# Topology in quasicrystals

Oded Zilberberg  Institute for Theoretical Physics, ETH Zurich, 8093 Zürich,
Switzerland<EMAIL_ADDRESS>
###### Abstract

Topological phases of matter have sparked an immense amount of activity in
recent decades. Topological materials are classified by topological invariants
that act as a non-local order parameter for any symmetry and condition. As a
result they exhibit quantized bulk and boundary observable phenomena,
motivating various applications that are robust to perturbations. In this
review, we explore such a topological classification for quasiperiodic
systems, and detail recent experimental activity in the field.

††journal: ome††articletype: Research Article

https://quest.phys.ethz.ch/

##  1 Introduction

The realization that solid structures form mostly as periodic crystalline
structures allowed us to describe numerous material properties in nature [1].
Specifically, as there is only a finite number of possible periodic
arrangements in each dimension, we can categorize the material spectral
properties (band structures) according to their spatial symmetries. Using this
general framework, we obtain an overview of material bulk responses. In recent
decades, concepts originating from the mathematical field of topology bear
growing impact on our understanding of material properties [2, 3]. Topology
generalizes the notion of symmetry by attributing non-local order parameters
(topological invariants) to the material, which are used to classify materials
into distinct families. These families cannot be continuously deformed into
one another, implying the existence of observable topological phase
transitions.

In many cases, the topological invariant is also observable through quantized
bulk responses with associated boundary phenomena at the open end of the
material. The most known example of such a response is the quantum Hall
effect: a 2D electron gas in the presence of a perpendicular magnetic field
exhibits a quantized transverse (Hall) conductance, where the quantization
originates from topological invariants (Chern numbers) associated with the
system’s bandstructure [4, 5, 6]. In correspondence to the quantized bulk
response, chiral edge modes appear at the boundary of the system [7, 8, 9, 2,
3, 10]. Note that the non-locality of the topological invariants implies a
robustness to perturbations such as disorder - a fact that promoted the
quantum Hall effect as a standard for metrology [11].

The topology in the quantum Hall effect originates from a length scale
competition between the magnetic field and the translation invariance of the
electron’s kinetic energy [12, 5]. When the electrons hop on a lattice, this
competition can also lead to incommensurate situations, where the system is no
longer invariant to translations [13]. Such is also the case in quasicrystals,
which are ordered but non-periodic solid structures [14, 15]. The long-range
order of quasicrystals can be projected from periodic higher-dimensional
systems [16, 17, 18], and thus exhibits sharp diffraction peaks [19, 20, 21].
Spectral characteristics of such quasiperiodic patterns have been studied in a
variety of platforms, revealing highly fractal energy levels with infinitely
many energy gaps, self-similar eigenstates and critical localization [22, 23,
24, 25, 26, 27]. Interestingly, the topological characterization of
quasicrystals can also be mapped to higher-dimensions using a mapping to the
quantum Hall effect, and associating bulk gaps with Chern numbers [28, 29, 30,
31, 32]. This mapping complements earlier topological classifications of
quasiperiodic systems by gap labeling approach [33, 34, 35].

Here, we review the mapping between quasiperiodic models and quantum Hall
systems. In section 2, we introduce the concept of quasicrystals and their
corresponding quasiperiodic models. Section 3 is devoted to the relation of
some families of quasiperiodic models to the quantum Hall effect, whereas in
section 4, we present the proof from Ref. [28] on how to associate Chern
numbers to gaps of the quasiperiodic models. In section 5, we then shortly
review recent experiments in various optical setups that explore the
topological properties of quasicrystals.

##  2 Quasiperiodic order and approximants

Quasicrystals are materials with atoms arranged in a non-periodic but ordered
pattern. To describe their electronic properties, we consider hybridization of
their valence electrons similar to what is done for periodic materials, e.g.,
by constructing a Kronig–Penney model for nearly-free electrons in the
corresponding quasiperiodic potential energy landscape [1], see Fig. 1 for an
illustration of a quasiperiodic 1D chain. As we discuss below, such a
construction lends itself also to metamaterial structures, where waves
interfere in the presence of a patterned medium. Assuming sufficiently-spaced
and sufficiently-deep potential wells allows labeling of the resulting
spectral band structures according to the separated orbits of the individual
potential wells, e.g., by s-, p-, and d-like orbital bands. On top of this
orbital separation, the longer-ranged quasiperiodic spatial arrangement of the
“atoms” leads to splitting of the bands into infinitely-many mini-bands.

Figure 1: A segment in a quasiperiodic arrangement of potential wells. (a) A
1D quasiperiodic arrangement of potential wells. Each well has a specific
height and width, and corresponding bound modes. A quasiperiodic tight-binding
limit model is constructed, for example, from the lowest state in each well
$n$ with energy $E_{n}$. Through evanescent wave overlap of the states (dashed
blue line example), we obtain variable-range hopping amplitudes
$t_{i,j}^{\phantom{\dagger}}$, e.g., shown are nearest-neighbor (red arrows)
and next-nearest-neighbor hopping (green arrow). The quasiperiodicity in the
potential shape translates to a quasiperiodic modulation in the tight-binding
model, cf. Eq. (1). (b) A similar construction of a quasiperiodic 1D chain for
waves moving in a sculptured 2D medium, where the lowest bound modes of each
box are shown (red gradients). Such a construction readily extends to other
quasiperiodic arrangements in 2D and 3D.

Considering, for simplicity, the resulting tight-binding Hamiltonian
description of the s-orbits hybridization along a 1D quasiperiodic chain, we
have

$\displaystyle
H_{s}=\sum_{n}\left\\{\lambda^{\text{d}}V^{\text{d}}_{n}c_{n}^{\dagger}c^{\phantom{{\dagger}}}_{n}+\sum_{m>0}\left[\left(t^{\phantom{{\dagger}}}_{m}+\lambda^{\text{od}}_{m}V^{\text{od}}_{n,m}\right)c_{n}^{\dagger}c^{\phantom{{\dagger}}}_{n+m}+h.c.\right]\right\\}\,,$
(1)

where $n,m$ are integers iterating over the atomic sites along the chain,
$c_{n}^{\dagger}$ is a second-quantized creation operator at site $n$,
$V^{\text{d}}_{n}$ is an on-site quasiperiodic potential corresponding to the
height of the potential-well at each site with modulation amplitude
$\lambda^{\text{d}}$. The kinetic energy matrix elements are decomposed into
hopping amplitudes $t_{m}$ associated with the mean hybridization between
orbits separated by $m$ sites, on top of which a quasiperiodic function
$V^{\text{od}}_{n,m}$ of amplitude $\lambda^{\text{od}}_{m}$ encodes the
modulated inter-atomic distances. Note that generally we can also consider
additional quasiperiodic potentials, e.g., spin-dependent and spin-orbit-
dependent terms.

The quasiperiodic 1D functions $V^{\text{d}}_{n}$ and $V^{\text{od}}_{n,m}$
can be constructed, for example, by primitive substitutions [14, 36] to
iteratively generate a non-repeating sequence of discrete values. Known
examples involving two discrete values $a$ and $b$ are the Fibonacci
($a\rightarrow ab$, $b\rightarrow a$) [23, 37, 20], Thue-Morse ($a\rightarrow
ab$, $b\rightarrow ba$) [35] and Period-doubling sequences ($a\rightarrow ab$,
$b\rightarrow aa$) [35, 38], see Fig. 2(a) for a 2D example. Alternatively,
the method of “cut-and-project” [see Fig. 2(b)] can be employed to generate
step-wise (Sturm) functions of the form $V_{\rm
Sturm}(x)=2(\lfloor(x+2)/\tau\rfloor-\lfloor(x+1)/\tau\rfloor)-1=\pm 1$, where
$\tau$ is an irrational number, $\lfloor x\rfloor$ is the floor function, and
the potentials in Eq. (1) sample the continuous potential at integer values
$V(n)$, thus, enforcing the incommensuration between the periodic lattice and
the potential modulation lengthscales [14, 39]. When $\tau=(1+\sqrt{5})/2$ is
the golden ratio, this method also yields the Fibonacci sequence. As another
example, quasiperiodic sequences with infinitely-many values can be obtained
by an incommensurate sampling of a periodic function, e.g., $f(n/\tau+\phi)$,
which is periodic under a scan of $\phi$ from 0 to $2\pi$, see Fig. 2(c). A
useful example for our discussion is the interpolating Aubry-André-Fibonacci
(IAAF) function [39, 40, 41]

$V_{\text{IAAF}}(\beta,\tau,x,\phi)=-\frac{\tanh{\left[\beta(\cos{(2\pi
x/\tau+\phi)}-\cos{(\pi/\tau)})\right]}}{\tanh\beta}\,,$ (2)

where $\phi$ encodes a ‘phasonic’ continuous shift of the potential, and
$\beta$ interpolates between two known limiting cases, namely, for
$\beta\rightarrow 0$, we obtain the Harper-Aubry-André cosine modulation [42,
22], up to a constant shift $\left[\cos{(2\pi
x/\tau+\phi)}-\cos{(\pi/\tau)}\right]$, and for $\beta\rightarrow\infty$ the
Fibonacci sequence is reached with alternating values of $\pm 1$ [23, 37, 39].
Note that in dimensions $D>1$ the quasiperiodic potential functions become
more complex and include also rotational symmetries that are forbidden for
periodic crystals [14], see Fig. 2(a).

Figure 2: Methods to generate quasiperiodic tiling and potentials. (a)
Substitution rules for rhombic tiles (upper right) are applied to create a 2D
Penrose tiling. (b) The cut-and project method for generating 1D quasiperiodic
chains. A cut strip (gray) is tilted in an angle with tangent $\tau$ relative
to a 2D square lattice. The lattice points within the cut’s acceptance window
are projected onto a 1D line, with two possible spacings (red and blue). (c)
Sampling of the Interpolating Aubry-André-Fibonacci potential at integer
values, cf. Eq. (2). Periodicity with respect to $\phi$ in (c) is mappable to
periodicity with respect to displacement of the cut strip in (b) [39].

From Eq. (1), we can identify several branching points associated with how
different fields analyze quasicrystals. For example, we can either (i) study
directly the ab-initio atomic arrangement and hybridization as discussed
above, (ii) analyze the most general spectral properties that can be obtained
from a specific quasiperiodic atomic arrangement, i.e., without turning to ab-
initio extraction of parameters; this is similar to symmetry group analysis of
spectral features of periodic crystals, and (iii) analyze specific sub-cases
of the general case (1), e.g., by considering solely short-ranged potentials.
In the spirit of (ii), much research is pursued within the mathematical
physics community, aiming to answer (a) whether the spectrum has a well-
defined structure with a countable number of bands and gaps [see, e.g., Ref.
[35]], (b) whether the eigenstates of the quasicrystals are spatially-extended
or localized [see, e.g., Refs. [22, 43, 44]], and (c) whether a topological
classification of spectral bands of the quasicrystals is possible by
harnessing K-theoretical tools [see e.g., Refs. [35, 45] and references
therein]. Note that (a) and (c) are often combined, as specific quasiperiodic
sequences fix the integrated density of states as a topological index that
allows for labeling of spectral gaps of the material, and thus their
enumeration [35]. In the context of (iii), different approximations are taken,
such as assuming only short-ranged nearest-neighbor hopping
($t^{\phantom{{\dagger}}}_{m\neq 1}=\lambda^{\text{od}}_{m\neq 1}=0$), on top
of which only diagonal
($t^{\phantom{{\dagger}}}_{1}=\lambda^{\text{od}}_{1}=0$) or off-diagonal
($\lambda^{\text{d}}=0$) models are explored. Note that the large variety of
such approximations in dimensions larger than 1 leads to different models
exhibiting different effects, while describing the same long-ranged
quasiperiodic spatial modulation.

Regardless of the approach or approximation discussed above, the long-range
nature of the quasiperiodic potentials does not allow us to apply Bloch’s
theorem to obtain quasimomenta as good quantum numbers and thus simplify the
solution of the problem. Similarly, we cannot solve exactly the spectrum of
the system on a computer as the quasiperiodicity manifests as a deviation from
periodicity at all length scales, i.e., a true quasiperiodic system is
infinitely large. Similar challenges also occur in the study of disordered
systems. There the thermodynamic properties of the disordered material are
approached by studying the impact of the disorder on a finite system and
observing how the effect scales with the system’s size. Crucially, a disorder-
average is taken in the analysis, such that the general (noise) ensemble
properties are explored, and not specific (sample path) cases. In comparison,
for quasiperiodic systems, we can also study the spectral properties of
finite-sized samples from the long-ranged system, and average over them, with
a reduced statistical averaging arising from the self-similar nature of
quasicrystalline eigenstates, see, e.g., Ref. [44] and references therein.

Figure 3: Approximant path and its spectrum for the 1D diagonal model (1) with
an on-site quasiperiodic potential (2) at $\beta=2$ and approaching
$\tau=(1+\sqrt{5})/2$. (left) The approximant model is placed on a ring with
periodic boundary conditions. Red dots mark atomic sites, which sample the on-
site potential (blue line) at the blue dots. Shown are the approximants with
unit cells of length $q_{n}=\\{3,5,8,13\\}$. The bulk spectrum of the
approximant is spanned by the phase twist $\theta$ at the periodic boundary,
which is equivalent to threading a flux through the ring. The potential (2)
can be displaced as a function of the phason $\phi$. (right) The corresponding
spectrum of the approximants with potential amplitude
$\lambda^{\text{d}}/t_{1}=0.9$ as a function of $\phi$ for $\theta=0$. We
observe splitting of the states into mini-bands, where large features (gaps)
are clearly robust along the approximant path while new (smaller) features
appear.

We can study the bulk spectral properties of the quasiperiodic model along a
so-called approximant path, where we explore finite periodic models that
capture well the incommensurate length scales competition within their
enlarged unit cell [14, 46]. Beyond the finite unit cell length scale, we use
a periodic approximation and apply Bloch’s theorem. We recall that when the
approximant length scale in real space is equivalent to $q$, the corresponding
first Brillouin zone in momentum space is $q$ times smaller than for a simple
periodic lattice. As a consequence, the energy dispersion $E(k)$ of the
periodic chain folds from the larger to the reduced 1st Brillouin zone, thus
forming $q$ different energy bands. Note that the bands may overlap depending
on the specific model, e.g., if a symmetry persists under the folding. As an
example for such an approximant approach, we consider the interpolating
quasiperiodic function (2), see Fig. 3. The irrational modulation $\tau$ has a
unique continued fraction representation

$\displaystyle\frac{1}{\tau}=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{\ddots+\cfrac{1}{a_{n}}}}}\,,$
(3)

where $a_{i}$ are integer numbers. Thus, while scaling the system size, at the
$n^{\rm th}$ iteration, we can take a rational modulation
$\tau_{n}=q_{n}/p_{n}$, that leads to a periodic chain with $q_{n}>q_{n-1}$
bulk bands. Clearly, the difference between the $n-1$ and $n$ iteration occurs
at relatively longer length scales, implying that the differences between the
bulk bands of different approximants will occur on small energy scales, i.e.,
by splitting of bulk bands into smaller bands. For the case of the Fibonacci
sequence, the approximant path grows as the Fibonacci numbers with $a_{0}=0$
and $a_{i\neq 0}=1$, corresponding to denominators
$q_{n}\in\\{1,2,3,5,8,13,21,34,\ldots\\}$. Approximant paths exist also for
other quasiperiodic orders, as well as in higher-dimensional quasicrystals,
where the most important challenge is to identify properties that survive the
scaling to the thermodynamic limit.

A prominent topological classification approach for quasicrystals involves the
labeling of spectral gaps that are robust along the approximant path. Once a
spectral gap appears, at sufficiently later approximants along the scaling,
the formation of small mini-bands cannot close the robust (larger) energy gap,
and thus the integrated density of states remains constant under the scaling
and acts as a topological index, which cannot be changed unless a perturbation
is introduced that closes that gap.

##  3 Dimensional extension and relation to the quantum Hall effect

We now focus on the specific family of models including only an on-site
(diagonal) interpolating quasiperiodic potential (2), and study their
dependence on the parameter $\phi$. For any given $\phi$, we can understand
the potential to be originating from a quasimomentum in some perpendicular
(non-physical) direction. In other words, the combined real space of the chain
with the compact parameter $\phi$ is two-dimensional. It is illuminating to
extend our description to this additional perpendicular axis by performing an
inverse Fourier map of the model with respect to the parameter $\phi$. We thus
obtain a 2D real space ancestor model [39]

$\displaystyle H_{\rm 2D}=\sum_{n,m}$
$\displaystyle\Big{[}tc_{n,m}^{\dagger}c_{n+1,m}+\lambda^{\text{d}}\sum_{l=0}^{\infty}v^{\text{S}}_{l}(\beta)e^{i\pi(2n+3)l/\tau}c_{n,m}^{\dagger}c_{n,m+l}+h.c.\Big{]}\,,$
(4)

where
$v^{\text{S}}_{l}(\beta)=\int_{0}^{2\pi}(d\phi/2\pi)e^{il\phi}\,V_{\text{IAAF}}(\beta,\tau,x,\phi)$.
The model describes particles hopping on a 2D square lattice in the presence
of a perpendicular magnetic field with $2\pi/\tau$ flux threading each square
plaquette written in the Landau gauge using Peierls substitution [47, 48], see
Fig. 4(a). Along the $x$-axis, we have nearest-neighbor hopping as in the
descendent 1D model. Along the $y$-axis, we have long-ranged hopping depending
on $\beta$, where in the limit of $\beta\rightarrow 0$, it also becomes short-
ranged and we obtain the 2D model Hofstader model [12, 49, 13]

$H_{\rm
Hof}=\sum_{n,m}\Big{[}tc_{n,m}^{\dagger}c_{n+1,m}+\frac{\lambda^{\text{d}}}{2}e^{i2\pi
n/\tau}c_{n,m}^{\dagger}c_{n,m+1}+h.c.\Big{]}\,.$ (5)

Figure 4: Two-dimensional quantum Hall effect on a lattice. (a) Illustration
of the 2D model obtained by dimensional extension (4) with nearest-neighbor
hopping along the $x$\- and $y$-directions, as well as longer-ranged hopping
along $y$ depending on $\beta$. A magnetic flux $2\pi/\tau$ pierces each
square plaquette, leading to an enlarged magnetic unit cell (blue box for
$q_{n}=5$). (b) Hofstadter butterfly spectrum as a function of $1/\tau$ for
the diagonal model with $\beta=2$ and $\lambda^{\text{d}}/t=0.9$. The Chern
number of the large gaps is marked (red), cf. Eq. (6). (c) The spectrum of the
same system for $\tau=5/3$ [marked with gray dashed line in (b)] with open
boundary conditions, exhibiting 5 bands and 4 gaps. Topological edge modes
cross the bulk gaps in accordance with the bulk-boundary correspondence. As
the quasimomentum $k_{y}$ of the quantum Hall system (4) is equivalent to the
pump parameter $\phi$, we observe the boundary phenomena of topological pumps,
namely, at the 0D boundary of the 1D pump, boundary modes will appear and
thread across the gap.

Interestingly, we now mapped the length scales competition between the 1D
lattice and the quasiperiodic potential in Eq. (1) to a competition between
the 2D lattice and the magnetic length scale in Eqs. (4) and (5). In the
absence of the magnetic field, $\tau\rightarrow\infty$, the Hamiltonian (5)
commutes with the group of lattice translations, whereas when the magnetic
field breaks this symmetry, the notion of the magnetic translation group was
introduced [50]. It is generated by the operators $T_{\hat{m}}$ and
$T_{\hat{n}}$, where $T_{\hat{m}}c_{n,m}T_{\hat{m}}^{-1}=c_{n,m+1}$ and
$T_{\hat{n}}c_{n,m}T_{\hat{n}}^{-1}=e^{-i2\pi m/\tau}c_{n+1,m}$. For an
irrational $\tau$, these operators commute with the Hamiltonian but not with
each other. Nevertheless, similar to the approximant discussion above, for a
rational flux, $\tau=q/p$, the operator $T_{q\hat{n}}=(T_{\hat{n}})^{q}$
commutes with $T_{\hat{m}}$. We can therefore simultaneously diagonalize
$H_{\rm 2D}$, $T_{q\hat{n}}$, and $T_{\hat{m}}$. Here, too, the enlarged real
space magnetic unit cell implies folding of the corresponding quasimometa (now
onto a 2D torus) and the formation of $q$ bands in the spectrum, thus
generating Hofstadter butterfly spectra, see Fig. 4(b) [13].

Using the 2D vantage point, the magnetic length scale competition allows for
another topological characterization of the system. Specifically, each gap $r$
in the spectrum of this model is associated with a quantized and nontrivial
Chern number [5, 51, 52, 53, 54]

$\displaystyle\nu_{r}$ $\displaystyle=\frac{1}{2\pi i}\int_{0}^{2\pi}d\phi
d\theta\,C_{r}(\theta,\phi)\,,$ (6)

where $\phi$ and $\theta$ are phase twists (fluxes) associated with the
periodic boundary conditions along the $y$\- and $x$-directions, and

$\displaystyle C_{r}(\phi,\theta)$
$\displaystyle=\textrm{Tr}\left\\{P_{r}\left[\partial_{\phi}P_{r},\partial_{\theta}P_{r}\right]\right)\,,$
(7)

is the Chern density (curvature) with $[a,b]$ the commutator of $a$ and $b$
and the projector on the states below the chosen spectral gap

$\displaystyle P_{r}(\phi,\theta)$
$\displaystyle=\sum_{E_{n}<E_{\text{gap}}}|n\rangle\langle n|\,.$ (8)

Here, $|n\rangle$ is an eigenstate of $H_{\rm 2D}$ with energy $E_{n}$, and
$E_{\text{gap}}$ is the energy at the center of the gap. For a sufficiently
large system size, such that the gaps are open for any $\phi$ and $\theta$,
$P_{r}(\phi,\theta)$ is a well-defined quantity, see Fig. 4(b). Importantly,
this topological characterization manifests as a quantized bulk response of
the system, namely, quantized Hall conductance [4, 55, 5, 56]. Furthermore,
the quantized bulk response also implies the appearance of chiral edge modes
that cross the spectral gap and are localized at the boundary of the system,
known as the bulk-boundary correspondence, see Fig. 4(c) [7, 8, 9, 2, 3, 10].

Recall that for any rational $\tau=q/p$, we have $q$ bands and maximally $q-1$
gaps in the spectrum, cf. discussion on approximants above and Fig. 4(b). Due
to the symmetry properties of the magnetic translation group, there is an
interesting connection between number theory and the Chern number $\nu_{r}$
associated with a gap number $r=1,\ldots,(q-1)$, namely it satisfies the
Diophantine equation $r=\nu_{r}p+t_{r}q$, where $\nu_{r}$ and $t_{r}$ are
integers [5, 51, 54, 57]. This condition has infinitely many possible
solutions, as $\nu_{r}$ is only well defined modulo $q$. For specific models,
the Chern numbers are constrained to provide a single solution for the
Diophantine equation, e.g., for the Hofstadter model, where $0<|\nu_{r}|<q/2$
[5]. For other models, finding the correct Chern numbers requires evaluation
of Eq. (6) and is called “coloring Hofstadter butterflies” due to the
association of a color to each value of the Chern number in a spectral gap
[58, 59]. Crucial for our discussion, we note also that we can divide the
Diophantine equation by $q$ to obtain $\rho_{r}=\nu_{r}/\tau+t_{r}$, which
defines the Chern number for any particle density (filling factor) $\rho_{r}$
within a spectral gap. The latter result does not rely on approximants, it can
be applied also for quasiperiodic models, and is sometimes referred to as
Streda’s formula [51, 2]. Note that the filling factor (a.k.a. integrated
density of state) $\rho$ is fixed for quasiperiodic models in each gap by the
above condition, i.e., it cannot change without a change in $\tau$.

Following the approximant path and moving back towards quasiperiodic models,
we approach an irrational magnetic flux by taking an appropriate rational
limit with $p,q\rightarrow\infty$. In this limit, the spectrum becomes a
Cantor set fractal [13]. At each approximant generation, new gaps can open,
exhibiting quantized Chern numbers. It is important to note that depending on
the model, the gaps can close and reopen under some perturbation that does not
break the unit cell, see Fig. 5. In such a case, the Diophantine equation
still holds and the corresponding Chern numbers can change by a multiple of
$q$ modulo $q$. Nevertheless, at later approximants, due to the changed
denominator $q_{n+l}>q_{n}$, such a exchange of Chern numbers is no longer
possible and the gap may close but without a change to the Chern number.
Crucially, in order to change a topological index of the gap, a phase
transition must occur and the gap must close, i.e., there is no way to deform
between the two distinct topological states without the phase transition.
However, picking a specific deformation [as for example done in the IAAF model
(2)], gaps can close and reopen, but it does not necessitate a change in the
topological index. Combining the two arguments in this paragraph, alongside
the fact that all gaps are known to be open in both limits of the IAAF
defomration, we could color the gaps of the Fibonacci limit of the model [39].

Figure 5: Phase transitions along the approximant path and interpolation
deformation (2). Zoom in on the upper bands in the spectrum of the diagonal
IAAF, as a function of $\beta$ for approximants (a) $\tau=5/3$, (b)
$\tau=8/5$, and (c) $\tau=13/8$. Gap closings can occur as a function $\beta$,
but a change of the Chern number (red labels) is only allowed in specific
cases when it complies with the Diophantine equation.

We can employ dimensional extension of quasiperiodic models in $D>1$ with
spatial modulations along orthogonal axes to obtain quantum Hall effects in
$2\times D>2$, which are then characterized by higher dimensional Chern
numbers, families of associated bulk responses, as well as, associated
boundary effects with co-dimension $d\in\\{1,\ldots,D\\}$ [29, 30, 60, 61, 62,
63, 64, 65, 66]. For such cases, similar approximant paths apply with
generalized translation operators as discussed above. At the same time, it
becomes much harder to identify gaps that remain open, and the gap labeling
involves bookkeeping that tracks the labels from the orthogonal axes.
Furthermore, introducing terms that mix the orthogonal spatial axes, as is
expected for more general quasicrystals [cf. Fig. 2(a)], implies that a
mapping to a $2\times D$ ancestor model may be possible [see, e.g., recent
work with hexagonal moiré structures [67]], but the gap labeling approach
detailed here will no longer hold.

##  4 Topological pumps, dimensional reduction, and Chern numbers
of the quasiperiodic model

We have seen a relationship between the family of 1D models (1) parametrized
by $\phi$ and the 2D quantum Hall effect (4). Such a relationship allows for
the prediction of quantized charge transport in temporally-modulated chains,
i.e., topological charge pumping [55, 68, 28, 69]. Here, the time-dependent
and adiabatic threading over the parameter $\phi$ allows for the 1D family of
models to cover the 2D space spanned by both $\phi$ and $\theta$, cf. Eq. (6).
This, in similitude to the quantized Hall conductance, leads to a quantized
number of charges being pushed across the chain per cycle of $\phi$ from 0 to
$2\pi$, with the quantization originating from the Chern number of the gap.
Note that such a topological bulk response is associated with the whole pump
cycle, and implies a specific boundary phenomena: at the 0D boundary of the
pump, subgap modes cross the topological gap to carry the quantized charge
across the system [7, 28, 70, 3]. This can be thought of as an iteration with
$\phi$ over the dispersive and chiral 1D edge modes of the 2D quantum Hall
effect, see Fig. 4(c).

Similar to the 2D quantum Hall effect on a lattice, the topological pump is
characterized by a 2D Chern number and exhibits quantized charge transport for
both rational and irrational $\tau$. We, now, focus instead on the direct
topological characterization of the quasiperiodic chain using Chern numbers,
i.e., without the need to use the parameter $\phi$, where we summarize and
repeat the proof shown in Ref. [28], cf. also Refs. [31, 71]. To do so, we
employ the approximant path described above, and place at each iteration the
chain on a ring (periodic boundary conditions) of length $q_{n}$. We also
introduce a flux $\theta$ (phase twist) threading the ring, cf. Fig. 3. The
periodic model has the same spectral gaps as those of the original chain up to
small corrections [56], whereas in the thermodynamic limit the effect of
$\theta$ completely vanishes [56, 72].

We now show that the impact of $\phi$ is also vanishing for quasiperiodic
chains. Consider the impact of a shift of the parameter
$\phi\rightarrow\phi+\epsilon$, where $\epsilon=2\pi l/q_{n}$ and
$l=0,1,...,L-1$. Under this choice of $\epsilon$, the magnetic translation
symmetry discussed above implies that there is always a spatial translation of
$n\rightarrow n+n_{\epsilon}$ lattice sites, where $n_{\epsilon}\in
0,1,...,q_{n}-1$, such that the potential (2), has $V_{\rm
IAAF}(x,\phi+\epsilon)=V_{\rm IAAF}(x+n_{\epsilon},\phi)$. Therefore, for each
shift $\epsilon$ of $\phi$, we can associate a spatial translation operator
$\hat{T}_{\epsilon}$, corresponding to a translation by $n_{\epsilon}$ sites.
The equivalence between these discrete phase shifts and spatial translations
implies that the shifts have no physical consequences, such as closing of
energy gaps. Indeed, we solely observe modulation of the spectrum with
periodicity $2\pi/q_{n}$, see Figs. 3 and 4(c), which means that along the
approximant path, $\epsilon\rightarrow 0$, the spectrum becomes independent of
$\phi$.

Since the spectral projector $P_{r}$ [cf. Eq. (8)] is composed out of the
eigenstates on the model Hamiltonian, we readily obtain that
$P(\phi+\epsilon)=T_{\epsilon}P(\phi)T_{\epsilon}^{-1}$ and
$\partial_{\phi}P(\phi+\epsilon)=T_{\epsilon}\partial_{\phi}P(\phi)T_{\epsilon}^{-1}$.
Similarly, the Chern density is periodic in $1/q_{n}$,
$C(\phi+\epsilon)=C(\phi)$, implying that the Chern number can be written as

$\displaystyle\int_{0}^{2\pi}d\phi\,C(\phi)=q_{n}\int_{0}^{2\pi/q_{n}}d\phi\,C(\phi)=2\pi
C(\phi=0)+O(1/q_{n})\,,$ (9)

where in the first equality we used the above discussed periodicity of the
Chern density, and in the second equality we expand the integral in powers of
$1/q_{n}$. In the limit of $q_{n}\rightarrow\infty$, the approximation becomes
exact and the integral becomes invariant of $\phi$. In other words, the Chern
number associated with a topological pump family (as a function of $\phi$), is
equivalent, when the spatial modulation is quasiperiodic, to the Chern density
of a single chain at a given $\phi$ times a factor of $2\pi$. Note that the
dependence of $\phi$ is required for the evaluation of the Chern density,
i.e., the latter can be seen as the response of the chain to a potential
shift, cf. Eq. (2).

Using this argument, we can associate a 2D gap topological index (Chern
number) to a gap of a 1D quasiperiodic model. In other words, using the
quasiperiodic order, we quantize/flatten the Chern density such that it serves
as a topological index for the 1D chain. This construction locks the
Diophantine/Streda relation described above, such that at a given filling
factor, when a gap occurs, an associated single-valued Chern density appears
that corresponds to the Chern number of the related topological pump divided
by $2\pi$. This discussion is transferable to other quasicrystals that have a
Hamiltonian that is periodic with a phase shift $\phi$, which can be
compensated by a translation, including in higher-dimensions [29].

This topological characterization has physical implications both in the bulk
and at the boundary of the quasiperiodic system: (i) moving from one
quasiperiodic modulation $\tau_{1}$ to another $\tau_{2}$ will induce
rearrangements of the bulk gaps (phase transitions), at energy scales
corresponding to the approximant at which the two modulation separate from one
another [35]; (ii) at any periodic approximant level the topological phase
transitions will be avoided with a correction of $O(1/q_{n})$ [28, 73, 30];
(iii) the above two points manifest at a smoothly-interpolating spatial
boundary; there, subgap boundary modes with increasing density appear as a
function of the boundary length [40]; (iv) the filling-factor gap labels can
appear in bulk diffraction experiments [74]; (v) the relation to the
topological pump implies that at a finite boundary to the vacuum, the bulk gap
will be densely filled by subgap modes as a function of $\phi$ or
equivalently, as a function of removing the last site of a half-infinite chain
[28, 41, 30, 75, 76], which can also be understood as a result of the
mechanical work associated with the bulk Chern density, manifesting at the
boundary [45]; and (vi) at higher dimensions $D>1$, the boundary manifests a
variety of effects with co-dimension up to $D$, corresponding to bulk
responses of higher Chern numbers [29, 63, 65, 66].

Before moving to discuss experimental realizations of the topological
consequences, we now highlight once more the assumptions that we relied on. We
thus discuss caveats regarding the order of limits taken in our analysis, and
challenges in generalizing the results to other types of quasicrystals.
Whereas labeling of gaps using filling factors is possible for various
quasiperiodic models [35], here we discuss also another index (related to the
Chern number of a higher dimensional quantum Hall state), which is associated
with physical observables. This index arises as a response to a continuous
potential shift $\phi$. Considering discrete quasiperiodic potentials,
however, will show discontinuities that will complicate such a construction.
Specifically, the interpolating potential (2) supports our analysis everywhere
except at $\beta=\infty$. Therefore, we must first take the approximant path
to infinity before we can take the discrete step limit. Crucially, along the
path, the characterization strictly relies on the fact that the gap stays open
for all $\phi$ and $\theta$, and as a function of the approximant scaling, cf.
Fig. 5.

Due to these caveats, it remains to be explored how to associate Chern numbers
to discrete chains such as the Thue-Morse and Period-Doubling chains, where a
relation to topological boundary effects and pumping was not established.
Considering cut-and-project quasicrystals, the interpolating potential (2) is
equivalent to “smoothening” the cut’s acceptance window, whereas we study the
response system to a phason shift of the window, cf. Fig. 2(b) and Ref. [39].
Such a smoothening in higher-dimensional quasicrystals is challenging, e.g.,
in the Amman-Beekner tiling the cut-window is octagonal, and can be modified
in different ways. Additionally, in the presence of several quasiperiodic
potentials, the length scales competition can become more elaborate than
discussed here, and the gaps can close as a function $\phi$ and $\theta$.

We would like to also emphasize that we have dealt here with single-particle
Hamiltonian models. The delicate length scale competition required for the
topological characterization of quasicrystals can be easily perturbed. For
example, disorder or dissipation produce a self-energy broadening that will be
able to close small spectral gaps associated with long-ranged deviations from
periodicity, i.e., they truncate long-ranged length scale. Alternatively,
perturbing quasicrystals with strong potentials that close spectral gaps
arising from the quasiperiodic length scales competition can generate real-
space curvatures and observed chiral edge modes, which are not related to the
higher-dimensional topology discussed here [77]. Similarly to disorder and
dissipation, many-body interactions have also been shown to act as a cut-off
on the length scale at which the incommensuration prevails [78, 79]. At the
same time, new results recently appeared, where strongly-correlated
quasiperiodic spin chains exhibit topological excitation gaps [80] or robust
topological edge modes that are strongly shaped by the interaction [81].
Furthermore, anomalous inverse compressibility is predicted to occur due to
inter-band interactions [82].

Figure 6: Optical platforms for exploring the topology of quasicrystals. (a)
Illustration of a coupled photonic waveguide array [cf. parallel review by
Szameit], where light is injected at the input facet and emulates a tight-
binding Hamiltonian evolution as it propagates through the array. Using such a
platform, the dependence of the topological boundary states of quasicrystals
as a function of the phason degree of freedom $\phi$ was explored [28, 41,
63]. Here, the red, green, and blue waveguides mark two different bulk
quasiperiodic chains (red and blue) with an interpolating region (green) used
to explore topological equivalence and phase transitions between different
quasiperiodic models [40]. (b) A scanning electron micrograph image of
sculptured polariton semiconductor microcavities used to explore the
interpolating model 2, taken from Ref. [44], cf. Fig. 1(b) and [parallel
review on Exciton-polaritons for topology]. Such a setup and a similar one for
acoustic waves were used to explore also the topological boundary states of
the Fibonacci chain [75, 76]. (c) Using a collimated laser beam impinging on a
digital micromirror device (DMD) that encodes the Fibonacci chain spacing, the
far-field diffraction pattern reveal the bulk gaps and their labels [74].

##  5 Optical experiments

Contemporary technology allows for the design of structured mediums for waves,
in the spirit of Fig. 1(b). The chosen pattern can be thought of as a spatial
modification of the effective refractive index of the medium, cf. the variety
of technological platforms covered in this special issues [cross reference
here to the other review work in this issue]. Such a spatial modification
generates an effective metamaterial for the waves propagating through the
medium, which are then readily probed using input-output experiments.
Furthermore, in the limit of deep and well-separated effective potential
wells, the waves behave similarly to electrons hopping between atoms, i.e.,
akin to the model (1).

With such platforms, both the topological bulk and boundary effects described
above have been explored in 1D quasiperiodic chains, see Fig. 6. Specifically,
using coupled photonics waveguide arrays, [cf. review by Szameit], topological
phase transitions and topological equivalence between different quasiperiodic
chains at smooth boundaries were explored, see Fig. 6(a) and Ref. [40]. Using
a similar platform, (i) the relation of 1D quasiperiodic chains to 1D
topological pumps and the associated associated topological boundary states of
quasiperiodic chains, were explored [28, 41]; (ii) The 0D topological boundary
states were used to generate a high-fidelity on-chip Hong-Ou-Mandel
interference of single-photons [83] [cf. contribution on quantum]; (iii)
Moving to 2D topological pumps, their predicted boundary phenomena with co-
dimension 1 and 2 [29], was also observed using coupled photonic waveguide
arrays [63], and their relationship to high-order topology was elucidated [84,
66]. In the spirit of Fig. 1(b), topological boundary modes of quasiperiodic
chains were also observed in polariton semiconductor microcavities [75] [see
Fig. 6(b), and acoustic chambers [76]. In diffraction experiments, bulk gap
labels of the 1D Fibonacci have been observed [74], see Fig. 6(b).

##  6 Conclusions

We have seen that the topology of quasicrystals originates from a competition
between length scales. Following the approximant path, we understood that this
competition “locks” the spectrum of the system into a very specific
arrangement with well-defined integrated density of states, as well as
associated bulk responses. The latter are quantized and serve for a
topological classification of quasicrystals with corresponding bulk and
boundary observable effects. In this sense, quasiperiodic order can be thought
of as a very specific crystalline symmetry that allows for an enriched
topological classification, akin to other crystalline symmetries [85]. At the
same time, the topological characterization discussed here has only been
possible for certain types of quasicrystals and under specific assumptions.
There is, therefore, still much to explore in this rich zoo of quasiperiodic
models. Furthermore, with the advent of controllable shaping of various
metamaterials for optical, acoustic, and matter waves the topological
properties of quasicrystals are now observable, allowing for new discoveries
and applications.

## Funding

We acknowledge financial support from the Swiss National Science Foundation
through grants PP00P2P2_1163818 and PP00P2_190078.

## Acknowledgments

I would like to thank Y. E. Kraus, Z. Ringel, M. Verbin, and Y. Lahini for
starting this activity with me; J. L. Lado, A. Štrkalj, I. Petrides, T. M. R.
Wolf, J. Bloch, and H. M. Price for collaboration along this path; and E.
Prodan, J. Kellendonk, A. Eichler, R. Lifshits, and Y. Avron for fruitful
discussions.

## Disclosures

The author declares no conflicts of interest.

## References

  * [1] N. W. Ashcroft and N. D. Mermin, _Solid State Physics_ (Holt-Saunders, 1976).
  * [2] B. A. Bernevig and T. L. Hughes, _Topological insulators and topological superconductors_ (Princeton university press, 2013).
  * [3] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg _et al._ , “Topological photonics,” Reviews of Modern Physics 91, 015006 (2019).
  * [4] K. v. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance,” Physical Review Letters 45, 494 (1980).
  * [5] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized hall conductance in a two-dimensional periodic potential,” Physical review letters 49, 405 (1982).
  * [6] M. E. Cage, K. Klitzing, A. Chang, F. Duncan, M. Haldane, R. Laughlin, A. Pruisken, and D. Thouless, _The quantum Hall effect_ (Springer Science & Business Media, 2012).
  * [7] B. I. Halperin, “Quantized hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential,” Physical Review B 25, 2185 (1982).
  * [8] A. MacDonald and P. Středa, “Quantized hall effect and edge currents,” Physical Review B 29, 1616 (1984).
  * [9] P. Streda, J. Kucera, and A. MacDonald, “Edge states, transmission matrices, and the hall resistance,” Physical review letters 59, 1973 (1987).
  * [10] J. Fröhlich, “Chiral anomaly, topological field theory, and novel states of matter,” Reviews in Mathematical Physics 30, 1840007 (2018).
  * [11] K. von Klitzing, “Essay: Quantum hall effect and the new international system of units,” Physical review letters 122, 200001 (2019).
  * [12] P. G. Harper, “Single band motion of conduction electrons in a uniform magnetic field,” Proceedings of the Physical Society. Section A 68, 874 (1955).
  * [13] D. R. Hofstadter, “Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields,” Physical review B 14, 2239 (1976).
  * [14] M. Senechal, _Quasicrystals and geometry_ (Cambridge University Press, Cambridge ; New York, 1995).
  * [15] R. Lifshitz, “Quasicrystals: A matter of definition,” Foundations of Physics 33, 1703–1711 (2003).
  * [16] H. Bohr _et al._ , “Zur theorie der fastperiodischen funktionen: Ii. zusammenhang der fastperiodischen funktionen mit funktionen von unendlich vielen variabeln; gleichmässige approximation durch trigonometrische summen,” Acta Mathematica 46, 101–214 (1925).
  * [17] H. Bohr _et al._ , “Zur theorie der fastperiodischen funktionen: Iii. dirichletentwicklung analytischer funktionen,” Acta Mathematica 47, 237–281 (1926).
  * [18] R. Penrose, “The role of aesthetics in pure and applied mathematical research,” Bull. Inst. Math. Appl. 10, 266–271 (1974).
  * [19] D. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, “Metallic phase with long-range orientational order and no translational symmetry,” Physical review letters 53, 1951 (1984).
  * [20] D. Levine and P. J. Steinhardt, “Quasicrystals: a new class of ordered structures,” Physical review letters 53, 2477 (1984).
  * [21] A. Yamamoto, “Crystallography of quasiperiodic crystals,” Acta Crystallographica Section A: Foundations of Crystallography 52, 509–560 (1996).
  * [22] S. Aubry and G. André, “Analyticity breaking and anderson localization in incommensurate lattices,” Ann. Israel Phys. Soc 3, 18 (1980).
  * [23] M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: Mapping and escape,” Phys. Rev. Lett. 50, 1870–1872 (1983).
  * [24] M. Kohmoto and Y. Oono, “Cantor spectrum for an almost periodic Schrödinger equation and a dynamical map,” Physics Letters A 102, 145–148 (1984).
  * [25] M. Kohmoto and J. R. Banavar, “Quasiperiodic lattice: Electronic properties, phonon properties, and diffusion,” Phys. Rev. B 34, 563–566 (1986).
  * [26] M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020–1033 (1987).
  * [27] D. Tanese, E. Gurevich, F. Baboux, T. Jacqmin, A. Lemaître, E. Galopin, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Fractal energy spectrum of a polariton gas in a fibonacci quasiperiodic potential,” Physical review letters 112, 146404 (2014).
  * [28] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, “Topological States and Adiabatic Pumping in Quasicrystals,” Phys. Rev. Lett. 109, 106402 (2012).
  * [29] Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Four-Dimensional Quantum Hall Effect in a Two-Dimensional Quasicrystal,” Phys. Rev. Lett. 111, 226401 (2013).
  * [30] Y. E. Kraus and O. Zilberberg, “Quasiperiodicity and topology transcend dimensions,” Nature Physics 12, 624–626 (2016).
  * [31] E. Prodan, “Virtual topological insulators with real quantized physics,” Physical Review B 91, 245104 (2015).
  * [32] E. Prodan and H. Schulz-Baldes, “Bulk and boundary invariants for complex topological insulators,” K (2016).
  * [33] J. Bellissard, B. Iochum, E. Scoppola, and D. Testard, “Spectral properties of one dimensional quasi-crystals,” CMaPh 125, 527–543 (1989).
  * [34] J. Bellissard, B. Iochum, and D. Testard, “Continuity properties of the electronic spectrum of 1D quasicrystals,” Communications in Mathematical Physics 141, 353–380 (1991).
  * [35] J. Bellissard, A. Bovier, and J.-M. Ghez, “GAP LABELLING THEOREMS FOR ONE DIMENSIONAL DISCRETE SCHRÖDINGER OPERATORS,” Reviews in Mathematical Physics 04, 1–37 (1992).
  * [36] N. P. Fogg, V. Berthé, S. Ferenczi, C. Mauduit, and A. Siegel, eds., _Substitutions in Dynamics, Arithmetics and Combinatorics_ , vol. 1794 of _Lecture Notes in Mathematics_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2002).
  * [37] S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, and E. D. Siggia, “One-dimensional schrödinger equation with an almost periodic potential,” Phys. Rev. Lett. 50, 1873–1876 (1983).
  * [38] D. Damanik, “Singular continuous spectrum for the period doubling hamiltonian on a set of full measure,” Communications in mathematical physics 196, 477–483 (1998).
  * [39] Y. E. Kraus and O. Zilberberg, “Topological equivalence between the fibonacci quasicrystal and the harper model,” Phys. Rev. Lett. 109, 116404 (2012).
  * [40] M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, and Y. Silberberg, “Observation of Topological Phase Transitions in Photonic Quasicrystals,” Phys. Rev. Lett. 110, 076403 (2013).
  * [41] M. Verbin, O. Zilberberg, Y. Lahini, Y. E. Kraus, and Y. Silberberg, “Topological pumping over a photonic Fibonacci quasicrystal,” Phys. Rev. B 91, 064201 (2015).
  * [42] P. Harper, “Pg harper, proc. phys. soc., london, sect. a 68, 874 (1955).” Proc. Phys. Soc., London, Sect. A 68, 874 (1955).
  * [43] S. Y. Jitomirskaya, “Metal-insulator transition for the almost mathieu operator,” Annals of Mathematics 150, 1159–1175 (1999).
  * [44] V. Goblot, A. Štrkalj, N. Pernet, J. L. Lado, C. Dorow, A. Lemaître, L. L. Gratiet, A. Harouri, I. Sagnes, S. Ravets, A. Amo, J. Bloch, and O. Zilberberg, “Emergence of criticality through a cascade of delocalization transitions in quasiperiodic chains,” Nature Physics 16, 832–836 (2020).
  * [45] J. Kellendonk and E. Prodan, “Bulk–boundary correspondence for sturmian kohmoto-like models,” in _Annales Henri Poincaré,_ vol. 20 (2019), pp. 2039–2070.
  * [46] S. Beckus, J. Bellissard, and G. De Nittis, “Spectral continuity for aperiodic quantum systems I. General theory,” Journal of Functional Analysis 275, 2917–2977 (2018).
  * [47] R. Peierls, “Zur theorie des diamagnetismus von leitungselektronen,” Zeitschrift für Physik 80, 763–791 (1933).
  * [48] J. Luttinger, “The effect of a magnetic field on electrons in a periodic potential,” Physical Review 84, 814 (1951).
  * [49] M. Y. Azbel, “Energy spectrum of a conduction electron in a magnetic field,” Sov. Phys. JETP 19, 634–645 (1964).
  * [50] J. Zak, “Magnetic translation group,” Physical Review 134, A1602 (1964).
  * [51] P. Streda, “Theory of quantised hall conductivity in two dimensions,” Journal of Physics C: Solid State Physics 15, L717 (1982).
  * [52] J. E. Avron, R. Seiler, and B. Simon, “Homotopy and quantization in condensed matter physics,” Physical review letters 51, 51 (1983).
  * [53] M. Kohmoto, “Topological invariant and the quantization of the hall conductance,” Annals of Physics 160, 343–354 (1985).
  * [54] I. Dana, Y. Avron, and J. Zak, “Quantised hall conductance in a perfect crystal,” Journal of Physics C: Solid State Physics 18, L679 (1985).
  * [55] R. B. Laughlin, “Quantized hall conductivity in two dimensions,” Physical Review B 23, 5632 (1981).
  * [56] Q. Niu, D. J. Thouless, and Y.-S. Wu, “Quantized hall conductance as a topological invariant,” Physical Review B 31, 3372 (1985).
  * [57] Y. Avron, R. Seiler, and B. Shapiro, “Generic properties of quantum hall hamiltonians for finite systems,” Nuclear Physics B 265, 364–374 (1986).
  * [58] J. Avron, O. Kenneth, and G. Yehoshua, “A study of the ambiguity in the solutions to the diophantine equation for chern numbers,” Journal of Physics A: Mathematical and Theoretical 47, 185202 (2014).
  * [59] A. Agazzi, J.-P. Eckmann, and G. M. Graf, “The colored hofstadter butterfly for the honeycomb lattice,” Journal of Statistical Physics 156, 417–426 (2014).
  * [60] H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, “Four-dimensional quantum hall effect with ultracold atoms,” Physical review letters 115, 195303 (2015).
  * [61] H. M. Price, O. Zilberberg, T. Ozawa, I. Carusotto, and N. Goldman, “Measurement of chern numbers through center-of-mass responses,” Physical Review B 93, 245113 (2016).
  * [62] T. Ozawa, H. M. Price, N. Goldman, O. Zilberberg, and I. Carusotto, “Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum hall physics,” Physical Review A 93, 043827 (2016).
  * [63] O. Zilberberg, S. Huang, J. Guglielmon, M. Wang, K. P. Chen, Y. E. Kraus, and M. C. Rechtsman, “Photonic topological boundary pumping as a probe of 4d quantum hall physics,” Nature 553, 59–62 (2018).
  * [64] M. Lohse, C. Schweizer, H. M. Price, O. Zilberberg, and I. Bloch, “Exploring 4d quantum hall physics with a 2d topological charge pump,” Nature 553, 55–58 (2018).
  * [65] I. Petrides, H. M. Price, and O. Zilberberg, “Six-dimensional quantum hall effect and three-dimensional topological pumps,” Physical Review B 98, 125431 (2018).
  * [66] I. Petrides and O. Zilberberg, “Higher-order topological insulators, topological pumps and the quantum hall effect in high dimensions,” Physical Review Research 2, 022049 (2020).
  * [67] M. Rosa, M. Ruzzene, and E. Prodan, “Topological gaps by twisting,” arXiv preprint arXiv:2006.10019 (2020).
  * [68] D. J. Thouless, “Quantization of particle transport,” Physical Review B 27, 6083 (1983).
  * [69] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, “A thouless quantum pump with ultracold bosonic atoms in an optical superlattice,” Nature Physics 12, 350–354 (2016).
  * [70] Y. Hatsugai and T. Fukui, “Bulk-edge correspondence in topological pumping,” Physical Review B 94, 041102 (2016).
  * [71] C. Bourne and A. Rennie, “Chern numbers, localisation and the bulk-edge correspondence for continuous models of topological phases,” Mathematical Physics, Analysis and Geometry 21, 16 (2018).
  * [72] Z. Ringel and Y. E. Kraus, “Determining topological order from a local ground-state correlation function,” Physical Review B 83, 245115 (2011).
  * [73] Y. E. Kraus, Z. Ringel, and O. Zilberberg, “Comment on “Topological equivalence of crystal and quasicrystal band structures”,” arXiv e-prints arXiv:1308.2378 (2013).
  * [74] A. Dareau, E. Levy, M. B. Aguilera, R. Bouganne, E. Akkermans, F. Gerbier, and J. Beugnon, “Revealing the topology of quasicrystals with a diffraction experiment,” Phys. Rev. Lett. 119, 215304 (2017).
  * [75] F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. Le Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95, 161114 (2017).
  * [76] D. J. Apigo, W. Cheng, K. F. Dobiszewski, E. Prodan, and C. Prodan, “Observation of topological edge modes in a quasiperiodic acoustic waveguide,” Phys. Rev. Lett. 122, 095501 (2019).
  * [77] M. A. Bandres, M. C. Rechtsman, and M. Segev, “Topological photonic quasicrystals: Fractal topological spectrum and protected transport,” Physical Review X 6, 011016 (2016).
  * [78] J. Vidal, D. Mouhanna, and T. Giamarchi, “Correlated fermions in a one-dimensional quasiperiodic potential,” Physical review letters 83, 3908 (1999).
  * [79] J. Vidal, D. Mouhanna, and T. Giamarchi, “Interacting fermions in self-similar potentials,” Physical Review B 65, 014201 (2001).
  * [80] J. L. Lado and O. Zilberberg, “Topological spin excitations in Harper-Heisenberg spin chains,” Physical Review Research 1, 033009 (2019).
  * [81] Y. Liu, L. F. Santos, and E. Prodan, “Topological gaps in quasi-periodic spin chains: A numerical and k-theoretic analysis,” arXiv preprint arXiv:2009.03752 (2020).
  * [82] Y. E. Kraus, O. Zilberberg, and R. Berkovits, “Enhanced compressibility due to repulsive interaction in the Harper model,” Phys. Rev. B 89, 161106 (2014).
  * [83] J.-L. Tambasco, G. Corrielli, R. J. Chapman, A. Crespi, O. Zilberberg, R. Osellame, and A. Peruzzo, “Quantum interference of topological states of light,” Science advances 4, eaat3187 (2018).
  * [84] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, “Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators,” Physical Review B 96, 245115 (2017).
  * [85] L. Fu, “Topological crystalline insulators,” Physical Review Letters 106, 106802 (2011).