# Fate of Quadratic Band Crossing under quasiperiodic modulation

Raul Liquito1, Miguel Gonçalves1,2, Eduardo V. Castro1,3 1Centro de Física da
Universidade do Porto, Departamento de Física e Astronomia, Faculdade de
Ciências, Universidade do Porto, 4169-007 Porto, Portugal  2CeFEMA, Instituto
Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa,
Portugal  3Beijing Computational Science and Research Center, Beijing 100084,
China

###### Abstract

We study the fate of two-dimensional quadratic band crossing topological
phases under a one-dimensional quasiperiodic modulation. By employing
numerically exact methods, we fully characterize the phase diagram of the
model in terms of spectral, localization and topological properties. Unlike in
the presence of regular disorder, the quadratic band crossing is stable
towards the application of the quasiperiodic potential and most of the
topological phase transitions occur through a gap closing and reopening
mechanism, as in the homogeneous case. With a sufficiently strong
quasiperiodic potential, the quadratic band crossing point splits into Dirac
cones which enables transitions into gapped phases with Chern numbers $C=\pm
1$, absent in the homogeneous limit. This is in sharp contrast with the
disordered case, where gapless $C=\pm 1$ phases can arise by perturbing the
band crossing with any amount of disorder. In the quasiperiodic case, we find
that the $C=\pm 1$ phases can only become gapless for a very strong potential.
Only in this regime, the subsequent quasiperiodic-induced topological
transitions into the trivial phase mirror the well-known “levitation and
annihilation” mechanism in the disordered case.

##  I Introduction

The unique characteristics of topological insulators compared to conventional
band insulators have rendered them a central topic of current research [1, 2,
3, 4]. The groundbreaking discovery of the quantum Hall effect [5] and its
subsequent theoretical explanation through the lens of topology [6, 7], paved
the way for the proposal of the quantum anomalous Hall effect [8]. Remarkably,
this effect allows for the appearance of a topological phase in the absence of
a uniform magnetic field, and was experimentally realized in various platforms
[9, 10, 11, 12]. These so called Chern insulators [4] arise from opening non-
trivial gaps in systems with band crossing points carrying a finite quantized
Berry phase, which can be accomplished by breaking time-reversal symmetry. The
simplest and more extensively studied case of a band crossing point is that of
a Dirac point, that hosts a low-energy linear dispersion described by a Dirac
Hamiltonian, as it is the case for nodal superconductors and graphene [13].
Dirac points contain Berry phases of $\pm\pi$, but there are other
possibilities for band crossings and associated Berry phases, as is the case
of quadratic band crossing points (QBCPs). Two-dimensional systems with QBCPs
are also very interesting because on the one hand they are associated with a
finite Berry phase of $\pm 2\pi$ and on the other, a finite density of states
at QBCP, contrary to Dirac points, renders these systems unstable to
interactions [14]. In QBCP systems, interactions can induce nematic phases
with two Dirac cones, each carrying half of the QBCP’s Berry phase; or gap
openings that may give rise to topological insulating phases [15, 16, 17, 18].

Even though topological band theory is usually studied in momentum-space for
systems that have translational invariance, topological insulators are known
to be robust to the effects of uncorrelated disorder [19]. In fact, disorder
is a key ingredient for the observation of quantized Hall conductivity for
quantum Hall systems since it localizes every state except those responsible
for the quantized Hall current, that live at very narrow energy windows [20,
21, 22]. In this way, varying the Fermi level in the gap filled with localized
states cannot change the Hall conductivity, giving rise to a filling-
independent plateau. Sufficiently large disorder, however, generically
suppresses the topological properties by inducing a topological phase
transition into a trivial phase, where the topological extended states meet
and become suppressed through the “levitation and annihilation” mechanism [23,
24, 22, 25]. However, numerous instances of disorder-induced topological
phases, known as topological Anderson insulators, have also been found [26,
27, 28, 29, 30, 31].

A different class of systems that break translational invariance, where more
exotic localization properties can occur, are quasiperiodic systems. Contrary
to disordered systems, extended, localized and critical multifractal phases
can arise even in one dimensional (1D) [32, 33, 34, 35, 36, 37, 38, 39, 40].
In higher dimensions, these systems have received considerable attention,
mostly on the interplay between moiré physics and localization [41, 42, 43,
44, 45, 46, 47, 48, 49]. Systems with quasiperiodic modulations can be
realized in widely different platforms, including optical lattices [50, 51,
52, 53, 54, 55, 56, 57, 58, 59], photonic [60, 61, 62, 63, 48, 64, 65] and
phononic [66, 67, 68, 69, 70, 71] metamaterials, and moiré materials [72, 73,
74]. The impact of quasiperiodic modulations on parent topological systems has
also been previously studied [75, 76, 77, 78], and was found to give rise to
interesting topological phases with richer localization properties than in the
disordered cases.

The fate of a QBC in the presence of disorder was numerically studied in Ref.
[79], in the non-interacting limit, where it was found to be unstable to the
formation of topological insulating phases with Chern numbers $C=\pm 1$, not
present in the model’s clean limit. The interplay between disorder and
interactions was also studied in Refs. [80, 81] by renormalization-group
methods, where the interaction-induced topological insulating phases were
found to be suppressed at strong disorder. The influence of quasiperiodic
modulations on QBCP has, however, remained poorly explored so far, with the
notable exception of Ref. [82]. In this reference, a quasiperiodic potential
was applied to a QBCP system, that was found to be stable up to a flatband
regime at which the wave function becomes delocalized in momentum-space,
closely related to the incommensurate magic-angle regime found in moiré
systems [46, 47, 83]. The study of the topological phases that can be induced
by applying quasiperiodic modulations to a QBCP system has however remained
unexplored up to now.

In this work we study the topological and localization properties of a QBCP
model with an applied 1D quasiperiodic modulation. The main results are shown
in Fig. 1. The QBCP is robust for a small quasiperiodic potential, with the
phase diagram remaining essentially unchanged with respect to the homogeneous
limit, as seen in Fig. 1(a). For a large quasiperiodic potential, phases with
Chern numbers $C=\pm 1$, not present in the homogeneous limit, arise. The
topological transitions into these phases were found to occur through the gap
closing and reopening mechanism, as shown in in Fig. 1(b), being therefore of
different nature from the ones found in the disordered case, in Ref. [79].
Interestingly, in the majority of the studied topological transitions, the gap
closes at the same momenta as in the homogeneous case: at the center and
corners of the first Brillouin zone (FBZ), defined in the translationally
invariant direction. In Fig. 1(c), the localization length at the Fermi
energy, obtained by the transfer matrix method, is seen to diverge at the gap
closing phase transitions. Away from the Fermi energy, we found large clusters
of bulk extended states both in topological and in trivial phases. This is in
contrast with the disordered case, where all eigenstates are localized within
the topological phases except for topologically non-trivial states arising at
special energies. Only for an even larger potential, the $C=\pm 1$ phases can
become gapless and exhibit similar physics to the disordered case. In fact,
further increasing the potential induces a transition into a trivial phase
that mirrors typical topological transitions in the presence of disorder:
close to the transition, topologically non-trivial extended states arise only
at very narrow energy windows, merging at the transition, after which all
states become trivial and localized, as in the “levitation and annihilation”
mechanism [23, 24, 22, 25].

Figure 1: (a) Chern number as a function of the model parameter $B$ and
quasiperiodic potential $W$ for a system size of $L=21$ and $20$ different
twists realizations. The greyed out areas are gapless. (b) Energy gap as a
function of $B$ and $W$ for $L_{y}=1597$. No twist realizations were
considered. The grey lines represent the lines at which a topological phase
transition occurs. (c) Localization length ($\xi$) at the Fermi level ($E=0$)
for the $k_{x}$ values where the gap closes. A 1D system size of $N=30000$ was
used. The grid discretization is $300\times 300$.

The paper is organized as follows: In Sec. II, we introduce the tight-binding
model used to describe the electronic properties of the quasiperiodiced QBC
system and the methods to analyze its properties. The topological, spectral,
and localization properties are discussed in Sec. III. A thorough discussion
of the obtained results is given in Sec. IV. We also include three appendices:
in Appendix A we provide the real space TB Hamiltonian and the
$k_{x}$-dependent model; in Appendix B we present the Golden Ratio method used
to compute the extrema of the energy spectrum; finally IPR and fractal
dimension results are presented in Appendix C.

##  II Model and methods

We consider a QBCP model on a square lattice with two orbitals per unit cell
[79]. This model is defined by the following clean limit $\mathbf{k}$-space
Hamiltonian

$\mathcal{H}_{0}=\sum_{\mathbf{k}}\boldsymbol{\Psi}_{\bm{k}}^{\dagger}\mathcal{H_{\mathbf{k}}}\boldsymbol{\Psi}_{\bm{k}},$
(1)

where
$\boldsymbol{\Psi}_{\bm{k}}^{\dagger}=\left(\begin{array}[]{cc}c_{\mathbf{k},A}^{\dagger}&c_{\mathbf{k},B}^{\dagger}\end{array}\right)$,
with $c_{\mathbf{k},\alpha}^{\dagger}$ the creation operator of a state with
crystal momentum $\mathbf{k}$ in the $\alpha$ orbital and

$\mathcal{H_{\mathbf{k}}}=\mathbf{h}(\mathbf{k})\cdot\boldsymbol{\sigma},$ (2)

with $\boldsymbol{\sigma}$ the vector of Pauli matrices and the vector
$\mathbf{h}(\mathbf{k})$ given by

$\displaystyle h_{x}=2t_{x}\sin\left(k_{x}\right)\sin\left(k_{y}\right)$
$\displaystyle h_{y}=0$ $\displaystyle
h_{z}=2t_{z}\left(\cos\text{$\left(k_{x}\right)$}-\cos\left(k_{y}\right)\right).$
(3)

From this point on, we set $t_{x}=t_{y}=t=1$, so that every physical
observable with energy units come in units of $t$.

QBCPs with symmetric Berry curvatures occur at $\Gamma=(0,0)$ and
$M=(\pm\pi,\pm\pi)$ points of the FBZ. For $h_{y}=0$, the system is gapless,
having a finite DOS at the Fermi level ($E=0$). The introduction of a non-zero
$h_{y}$ perturbation opens a gap that can give rise to Chern insulating
phases. This can be accomplished by introducing the following $h_{y}$ term,

$h_{y}=1+\frac{B+1}{2}\left(\cos\left(k_{x}\right)+\cos\left(k_{y}\right)\right).$
(4)

For this model, a single QBCP occurs at $M$ for $B=0$, and for $B=-2$ at
$\Gamma$. These are the only values of $B$ for which the system becomes
gapless, where topological phase transitions between phases with
$C=2\leftrightarrow C=0$ ($B=-2$) and $C=0\leftrightarrow C=-2$ ($B=0$) take
place (see Fig. 1(a), for $W=0$).

In this work, we will study the fate of the model’s phase diagram upon the
addition of a 1D quasiperiodic potential. We consider the Aubry-André
unidirectional potential given by the real-space Hamiltonian,

$\mathcal{H}_{W}=\frac{W}{2}\sum_{\mathbf{R},\alpha}\cos\left(2\pi\beta
n\right)c_{\mathbf{R},\alpha}^{\dagger}c_{\mathbf{R},\alpha},$ (5)

where $\mathbf{R}=ma\hat{\mathbf{e}}_{x}+na\hat{\mathbf{e}}_{y}$ is a lattice
vector and $\beta$ is the potential frequency. In the thermodynamic limit we
choose $\beta$ to be an irrational number in order to break translational
invariance; we choose the golden ratio
$\beta=\phi_{\textrm{GR}}=(1+\sqrt{5})/2$. We carried out numerical
simulations for finite systems with $L_{x}=L_{y}=L$ (with $L$ the number of
unit cells in each direction) and periodic/twisted boundary conditions. In
order to avoid boundary defects we have chosen system sizes $L\rightarrow
L_{n}=F_{n}$, where $F_{n}$ is the $n$-th order Fibonacci number, and
approximated $\beta$ with rational approximants of the golden ratio
$\beta\rightarrow\beta_{n}=F_{n+1}/F_{n}$. This choice ensures that the
system’s unit cell is of size $L$, which guarantees that the system always
remains incommensurate as $L$ is increased. The potential in Eq. (5) keeps the
system translational invariant along the $x$ direction, and we take the
Fourier transform along this direction to obtain $\mathcal{H}_{k_{x}}$, an
Hamiltonian diagonal in Bloch momentum $k_{x}$ (see Appendix A).

In what follows, we carry out an extensive study of spectral, topological and
localization properties of the model. We compute the Chern number phase
diagram of the system with an implementation of the coupling matrix method of
Ref. [84] as implemented in Ref. [85]. Spectral properties were studied using
exact diagonalization (ED) and the kernel polynomial method [86].

The transfer matrix method (TMM), introduced in [87, 88], can be used to
compute the localization length and therefore to study bulk localization
properties. At topological phase transitions, the states are extended at the
Fermi level, while in topological phases the system is either gapped or
populated with localized states around the Fermi level. For the two last
cases, the localization length at Fermi level is finite, while in the former,
it diverges in the thermodynamic limit. The TMM method was then also used to
cross-check the topological phase diagram obtained through Chern number
calculations.

Throughout this work averages over twisted boundary conditions were realized
such that the phase twist follow a random uniform distribution in the interval
$\theta_{i}\in[0,2\pi[$. To apply phase twists, the boundaries are
periodically closed (as for periodic boundary conditions), but with an
additional phase twist, so that:

$\psi_{\alpha}(\boldsymbol{R}+L\mathbf{a}_{i})=e^{i\text{{\hbox{\theta_{i}}}}}\psi_{\alpha}(\boldsymbol{R}),$
(6)

where
$\psi_{\alpha}(\boldsymbol{R})=c_{\mathbf{R},\alpha}^{\dagger}\left|0\right\rangle$.
As the system approaches thermodynamic limit, any dependence on phase twists
should vanish.

##  III Results

###  III.1 Topological Properties

We start by characterizing the topological phase diagram, shown in Fig. 1(a).
The phase diagram was obtained for a system with $21\times 21$ unit cells
averaged over $20$ random phase twists with twisted boundary conditions.

The clean limit presents topological transitions from $C=\pm 2$ to $C=0$, with
the gap closing at a QBCP ($B=\\{-2,0\\}$). These transitions are robust for
small $W$, still occurring at $B=\\{-2,0\\}$. The observed behavior contrasts
with the case of uncorrelated disorder, studied in Ref. [79], where it is
shown that for infinitesimal disorder strength $W$, $C=\pm 1$ phases appear.
For the present case of an incommensurate potential, $C=\pm 1$ phases only
appear at large enough potential strength, indicating a rather different
phenomenology. Interestingly, along the lines of fixed $B=-2,0$, when the QBCP
occurs, we see the appearance of very small $C=\pm 1$ bubbles, indicated by
the arrows in Fig. 1(a). These bubbles are also seen in Fig. 1(b) for the gap
and 1(c) for the localization length, which were obtained with very different
system sizes, confirming they are not merely a result of finite-size effects.

Increasing the quasiperiodic potential along the lines of fixed
$B=\\{-2,0\\}$, it will be shown below that for $W\in[3.0,4.5]$ the QBCP
splits into two Dirac cones, and eventually at $W\approx 4.5$ a topological
phase transition into $C=\pm 1$ phase occurs. For a large enough potential
strength, all topological phases are suppressed and a trivial phase is
eventually reached. Unlike in the disordered case, there are both gapped and
gapless $C=\pm 1$ phases separated by the thin black lines in Fig. 1(a), as we
will detail below. These phases are induced by quasi-periodicity, and are
different from the topological Anderson insulators that exist for uncorrelated
disorder due to their distinct localization properties, as we will show.

###  III.2 Gapped/Gapless Regions

In order to study the spectral gap of the system, we computed the closest
eigenvalue to $E=0$, which we call $\epsilon_{0^{+}}$, using the
$k_{x}$-dependent model presented in Appendix A. We then estimate the gap of
the system as $2\epsilon_{0^{+}}$, which is well justified due to the particle
hole symmetry of the model, that ensures the spectrum is symmetric around
$E=0$.

In Fig. 1(b) we present the gap of the system. These results were obtained
using Lanczos decomposition alongside with the shift invert method for
$L_{y}=1597$ at the value of $k_{x}$ where the gap closes. Generally we do not
know the value of $k_{x}$ at which the gap closes 111At general $k_{x}$ the
spectrum of $\mathcal{H}_{k_{x}}$ is gapped, thus one might conclude that the
system is gapped if the proper $k_{x}$ is not chosen. For this reason a golden
ratio search algorithm (see Appendix. B) was implemented to compute the value
of $k_{x}$ that maximizes the valence band energy, that corresponds to the gap
closing point.

We can see in Fig. 1(b) that the gap closes and reopens in most of the
topological phase transitions. We also see the appearance of gapless regions
for strong potential, seen as wider blackish regions at high $W$. In Fig. 1(a)
the greyed out regions were computed using the data in Fig. 1(b). Since the
spectrum of a finite system is always discrete, we need to properly define a
threshold gap value $\delta$ below which we consider the system to be gapless.
This threshold was fixed as the minimum value of the gap at well defined gap
closing and reopening topological transitions. For the system size considered
this value is $\delta\approx 0.015$.

###  III.3 Localization Properties

To complete the characterization of the phase diagram, we now turn to the
study of the localization properties. We start by discussing the localization
length ($\xi$) at the Fermi level ($E=0$), obtained through the TMM, and shown
in Fig. 1(c).

From the gap results presented in Fig. 1(b), we know that for most of the
topological phase transitions the gap closes and reopens. Since the gap
closing point should contain extended states at a topological phase
transition, the localization length $\xi$ should diverge at these points,
while being finite at the gapped regions. We can therefore capture the
topological transitions by computing $\xi$ as a function of $B$ and $W$, as
shown in Fig. 1(c).

Using the $k_{x}$-dependent model, we can reach closer to thermodynamic limit
reaching system sizes of the order of $L\approx 10^{4}$ with a relative error
of approximately $\epsilon\approx 1\%$, where $\epsilon$ is the relative error
of the average localization length after $N$ TMM iterations. In order to
compute the localization length at the Fermi level, TMM was implemented for
the $k_{x}$ at which the gap closes (calculated using the golden ratio search
algorithm discussed in Appendix B ). We have noticed that the valence band
presents two local maxima, one around the $\Gamma$-piont and the other around
the $M$-point, however the gap only closes around one of them. The TMM results
in Fig. 1(c) are a sum of the localization lengths for the two local maxima, a
quantity that diverges when either one of the localization length does. For
any other value of $k_{x}$, TMM will always give a finite but small $\xi$
since the spectra of $\mathcal{H}_{k_{x}}$ is gapped. With this approach, we
compute the localization length $\xi$ at $E=0$ for a set of points $(B,W)$,
thus obtaining in Fig. 1(c) the contour of the original topological phase
diagram shown in Fig. 1(a). For high enough quasiperiodic potential strength,
when $B\lesssim-3$ or $B\gtrsim 2$, this approach fails since the gap closes
and does not reopens. The method of golden ratio search also fails in gapless
regions since the gap closes at a continuous region of $k_{x}$.

Figure 2: Localization length as a function of energy $E$ and momentum $k_{x}$
at $B=-3$ for the entire range of the valence band. The grid discretization is
$300\times 300$. The insets follow the same procedure for a smaller region
($E\in[-1,0]$ and $k_{x}\in[-\pi,-\pi+0.2]$) with a grid discretization of
$600\times 600$.

To better understand the localization properties, we have performed an energy
resolved study including energies away from the Fermi level. Since localized
bulk states do not contribute to the system’s topological properties,
analysing the energy position of the extended bulk states with increasing
quasiperiodic potential should give us some intuition on how they affect the
topological phases. For regular Anderson disorder in 1D and 2D, any amount of
disorder fully localizes the spectrum for systems belonging to the orthogonal
symmetry class A [90]. As previously discussed, the same cannot be said about
incommensurate structures for which finite fractions of extended states can
exist. To study the localization properties of the bulk states as a function
of the energy $E$ we computed the $k_{x}$ and $E$ resolved localization
length, $\xi(k_{x},E)$ for different values of the quasiperiodic potential.

In Fig. 2 the $k_{x}$-dependent results for the localization length can be
seen for $B=-3$. For the first topological phase transition from $C=2$ to
$C=1$ (upper panel) there are a set of extended states around $k_{x}=0$, where
the gap closes. We can clearly see that the higher the quasiperiodic potential
$W$ the larger is the fraction of localized states. On the lower panel of Fig.
2, we see that right before the second transition from $C=1$ to $C=0$, most of
the bulk states are localized except for a few extended states that can be
observed if we use a fine enough grid discretization, as seen in the inset.
After the transition, for $W>9$ all the states are Anderson localized, and the
system is in a trivial Anderson insulating phase.

Figure 3: Localization length as a function of energy $E$ and momentum $k_{x}$
at $B=-3$ for the entire range of the valence band. The grid discretization is
$300\times 300$.

Figure 2 shows the localization length $\xi(k_{x},E)$ for $B=-2$. For small
$W$ we can see that the QBCP is robust, as expected from previous results.
Around $W=3.5$, we see that the QBCP splits into two Dirac points around the
$\Gamma$ point. It is still clear that under increasing quasiperiodic
potential $W$ bulk states become increasingly more localized. In the lower
panel of Fig. 2, we show the localization length results for a topological
phase transition from $C=1$ into a trivial phase. We see that the gap closes
at the transition, after which all the states fully localize. Overall the same
behavior can be observed for other values of $B$. On the the other side of the
diagram, $B>-1$, the behavior is similar, with a shift of $\Delta k_{x}=\pi$.

##  IV Discussion

In this work we unveiled the full phase diagram of a 2D system with a QBCP
under an applied 1D quasiperiodic potential. In the following, we discuss our
main results and compare with previous findings for the disordered QBCP
system.

In the absence of quasiperiodicity, topological phase transitions occur when
the gap closes at a QBCP. When a small quasiperiodic potential is applied,
even though translational invariance is broken along the $y$-direction, we
observe that the topological transitions are accompanied by a gap closing with
a quadratic dispersion along $k_{x}$, as is obvious from Figs. 2 and 3.
Moreover, these transitions occur through a gap closing and reopening
mechanism as in the clean case, which strongly indicates that the QBCP is
stable to the addition of quasiperiodic modulations. This is in sharp contrast
with the disordered case, for which the QBCP is unstable to the formation of
gapless phases with Chern numbers $C=\pm 1$, not present in the clean limit
[79]. Interestingly, in the quasiperiodic case, topological transitions into
$C=\pm 1$ phases were also observed for larger quasiperiodic potentials.
However, again in contrast to the disordered case, these transitions also
occur via a gap closing and reopening mechanism. For the topological
transitions from $C=\pm 2$ to $C=\pm 1$, one of the QBCP is split in two
gapped Dirac cones (see, for instance, the upper right panel in Fig. 3), and
the gap closes only at one of them (otherwise the Chern variation would be
larger than one) 222While each QBCP gives contributions $\Delta C=\pm 1$ to
the Chern number, each Dirac cone only contributes with $\Delta C=\pm 1/2$.
When the gap closes and reopens at a given Dirac cone, the sign of its
contribution changes, canceling with the unchanged contribution of the
remaining cone. As a result, there is a net change of $\Delta C=\pm 1$ and we
enter the $C=\pm 1$ phases.. The Dirac cone at a gap closing point is clearly
seen in Fig. 2 (upper, middle panel). Once again in sharp contrast with the
disordered case, only for very large potential strengths ($W\gtrsim 9$) the
$C=\pm 1$ phases becomes gapless (with localized states at the Fermi level).

The localization properties were also found to be very distinct from the
disordered case at small $W$. In the presence of disorder all the eigenstates
are always localized except for extended states that live on very narrow
energy windows. This contrasts with the quasiperiodic case where considerably
large clusters of bulk states remain extended. Furthermore, at large potential
strengths, an analogous of the “levitation and annihilation” mechanism
observed in disordered systems can be observed close to the transitions from
the $C=\pm 1$ phases into the trivial phase. In fact, in this large potential
limit, the $C=\pm 1$can become gapless, with extended states only existing at
very narrow energy windows 333Interestingly, these narrow extended states
arise even slightly before the gap closes.. After the topological transition
into the trivial phase, all states become localized, as in the disordered
case. The topological and localization properties therefore show a mixture of
features present in the homogeneous and disordered cases, as it was also
previously observed for quasiperiodic Chern insulators in Ref. [78]. The small
potential results are also in agreement with Ref. [82], where a different QBCP
system was found to be robust to the application of a (in this case fully two-
dimensional) quasiperiodic potential. Our findings can be verified in
ultracold atoms and trapped ions experiments, where quasiperiodic potentials
are routinely implemented [50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. By keeping
translational invariance along one direction, our study of a 1D quasiperiodic
potential was a suitable starting point to address the interplay between
quasiperiodicity and quadratic band crossings. However, an equally interesting
question lies on the fate of the QBCP upon the application of a fully two-
dimensional quasiperiodic potential, that we leave for future exploration.

###### Acknowledgements.

The authors acknowledge partial support from Fundação para a Ciência e
Tecnologia (FCT-Portugal) through Grant No. UIDB/04650/2020. MG acknowledges
partial support from Fundação para a Ciência e Tecnologia (FCT-Portugal)
through Grant No. UID/CTM/04540/2019. MG acknowledges further support from
FCT-Portugal through the Grant SFRH/BD/145152/2019.

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##  Appendix A Real Space and $k_{x}$-dependent Hamiltonian

In this section we present the $k_{x}$-dependent Hamiltonian. We start from
the Hamiltonian written in the real space basis and take plane wave solutions
along the invariant direction $x$, which enables us to reach far closer to
thermodynamic limit This transformation comes with some caveats, since to have
the full picture of the 2D system one needs to consider the entirety FBZ along
the invariant direction, and so when computing gaps, for example, one needs to
know exactly at which $k_{x}$ does the gap close. The real space Hamiltonian
is: II.

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{B}+\mathcal{H}_{\text{1D}}$

Where each term is defined by:

$\mathcal{H}_{0}=\sum_{\boldsymbol{R}}\left\\{t_{z}\left(\left|\boldsymbol{R}\pm
a\hat{e}_{x},A\left\rangle\right\langle\boldsymbol{R},A\right|-\left|\boldsymbol{R}\pm
a\hat{e}_{y},A\left\rangle\right\langle\boldsymbol{R},A\right|\right)\right.-\\\
-t_{z}\left(\left|\boldsymbol{R}\pm
a\hat{e}_{x},B\left\rangle\right\langle\boldsymbol{R},B\right|+\left|\boldsymbol{R}\pm
a\hat{e}_{y},B\left\rangle\right\langle\boldsymbol{R},B\right|\right)+\\\
+\frac{t_{x}}{2}\left(\left|\boldsymbol{R}\pm a\hat{e}_{x}\pm
a\hat{e}_{y},A\left\rangle\right\langle\boldsymbol{R},B\right|-\left|\boldsymbol{R}\pm
a\hat{e}_{x}\mp
a\hat{e}_{y},A\left\rangle\right\langle\boldsymbol{R},B\right|\right)+\\\
+\left.\frac{t_{x}}{2}\left(\left|\boldsymbol{R}\pm a\hat{e}_{x}\pm
a\hat{e}_{y},B\left\rangle\right\langle\boldsymbol{R},A\right|-\left|\boldsymbol{R}\pm
a\hat{e}_{x}\mp
a\hat{e}_{y},B\left\rangle\right\langle\boldsymbol{R},A\right|\right)\right\\}$

$\mathcal{H}_{B}=\sum_{\boldsymbol{R}}\left\\{i\left(\left|\boldsymbol{R},B\left\rangle\right\langle\boldsymbol{R},A\right|-\left|\boldsymbol{R},A\left\rangle\right\langle\boldsymbol{R},B\right|\right)\right.\\\
-i\frac{B+1}{4}\left(\left|\boldsymbol{R}\pm
a\hat{e}_{x},A\left\rangle\right\langle\boldsymbol{R},B\right|+\left|\boldsymbol{R}\pm
a\hat{e}_{y},A\left\rangle\right\langle\boldsymbol{R},B\right|\right)\\\
+\left.i\frac{B+1}{4}\left(\left|\boldsymbol{R},B\left\rangle\right\langle\boldsymbol{R}\pm
a\hat{e}_{x},A\right|+\left|\boldsymbol{R},B\left\rangle\right\langle\boldsymbol{R}\pm
a\hat{e}_{y},A\right|\right)\right\\}$

$\mathcal{H}_{1D}=\frac{W}{2}\sum_{\mathbf{R}}\cos\left(2\pi\beta
na\right)\left\\{\left|\boldsymbol{R},A\left\rangle\right\langle\boldsymbol{R},A\right|+\left|\boldsymbol{R},B\left\rangle\right\langle\boldsymbol{R},B\right|\right\\}$

We can now make a change of basis along the invariant direction, using
$\left|m,\alpha\right\rangle=\frac{1}{\sqrt{L_{x}}}\sum_{k_{x}}e^{-ik_{x}ma}\left|k_{x},\alpha\right\rangle$
and obtain the 1D hamiltonian
$\mathcal{H}=\sum_{k_{x}}\mathcal{H}(k_{x})\left|k_{x}\left\rangle\right\langle
k_{x}\right|$

$\mathcal{H}(k_{x})=\mathcal{H}_{0}(k_{x})+\mathcal{H}_{B}(k_{x})+\mathcal{H}_{\text{1D}}(k_{x})$

where:

$\displaystyle\mathcal{H}_{0}\left(k_{x}\right)$
$\displaystyle=\sum_{n}\left\\{t_{z}\left(2\cos\left(k_{x}a\right)\left|n,A\left\rangle\right\langle
n,A\right|-\left|n\pm 1,A\left\rangle\right\langle n,A\right|\right)\right.+$
$\displaystyle+t_{z}\left(-2\cos\left(k_{x}a\right)\left|n,B\left\rangle\right\langle
n,B\right|+\left|n\pm 1,B\left\rangle\right\langle n,B\right|\right)+$
$\displaystyle+it_{x}\sin\left(k_{x}a\right)\left(\left|n+1,A\left\rangle\right\langle
n,B\right|-\left|n-1,A\left\rangle\right\langle n,B\right|\right)+$
$\displaystyle+\left.it_{x}\sin\left(k_{x}a\right)\left(\left|n+1,B\left\rangle\right\langle
n,A\right|-\left|n-1,B\left\rangle\right\langle n,A\right|\right)\right\\}$

$\displaystyle\mathcal{H}_{B}(k_{x})$
$\displaystyle=\sum_{n}\left\\{i\left(\left|n,B\left\rangle\right\langle
n,A\right|-\left|n,A\left\rangle\right\langle n,B\right|\right)\right.-$
$\displaystyle-i\frac{B+1}{4}\left(2\cos\left(k_{x}a\right)\left|n,A\left\rangle\right\langle
n,B\right|+\left|n\pm 1,A\left\rangle\right\langle n,B\right|\right)+$
$\displaystyle+\left.i\frac{B+1}{4}\left(2\cos\left(k_{x}a\right)\left|n,B\left\rangle\right\langle
n,A\right|+\left|n,B\left\rangle\right\langle n\pm 1,A\right|\right)\right\\}$

$\mathcal{H}_{\text{1D}}(k_{x})=\frac{W}{2}\sum_{n}\cos\left(2\pi a\beta
n\right)\left\\{\left|n,A\left\rangle\right\langle
n,A\right|+\left|n,B\left\rangle\right\langle n,B\right|\right\\}$

##  Appendix B Golden ratio Search Method

Here we present the Golden Ratio Search method, implemented to compute the
local maximum/minimum of a function of one variable $f(x)$. The simplest
implementation of the method only computes local or global minima. However
taking $f(x)\to-f(x)$ one can compute the maxima of the function $f(x)$.

  1. 1.

Choose two initial outside points $x_{1}$ and $x_{4}$, compute two interior
points $x_{2}=x_{4}-(x_{4}-x_{1})/z$ and $x_{3}=x_{1}+(x_{4}-x_{1})/z$ (Golden
Ratio Rule), then evaluate $f(x)$ at each of the points and set a target
accuracy $\epsilon$ for the position of the minimum.

  2. 2.

If $f(x_{2})<f(x_{3})$ then the minimum lies between $x_{1}$ and $x_{3}$. Each
point is shifted such that $x_{3}$ becomes the new $x_{4}$, $x_{2}$becomes the
new $x_{3}$ and the new $x_{2}$ is computed using the Golden Ratio Rule.
Evaluate $f(x)$ at the new point.

  3. 3.

Otherwise, the minimum lies between $x_{2}$ and $x_{4}$. Each point is shifted
such that $x_{2}$ becomes the new $x_{1}$, $x_{3}$becomes the new $x_{2}$ and
the new $x_{3}$ is computed using the Golden Ratio Rule. Evaluate $f(x)$ at
the new point.

  4. 4.

If $x_{4}-x_{1}>\epsilon$ repeat from step $2$. Otherwise the final estimate
of the position of the minimum is $\frac{1}{2}(x_{2}+x_{3})$.

The value of $z$ on the Golden Ratio Rule can be any value from $]0,1[$,
however the value of $z$ that lets the method converge in the least amount of
iterations is $z=\frac{1+\sqrt{5}}{2}$, the Golden Ratio. One can now
implement this algorithm to maximize $E_{-}(k_{x})$.

##  Appendix C IPR and Fractal Dimension

Here we corroborate the results obtained via TMM at the insets of figure 2. We
consider the IPR scaling with system size $L$ and compute the fractal
dimension $\nu$ for the three eigenstates with bigger localization length.

Figure 4: IPR values of the three states with the biggest localization length
obtained with TMM for $B=-3$. The fractal dimension was estimated for each
pair of $(E,k_{x})$. (a) $W=8.0$; (b) $W=8.5$; (c) $W=9.0$

We confirm that overall these states are always fully extended, yielding
$\nu\approx 1$. For the states that have smaller fractal dimension we still
conclude that they are extended since $\nu$ approaches $1$ for the larger
system sizes.