effects – due solely to the Pauli principle – on the conductance of Fibonacci
chains Varma  _et al._ (2016). They computed the effective localization length
$\Lambda$ for various band fillings, using a formalism developed by Kohn in
his theory of the insulating state Kohn (1964). This localization length,
which is related to the real part of the conductivity tensor, is more
sensitive to spectral gaps, and to transport properties than the more familiar
single particle localization length defined in terms of the decay of the
envelope of a given wavefunction. In a conductor, this many body localization
length scales to infinity, while in an insulator, this quantity saturates with
the system size. Varma et al showed that in the hopping model, there are
special values of the band filling corresponding to an insulating state. These
fillings correspond to values of IDOS of $\omega^{2}$ and $\omega^{3}$
correspond to the positions of the large gaps. This is shown in Fig.35 where
$\Lambda^{-2}$ is plotted versus inverse system size $1/L$. The fillings
$\frac{1}{2}$ and $\frac{1}{4}$ for which the trend is decreasing with
increasing system size most likely correspond to a metallic state.

Figure 35: Scaling of the many body localization length $\Lambda$ with system
size $L$, for different band fillings. One sees a metallic trend ($\Lambda$
decreasing with $L$) for fillings of 1/2 and 1/4, and an insulating behavior
($\Lambda$ constant with $L$) for fillings $g^{-2}$ and $g^{-3}$ where $g$ is
the golden mean. Red (grey) and black symbols correspond to curves for hopping
ratios equal to 0.5 and $\omega$ respectively. (Figure reproduced with
permission from Varma  _et al._ (2016)).

##  VIII Disorder and boundary effects

The question of perturbations and their effects on critical states has been
much discussed. The interesting conceptual problem raised has, of course,
experimental implications. Disorder can be expected to play an important role
for the electronic transport in real quasicrystals for the following reason.
We have seen in the previous section that, for a perfect FC, transport at
$T=0$ can be described by different power laws, and can either scale towards
an insulating or to a metallic state in the thermodynamic limit, depending on
the Fermi energy. Adding disorder of any form – be it structural defects or
chemical substitutions or phonons – introduces a cut-off timescale, $\tau$,
whose value depends on disorder. One can therefore have a conductance which is
finite in a quasicrystal that is weakly disordered. For increasing disorder,
one expects that the quantum interference phenomena which lead, in a perfect
quasicrystal to the multifractality of the density of states and the
multifractality of states, would be progressively suppressed. Combined, these
effects might contribute to improving transport as disorder increases,
however, these are still open questions.

The problem of a single impurity in a FC was studied by several authors. The
trace map method was used to compute localization lengths of an impurity state
represented by a delta-function potential in Naumis (1999). The effect of
structural defects, namely a single phason defect (in which a pair of bonds is
locally exchanged, as for example $AB\rightarrow BA$ has been studied Naumis
and Aragón (1996). This study concluded that the presence of a single impurity
affects all of the states and leads to an increase of the fractal dimension of
the spectrum. A weak form of structural disorder was considered in Velhinho
and Pimentel (2000), by allowing randomness in the substitution rules for
building chains. The conclusion reached in this and a later study Huang and
Huang (2004) is that this type of disorder is irrelevant, in that the Lyapunov
exponents of states were not changed. To modify the critical states of the
pure system, the disorder must break some symmetries of the Fibonacci
Hamiltonian as in the models we discuss in the following section.

###  VIII.1 Finite systems and approach to Anderson localization

We now focus on the effects of adding a finite bulk disorder to the two models
under discussion, Eqs.13 and 14. Quite general rigorous arguments show that,
in one dimensional models an infinitesimal disorder leads to localized states
Delyon  _et al._ (1985). This has been confirmed by a variety of numerical
studies Naumis (1999); Liu and Riklund (1987). Das Sarma and Xie Das Sarma and
Xie (1988) studied the effect of randomness in a Fibonacci QC using a
scattering model for a system in which the scatterers of constant height were
placed in a Fibonacci sequence of two distances $a$ and $b$. The system was
coupled to leads and conductance computed using the Landauer formula
$G=\frac{2e^{2}}{\hbar}T/(1-T)$, where $T$ is the transmission coefficient of
$F_{n}$ site system. They reported that, while disorder (in the positions of
the scatterers) eventually localizes all states, small disorder did not change
the physics qualitatively. Introducing a shuffling of the sequence of Kronig-
Penney type scatterers leads as well to localization of states and
consequently an exponential decay of the conductance.

There is no doubt that sufficiently large disorder strength leads to strong
localization. Interestingly, however, the approach to localization can be
quite complicated and state-dependent. It has been shown in Jagannathan  _et
al._ (2019) that there are interesting crossover phenomena going from the pure
system to the localized system as the strength of disorder is increased. The
authors studied the change of critical states of the pure hopping model when
hopping amplitudes are randomly perturbed from their initial values $t_{B}$
and $t_{A}=\rho t_{B}$. While most states tend to become more and more
localized as the disorder strength is increased, some states go the other way
initially, becoming delocalized before turning over and localizing like the
other states. This can be seen in Fig.36a which shows the change of the IPR
for the 20 lowest lying states $\alpha=1,..,20$ of the $n=10$ approximant
taking a hopping ratio of 0.5. The IPR values for the pure chain are shown by
filled circles, while the disorder averaged values for weak disorder $W=0.05t$
are shown by open circles. It can be seen that, for most of the states,
disorder results in increased IPR, as expected. However some states behave
anomalously – these states are indicated $\alpha=3,7,11,15,...$ in the figure.
The anomalous behavior is observed for arbitrarily large system size, albeit
for weaker disorder for longer approximants.

When the averaged IPR of each state is plotted versus disorder strength $W$,
as is shown for four low-lying states in Fig.36b), the anomalous states in
Fig.36a) show a marked initial decrease of IPR, followed by an upturn. The
perturbative RG method provides an explanation of the observed behaviors in
terms of the renormalization path of the states. Fig.36b) shows the RG path of
each of the states, and one sees that the band edge state $\alpha=1$ (shown in
red, RG path: $mmmmm$) has a monotonic increase of the IPR. In contrast, the
state $\alpha=3$ (shown in blue, RG path: $mmma$) has a minimum of the IPR –
the state first delocalizes under weak disorder before the upturn sets in. One
can show that the states which have “atomic” character in the final step of RG
have such re-entrant localization behavior and that they occur all through the
spectrum. In simple terms, using the picture for small values of $\rho$, for
atomic states IPR tends to decrease since disorder acts to “smear” the
wavefunction onto neighboring sites. In contrast, for a wavefunction composed
of molecular states, the change of IPR has the opposite sign (for details see
Jagannathan and Tarzia (2020b)).

Figure 36: (a) IPRs plotted for the lowest 20 states of a $n=10$ Fibonacci
chain for $\rho=0.5$. Filled circles give IPR values of the pure chain while
open circles give the sample averaged IPR values for weak disorder
($W=0.05t_{B}$). States with $\alpha=3,7,..$ indicated are those showing an
anomalous behavior. (b) Plot of the averaged IPRs versus disorder strength $W$
for four low-lying states of $n=10$ approximant. The RG path is indicated next
to each curve. Figure reproduced from Jagannathan  _et al._ (2019)

The changes in the IPRs can be described in terms of scaling functions – i.e.
for a $given$ state one can collapse the data for different system sizes $L$
and different disorder strengths $W$ onto a single curve. The finite size
analysis in Jagannathan  _et al._ (2019) showed that despite their different
approaches to localization, ALL states are described by a single critical
exponent, $\nu$. The value of $\phi$ depends on the ratio $t_{A}/t_{B}$ and
was found numerically from finite size scaling plots as in Fig.37 – $\nu=0.53$
for $\rho=0.33$. The re-entrant behavior of the IPR can be explained in terms
of the perturbative RG theory (see Jagannathan  _et al._ (2019); Jagannathan
and Tarzia (2020a) for details). However the phenomenon seems to be more
generic, and the re-entrance behavior is observed for the diagonal model, as
well as for other types of generalizations of the Hamiltonian.

Figure 37:  Data for the normalized IPR of several low lying states plotted
versus the scaling variable $WL^{1/\nu}$ showing the data collapse for
different disorder and system sizes. The top and bottom figures show results
for two values of the hopping ratio $t_{A}/t_{B}$ – note the changes in the
scaling functions, which are nonuniversal. (from Jagannathan and Tarzia
(2020a))

###  VIII.2 The proximity effect

It is well-known that it is possible to induce superconducting correlations in
a non-interacting conducting system (N) by coupling it to a superconductor (S)
– the so-called proximity effect. The proximity effect provides a way to
observe experimentally the unique properties of critical states in the FC. A
first step consists of examining the proximity induced local pairing order
parameter (OP) as a function of distance from the N-S interface.

Connecting the FC to a bulk superconductor and using a mean field theory, Rai
et al Rai  _et al._ (2019b) computed the distribution of $\Delta_{i}$, the
induced local superconducting order parameter on site $i$. There are very
large spatial fluctuations of $\Delta_{i}$, due precisely to critical states.
Fig.38 (top) shows the profile of the order parameter (OP) as a function of
the site number. One sees here the characteristic multifractal properties
reflected in the variations of the order parameter. Fitting the average curve
obtained by changing the phason angle parameter $\phi$, one sees that the OP
decays as a power law in the distance from the N-S interface Rai  _et al._
(2020). The power, which varies with $t_{A}/t_{B}$, is expected to depend both
on the exponent of the density of states, and the averaged fractal dimensions
of the wavefunctions near the Fermi energy, here taken to be at $E=0$.

Figure 38: The superconducting order parameter $\Delta(i)$ in a Fibonacci
chain placed in contact with a BCS superconductor at both ends, plotted versus
position showing power law decay ($t_{A}/t_{B}=0.8$, decay exponent=0.6)

Along with the other states, edge modes contribute to the induced order
parameter on each site. In fact, if one cycles through chains as a function of
the phason angle parameter $\phi$ in Eq.11, the induced order parameter
$\Delta_{i}$ at a given site oscillates and the periods are just the
topological numbers of gaps close to the Fermi energy Rai  _et al._ (2019a).
This can be seen in the Fig.39 which shows the variations of the OP at the
midpoint of the chain as a function of the phason angle $\phi$. Two different
chains are shown to emphasize that the basic periods do not change when going
from smaller to larger systems, only additional periods appear. The lower
figure shows the power spectrum of the oscillations, and the periods which are
present in the curve of $\Delta_{mid}$. One sees the periods $4,17,9,21,6$, –
these correspond precisely to the $\mathrm{q}$ values of the largest gaps near
the Fermi energy, in this case situated in the band center, $E_{F}=0$.

Figure 39: (top) The superconducting order parameter $\Delta(mid)$ at the mid-
point of a Fibonacci chain versus the phason angle $\phi$ showing
oscillations. (bottom) Fourier spectrum of the above plot, showing the main
periods showing the peaks at values equal to topological labels $\mathrm{q}$ .
(reproduced with permission from Rai  _et al._ (2019a))

##  IX Generalized Fibonacci models

###  IX.1 Phonon models

Phonon modes in a quasiperiodic chain can be studied by considering the set of
equations

$\displaystyle H\psi_{n}=K_{n-1}\psi_{n-1}+K_{n}\psi_{n+1}$
$\displaystyle-(K_{n-1}+K_{n})\psi_{n}=E\psi_{n}$ (75)

where $\psi_{n}$ denotes the displacement of atom $n$ of mass $m$ with respect
to its equilibrium position, and the couplings $K_{n}$ can take two values,
$K_{A}$ or $K_{B}$. One can consider, alternatively, another version of the
model in which the masses $m$ can vary, and the couplings are constant. The
operator $H$ generalizes the discretized Laplacian operator, and its
eigenvalues yield the frequencies of phonon modes. The phonon problem is
tackled using the methods we have seen already for the closely related
electron problem Kohmoto  _et al._ (1983a); Luck and Petritis (1986); Ostlund
_et al._ (1983); Nori and Rodriguez (1986); Lu  _et al._ (1986); Ashraff and
Stinchcombe (1989). The trace map equation is the same as in the electron
problem, namely Eq.25, and the same kind of analysis applies. The spectrum of
energies $E$ has a Cantor-set structure as can be seen in Fig.40 (taken from
Luck and Petritis (1986)) which shows the IDOS versus energy (these quantities
are denoted in the figure by $H$ and $z$). In contrast to the electronic case,
for phonons, the scaling is non-uniform as a function of energy, and gaps
become very small as $E$ tends to zero. At the lowest frequencies, the
integrated density of states, which is plotted in Fig.40 looks almost
indistinguishable from that of a periodic chain where the coupling is given by
the appropriately defined average value of the two Fibonacci couplings. Like
the periodic chain, the IDOS has a van Hove singularity, $IDOS\sim\sqrt{E}$,
for small $E$. This behavior seems to indicate at first glance that the long
wavelength Goldstone modes are robust with respect to the quasiperiodic
modulation. However Luck and Petritis presented a rigorous argument to show
that the spectrum does not have any absolutely continuous component, even for
frequency tending to zero.

The gap labeling theorem is seen to hold, as expected, and the letters A, B
and C indicate three important gaps having gap labels given by the three
smallest Fibonacci numbers Luck and Petritis (1986). The IDOS at these
plateaux are given by $H_{k}=1-\omega^{k}$ for $k=1,2$ and 3 where $\omega$ is
the inverse of the golden mean. Luck and Petritis showed that the upper edge
of the spectrum is described by a new 6-cycle
$a\rightarrow-b\rightarrow-a\rightarrow b\rightarrow-a\rightarrow-b$. This
observation was then used to show that, close to the upper edge, the quantity
$1-N(E)$ where $N$ is the IDOS has a power law modulated by log-periodic
oscillations.

Figure 40: Plot of integrated phonon density of states (IDOS) $H(z)$, versus
the eigenvalues $z$ of the Laplacian operator on the Fibonacci chain for a
coupling ratio 0.5 (from Luck and Petritis (1986) reproduced with permission).
The three main plateaux are labeled A, B and C. The dashed line represents the
IDOS for the periodic system based on the average lattice,showing the
similarity of the two curves for small $z$.

Ashraff and Stinchcombe Ashraff and Stinchcombe (1989) computed the dynamic
structure factor for the Fibonacci chain. Quantum diffusion properties in this
model have been studied Kohmoto and Banavar (1986); Lifshitz and Dar Mandel
(2011) and found to share electronic properties of multifractal structure and
log periodic oscillations. For a more detailed discussion on phonon modes in
Fibonacci quasicrystals we refer the reader to the corresponding chapter in
Janssen  _et al._ (2018).

###  IX.2 Mixed Fibonacci models

The term mixed models is used to denote a general member of the family of
models Eq.12, where diagonal and off-diagonal quasiperiodic modulations are
both present. These are relevant for experiments, for example, as real systems
can be expected to have both forms of quasiperiodicity.

Many of the techniques, including the powerful transfer matrix analysis can be
extended to mixed models. Macia and Dominguez-Adame Macia and Dominguez-Adame
(1996); Maciá and Domínguez-Adame (1997) considered a mixed model in which the
A and B sites have onsite energies of $\alpha$ or $\beta$, following a
Fibonacci sequence. The hopping amplitudes are assumed to have two possible
values, $t_{AB}=t_{BA}$ and $t_{AA}=\gamma t_{AB}$. The initial step consists
of defining the basic transfer matrices. Choosing, without loss of generality,
units such that $\beta=-\alpha$ and $t_{AB}=1$ one obtains four different
transfer matrices in this model as follows :

$\displaystyle X$ $\displaystyle=$
$\displaystyle\begin{bmatrix}(E+\alpha)&-1\\\
1&0\end{bmatrix},Y=\begin{bmatrix}(E-\alpha)/\gamma&-1/\gamma\\\
1&0\end{bmatrix},$ $\displaystyle W$ $\displaystyle=$
$\displaystyle\begin{bmatrix}(E-\alpha)&-\gamma\\\
1&0\end{bmatrix},Z=\begin{bmatrix}(E-\alpha)&-1\\\ 1&0\end{bmatrix}$ (76)

Macia and Dominguez-Adame showed that after renormalization the global
transfer matrix in this model has a structure identical to the one for the
Fibonacci sequence for the diagonal model, Eq.22. This can be seen by defining
blocks of sites via $T_{A}=ZYX$ and $T_{B}=WX$. They showed that in finite
chains the energies of certain transparent states i.e. with transmission
coefficients of unity are of the form

$\displaystyle E(k)=\pm\sqrt{\alpha^{2}+4\cos^{2}(k\pi/N)}$ (77)

with $|\alpha|\leq 2$ and $k$ an integer such that $N\tau=k\pi$ with
$k=0,1,...$. The states with such energies $E(k)$ were confirmed as being
extended states via a multifractal analysis.

For this set of models one can determine the resonance energy $E_{*}$ – for
which the two elementary transfer matrices of the chain commute – as a
function of the parameters. Fig.41 adapted from Maciá (2017) shows the
transmission coefficient $T_{N}(E_{*})$ plotted as a function of resonance
energy $E_{*}$ versus the hopping amplitude $\gamma$ with fixed diagonal term
strength $1/2$ (both in units of $t_{AB}$) for three different models. The
dashed line represents results for the Fibonacci model for $N=41$ atoms, while
the blue and purple lines correspond to silver mean quasicrystal and the
period doubling cases respectively. One sees that transparent states occur
throughout the spectrum.

Figure 41: Transmission coefficients at resonance energies $E_{*}$ plotted
versus $\gamma$ (see text for definitions) for several mixed models. The
Fibonacci case is shown by the dashed line. The purple and blue curves
represent the silver mean and period doubling models respectively (figure
adapted with permission from Maciá (2017))

Sire and Mosseri have investigated a model for approximant chains in which the
hopping takes two values, $t_{A}$ or $t_{B}$, according to the Fibonacci
sequence Sire and Mosseri (1990). The onsite potentials whose values depend on
the nature of bonds to the left and right, were taken to be
$V_{AA}=-\lambda/2$ and $V_{AB}=V_{AB}=\lambda/2$. The model thus has two
parameters $\lambda$ and $\rho$. By considering $n$th generation approximant
chains, Sire and Mosseri showed that there are gap closings and quasi-extended
states for two different families of solutions. In particular, they showed
that as $n\rightarrow\infty$ there are Bloch-type extended wavefunctions that
can be described in terms of a wave-vector $k\in(0,\pi)$. The energies of such
states are distributed throughout the band, with the band edges corresponding
to $k=0$ and $k=\pi$. Similar conclusions as to the existence of such extended
Bloch-type states were reached by Kumar and Ananthakrishna Kumar and
Ananthakrishna (1987); Kumar (2017). Extended states with periodic envelopes
may exist in mixed systems can exist even for certain disordered cases, as
reported in Huang and Gong (1998).

###  IX.3 Interference and flux dependent phenomena

Transmission properties of chains of Aharonov-Bohm rings of two different
sizes, and connected in a Fibonacci sequence, have been studied Chakrabarti
_et al._ (2003). The transport properties now become flux dependent. It was
observed that transmission decreases as a power law in the number of rings and
that there are resonant states for specific flux values. An RG analysis using
the Landauer formalism and the trace map method shows that the transmission
coefficient possesses a self-similar structure Nomata and Horie (2007).

##  X Other quasiperiodic chains

Related models of particular interest include a class of aperiodic 1D chains
which can be obtained by generalizing the substitution rules we have
introduced in Sec.II.1. Higher dimensional lattices are briefly described.

###  X.1  Aperiodic substitutional chains

Although we have focused on a single quasiperiodic system, described by the
golden mean $\tau$, many of the methods used are generalizable to other
irrationals. The nature of the irrational number (algebraic or not) is of
primary importance for the geometric properties and in consequence for the
electronic properties as well. As we have said, quasicrystals are a special
class of structures based on Pisot numbers. The so-called metallic means which
are solutions of the equation $x^{2}-nx-1=0$ (n=1,2,..) belong in this
category. Of this series, the two best studied members are the gold (n=1) and
silver (n=2) mean quasicrystals, the latter also called the octonacci chain.
These chains which have inflation properties and electronic structures
analogous to those of the Fibonacci chain are reviewed in Yuan  _et al._
(2000); Thiem and Schreiber (2012, 2013, 2011). A study of the multifractal
exponents for the central $E=0$ state for metallic mean chains is done in Mace
_et al._ (2017). Energy spectra of generalized Fibonacci-type quasilattices
having self-similar as well as quasiperiodic structure were studied in Fu  _et
al._ (1997). A gap labeling theorem is shown to exist in these cases.

Figure 42: Schemas of 2D product lattices hosting two different types of
hopping models – (left) direct product Hamiltonian with $t_{A}$ (long
bonds)and $t_{B}$ (short bonds) (right) Labyrinth model (hopping along one of
the diagonals of each plaquette) (from Thiem and Schreiber (2013) reproduced
with permission)

Some well-known aperiodic, but not quasiperiodic, systems which can be
obtained by substitution rules are the Thue-Morse, the period-doubling and
Rudin-Shapiro sequences. We refer readers to Maciá (2005) for a discussion of
their electronic properties.

###  X.2 Products of chains

The 1D chain can be used as the basis for extensions to arbitrary dimensions
$d$. Taking $d=2$, for example, the direct product $C_{n}\times C_{n}$ of two
Fibonacci approximants aligned along $x$ and $y$ axes forms a 2D lattice of
squares and rectangles Lifshitz (2002). As can be seen in Fig.42a) its
connectivity is that of the square lattice. The energy spectrum and
wavefunctions for tight-binding models on these direct product lattices have
been studied. In the pure hopping vertex model, electrons can hop along the
two directions with amplitudes $t_{A}$ or $t_{B}$. The Hamiltonian is
separable into two independent Fibonacci chain problems. The energies are the
sum of two 1D energies $E_{ij}=E_{i}+E_{j}$, and the corresponding
wavefunctions given by the product $\psi_{ij}(x,y)=\psi_{i}(x)\psi_{j}(y)$
where $E_{i}$ and $\psi_{i}$ are solutions to the 1D problem. The properties
of the spectrum depend on the value of $\rho$. The spectrum of the $d=2$
product lattice is purely singular continuous for $\rho<\rho_{1}$ where
$\rho_{1}\approx 0.6$ in the product lattice. For $\rho>\rho_{1}$, the
spectrum has a continuous part Sire (1989); Mandel and Lifshitz (2008); Thiem
and Schreiber (2013).

The labyrinth model Sire  _et al._ (1989), a 2D variant based on the direct
product of chains (see Fig.42b), also has properties derivable from the 1D
solutions. The generalized dimensions describing multifractal properties of
wavefunctions in $d$ dimensional product lattices was investigated in Yuan
_et al._ (2000); Thiem and Schreiber (2011). The exponents for $d$-dimensions
are simply proportional to the 1D exponents, $D^{\psi,d}_{q}=dD^{\psi,1}_{q}$.
Dynamical exponents have been computed for these lattices Zhong and Mosseri
(1995); Thiem and Schreiber (2012). The return probability exponent shows a
$d$ dependence, $\gamma^{d}=d\gamma^{(1)}$. The diffusion exponent $\beta$ is
expected according to theory to be constant as the dimensionality $d$
increases and this is indeed found numerically, as can be seen in Fig.28. The
autocorrelation function exponent $\delta$ depends on the dimensionality, and
for higher $d$ increases faster as a function of the modulation strength
parameter $w=t_{A}/t_{B}$ as can be seen in Fig.43. A complete account of the
electronic properties of such $d$-dimensional generalizations is given in
Thiem and Schreiber (2013).

Figure 43: Plot of the autocorrelation function exponent $\delta$ as a
function of the strength of quasiperiodic modulation (the variable
$w=t_{A}/t_{B}$ corresponds to $\rho$ of our text) for product lattices of
dimensions 1,2 and 3 (from Thiem and Schreiber (2013) reproduced with
permission)

$d$-dimensional tilings The spatial connectivity of the product lattice
systems is simple in that they have an underlying average periodic structure:
$d$-dimensional hypercubic lattices. The electronic properties of such product
lattices are seen to be “inherited” from the parent chains. The situation is
different for other quasiperiodic tilings. For 2D and 3D tilings, such as the
Penrose tiling, spectra and wavefunctions remain difficult to compute
analytically, with the exception of the ground state Mace  _et al._ (2017). As
for the Fibonacci quasicrystal, nontrivial topological properties are to be
expected in these higher dimensional cases. The possibility of higher order
topological insulators based on the Penrose and octagonal tilings has been
discussed, for example, in Fulga  _et al._ (2016); Chen  _et al._ (2020).

##  XI Interactions and quasiperiodicity

The topic of interacting quasiperiodic systems requires a separate review.
This section will be restricted to giving brief outlines of some of the main
contributions, along with a non-exhaustive list of references.

The effects of quasiperiodic perturbations in interacting fermionic chains was
investigated by Vidal et al for continuum models using renormalization group
in Vidal  _et al._ (1999, 2001). Considering in particular the case of
metallic mean chains, they found that there was a metal-insulator transition
Vidal  _et al._ (2001) for repulsive interactions. Hiramoto Hiramoto (1990)
did a Hartree-Fock analysis to study the effect of weak interaction U. The
study showed that the singular continuous single particle spectrum persists in
the presence of interactions, in contrast to the critical Harper model where
the singular continuous behavior is destroyed by U.

In a study of the Hubbard model on a Fibonacci chain by weak-coupling
renormalization group and density matrix renormalization group methods Hida
(2001), Hida showed that, for the diagonal Fibonacci model, weak Coulomb
repulsion is irrelevant in the sense of RG and the system will behave as a
free Fibonacci chain. For strong Coulomb repulsion the system becomes a Mott
insulator and, in the spin sector, can be modeled in terms of a uniform
Heisenberg antiferromagnetic chain. For the off-diagonal case, he obtained a
Mott insulator with a low energy sector that could be described in terms of a
Fibonacci antiferromagnetic Heisenberg chain. Gupta et al Gupta  _et al._
(2005) studied the DC electrical conductivity for half filling, using Hartree-
Fock mean field theory, to see the interplay of interactions and
quasiperiodicity. They concluded that, while each of these factors taken
individually tend to $decrease$ the conductivity, there may be enhancement of
the conductivity due to competition between them.

The evolution of multifractality in an interacting fermion chain was studied
in Macé  _et al._ (2019). Contrary to naive expectations, these authors found
that adding repulsive interactions did not lead to enhanced delocalization.
Fig.44 shows the half chain von Neumann entropy plotted against time for
different strengths of the quasiperiodic potential (controlled by a parameter
$h$, with $h=0$ being the periodic case, and $h=1$ the strongly quasiperiodic
chain). For a periodic chain, (black curve) the entanglement entropy grows as
a power-law in the time, $S(t)\sim t^{1/z}$. The exponent $z$ increases as the
strength of the quasiperiodicity $h$ is increased. This further confirms that
the free Fibonacci chain is intermediate between the free delocalized state
and an Anderson localized state, from the point of view of its transport
properties. One also sees log-periodic oscillations superposed on the power-
law behavior, especially visible for the yellow curve (strong quasiperiodicity
limit).

Figure 44: Entanglement entropy as a function of time for different values of
the strength of quasiperiodic modulation $h$ (black curve: periodic chain,
yellow (light grey) chain: strongly quasiperiodic chain). The inset shows the
dependence of the power $z$ on $h$ (reproduced with permission from Macé  _et
al._ (2019))

###  XI.1 Heisenberg and XY chains

The properties of spin chains with Fibonacci couplings have been investigated
by a number of methods. Hida analyzed the spin 1/2 Heisenberg model using
density matrix renormalization group finding that quasiperiodic modulations
are relevant in this case and that the ground state is in a different
universality class from that of the XY chain (or free particle) problem
Hermisson (2000); Hida (1999). The entanglement entropy $S$ of aperiodic
critical chains was studied by Igloi et al Iglói  _et al._ (2007). For these
chains, the half chain entanglement increases as
$S\sim\frac{c}{3}\ln_{2}L+cst$, where $L$ is the chain length and $c$ is the
central charge (=0.5 for the periodic case). They found that for Fibonacci XY
chains the quasiperiodic modulation is marginal, in that the central charge
$c$ in this case is non-universal and depends on $\rho=J_{A}/J_{B}$ (the ratio
of spin-spin couplings). For Fibonacci Heisenberg chains, the quasiperiodicity
is strongly relevant and the prefactor is given by $c(0)\approx 0.8$.

###  XI.2 Many body localization

The question of many body localization due to quasiperiodic potentials was
raised in Iyer  _et al._ (2013); Khemani  _et al._ (2017). These studies asked
whether the MBL transitions are different in the presence of quasiperiodic
potentials compared to random potentials, and if so in what ways. The MBL
transition in the quasiperiodic AAH model, which can be experimentally
realized in interacting boson and fermions cold atoms systems, has been
studied, and shown to lead to a new type of “non-random” universality class
Khemani  _et al._ (2017). It is interesting to ask whether there are any
significant differences between many body localization in AAH and in Fibonacci
chains. Details of the transition have been discussed in Macé  _et al._
(2019); Varma and Žnidarič (2019). Density-density correlations at infinite
temperature have been investigated using the dynamical quantum typicality
approach. This allows to study the evolution of dynamical properties of the
Fibonacci model as interaction strength is ramped up, and the crossover to a
MBL state.

###  XI.3 Anomalous diffusion properties

Settino et al Settino  _et al._ (2020) have studied dynamics in interacting
aperiodic many body systems including the FC . They showed that, for the
onsite Fibonacci model, the singular continuous spectrum for the non-
interacting problem remains and induces an anomalous dynamics. Lo Gullo et al
Lo Gullo  _et al._ (2017) studied aperiodic discrete time quantum walk
problems, relevant in quantum computing, and which could be realized using
optical fibers Nguyen  _et al._ (2020). They computed the energy spectra and
the spreading of an initially localized wave packet for different cases,
finding that in the case of Fibonacci and Thue-Morse chains the system is
superdiffusive, whereas for Rudin-Shapiro, another substitutional chain, it is
strongly subdiffusive. They propose that the different dynamics are linked to
the nature of the spectra in the two cases – singular continuous for the
former, discrete for the latter.

##  XII Experimental systems

Fibonacci sequences occur $naturally$ in 3D icosahedral quasicrystals and also
in dodecagonal quasicrystals. These are structures which are based on the
golden mean. Fig.45a which shows an STM image of copper adatoms deposited on
an icosahedral AlPdMn quasicrystal McGrath  _et al._ (2012) provides a
striking illustration of this connection. As the height profile in Fig.45b
shows, the distances between columns is a Fibonacci sequence. These rows of
aperiodically spaced layers are coupled to the bulk, and the resulting
Hamiltonians are likely to be fairly complicated. For experimental
investigations of the 1D model, it is therefore useful to fabricate artificial
systems in order to study the Fibonacci chain, and we will now describe a few
such systems.

Figure 45: a) STM image of a $100\times 100$A square zone of copper atoms
deposited on the 5-fold surface of i-AlPdMn b) height profile between points
marked X showing an alternating sequence of distances L(7.3 A) and S(4.5A)
(figure reproduced from McGrath  _et al._ (2012)

The off-diagonal tight-binding Fibonacci model has been experimentally
realized in a polaritonic gas in a quasi-1D cavity Tanese  _et al._ (2014);
Baboux  _et al._ (2017). Some of the theoretical predictions for the energies
and the eigenmodes of this system were observed. The discrete scale invariance
of the spectrum and the gap labeling theorem were thus experimentally
verified. The existence of gap states was checked, and their spatial structure
mapped out. Most interestingly, their topological winding numbers could be
experimentally measured by varying the phason angle $\phi$.

Optical waveguides fabricated by using a femtosecond laser beam to inscribe
quasiperiodic spatial modulation in a bulk glass have been studied in Kraus
_et al._ (2012). Injecting light into these allowed to study propagating and
localized modes, and to demonstrate the topologically protected edge modes
predicted by theory. Topological pumping of photons was demonstrated in Verbin
_et al._ (2015), where the topological equivalence between the Harper and the
Fibonacci models was experimentally verified. Related subjects of recent
investigations, outside the scope of this review, are higher order topological
insulators using quasicrystals Chen  _et al._ (2020) and topological quantum
computation using Fibonacci anyon chains Feiguin  _et al._ (2007); Chandran
_et al._ (2020).

Merlin et alMerlin  _et al._ (1985); Bajema and Merlin (1987) fabricated
semiconductor superlattices following a Fibonacci sequence and studied their
properties using Raman spectroscopy. The samples were composed of two types of
films : 27 nm-thick layers of GaAs and 43 nm-thick layers of GaAlAs (such that
the ratio of thicknesses is close to the golden mean). These layers were piled
on top of each other in a quasiperiodic sequence along the $z$ direction. The
Raman frequency shifts were compared for periodic and for quasiperiodic
structures. In addition to the acoustic phonons in this system, plasmon-
polariton modes are argued to play an important role in Albuquerque and Cottam
(2003), where their Raman cross-sections for the samples are discussed.
Hawrylak et al Hawrylak  _et al._ (1987) have computed the plasmon spectrum
for such superlattices, described its scaling properties and explicitly
computed the $f(\alpha)$ spectrum. For a wide-ranging discussion of photonic
and phononic heterostructures see the review by Steurer and Sutter-Widmer
(2007).

A variety of photonic structures can be obtained by, for example, coupling
single mode waveguides to form lattices, through dynamical coupling or using
nonlinearities. It can be useful to consider these in terms of effective
models in a higher dimensional “synthetic space” in which the external
parameters play a role of extra dimensions. These systems offer in particular
a means to investigate the topological properties of the Su-Schreiffer-Heeger
(SSH) and the AAH models Yuan  _et al._ (2018), and leave open the possibility
of extensions including the Fibonacci chain.

Metamaterials in the nanoscale also provide a wide array of possibilities for
aperiodic structures. Optical transmission spectra of photonic band-gap
Fibonacci quasiperiodic nanostructures composed of both positive (SiO2) and
negative refractive index materials are discussed by de Medeiros et al de
Medeiros  _et al._ (2007). In another direction, there are possibilities to
make quasiperiodic sequences in 1D biomaterial (DNA-based) systems and study
the consequences for transport Albuquerque  _et al._ (2005).

Magnetic multilayers composed of Fe/Cr layers and studied by MOKE (magneto-
optic Kerr effect) and FMR (ferromagnetic resonance) should display self-
similar magnetization versus curves and interesting thermodynamic signatures
Bezerra  _et al._ (2001). Another promising system is composed of epitaxially
grown layers of Fe and Au using ultra-high-vacuum vapor deposition Suwa  _et
al._ (2017) which were theoretically predicted to have anomalous
magnetoresistance Machado  _et al._ (2012).

Fibonacci nanowire arrays have been studied recently by Lisiecki et al who
discuss their magnonic properties and interest for possible applications
Lisiecki  _et al._ (2019).

Quantum dots can be used to make artificial crystals and quasicrystals
Kouwenhoven  _et al._ (1990). They have already been used to study
magnetotransport in a periodic crystal. It may therefore be possible to study
transport in artificial Fibonacci chains made with quantum dots.

On a macroscopic length scale, microwave propagation in dielectric resonators
have been used with success to simulate the tight-binding model for graphene
Bellec  _et al._ (2013). Preliminary work 777F. Piéchon and F. Mortessagne,
private communication. shows that this may provide an extremely versatile
system in which to study electronic properties of Fibonacci chains, including
effects of various forms of disorder or interactions.

Cold atoms in optical potentials constitute a particularly fertile ground to
realize quasiperiodic models and study their properties under controlled
conditions. A number of recent theoretical studies have thus looked at
generalizations of the tight-binding models that are relevant to cold atom
experiments. The discrete-valued Fibonacci potential is more difficult to
realize experimentally, compared to the Harper model, which can be realized by
applying an incommensurate laser potential Lye  _et al._ (2007); Fallani  _et
al._ (2007). Recently, however, Singh et al Singh  _et al._ (2015) proposed a
means of realizing generalized Fibonacci models on chains based on the cut-
and-project method, in a 2D optical lattice. If realized, this would provide
opportunities to study experimentally multifractal states and probe the
multiscale dynamics in the Fibonacci quasicrystal. An interesting direction
discussed in Sagi and Nussinov (2016) concerns emergent quasiperiodic
structures in interacting systems. The authors show that fractional quantum
Hall systems for irrational filling fractions can result in quasiperiodic
electronic configurations which include, in 1D, the case of an emergent
Fibonacci ordering. The authors argue that these could be realized with
ultracold Rydberg atoms on optical lattices. Such structures, although not
expected to be stable under disorder or quantum/thermal fluctuations, could
nevertheless persist on intermediate length scales in the form of
“quasicrystalline puddles”.

##  XIII Summary and Outlook

We have introduced some of the main concepts and techniques relevant to the 1D
Fibonacci tight-binding Hamiltonians. This class of model is important from a
fundamental viewpoint in its own right, but also as a starting point for
understanding higher dimensional quasicrystals. The fascinating topological
characteristics of the 1D models arising from their “hidden dimension” can be
generalized to higher dimensional quasicrystals. Thus, just as the Fibonacci
chain has topological properties inherited from a parent 2D system, similarly,
certain 2D quasicrystals have been shown to have topological invariants
corresponding to a 4D quantum Hall system Kraus  _et al._ (2013) with
associated edge modes which are symmetry protected.

Interpolating between the AAH and Fibonacci models offers a possibility of
realizing topological pumps, as in the optical waveguide systems of Verbin
_et al._ (2015). More recently it was shown that polaritonic quasicrystals,
also allow this type of tunability Goblot  _et al._ (2020). For these systems
one can control localization-delocalization transitions for specific bands,
suggesting the possibility of their use in selective band-pass filters.

One of the distinctive characteristics of the quasicrystal is the existence of
multifractalities as a function of the energy, of the space coordinates and of
temporal correlations. We have discussed these properties, along with explicit
calculations for specific examples. We have mentioned a few consequences of
these multifractal states for physical properties : transport, disorder
induced localization/delocalization and the proximity effect. Critical states
persist in higher dimensional models. There exists notably an exact solution
for ground states which is a 2D analogue of the $E=0$ solution on the FC that
was discussed here Kalugin and Katz (2014); Mace  _et al._ (2017).

The Fibonacci family of models is interesting from the point of view of
applications as well : in electronic devices or for their mechanical
properties. Although we focused here on electronic properties, there are many
interesting and closely related problems for electromagnetic wave modes in
aperiodic media. The unique optical reflectivity properties of aperiodic
multilayers suggest applications as perfect mirrors with omnidirectional
reflectivity for all polarizations of incident light over a wide range of
wavelengths Axel and Peyrière (2010), and more generally in nanodevices
Steurer and Sutter-Widmer (2007); Macia (2012). The experimental possibilities
to create phononic systems suggest their use in thermal and acoustic shields,
or in acoustic lenses, for example. In superconductors, the critical currents
versus field of quasiperiodically spaced pinned vortices were computed by
Misko  _et al._ (2006) with a view to applications, and it was observed that
for a 1D Fibonacci vortex array, the critical currents have a self-similar
structure. More speculatively, one can conceive of applications on larger
scale suggested by some of the properties discussed here. The large number of
spectral gaps generically present in this family of models suggests the
possibility of dissipating ocean waves and even reflecting tsunamis on the
shoreline using quasiperiodic arrays of scatterers. Seismic barriers against
propagating Rayleigh waves (surface seismic modes) using, for example
quasiperiodic trenches, are another interesting possibility.

The models discussed here hopefully already provide a useful framework for
understanding 1D quasiperiodic systems. Many questions still remain, however.
The multifractal dynamical properties of quasiperiodic chains remain to be
elucidated in more detail both theoretically and by experiment. The effects of
interactions and their interplay with quasiperiodicity is an important problem
requiring more investigations. The extension to higher dimensional
quasicrystals is a major challenge. As far as understanding and controlling
physical properties of quasicrystals is concerned, we have only scratched the
surface.

###### Acknowledgements.

I gratefully acknowledge many valuable discussions over the years with Michel
Duneau, Pavel Kalugin, Jean-Marc Luck, Nicolas Macé and Frédéric Pièchon.

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