in systems with chiral symmetry. For the Anderson model in $2-$D [88]
and $3-$D [89], chiral symmetry is known to induce multifractal states near
the band centre, the origin of which can be traced back to the power-law decay
of the eigenstates. It has been observed that chiral symmetry tends to
delocalize eigenstates close to the origin [90]. In $1-$D [91], it has been
proved that eigenstates at the band centre are not localized exponentially.
Chiral symmetry combined with strong disorder induces power-law localization
and multifractal states near the origin [89].

##  V Multifractality in antisymmetric case: Analytical Treatment

In this section, we analytically uncover the origin of the multifractality in
the antisymmetric case. The diamond lattice in the disorder-free limit can be
described through matrices $V$ and $T$ which capture the intra-cell and inter-
cell information respectively:

$V=\begin{pmatrix}0&0&-1\\\ 0&0&1\\\ -1&1&0\end{pmatrix},\hskip
28.45274ptT=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 1&1&0\end{pmatrix}.$ (14)

For the on-site disorder, we introduce another matrix $W_{n}$ given by:

$W_{n}=\begin{pmatrix}\zeta_{n}^{u}&0&0\\\ 0&\zeta_{n}^{d}&0\\\
0&0&\zeta_{n}^{c}\end{pmatrix}.$ (15)

The unitary matrix

${\color[rgb]{0,0,0}U_{1}}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&1\\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&1\\\ 1&1&0\end{pmatrix},$ (16)

diagonalizes the $V$ matrix when we carry out the transformation
$V_{1}={\color[rgb]{0,0,0}U_{1}}V{\color[rgb]{0,0,0}U_{1}}^{\dagger}$. The
same transformation can be applied to the $T$ matrix to obtain
$T_{1}={\color[rgb]{0,0,0}U_{1}}T{\color[rgb]{0,0,0}U_{1}}^{\dagger}$. The
transformed matrices are:

$V_{1}=\left(\begin{array}[]{ccc}-\sqrt{2}&0&0\\\ 0&\sqrt{2}&0\\\ 0&0&0\\\
\end{array}\right),\hskip 7.11317ptT_{1}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&1\\\ 0&0&0\\\ \end{array}\right).$ (17)

Applying the same transformation on the matrix $W_{n}$ we obtain $(W_{1})_{n}$
given by ${\color[rgb]{0,0,0}U_{1}}W_{n}{\color[rgb]{0,0,0}U_{1}}^{\dagger}$:

$(W_{1})_{n}=\left(\begin{array}[]{ccc}\frac{\zeta_{n}^{u}}{4}+\frac{\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&\frac{-\zeta_{n}^{u}}{4}+\frac{-\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&\frac{\zeta_{n}^{u}}{2\sqrt{2}}-\frac{\zeta_{n}^{d}}{2\sqrt{2}}\\\
\frac{-\zeta_{n}^{u}}{4}+\frac{-\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&\frac{\zeta_{n}^{u}}{4}+\frac{\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&\frac{-\zeta_{n}^{u}}{2\sqrt{2}}+\frac{\zeta_{n}^{d}}{2\sqrt{2}}\\\
\frac{\zeta_{n}^{u}}{2\sqrt{2}}-\frac{\zeta_{n}^{d}}{2\sqrt{2}}&\frac{-\zeta_{n}^{u}}{2\sqrt{2}}+\frac{\zeta_{n}^{d}}{2\sqrt{2}}&\frac{\zeta_{n}^{u}}{2}+\frac{\zeta_{n}^{d}}{2}\\\
\end{array}\right).$ (18)

In the antisymmetric case, $\zeta_{n}^{c}=0,$
$\zeta_{n}^{u}=-\zeta_{n}^{d}=\zeta_{n}$, yielding a simplification:

$(W_{1})_{n}=\frac{\zeta_{n}}{\sqrt{2}}\left(\begin{array}[]{ccc}0&0&1\\\
0&0&-1\\\ 1&-1&0\end{array}\right).$ (19)

The above transformation is a part of a series of lattice transformations
shown in Appendix A.1. Using the lattice equation:

$(W_{1})_{n}\psi_{n}-V_{1}\psi_{n}-T_{1}\psi_{n+1}-T_{1}^{\dagger}\psi_{n-1}=E\psi_{n},$
(20)

the corresponding equation for the components
$\psi_{n}=(u_{n},d_{n},c_{n})^{T}$ can be written as:

$\displaystyle
Eu_{n}=\frac{\zeta_{n}}{\sqrt{2}}c_{n}-\left(-\sqrt{2}u_{n}+c_{n+1}\right),$
(21) $\displaystyle
Ed_{n}=-\frac{\zeta_{n}}{\sqrt{2}}c_{n}-\left(\sqrt{2}d_{n}+c_{n+1}\right),$
(22) $\displaystyle
Ec_{n}=\frac{\zeta_{n}}{\sqrt{2}}\left(u_{n}-d_{n}\right)-\left(u_{n-1}+d_{n-1}\right).$
(23)

Substituting $u_{n}$ and $d_{n}$ into Eq. 23 we get,

$E\left[E^{2}-4\right]c_{n}=\zeta_{n}^{2}Ec_{n}-2\zeta_{n}c_{n+1}-2\zeta_{n-1}c_{n-1}.$
(24)

Substituting $\zeta_{n}=\lambda\cos(2\pi nb+\theta)$, we have

$\displaystyle\frac{2}{\lambda^{2}}\left[E^{2}-4-\frac{\lambda^{2}}{2}\right]c_{n}=$
$\displaystyle\cos(4\pi nb+2\theta)c_{n}-$ $\displaystyle-$
$\displaystyle\frac{4}{\lambda E}\Bigl{(}\lambda\cos(2\pi nb+\theta)c_{n+1}+$
$\displaystyle+$ $\displaystyle\lambda\cos(2\pi(n-1)b+\theta)c_{n-1}\Bigr{)}.$
(25)

The above equation resembles the extended Harper model [92], but with a
doubled frequency of the on-site potential. Following the recent preprint
[77], we compare the hopping amplitude $4/(\lambda|E|)$ with the level spacing
amplitude $\max_{n}\left[\cos(4\pi
nb+2\theta)-\cos(4\pi(n+1)b+2\theta)\right]\simeq 2\sin(2\pi b)$ and
conjecture that the transition between non-ergodic extended and localized
states should be at [93]

$\displaystyle\lambda_{c}=\frac{4}{|E|}\ .$ (26)

An alternative way to see the emergence of multifractality in this model is
provided by Thouless in his consideration of the Harper model [58, 94].
Indeed, by taking the discrete Fourier transform:

$c_{m,n}=\frac{1}{\sqrt{N}}\sum_{k}{c_{n}}_{\left(\theta=\frac{2\pi k}{N}+\pi
b\right)}e^{\frac{2\pi i}{N}km},$ (27)

we get a $2-$D model without on-site potential, but with an effective magnetic
flux, penetrating some of the plaquettes:

$\displaystyle E\left[E^{2}-4-\frac{\lambda^{2}}{2}\right]c_{n,m}=$
$\displaystyle\frac{E\lambda^{2}}{4}\left(e^{4i\pi
b(n+\frac{1}{2})}c_{n,m+2}+e^{-4i\pi
b(n+\frac{1}{2})}c_{n,m-2}\right)-\lambda\left(e^{2i\pi
b(n+\frac{1}{2})}c_{n+1,m+1}+e^{-2i\pi b(n+\frac{1}{2})}c_{n+1,m-1}\right)$
$\displaystyle-$ $\displaystyle\lambda\left(e^{2i\pi
b(n-\frac{1}{2})}c_{n-1,m+1}+e^{-2i\pi b(n-\frac{1}{2})}c_{n-1,m-1}\right),$
(28)

which is the $\frac{\pi}{4}$ rotated square lattice with nearest-neighbour and
selected next-nearest-neighbour hoppings and a staggered magnetic field (see
Fig. 6). Such models are also known to host multifractal states in an extended
region of parameter space [95, 96, 97].

Figure 6: A unit cell of the $\frac{\pi}{4}$ rotated square lattice with
nearest-neighbour and selected next-nearest-neighbour hopping and staggered
magnetic field showing the phases of the hopping on each link. The alternating
fluxes $0$ and $Q=4\pi b$ are derived by summing the phases in the clockwise
direction over each separate loop.

While our manuscript was under review, we came across a preprint [77] where
the effect of quasiperiodic perturbations on one-dimensional all-bands-flat
lattice models has been investigated. There the authors discuss the presence
of an energy-dependent critical-to-insulating transition where “fractality
edges” separate localized states from critical states. In the case of the ABF
diamond lattice with an antisymmetric application of disorder, they observed a
critical-to-insulating transition at $|E|=4/\lambda$. In the current version,
we have confirmed the above conjecture with the aid of our numerical data as
well as by a mapping of the Hamiltonian to other models known to host
multifractal states from the literature before. Further, the substitution of
$E=0$ in Eq. 25 gives:

$0=-2\lambda\cos(2\pi
nb+\theta)c_{n+1}-2\lambda\cos(2\pi(n-1)b+\theta)c_{n-1}.$ (29)

This model is equivalent to the off-diagonal Harper model [94, 98] where the
zero energy modes remain critical at all disorder strengths.

##  VI Conclusion

This paper explores the effects of a quasiperiodic $AA$ potential on a one-
dimensional ABF diamond lattice. We find that the fate of the compact
localized states is strongly dependent on the manner in which the potential is
applied. We discuss the consequences when the potential is applied in two
specific ways: symmetric and antisymmetric. When the diagonal disorder is
applied symmetrically, the chiral symmetry of the lattice is broken. Although
the large-scale degeneracy is destroyed, all the eigenstates remain compactly
localized. It is also observed that the compact localization is independent of
the precise nature of the applied perturbation, as long as it is applied in a
symmetric manner. The inclusion of the potential only on the $c$ sites is a
particular case where CLSs are observed and the degeneracy of the central
flatband is preserved.

In the antisymmetric case, the tiniest perturbation lifts the degeneracy, and
the eigenstates no longer remain compactly localized. An exploration of the
nature of the eigenvalues and eigenfunctions through various observables shows
the appearance of flatband-based multifractal states. Here all the wave
functions are extended but non-ergodic in the low disorder regime. All the
bands start to hybridize at a critical potential strength, leading to
conventional Anderson localization at higher magnitudes of $\lambda$. However,
a central band is observed whose states continue to display extended behaviour
at all strengths of $\lambda$. A systematic study of the spectrum and the
spacing between consecutive energy values is performed and compared with the
AAH model. The robust existence of multifractal states with increasing
disorder strength is a remarkable finding. Another interesting finding is that
the chiral symmetry of the Hamiltonian is preserved in the presence of
antisymmetric disorder when the total number of unit cells is even. In support
of this finding, we are able to write down explicitly the chiral operator
$\Gamma$.

The diamond lattice model in the zero-disorder limit can be converted through
a series of transformations into a new lattice with decoupled sites. We study
the effect of the same set of transformations for the diamond lattice model in
the presence of disorder. We find that in the symmetric case, these
transformations yield a lattice which displays an absence of inter-cell
hopping, indicating the preservation of compact localization. On the other
hand, the antisymmetric configuration of disorder supports the loss of CLSs
owing to inter-cell hopping in the transformed lattice. Further, we
demonstrate that our lattice transformations convert our Hamiltonian (with
antisymmetric disorder) into a close relative of the extended Harper model and
into a $\frac{\pi}{4}$ rotated square lattice with nearest-neighbour and
selective next-nearest-neighbour hopping and staggered magnetic field, both of
which support the multifractality observed in the antisymmetric case.

We have seen that the introduction of the $AA$ potential in a flat-band
diamond chain yields interesting results within a single-particle setup. An
exciting direction for research would be to explore the physics of such
systems in the presence of interactions, and in particular, to look for flat-
band-based many-body localization phenomena. We also look forward to future
studies that can extend these ideas to two and three dimensional systems.

###### Acknowledgements.

We are grateful to Carlo Danieli and Nilanjan Roy for discussions. A.A. is
grateful to the Council of Scientific and Industrial Research (CSIR), India,
for her PhD fellowship. A.R. acknowledges the financial support from Christ
College Irinjalakuda via CCRSF-2021. A.S. acknowledges financial support from
SERB via the grant (File Number: CRG/2019/003447) and from DST via the DST-
INSPIRE Faculty Award [DST/INSPIRE/04/2014/002461]. I. M. K. acknowledges the
support by the European Research Council under the European Union’s Seventh
Framework Program Synergy No. ERC-2018-SyG HERO-810451.

##  Appendix A Lattice transformation

In this section, a series of transformations are presented, whose application
to the Hamiltonian of the diamond chain gives a new lattice which has
decoupled sites in the zero-disorder limit. We utilize a similarity
transformation to obtain a new Hamiltonian whose spacial arrangement of the
sites significantly differs from the original lattice following the work of
Danieli et al. [99]. On the one hand, this process identifies new flat band
lattices and, on the other hand, helps in understanding the influence of the
symmetry of the applied potential. We first present the transformation of the
unperturbed ABF diamond chain. We then discuss the effect of the symmetric and
antisymmetric configuration of disorder on the ABF diamond chain with the help
of these transformations.

Figure 7:  Schematic representations of the transformation of the diamond
chain into the new lattice with decoupled sites.

###  A.1 Transformation of unperturbed ABF diamond chain

The unperturbed diamond chain shown in Fig. 7(a) consists of three sites per
unit cell. The intra-cell and inter-cell information of the Hamiltonian can be
represented by $V$ and $T$ respectively and is given by:

$V=\begin{pmatrix}0&0&-1\\\ 0&0&1\\\ -1&1&0\end{pmatrix},\hskip
28.45274ptT=\begin{pmatrix}0&0&0\\\ 0&0&0\\\ 1&1&0\end{pmatrix}.$ (30)

Thus, the lattice equation is

$-V\psi_{n}-T\psi_{n+1}-T^{\dagger}\psi_{n-1}=E\psi_{n},$ (31)

with $\psi_{n}=(u_{n},d_{n},c_{n})^{T}$ being the tight binding representation
of the wavefunction for the $n^{th}$ unit cell. Considering the unitary matrix

${\color[rgb]{0,0,0}U_{1}}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}&1\\\
-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&1\\\ 1&1&0\end{pmatrix},$ (32)

we perform the transformations
$V_{1}={\color[rgb]{0,0,0}U_{1}}V{\color[rgb]{0,0,0}U_{1}}^{\dagger}$ and
$T_{1}={\color[rgb]{0,0,0}U_{1}}T{\color[rgb]{0,0,0}U_{1}}^{\dagger}$ yielding
new matrices

$V_{1}=\left(\begin{array}[]{ccc}-\sqrt{2}&0&0\\\ 0&\sqrt{2}&0\\\ 0&0&0\\\
\end{array}\right),\hskip 7.11317ptT_{1}=\left(\begin{array}[]{ccc}0&0&1\\\
0&0&1\\\ 0&0&0\\\ \end{array}\right).$ (33)

The resulting lattice is shown in Fig. 7(b). A new unit cell can be identified
considering the connected lattice sites $(u_{n},c_{n+1},d_{n})$, which affirms
that the CLS stays in one unit cell and the class of the CLS is $U=1$ in this
representation. The corresponding lattice is shown in Fig. 7(c), and the
matrices $V_{2}$ and $T_{2}$ are given by

$V_{2}=\left(\begin{array}[]{ccc}-\sqrt{2}&1&0\\\ 1&0&1\\\ 0&1&\sqrt{2}\\\
\end{array}\right),\hskip 7.11317ptT_{2}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&0\\\ \end{array}\right).$ (34)

Identifying the CLS in the new lattice, an inverse transformation can be
performed to obtain the CLS in the diamond chain, which matches with Danieli
et al. [99]. Hence the class of CLS is not unique under a transformation.

A further transformation can be performed to represent the lattice into a Fano
defect form with decoupled sites. For this, the matrix $H_{2}$ will be defined
as $H_{2}=-V_{2}-T_{2}e^{ik}-T_{2}^{\dagger}e^{-ik}$ for the lattice in Fig.
7(c). We obtain the transformation matrix ${\color[rgb]{0,0,0}U_{2}}$ here
from the eigenvectors of the matrix $H_{2}$, which is given by

${\color[rgb]{0,0,0}U_{2}}=\left(\begin{array}[]{ccc}\frac{3-2\sqrt{2}}{2\sqrt{6-4\sqrt{2}}}&\frac{2-\sqrt{2}}{2\sqrt{6-4\sqrt{2}}}&\frac{1}{2\sqrt{6-4\sqrt{2}}}\\\
\frac{3+2\sqrt{2}}{2\sqrt{6+4\sqrt{2}}}&\frac{-2-\sqrt{2}}{2\sqrt{6+4\sqrt{2}}}&\frac{1}{2\sqrt{6+4\sqrt{2}}}\\\
-\frac{1}{2}&-\frac{1}{\sqrt{2}}&\frac{1}{2}\\\ \end{array}\right).$ (35)

The transformed matrices
$V_{3}={\color[rgb]{0,0,0}U_{2}}V_{2}{\color[rgb]{0,0,0}U_{2}}^{\dagger}$ and
$T_{3}={\color[rgb]{0,0,0}U_{2}}T_{2}{\color[rgb]{0,0,0}U_{2}}^{\dagger}$ are:

$V_{3}=\left(\begin{array}[]{ccc}2&0&0\\\ 0&-2&0\\\ 0&0&0\\\
\end{array}\right),\hskip 7.11317ptT_{3}=\left(\begin{array}[]{ccc}0&0&0\\\
0&0&0\\\ 0&0&0\\\ \end{array}\right).$ (36)

The resulting lattice (see Fig 7(d)) consists of three linear chains
$(x_{n},y_{n},z_{n})$ without any hopping between sites. The isolated sites
represent the flat band lattice in this representation, with the eigenstates
strictly localized on only one site.

###  A.2 Transformation of ABF diamond chain with on-site potential

In the presence of on-site disorder, in addition to the intra-cell and
intercell matrices, we introduce another matrix $W_{n}$ given by

${\color[rgb]{0,0,0}W_{n}}=\begin{pmatrix}\zeta_{n}^{u}&0&0\\\
0&\zeta_{n}^{d}&0\\\ 0&0&\zeta_{n}^{c}\end{pmatrix},$ (37)

which is added in the general Hamiltonian:
$H={\color[rgb]{0,0,0}W}-V-Te^{ik}-T^{\dagger}e^{-ik}$. The transformation of
the matrix $W_{n}$ given by
${\color[rgb]{0,0,0}U_{1}}{\color[rgb]{0,0,0}W_{n}}{\color[rgb]{0,0,0}U_{1}}^{\dagger}$
results in:

$({\color[rgb]{0,0,0}W_{1}})_{n}=\left(\begin{array}[]{ccc}\frac{\zeta_{n}^{u}}{4}+\frac{\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&\frac{-\zeta_{n}^{u}}{4}+\frac{-\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&\frac{\zeta_{n}^{u}}{2\sqrt{2}}-\frac{\zeta_{n}^{d}}{2\sqrt{2}}\\\
\frac{-\zeta_{n}^{u}}{4}+\frac{-\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&\frac{\zeta_{n}^{u}}{4}+\frac{\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&\frac{-\zeta_{n}^{u}}{2\sqrt{2}}+\frac{\zeta_{n}^{d}}{2\sqrt{2}}\\\
\frac{\zeta_{n}^{u}}{2\sqrt{2}}-\frac{\zeta_{n}^{d}}{2\sqrt{2}}&\frac{-\zeta_{n}^{u}}{2\sqrt{2}}+\frac{\zeta_{n}^{d}}{2\sqrt{2}}&\frac{\zeta_{n}^{u}}{2}+\frac{\zeta_{n}^{d}}{2}\\\
\end{array}\right),$ (38)

in addition to $V_{1}$ and $T_{1}$ for the transformed lattice. As before, re-
arranging the unit cell, new matrices incorporating the on-site contributions
can be obtained as:

$\displaystyle V_{({\color[rgb]{0,0,0}W_{2}})_{n}}=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\zeta_{n}^{u}}{4}+\frac{\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}&0&\frac{\zeta_{n}^{u}}{2\sqrt{2}}-\frac{\zeta_{n}^{d}}{2\sqrt{2}}\\\
0&\frac{\zeta_{n+1}^{u}}{2}+\frac{\zeta_{n+1}^{d}}{2}&0\\\
\frac{\zeta_{n}^{u}}{2\sqrt{2}}-\frac{\zeta_{n}^{d}}{2\sqrt{2}}&0&\frac{\zeta_{n}^{u}}{4}+\frac{\zeta_{n}^{d}}{4}+\frac{\zeta_{n}^{c}}{2}\\\
\end{array}\right)$ (39d) $\displaystyle T_{({\color[rgb]{0,0,0}W_{2}})_{n}}=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&0\\\
\frac{\zeta_{n}^{u}}{2\sqrt{2}}-\frac{\zeta_{n}^{d}}{2\sqrt{2}}&0&-\frac{\zeta_{n}^{u}}{2\sqrt{2}}+\frac{\zeta_{n}^{d}}{2\sqrt{2}}\\\
0&0&0\\\ \end{array}\right),$ (39h)

in addition to $V_{2}$ and $T_{2}$.

Similarly, we can obtain
$V_{({\color[rgb]{0,0,0}W_{3}})_{n}}={\color[rgb]{0,0,0}U_{2}}V_{({\color[rgb]{0,0,0}W_{2}})_{n}}{\color[rgb]{0,0,0}U_{2}}^{\dagger}$
and
$T_{({\color[rgb]{0,0,0}W_{3}})_{n}}={\color[rgb]{0,0,0}U_{2}}T_{({\color[rgb]{0,0,0}W_{2}})_{n}}{\color[rgb]{0,0,0}U_{2}}^{\dagger}$
in addition to $V_{3}$ and $T_{3}$ as:

$\displaystyle V_{({\color[rgb]{0,0,0}W_{3}})_{n}}=$
$\displaystyle\left(\begin{array}[]{ccc}\zeta_{1}&\zeta_{2}&\zeta_{3}\\\
\zeta_{2}&\zeta_{1}&-\zeta_{3}\\\ \zeta_{3}&-\zeta_{3}&\zeta_{4}\\\
\end{array}\right),$ (43) $\displaystyle T_{{\color[rgb]{0,0,0}(W_{3})_{n}}}=$
$\displaystyle\left(\begin{array}[]{ccc}\zeta_{5}&-\zeta_{5}&\zeta_{6}\\\
-\zeta_{5}&\zeta_{5}&-\zeta_{6}\\\
\zeta_{6}&-\zeta_{6}&-\frac{1}{2}\zeta_{5}\\\ \end{array}\right),$ (47)

where
$\zeta_{1}=\frac{1}{8}(\zeta_{n}^{u}+\zeta_{n}^{d}+4\zeta_{n}^{c}+\zeta_{n+1}^{u}+\zeta_{n+1}^{d})$,
$\zeta_{2}=\frac{1}{8}(-\zeta_{n}^{u}-\zeta_{n}^{d}+4\zeta_{n}^{c}-\zeta_{n+1}^{u}-\zeta_{n+1}^{d})$,
$\zeta_{3}=\frac{1}{4\sqrt{2}}(\zeta_{n}^{u}+\zeta_{n}^{d}-\zeta_{n+1}^{u}-\zeta_{n+1}^{d})$,
$\zeta_{4}=\frac{1}{4}(\zeta_{n}^{u}+\zeta_{n}^{d}+\zeta_{n+1}^{u}+\zeta_{n+1}^{d})$,
$\zeta_{5}=\frac{1}{8}(-\zeta_{n+1}^{u}+\zeta_{n+1}^{d})$,
$\zeta_{6}=\frac{1}{4\sqrt{2}}(-\zeta_{n+1}^{u}+\zeta_{n+1}^{d})$.

(a) (b) (c)

Figure 8:  Schematic representations of the new lattice with disorder in the
symmetric case: (a) $\zeta_{n}^{u}=\zeta_{n}^{d}\neq 0,\zeta_{n}^{c}=0$, (b)
$\zeta_{n}^{u}=\zeta_{n}^{d}=0,\zeta_{n}^{c}\neq 0$ and (c) the antisymmetric
case: $\zeta_{n}^{u}=-\zeta_{n}^{d}\neq 0,\zeta_{n}^{c}=0$

#### Symmetric case

##### Disorder on u and d sites:

First, we consider the case with $\zeta_{n}^{u}=\zeta_{n}^{d}\neq 0$ and
$\zeta_{n}^{c}=0$.

Here the effects of on-site disorder incorporated through Eq. 43 and Eq. 47
together with Eq. 36 of the unperturbed lattice, result in matrices
$V_{S_{1}}$ and $T_{S_{1}}$ [100] which represent the intracell and intercell
hopping of the lattice:

$V_{S_{1}}=\left(\begin{array}[]{ccc}\frac{1}{8}(2\zeta_{n}^{u}+2\zeta_{n+1}^{u})+2&\frac{1}{8}(-2\zeta_{n}^{u}-2\zeta_{n+1}^{u})&\frac{1}{4\sqrt{2}}(2\zeta_{n}^{u}-2\zeta_{n+1}^{u})\\\
\frac{1}{8}(-2\zeta_{n}^{u}-2\zeta_{n+1}^{u})&\frac{1}{8}(2\zeta_{n}^{u}+2\zeta_{n+1}^{u})-2&\frac{1}{4\sqrt{2}}(-2\zeta_{n}^{u}+2\zeta_{n+1}^{u})\\\
\frac{1}{4\sqrt{2}}(2\zeta_{n}^{u}-2\zeta_{n+1}^{u})&\frac{1}{4\sqrt{2}}(-2\zeta_{n}^{u}+2\zeta_{n+1}^{u})&\frac{1}{4}(2\zeta_{n}^{u}+2\zeta_{n+1}^{u})\\\
\end{array}\right),\hskip
7.11317ptT_{S_{1}}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\\\
\end{array}\right).$ (48)

The corresponding lattice (see Fig. 8(a)) shows that the symmetric nature of
the disorder decouples the unit cells of the system. The transformed lattice
is a linear chain of uncoupled unit cells, implying that the states do not
hybridize. At the same time, their probability amplitudes may be re-arranged
along the sites to satisfy the lattice equation.

##### Disorder on c sites:

We next consider another type of symmetric configration where disorder is
introduced only on the $c$ sites i.e. $\zeta_{n}^{u}=\zeta_{n}^{d}=0$ and
$\zeta_{n}^{c}\neq 0$. Here Eq. 43 and Eq. 47 together with Eq. 36 for the
unperturbed lattice, give matrices $V_{S_{2}}$ and $T_{S_{2}}$:

$V_{S_{2}}=\left(\begin{array}[]{ccc}\frac{\zeta_{n}^{c}}{2}+2&\frac{\zeta_{n}^{c}}{2}&0\\\
\frac{\zeta_{n}^{c}}{2}&\frac{\zeta_{n}^{c}}{2}-2&0\\\ 0&0&0\\\
\end{array}\right),\hskip
7.11317ptT_{S_{2}}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&0&0\\\
\end{array}\right).$ (49)

As observed from Eq. 49, there is no intercell hopping here. However, we find
intracell hopping between two sites and an uncoupled site in a unit cell (see
Fig. 8(b)).

#### Antisymmetric case

In the antisymmetric case, $\zeta_{n}^{u}=-\zeta_{n}^{d}$ while
$\zeta_{n}^{c}=0$. Then Eq. 43 and Eq. 47 in addition to $V_{3}$ and $T_{3}$
(see Eq. 36) for the unperturbed lattice, give matrices $V_{A}$ and $T_{A}$
which represent the intracell and intercell hopping in the lattice:

$V_{A}=\left(\begin{array}[]{ccc}2&0&0\\\ 0&-2&0\\\ 0&0&0\\\
\end{array}\right),\hskip
7.11317ptT_{A}=\left(\begin{array}[]{ccc}\frac{-\zeta_{n+1}^{u}}{4}&\frac{\zeta_{n+1}^{u}}{4}&\frac{-\zeta_{n+1}^{u}}{2\sqrt{2}}\\\
\frac{\zeta_{n+1}^{u}}{4}&\frac{-\zeta_{n+1}^{u}}{4}&\frac{\zeta_{n+1}^{u}}{2\sqrt{2}}\\\
\frac{\zeta_{n+1}^{u}}{2\sqrt{2}}&\frac{-\zeta_{n+1}^{u}}{2\sqrt{2}}&\frac{\zeta_{n+1}^{u}}{2}\\\
\end{array}\right).$ (50)

The lattice corresponding to it is shown in Fig. 8(c). We observe that the
resulting lattice is a $3-$leg cross-stitch chain. The inter-cell hopping in
the lattice results in the non-existence of the CLSs, unlike the symmetric
case. We observe from the transformed lattice that the potential in the
antisymmetric case leads to coupling between the adjacent unit cells. Still,
the sites in a single unit cell remain decoupled. At low potential strengths,
in the absence of intra-cell coupling, the nearest neighbour (NN) inter-cell
coupling leads to the multifractal nature of the eigenstates. As the strength
of $\lambda$ increases, the stronger coupling leads to Anderson localization.

##  Appendix B Complementary quantities

In this section, we point out the usefulness of discussing several
complementary quantities when the Aubry-Andrè potential is applied in an
antisymmetric manner. We also analyze the effect of the antisymmetric
application of the uniform disorder through various measures.

(a) (b) (c)

Figure 9: In the antisymmetric case (a) IPR and PR averaged of all the single-
particle eigenstates separated by the fractal mobility edge $\lambda=4/|E|$
(solid lines with filled symbols for states in the inner region and dashed
lines with open symbols for states in the outer region) with increasing
strength of $AA$ potential $\lambda$ for $N=6000$ averaged over $50$ values of
$\theta_{p}$. The shaded region signifies the transition of the eigenstates
from extended to Anderson localization. (b) Single particle von Neumann
entropy $S^{\mathrm{sp}}$ averaged over the eigenstates separated by the
fractal mobility edge (solid lines with solid symbols for states in the inner
region and dashed lines with open symbols for states in the outer region) with
increasing strength of $AA$ potential $\lambda$ for various system sizes.
Averaging has been done over $50$ values of $\theta_{p}$ for all cases. (c)
The spectrum of the diamond lattice, where the color denotes the value of
fidelity $F$ with respect to eigenstates at $\lambda=0.1$. Here the system
size considered is $N=6000$. The black solid line in panel (c) shows the
transition between multifractal and localized states, conjectured in a recent
preprint [77] using an analogy to the extended Harper problem (see Sec. V for
more details).

###  B.1 AA disorder

The normalized participation ratio (PR) [101] is closely related to the
inverse participation ratio and is given by:

$P_{k}=\left[N\sum_{n=1}^{\frac{N}{3}}\sum_{\alpha=u,c,d}\left|\psi_{k}({\color[rgb]{0,0,0}\alpha_{n}})\right|^{4}\right]^{-1}.$
(51)

It vanishes for a perfectly localized eigenstate and goes to unity for a
perfectly delocalized eigenstate. In Fig. 9(a), we have plotted the IPR, and
the PR averaged over all the eigenstates in the inner part (shown with solid
line) and the outer part (shown with dashed line) of the fractal mobility edge
$\lambda=4/|E|$ [77], with increasing strength of $\lambda$. These two
quantities together help in the identification of the transition region. The
transition from the extended ($0<\text{PR}<1$) to the Anderson localized
regime lies around $\lambda\simeq 1.5$. This is because, in the zero disorder
limit, the gaps between the flat bands are precisely of size $2$, and thus a
disorder strength of around $1.5-2$ allows an inter-band hybridization.

Another measure that provides an understanding of the extent of localization
in a system is the single-particle von Neumann entanglement entropy [102]. The
von Neumann entropy associated with site $\alpha$ of the $n^{th}$ unit cell in
the $k^{th}$ eigenstate is given by [103]:

$\begin{split}S_{k}^{\alpha_{n}}=&-\left|\psi_{k}(\alpha_{n}))\right|^{2}\log_{2}\left(\left|\psi_{k}(\alpha_{n})\right|^{2}\right)\\\
&-\left(1-\left|\psi_{k}(\alpha_{n})\right|^{2}\right)\log_{2}\left(1-\left|\psi_{k}(\alpha_{n})\right|^{2}\right).\end{split}$
(52)

For a delocalized eigenstate $\left|\psi_{k}(\alpha_{n})\right|^{2}=1/N$ and
hence $S_{k}^{\alpha_{n}}\approx\frac{1}{N}\log_{2}N+\frac{1}{N}$ for large
values of $N$ whereas for an eigenstate localized on a single-site
$S_{k}^{\alpha_{n}}=0$. The contributions from all sites for a particular
eigenstate are given by $S_{k}=\sum_{n,\alpha}S_{k}^{\alpha_{n}}$. Thus the
average von Neumann entropy over all the eigenstates is defined as:

$S^{\mathrm{sp}}=\frac{\sum_{k=1}^{N}S_{k}}{N}.$ (53)

For large values of $N$, $S^{\mathrm{sp}}\approx\left(\log_{2}N+1\right)$ in
the delocalized phase whereas $S^{\mathrm{sp}}\approx 0$ in an extremely
(single-site) localized phase. Figure 9(b) shows the single-particle
entanglement entropy averaged over all the eigenstates in the inner part
(shown with solid line) and the outer part (shown with dashed line) of the
fractal mobility edge $\lambda=4/|E|$ [77], with increasing strength of
$\lambda$ for various system sizes. Here, we observe a system size dependence
in $S^{\mathrm{sp}}$ for the states in the inner region with its magnitude
being marginally less than its maximum value, which is a sign of the extended
nature of the eigenstates. $S^{\mathrm{sp}}$ is largely system-size
independent with its magnitude approaching $O(10^{-1})$, for the states
outside the fractal mobility edge indicating Anderson localization.

We have also plotted the fidelity or overlap between the eigenstates, which
helps distinguish extended and localized regions in the spectrum. Fidelity
between the $k^{\text{th}}$ eigenstates corresponding to two values of
$\lambda$ is given by

$F_{12}^{k}=\left|\left\langle\psi_{k}\left(\lambda_{1}\right)\mid\psi_{k}\left(\lambda_{2}\right)\right\rangle\right|^{2}.$
(54)

By choosing the first parameter $\lambda_{1}$ as the reference point, the
second parameter $\lambda_{2}$ is varied in Fig. 9(c). We have previously
shown that all the states below $\lambda\approx 1.5$ have multifractal nature.
We observe that for larger $\lambda$ the magnitude of fidelity is close to
zero for all the states (except the central band) indicating localization. For
the central band, for all values of $\lambda$, the fidelity $0<F<1$. This
indicates that though the states are multifractal in the central band, the
amplitude distribution on the lattice sites does not remain fixed with
increasing strength of the potential $\lambda$.

####  B.1.1 Scaling Analysis

The scaling analysis of the distribution of the IPR logarithm
$P(\text{ln}I_{2})$ at the critical point shows invariance of shape or width
with increasing system size $N$ [104]. After shifting the curves along the
x-axis, they all lie on top of each other, forming a scale-invariant IPR
distribution. We utilize this measure and analyze all the non-ergodic extended
states at low $-\lambda=0.01$ (see Fig. 10(a)) and those separated by the
fractal mobility edge [77] $\lambda_{c}=\frac{4}{|E|}$ at high $-\lambda=10$
(see Fig. 10(b)), and observe that the distributions of IPR are indeed scale
invariant. However for the states outside the fractality edge at a higher
disorder strength (like $\lambda=10$), we observe that the distribution is
independent of $N$, confirming that the states are localized (see Fig. 10(c)).

(a) (b) (c)

(d) (e) (f)

Figure 10: In the anti-symmetric case, distribution of $P(\ln I_{2})$ for
different system sizes computed using (a) all the eigenstates at
$\lambda=0.01$ (b) multifractal states comprising the inner region of the
fractal mobility edge $\lambda=4/|E|$, conjectured in [77] at $\lambda=10$ and
(c) localized states that exist in the outer region of the fractal mobility
edge at $\lambda=10$. The change of $\ln\chi$ as a function of $\ln l$ for
different system sizes computed using (d) all the eigenstates at
$\lambda=0.01$ (e) multifractal eigenstates comprising the inner region of the
fractal mobility edge $\lambda=4/|E|$, at $\lambda=10$ and (f) localized
states that exists in the outer region of the fractal mobility edge at
$\lambda=10$.

We also perform a multifractal analysis of the wave functions using the box-
counting method. For the AAH model, it was observed that the wave functions
exhibit multifractal behaviour extending to all length scales at the critical
point. For localized states, multifractal features are observed up to the
localization length and for extended states they are observed up to the
correlation length [105]. We analyze the multifractal properties here by
coarse-graining the system into boxes of length $l$. Given a normalized wave
function $\left|\psi_{k}\right\rangle=$ $\sum_{i=1}^{N}\psi_{k}(i)|i\rangle$
defined over a lattice of size $N$, we divide the lattice into $N/l$ segments
of length $l$ [106]:

$\chi_{j}(q)=\sum_{p=1}^{N/l}\left[\sum_{i=(p-1)l+1}^{pl}|\psi_{k}(i)|^{2}\right]^{q}.$
(55)

The average is then considered over the total number of states in the central
band (CB) $j_{\text{CB}}$:

$\chi(q)_{\text{CB}}=\frac{1}{j_{\text{CB}}}\sum_{j}^{j_{\text{CB}}}\chi_{j}(q),$
(56)

and the total number of states of the sidebands (SB) $j_{\text{SB}}$:

$\chi(q)_{\text{SB}}=\frac{1}{j_{\text{SB}}}\sum_{j}^{j_{\text{SB}}}\chi_{j}(q).$
(57)

Multifractality is characterized by a power-law behavior of
$\chi(q)\sim(l/L)^{\tau(q)}$ with the exponent $\tau(q)$ determining the
multifractal dimension $D_{q}=\tau(q)/(q-1)$ where $q=2$ gives
$\tau(2)=D_{2}$. In Fig. 10(d–f), we display the change of $\ln\chi$ as a
function of $\ln l$ for different system sizes. In the critical regime, it is
found that $\ln\chi$ is a linear function of $\ln l$ described by a series of
parallel lines for different $N$ with the same slope $D_{2}$. We observe this
behavior for the all states at low$-\lambda=0.01$ (see Fig. 10(d)) with
${\color[rgb]{0,0,0}D_{2}}=0.63\pm 0.005$ indicating that they are indeed non-
ergodic extended. The same is observed at high $-\lambda=10$, for the states
comprising the inner band of the fractal mobility edge [77]
$\lambda_{c}=\frac{4}{|E|}$ with ${\color[rgb]{0,0,0}D_{2}}=0.61\pm 0.005$
(see Fig. 10(e)). In the localized region, for lengths less than the
localization length $l<$ $l_{c}$, $\ln\chi$ is a linear function of $\ln l$,
which completely superposes together for different $N$ with an identical slope
of $D_{2}$ \- so one might naively conclude that the system exhibits
multifractal behaviour. However for $l>l_{c}$ the slopes decrease to $0$. The
same can be observed for all the states outside the fractal mobility edge at
high $-\lambda$ (see Fig. 10(f)) with $D_{2}\approx 0.11$ (fitted with the
dotted curves) for $l<l_{c}$ while the slopes decrease to $0$ (fitted with the
magenta lines) for $l>l_{c}$, thus showing that in fact these states are
localized.

(a) (b) (c)

(d) (e) (f)

Figure 11: In the anti-symmetric case (a) gap ratio $r$ [107, 108] and (c) IPR
averaged over all the eigenstates for uniform uncorrelated random disorder
with increasing strength $\Delta$ for various system sizes. The averaging has
been done over $50$ values of $\theta_{p}$. The spectrum of system size
$N=6000$ with increasing strength of disorder $\Delta$, where the colour
denotes the value of (b) IPR and (e) fractal dimension $D_{2}$ for all the
single-particle eigenstates. $D_{2}$ averaged over all the eigenstates with
increasing system size when applied disorder is (d) uniform uncorrelated
random disorder and (f) quasiperiodic $AA$ disorder.

###  B.2 Uniform uncorrelated random disorder

We have also analyzed the application of the uniform uncorrelated random
disorder in an antisymmetric manner. Here $\Delta$ is the disorder strength.
We observe from Fig. 11(a) that the average gap ratio $r$ [107, 108] remains
around the Poisson value ($\approx 0.39$) at all strengths of the disorder.
Further from the energy-resolved IPR study (see Fig. 11(b)), it can be
observed that all the eigenstates exhibit low IPR below $\Delta\approx 2$. The
presence of a mobility edge is also observed here. Dividing Eq. (24) by
$\Delta$ we get:

$\frac{E}{\Delta}\left[E^{2}-4\right]c_{n}=w_{n}\frac{\Delta
E}{2}c_{n}-t_{n}c_{n+1}-t_{n-1}c_{n-1},$ (58)

where $t_{n}=\zeta_{n}/\Delta$ are i.i.d. random numbers, homogeneously
distributed in a unit interval $|t_{n}|<1$, while
$w_{n}=4\zeta_{n}^{2}/\Delta^{2}$ are i.i.d. random numbers, whose
distribution $P(w_{n})\sim\theta(1-w_{n})/(2\sqrt{w_{n}})$ is singular but
integrable due to the cut tail for $|w_{n}|>1$. From this one can estimate the
finite-size mobility edge as the line where the localization length $\xi(E)$,
determining the eigenstate exponential decay

$\displaystyle|\psi_{E}(r)|\sim e^{-|r-r_{E}|/\xi_{E}}$ (59)

with respect to the random energy-dependent maximum $r_{E}$, is of the order
of the system size $\xi(E)\simeq N$.

The expression for $\xi(E)$ can be estimated as (see, e.g., [109])

$\displaystyle\xi_{E}\simeq
F\left(\frac{Et_{typ}}{w_{typ}^{2}}\right)\frac{t_{typ}^{2}}{w_{typ}^{2}}\ ,$
(60)

where a smooth function $F(x)\simeq O(N^{0})$ [110] can be replaced by a
constant as it changes by $10~{}\%$ from $x=0$ to $x=\infty$ and the typical
value $w_{typ}$ ($t_{typ}$) of the on-site disorder $w_{n}$ (hopping $t_{n}$)
is given by the typical value of the distributions of $P(\ln w_{n})$ ($P(\ln
t_{n})$). In our case

$\displaystyle w_{typ}=e^{\langle\ln w_{n}\rangle}=e^{-2},\quad
t_{typ}=e^{\langle\ln t_{n}\rangle}=e^{-1}\ ,$ (61)

i.e.,

$\displaystyle\xi_{E}(N)\simeq\left(\frac{2e}{\Delta|E|}\right)^{2}F\ .$ (62)

Thus from $\xi_{E}\simeq N$ we get:

$\displaystyle\Delta_{c}(E)=\frac{2e}{E}\sqrt{\frac{F}{N}}\ .$ (63)

This demonstrates that the mobility edge, shown in Fig. 11(b) for $N=6000$ and
$F=6000$ is a finite-size effect. However, on the other hand, it also shows
why the state at exactly zero energy $E=0$ (which exists for odd $N$) will not
localize. The latter is related to the conserved chiral symmetry in the
system, where the low energy states may keep their delocalized nature even in
the $1$d chiral Anderson model. In any case this regime deserves further
detailed investigations in future works.

From the IPR averaged over all the eigenstates (see Fig. 11(c)), we observe
that it is system size-independent, indicating localization. Thus while the
averaged gap ratio and IPR suggest localization for the entire disorder range,
the same is not observed from the energy-resolved IPR. The spectrum resolved
fractal dimension $D_{2}$ is plotted in Fig. 11(e). While the eigenstates
below $\Delta\approx 2$ and those belonging to the central band remain
multifractal with $D_{2}\approx 0.2$, they are less extended when the applied
perturbation is $AA$ ($D_{2}\approx 0.6$). Thus, we infer that the
localization characteristics of the eigenstates in the low-disorder regime
depends on the nature of the applied potential. On the other hand, when the
strength of the disorder is sufficient, the different bands hybridize,
conventional Anderson localization takes over, and the details of the form of
the disorder are not important. Figure 11(d) shows $D_{2}$ averaged over all
the eigenstates when the uniform disorder is applied. The magnitude of $D_{2}$
decreases both in the low and high disorder regime with increasing system
size, indicating a steady decline in the fraction of states exhibiting
multifractality. In contrast, the fraction of states exhibiting the
multifractal behaviour remains robust for the $AA$ potential Figure 11(f).
This suggests that the specific form of the $AA$ potential has an important
role in ensuring multifractality.

## References

  * Parameswaran  _et al._ [2013] S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum Hall physics in topological flat bands, Comptes Rendus Physique 14, 816 (2013), topological insulators / Isolants topologiques.
  * Bergholtz and Liu [2013] E. J. Bergholtz and Z. Liu, Topological flat band models and fractional Chern insulators, International Journal of Modern Physics B 27, 1330017 (2013).
  * Derzhko  _et al._ [2015] O. Derzhko, J. Richter, and M. Maksymenko, Strongly correlated flat-band systems: The route from Heisenberg spins to Hubbard electrons, International Journal of Modern Physics B 29, 1530007 (2015).
  * Flach  _et al._ [2014] S. Flach, D. Leykam, J. D. Bodyfelt, P. Matthies, and A. S. Desyatnikov, Detangling flat bands into Fano lattices, EPL (Europhysics Letters) 105, 30001 (2014).
  * 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).
  * Maimaiti  _et al._ [2017] W. Maimaiti, A. Andreanov, H. C. Park, O. Gendelman, and S. Flach, Compact localized states and flat-band generators in one dimension, Phys. Rev. B 95, 115135 (2017).
  * Sathe  _et al._ [2021] P. Sathe, F. Harper, and R. Roy, Compactly supported Wannier functions and strictly local projectors, Journal of Physics A: Mathematical and Theoretical 54, 335302 (2021).
  * Vidal  _et al._ [2000] J. Vidal, B. Douçot, R. Mosseri, and P. Butaud, Interaction induced delocalization for two particles in a periodic potential, Phys. Rev. Lett. 85, 3906 (2000).
  * Douçot and Vidal [2002] B. Douçot and J. Vidal, Pairing of Cooper pairs in a fully frustrated Josephson-junction chain, Phys. Rev. Lett. 88, 227005 (2002).
  * Mukherjee  _et al._ [2018] S. Mukherjee, M. Di Liberto, P. Öhberg, R. R. Thomson, and N. Goldman, Experimental observation of Aharonov-Bohm cages in photonic lattices, Phys. Rev. Lett. 121, 075502 (2018).
  * Bodyfelt  _et al._ [2014] J. D. Bodyfelt, D. Leykam, C. Danieli, X. Yu, and S. Flach, Flatbands under correlated perturbations, Phys. Rev. Lett. 113, 236403 (2014).
  * Tovmasyan  _et al._ [2018] M. Tovmasyan, S. Peotta, L. Liang, P. Törmä, and S. D. Huber, Preformed pairs in flat Bloch bands, Phys. Rev. B 98, 134513 (2018).
  * Gneiting  _et al._ [2018] C. Gneiting, Z. Li, and F. Nori, Lifetime of flatband states, Phys. Rev. B 98, 134203 (2018).
  * Vidal  _et al._ [1998] J. Vidal, R. Mosseri, and B. Douçot, Aharonov-Bohm cages in two-dimensional structures, Phys. Rev. Lett. 81, 5888 (1998).
  * Vidal  _et al._ [2001] J. Vidal, P. Butaud, B. Douçot, and R. Mosseri, Disorder and interactions in Aharonov-Bohm cages, Phys. Rev. B 64, 155306 (2001).
  * Naud  _et al._ [2001] C. Naud, G. Faini, and D. Mailly, Aharonov-Bohm cages in 2d normal metal networks, Phys. Rev. Lett. 86, 5104 (2001).
  * Kolovsky  _et al._ [2018] A. R. Kolovsky, A. Ramachandran, and S. Flach, Topological flat Wannier-Stark bands, Phys. Rev. B 97, 045120 (2018).
  * Wu  _et al._ [2007] C. Wu, D. Bergman, L. Balents, and S. Das Sarma, Flat bands and Wigner crystallization in the honeycomb optical lattice, Phys. Rev. Lett. 99, 070401 (2007).
  * Jacqmin  _et al._ [2014] T. Jacqmin, I. Carusotto, I. Sagnes, M. Abbarchi, D. D. Solnyshkov, G. Malpuech, E. Galopin, A. Lemaître, J. Bloch, and A. Amo, Direct observation of dirac cones and a flatband in a honeycomb lattice for polaritons, Phys. Rev. Lett. 112, 116402 (2014).
  * Masumoto  _et al._ [2012] N. Masumoto, N. Y. Kim, T. Byrnes, K. Kusudo, A. Löffler, S. Höfling, A. Forchel, and Y. Yamamoto, Exciton–polariton condensates with flat bands in a two-dimensional kagome lattice, New Journal of Physics 14, 065002 (2012).
  * Chern  _et al._ [2014] G.-W. Chern, C.-C. Chien, and M. Di Ventra, Dynamically generated flat-band phases in optical kagome lattices, Phys. Rev. A 90, 013609 (2014).
  * Jo  _et al._ [2012] G.-B. Jo, J. Guzman, C. K. Thomas, P. Hosur, A. Vishwanath, and D. M. Stamper-Kurn, Ultracold atoms in a tunable optical kagome lattice, Phys. Rev. Lett. 108, 045305 (2012).
  * Trescher and Bergholtz [2012] M. Trescher and E. J. Bergholtz, Flat bands with higher Chern number in pyrochlore slabs, Phys. Rev. B 86, 241111 (2012).
  * Mizoguchi and Udagawa [2019] T. Mizoguchi and M. Udagawa, Flat-band engineering in tight-binding models: Beyond the nearest-neighbor hopping, Phys. Rev. B 99, 235118 (2019).
  * Mielke [1999] A. Mielke, Ferromagnetism in single-band Hubbard models with a partially flat band, Phys. Rev. Lett. 82, 4312 (1999).
  * Tasaki [1998] H. Tasaki, From Nagaoka’s Ferromagnetism to Flat-Band Ferromagnetism and Beyond: An Introduction to Ferromagnetism in the Hubbard Model, Progress of Theoretical Physics 99, 489 (1998).
  * Guzmán-Silva  _et al._ [2014] D. Guzmán-Silva, C. Mejía-Cortés, M. A. Bandres, M. C. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, Experimental observation of bulk and edge transport in photonic lieb lattices, New Journal of Physics 16, 063061 (2014).
  * Mukherjee  _et al._ [2015] S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Öhberg, E. Andersson, and R. R. Thomson, Observation of a localized flat-band state in a photonic Lieb lattice, Phys. Rev. Lett. 114, 245504 (2015).
  * Zhang and Zhang [2013] Y. Zhang and C. Zhang, Bose-Einstein condensates in spin-orbit-coupled optical lattices: Flat bands and superfluidity, Phys. Rev. A 87, 023611 (2013).
  * Abrahams  _et al._ [1979] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Scaling theory of localization: Absence of quantum diffusion in two dimensions, Phys. Rev. Lett. 42, 673 (1979).
  * Evers and Mirlin [2008] F. Evers and A. D. Mirlin, Anderson transitions, Rev. Mod. Phys. 80, 1355 (2008).
  * Anderson [1958] P. W. Anderson, Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492 (1958).
  * Lee and Ramakrishnan [1985] P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57, 287 (1985).
  * Roy  _et al._ [2020] N. Roy, A. Ramachandran, and A. Sharma, Interplay of disorder and interactions in a flat-band supporting diamond chain, Phys. Rev. Research 2, 043395 (2020).
  * Aubry and André [1980] S. Aubry and G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc. 3, 18 (1980).
  * Modugno [2009] M. Modugno, Exponential localization in one-dimensional quasi-periodic optical lattices, New Journal of Physics 11, 033023 (2009).
  * Chalker  _et al._ [2010] J. T. Chalker, T. S. Pickles, and P. Shukla, Anderson localization in tight-binding models with flat bands, Phys. Rev. B 82, 104209 (2010).
  * Kravtsov  _et al._ [2015] V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini, A random matrix model with localization and ergodic transitions, New Journal of Physics 17, 122002 (2015).
  * Bogomolny and Sieber [2018] E. Bogomolny and M. Sieber, Eigenfunction distribution for the Rosenzweig-Porter model, Phys. Rev. E 98, 032139 (2018).
  * Facoetti  _et al._ [2016] D. Facoetti, P. Vivo, and G. Biroli, From non-ergodic eigenvectors to local resolvent statistics and back: A random matrix perspective, EPL (Europhysics Letters) 115, 47003 (2016).
  * Truong and Ossipov [2016] K. Truong and A. Ossipov, Eigenvectors under a generic perturbation: Non-perturbative results from the random matrix approach, EPL (Europhysics Letters) 116, 37002 (2016).
  * Tomasi  _et al._ [2019] G. D. Tomasi, M. Amini, S. Bera, I. M. Khaymovich, and V. E. Kravtsov, Survival probability in Generalized Rosenzweig-Porter random matrix ensemble, SciPost Phys. 6, 14 (2019).
  * Monthus [2017] C. Monthus, Multifractality of eigenstates in the delocalized non-ergodic phase of some random matrix models: Wigner–Weisskopf approach, Journal of Physics A: Mathematical and Theoretical 50, 295101 (2017).
  * Amini [2017] M. Amini, Spread of wave packets in disordered hierarchical lattices, EPL (Europhysics Letters) 117, 30003 (2017).
  * Kravtsov  _et al._ [2020] V. E. Kravtsov, I. M. Khaymovich, B. L. Altshuler, and L. B. Ioffe, Localization transition on the random regular graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble (2020), arXiv:2002.02979 .
  * Khaymovich  _et al._ [2020] I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe, Fragile extended phases in the log-normal Rosenzweig-Porter model, Phys. Rev. Research 2, 043346 (2020).
  * Khaymovich and Kravtsov [2021] I. M. Khaymovich and V. E. Kravtsov, Dynamical phases in a “multifractal” Rosenzweig-Porter model, SciPost Phys. 11, 45 (2021).
  * Roy and Sharma [2019] N. Roy and A. Sharma, Study of counterintuitive transport properties in the Aubry-André-Harper model via entanglement entropy and persistent current, Phys. Rev. B 100, 195143 (2019).
  * Deng  _et al._ [2019] X. Deng, S. Ray, S. Sinha, G. V. Shlyapnikov, and L. Santos, One-dimensional quasicrystals with power-law hopping, Phys. Rev. Lett. 123, 025301 (2019).
  * Voliotis [2019] D. Voliotis, Multifractality in aperiodic quantum spin chains, Journal of Physics A: Mathematical and Theoretical 52, 475001 (2019).
  * Serbyn  _et al._ [2017] M. Serbyn, Z. Papić, and D. A. Abanin, Thouless energy and multifractality across the many-body localization transition, Phys. Rev. B 96, 104201 (2017).
  * Pouranvari [2019] M. Pouranvari, Multifractality of the entanglement Hamiltonian eigenmodes, Phys. Rev. B 99, 155121 (2019).
  * Wang  _et al._ [2021] Y. Wang, C. Cheng, X.-J. Liu, and D. Yu, Many-body critical phase: Extended and nonthermal, Phys. Rev. Lett. 126, 080602 (2021).
  * Nosov  _et al._ [2019] P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov, Correlation-induced localization, Phys. Rev. B 99, 104203 (2019).
  * Das and Sharma [2022] P. Das and A. Sharma, Revisiting the phase transitions of the Dicke model, Phys. Rev. A 105, 033716 (2022).
  * Takada  _et al._ [2004] Y. Takada, K. Ino, and M. Yamanaka, Statistics of spectra for critical quantum chaos in one-dimensional quasiperiodic systems, Phys. Rev. E 70, 066203 (2004).
  * Machida and Fujita [1986] K. Machida and M. Fujita, Quantum energy spectra and one-dimensional quasiperiodic systems, Phys. Rev. B 34, 7367 (1986).
  * Thouless [1983] D. J. Thouless, Bandwidths for a quasiperiodic tight-binding model, Phys. Rev. B 28, 4272 (1983).
  * Castellani and Peliti [1986] C. Castellani and L. Peliti, Multifractal wavefunction at the localisation threshold, Journal of Physics A: Mathematical and General 19, L429 (1986).
  * Ramachandran  _et al._ [2017] A. Ramachandran, A. Andreanov, and S. Flach, Chiral flat bands: Existence, engineering, and stability, Phys. Rev. B 96, 161104 (2017).
  * Leykam  _et al._ [2017] D. Leykam, J. D. Bodyfelt, A. S. Desyatnikov, and S. Flach, Localization of weakly disordered flat band states, EPL (Europhysics Letters) 90 (2017).
  * Poli  _et al._ [2017] C. Poli, H. Schomerus, M. Bellec, U. Kuhl, and F. Mortessagne, Partial chiral symmetry-breaking as a route to spectrally isolated topological defect states in two-dimensional artificial materials, 2D Materials 4, 025008 (2017).
  * Štrkalj  _et al._ [2022] A. Štrkalj, E. V. H. Doggen, and C. Castelnovo, Coexistence of localization and transport in many-body two-dimensional Aubry-André models (2022), arXiv:2204.05198 .
  * Nosov and Khaymovich [2019] P. A. Nosov and I. M. Khaymovich, Robustness of delocalization to the inclusion of soft constraints in long-range random models, Phys. Rev. B 99, 224208 (2019).
  * Tang and Khaymovich [2021] W. Tang and I. M. Khaymovich, Non-ergodic delocalized phase with Poisson level statistics (2021), arXiv:2112.09700 .
  * Motamarri  _et al._ [2021] V. Motamarri, A. S. Gorsky, and I. M. Khaymovich, Localization and fractality in disordered Russian Doll model (2021), accepted to SciPost Phys., arXiv:2112.05066 .
  * Kutlin and Khaymovich [2021] A. G. Kutlin and I. M. Khaymovich, Emergent fractal phase in energy stratified random models, SciPost Phys. 11, 101 (2021).
  * Duthie  _et al._ [2022] A. Duthie, S. Roy, and D. E. Logan, Anomalous multifractality in quantum chains with strongly correlated disorder, Phys. Rev. B 106, L020201 (2022).
  * Cai  _et al._ [2013] X. Cai, L.-J. Lang, S. Chen, and Y. Wang, Topological superconductor to Anderson localization transition in one-dimensional incommensurate lattices, Phys. Rev. Lett. 110, 176403 (2013).
  * DeGottardi  _et al._ [2013] W. DeGottardi, D. Sen, and S. Vishveshwara, Majorana fermions in superconducting 1d systems having periodic, quasiperiodic, and disordered potentials, Phys. Rev. Lett. 110, 146404 (2013).
  * Wang  _et al._ [2016a] J. Wang, X.-J. Liu, G. Xianlong, and H. Hu, Phase diagram of a non-Abelian Aubry-André-Harper model with $p$-wave superfluidity, Phys. Rev. B 93, 104504 (2016a).
  * Danieli  _et al._ [2015] C. Danieli, J. D. Bodyfelt, and S. Flach, Flat-band engineering of mobility edges, Phys. Rev. B 91, 235134 (2015).
  * fn [1] Here we note that the choice of periodic boundary conditions is not very important, as the quasiperiodic potential anyway has a jump at the boundary. Unlike some other models, we do not put the rational approximants ${M/N}$ of $b$ by taking certain system sizes ${N}$ from the above approximations.
  * Asboth  _et al._ [2016] J. K. Asboth, L. Oroszlany, and A. Palyi,  _A Short Course on Topological Insulators_, Volume 919 (Springer, 2016, 2016).
  * Ryu  _et al._ [2010] S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, Topological insulators and superconductors: tenfold way and dimensional hierarchy, New Journal of Physics 12, 065010 (2010).
  * Ludwig [2015] A. W. W. Ludwig, Topological phases: classification of topological insulators and superconductors of non-interacting fermions, and beyond, Physica Scripta T168, 014001 (2015).
  * Lee  _et al._ [2022] S. Lee, A. Andreanov, and S. Flach, Critical-to-insulator transitions and fractality edges in perturbed flatbands (2022), arXiv:2208.11930 .
  * Lindinger  _et al._ [2019] J. Lindinger, A. Buchleitner, and A. Rodríguez, Many-body multifractality throughout bosonic superfluid and Mott insulator phases, Phys. Rev. Lett. 122, 106603 (2019).
  * Macé  _et al._ [2019] N. Macé, F. Alet, and N. Laflorencie, Multifractal scalings across the many-body localization transition, Phys. Rev. Lett. 123, 180601 (2019).
  * Mirlin and Evers [2000] A. D. Mirlin and F. Evers, Multifractality and critical fluctuations at the Anderson transition, Phys. Rev. B 62, 7920 (2000).
  * Bäcker  _et al._ [2019] A. Bäcker, M. Haque, and I. M. Khaymovich, Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems, Phys. Rev. E 100, 032117 (2019).
  * Meenakshisundaram and Lakshminarayan [2005] N. Meenakshisundaram and A. Lakshminarayan, Multifractal eigenstates of quantum chaos and the Thue-Morse sequence, Phys. Rev. E 71, 065303 (2005).
  * Evangelou and Pichard [2000] S. N. Evangelou and J.-L. Pichard, Critical quantum chaos and the one-dimensional Harper model, Phys. Rev. Lett. 84, 1643 (2000).
  * Janssen [1994] M. Janssen, Multifractal analysis of broadly-distributed observables at criticality, International Journal of Modern Physics B 08, 943 (1994).
  * Chen  _et al._ [2020] C. P. Chen, M. Szyniszewski, and H. Schomerus, Many-body localization of zero modes, Phys. Rev. Research 2, 023118 (2020).
  * Wu and Hou [2019] Y.-J. Wu and J. Hou, Symmetry-protected localized states at defects in non-Hermitian systems, Phys. Rev. A 99, 062107 (2019).
  * Leumer  _et al._ [2020] N. Leumer, M. Marganska, B. Muralidharan, and M. Grifoni, Exact eigenvectors and eigenvalues of the finite Kitaev chain and its topological properties, Journal of Physics: Condensed Matter 32, 445502 (2020).
  * Cerovski [2000] V. Z. Cerovski, Bond-disordered Anderson model on a two-dimensional square lattice: Chiral symmetry and restoration of one-parameter scaling, Phys. Rev. B 62, 12775 (2000).
  * García-García and Cuevas [2006] A. M. García-García and E. Cuevas, Anderson transition in systems with chiral symmetry, Phys. Rev. B 74, 113101 (2006).
  * Mudry  _et al._ [2000] C. Mudry, P. W. Brouwer, and A. Furusaki, Crossover from the chiral to the standard universality classes in the conductance of a quantum wire with random hopping only, Phys. Rev. B 62, 8249 (2000).
  * Dyson [1953] F. J. Dyson, The dynamics of a disordered linear chain, Phys. Rev. 92, 1331 (1953).
  * Avila  _et al._ [2017] A. Avila, S. Jitomirskaya, and C. A. Marx, Spectral theory of extended Harper’s model and a question by Erdös and Szekeres, Inventiones mathematicae 210, 283 (2017).
  * fn [2] The factor of $2$ (also mentioned in [77]) with respect to the extended Harper model comes from the two times larger level spacing given by the derivative of the potential.
  * Han  _et al._ [1994] J. H. Han, D. J. Thouless, H. Hiramoto, and M. Kohmoto, Critical and bicritical properties of Harper’s equation with next-nearest-neighbor coupling, Phys. Rev. B 50, 11365 (1994).
  * Rodriguez  _et al._ [2010] A. Rodriguez, L. J. Vasquez, K. Slevin, and R. A. Römer, Critical parameters from a generalized multifractal analysis at the Anderson transition, Phys. Rev. Lett. 105, 046403 (2010).
  * Chang  _et al._ [1997] I. Chang, K. Ikezawa, and M. Kohmoto, Multifractal properties of the wave functions of the square-lattice tight-binding model with next-nearest-neighbor hopping in a magnetic field, Phys. Rev. B 55, 12971 (1997).
  * Liu  _et al._ [2015] F. Liu, S. Ghosh, and Y. D. Chong, Localization and adiabatic pumping in a generalized Aubry-André-Harper model, Phys. Rev. B 91, 014108 (2015).
  * Kraus and Zilberberg [2012] Y. E. Kraus and O. Zilberberg, Topological equivalence between the Fibonacci quasicrystal and the Harper model, Phys. Rev. Lett. 109, 116404 (2012).
  * Danieli  _et al._ [2021] C. Danieli, A. Andreanov, T. Mithun, and S. Flach, Nonlinear caging in all-bands-flat lattices, Phys. Rev. B 104, 085131 (2021).
  * fn [3] Here and further the subscripts ${S}$ and ${A}$ correspond to the symmetric and antisymmetric cases.
  * Li  _et al._ [2017] X. Li, X. Li, and S. Das Sarma, Mobility edges in one-dimensional bichromatic incommensurate potentials, Phys. Rev. B 96, 085119 (2017).
  * Gong and Tong [2008] L. Gong and P. Tong, Fidelity, fidelity susceptibility, and von Neumann entropy to characterize the phase diagram of an extended Harper model, Phys. Rev. B 78, 115114 (2008).
  * Jia  _et al._ [2008] X. Jia, A. R. Subramaniam, I. A. Gruzberg, and S. Chakravarty, Entanglement entropy and multifractality at localization transitions, Phys. Rev. B 77, 014208 (2008).
  * Evers and Mirlin [2000] F. Evers and A. D. Mirlin, Fluctuations of the inverse participation ratio at the Anderson transition, Phys. Rev. Lett. 84, 3690 (2000).
  * Siebesma and Pietronero [1987] A. P. Siebesma and L. Pietronero, Multifractal properties of wave functions for one-dimensional systems with an incommensurate potential, Europhysics Letters (EPL) 4, 597 (1987).
  * Wang  _et al._ [2016b] Y. Wang, Y. Wang, and S. Chen, Spectral statistics, finite-size scaling and multifractal analysis of quasiperiodic chain with p-wave pairing, The European Physical Journal B 89, 254 (2016b).
  * Oganesyan and Huse [2007] V. Oganesyan and D. A. Huse, Localization of interacting fermions at high temperature, Phys. Rev. B 75, 155111 (2007).
  * Atas  _et al._ [2013] Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013).
  * Sanchez-Palencia  _et al._ [2007] L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, Anderson localization of expanding Bose-Einstein condensates in random potentials, Phys. Rev. Lett. 98, 210401 (2007).
  * fn [4] In a standard Anderson model ${F}\simeq 100$, but we consider it as a constant fitting parameter.