Electronic states of a disordered 2Dquasiperiodic tiling from critical states to Anderson localization Anuradha Jagannathan1and Marco Tarzia2 3

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Electronic states of a disordered 2Dquasiperiodic tiling: from critical
states to Anderson localization
Anuradha Jagannathan1and Marco Tarzia2, 3
1Laboratoire de Physique des Solides, Bât.510, Université Paris-Saclay, 91405 Orsay, France
2Laboratoire Théorique de la Matière Condensée,
Sorbonne Université, 75252 Paris Cédex 05, France
3Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France
(Dated: February 2, 2023)
We consider critical eigenstates in a two dimensional quasicrystal and their evolution as a function of
disorder. By exact diagonalization of finite size systems we show that the evolution of properties of a typi-
cal wave-function is non-monotonic. That is, disorder leads to states delocalizing, until a certain crossover
disorder strength is attained, after which they start to localize. Although this non-monotonic behavior is
only present in finite-size systems and vanishes in the thermodynamic limit, the crossover disorder strength
decreases logarithmically slowly with system size, and is quite large even for very large approximants. The
non-monotonic evolution of spatial properties of eigenstates can be observed in the anomalous dimensions
of the wave-function amplitudes, in their multifractal spectra, and in their dynamical properties. We com-
pute the two-point correlation functions of wave-function amplitudes and show that these follow power
laws in distance and energy, consistent with the idea that wave-functions retain their multifractal structure
on a scale which depends on disorder strength. Dynamical properties are studied as a function of disorder.
We find that the diffusion exponents do not reflect the non-monotonic wave-function evolution. Instead,
they are essentially independent of disorder until disorder increases beyond the crossover value, after which
they decrease rapidly, until the strong localization regime is reached. The differences between our results
and earlier studies on geometrically disordered “phason-flip” models lead us to propose that the two models
are in different universality classes. We conclude by discussing some implications of our results for transport
and a proposal for a Mott hopping mechanism between power law localized wave-functions, in moderately
disordered quasicrystals.
arXiv:2210.01762v2 [cond-mat.dis-nn] 1 Feb 2023
2
I. INTRODUCTION
The electronic properties of quasicrystals have continued to pose fundamental problems ever since the dis-
covery of these materials by Schechtman et al [1]. The question of what constitutes “intrinsic” behavior of
quasicrystals, and how it is affected by disorder is not yet clearly understood. The reasons for this deficit of
understanding lie in the fact that the complexity of real materials – both chemical and structural – render
numerical computations difficult, and that analytical tools lack for even the simplest two or three dimensional
cases. Many numerical studies have been carried out for simplified tight-binding models on quasiperiodic
tilings. These indicate that single particle electronic states are, typically, multifractal for undisordered tilings
(see review in [2]). The generalized dimensions can in fact be exactly computed for the ground states of two
well-known tilings [3]. However, it is not understood what happens to such multifractal states in the pres-
ence of disorder. The nature of electronic states in real quasicrystals, where a certain amount of disorder is
inevitably present, has remained an open question. While the transition to Anderson localization has been
studied in detail for a 1D quasicrystal, the Fibonacci chain [4, 5], the effects of disorder on eigenstates have
not been analyzed in detail for 2D and 3D quasicrystals. This paper fills the gap by presenting a comprehensive
numerical study of the effects of disorder on eigenstates of an eight-fold symmetric 2D tiling, the Ammann-
Beenker (AB) tiling [6]. A patch of this tiling, composed of squares and 45◦rhombuses is shown in Fig. 1,
along with a typical critical state, represented by colored circles whose radii are proportional to the amplitude
on each site. We will address here in detail the question of how such multifractal or critical states evolve as a
function of disorder into strongly Anderson localized states.
In the study of disordered quasiperiodic tilings the extensive literature on a variety of disordered systems
provides a valuable guide. Random matrix models [7]are important reference systems which predict certain
universal properties. In particular, as we will describe, the Gaussian orthogonal ensemble (GOE) which has
the symmetry of the Hamiltonians we will consider here describes some spectral properties of tilings for both
the pure case and for weakly disordered tilings of finite size. The best known reference system for condensed
matter is the Anderson model for localization in random lattices [8]. In the Anderson model, it is well-known
that a metal-insulator transition occurs for three dimensional lattices, in contrast to 2D or 1D lattices, where
any disorder, however small, suffices to localize all of the eigenstates [9]. These statements hold in an infinite
system. However, if one considers a finite system of linear dimension L, one can distinguish between two
regimes of disorder – a weak disorder regime, where states appear to be extended on the scale of the system
size, and a strong disorder regime, where the states are clearly localized. The crossover occurs for a disorder
strength W?which corresponds to the localization length ξloc(W?)∼L. Since in 2D the localization length
diverges exponentially as W→0, the apparent metallic-like regime at weak disorder extends up to very large
system sizes.
We show here that, within the weak disorder regime, most states display a non-monotonic behavior as a
function of the disorder strength W. That is, states tend at first to delocalize for very small W. Localization
occurs only when the disorder exceeds a crossover value W?, which depends on the energy and system size.
The non-monotonic evolution of the localization properties only exists for finite systems, since the value of
the cross-over disorder strength vanishes in the thermodynamic limit. However the characteristic disorder W?
decreases logarithmically with the system size and hence the non-monotonicity is observed also for very large
approximants. The non-monotonic behavior can be seen by plotting the inverse participation ratio (IPR) as a
function of disorder, but also in the f(α)curve, which describes multifractal properties of wave-functions, and
in the generalized dimensions of the wave functions Dψ
q. This non-monotonicity is particularly pronounced for
the states around zero energy, in the middle of the spectrum. This is due to the fact that there is a zero-width
band of localized states at E=0 in this model, analogous to the E=0 states present in the well-known Penrose
rhombus tiling [10, 11]. All of these degenerate states are mixed as soon as disorder is introduced, and they
delocalize rapidly for W6=0. In contrast, the states lying near band edges and the main pseudogap behave
differently. They are the fastest to localize and, upto numerical accuracy, their evolution under disorder is
monotonic.
To complete the description of electronic properties in the disordered quasicrystal, we have carried out
detailed studies of two-point spatial correlation functions C(r,ω;E). These describe correlations of the ampli-
tudes of two eigenstates of energies close to E, having an energy separation ω, as a function of spatial distance
r. We obtain these correlation functions for increasing values of disorder, and show that for a large range of
disorder strength across the crossover regime, correlations follow power laws over a broad range of distances.
3
FIG. 1. A patch of the Ammann-Beenker tiling illustrating a critical wave-function (for model Eq. 1 without disorder and
E=0.24t). The magnitude and sign of the wave-function amplitude on each site are represented by the circle radius and
color respectively.
In the crossover regime, the wave function correlations are seen furthermore to be enhanced by disorder.
Dynamical properties are then studied as a function of disorder strength. We consider the spreading of
wavepackets which are initially localized on clusters composed of a central site of connectivity k(3 ≤k≤8
for the AB tiling) and all of its nearest neighbors. The initial wavepacket energy is varied by a parameter θ,
which controls the initial wave function amplitudes on the center of the clusters and on the external nodes.
We study the probability of return to the origin (cluster center) of such wavepackets released at time t=0,
P(t), which is closely related to the Fourier transform of the correlation function C(0, ω;E)(computed for
amplitudes at the same spatial point, r=0), and is found likewise to exhibit different regimes of behaviors as
a function of t. After an initial fast decay for very short times, P(t)shows a power law decay in a wide range
of tand ultimately reaches a constant value at very long times. The onset of this power law regime moves
to shorter times with increasing disorder strength. The addition of quenched disorder has therefore the effect
of promoting the spreading of the wave packet in the crossover regime. This effect is particularly strong for
small k. We next computed the mean square distance, given by the average over all initial positions and over
disorder realizations of 〈r2(t)〉 ∼ t2β. We measured the effective diffusion exponents β(which depend on k
and θ) as a function of disorder strength. When disorder is turned on βas a function of Wappears initially to
remain constant. It then drops quickly to zero when the localization regime is reached for sufficiently strong
W¦W?.
A number of previous studies have investigated the effects of geometrically disordered tilings. These models
modify the quasiperiodic structure by making local permutations of the tiles at random locations, termed
“phason-flips”. Such models were studied both in 2D [12–16]and 3D [17]. These studies of spectral and
quantum diffusion properties concluded that phason disorder tends to delocalize the electronic states and
promote quantum diffusion. The conductivity in finite samples of the 2D Penrose tiling was computed using
Kubo formalism, and found to increase with a type of phason disorder [18]. Grimm and Roemer carried out a
recent study of spectral properties and found that both the pure and the phason-disordered tilings obey GOE
statistics, implying absence of localization [19]. The spectral properties of phason-disordered models agree
qualitatively with those of our model, in the weak disorder limit. However the diffusion exponent βin our
model does not show an increase for weak disorder, in contrast to the results reported in Ref. [13]. This may
imply that the effects of random on-site energies on quantum diffusion and transport are different from those
of phason disorder. For phason disordered models no localized phase has been observed, contrarily to the
onsite disorder model. It is an open question as to whether one can achieve an Anderson localized phase by
phason disorder alone.
As stated at the outset, one of the motivations for this study is to understand electronic properties of qua-
4
sicrystals, in particular, transport. Experimental measurements of electrical conductivity of quasicrystals show
that they are strikingly poor conductors compared to their constituent metals. One of the contributing factors
is that many quasicrystals display a strongly reduced density of states (DOS) at the Fermi level compared to
the parent materials – the “pseudogap”. The second factor has to do with the nature of the eigenstates, and
the resulting anomalous diffusion in quasicrystals. Weak localization theory and electron-electron interactions
theory have been invoked to explain the roughly linear increase of conductivity with temperature observed in
very high structural quality icosahedral quasicrystals such as i-AlCuFe [20–22]. In contrast, however, the low
temperature transport in i-AlCuRe samples obtained by fast-quench has given rise to much debate. On the
one hand, their DOS is very similar to that of samples obtained by slow cooling [23, 24]. Yet, there are huge
differences in transport – and the former have insulating behavior, as confirmed by recent transport measure-
ment which reported Mott variable range hopping (VRH) between localized states at low temperatures. The
answer evidently lies in the disorder strength and differences of the resulting electronic states in the two sets
of samples. Based on our study, we propose that for the more disordered samples it may be possible to observe
power law conductivity due to hopping between disorder-modified critical states. In addition there could occur
a crossover to the usual Mott exponential dependence for localized states at very low temperature Tco.
The paper is organized as follows: in Sec. II the model is introduced, and the DOS is shown for different
values of disorder. Sec. III presents the disorder dependence of the inverse participation-ratio of the eigenstates,
which show the non-monotonic behavior. In Sec. IV we focus on the anomalous dimensions of wave-functions’
amplitudes and on the multifractal analysis of the singularity spectrum. Sec. V discusses two-points correlation
functions. Sec. VI considers dynamical properties: the probability of return to the origin and the power law
spreading of wave packets. Sec. VII presents a discussion of transport based on the results. Sec. VIII ends with
a discussion and conclusions.
II. THE MODEL AND ITS DENSITY OF STATES
The tight-binding model we study is described by the Hamiltonian
H=X
i
εic†
ici+X
〈i,j〉
(ti j c†
icj+h.c.)(1)
where i=1, ..., Nare site indices, and 〈i,j〉denotes pairs of sites that are linked by an edge (see Fig. 1). Spin
indices are not written as they play no role aside from introducing a factor 2. The systems considered are square
approximants – tilings which are periodic but resemble the infinite quasiperiodic tiling in their local geometry
– generated from a 4D hypercubic lattice by the cut-and-project method [25]. In addition to the eight-fold
symmetry the tiling possesses a discrete scale invariance – a tiling of edge length acan be transformed into a
tiling of edge lengths which are smaller by a scale factor λ=1+p2 and vice-versa (called inflation/deflation
transformations, see [6]). In our computations, square approximants of total number of sites Nequal to 239,
1393, 8119 and 47321 were considered. Periodic boundary conditions were imposed in both directions.
In the pure limit, all the onsite energies are taken to be equal εi=ε0, and all hopping amplitudes to be
equal ti j =t. Without loss of generality, the origin of the energy is chosen such that ε0=0, and energy units
are chosen so that t=1. Since the tiling is bipartite (made from quadrilaterals) the energy spectrum of this
model is symmetric. Thus all plots are shown henceforth for the positive half of the spectrum only. The scale
invariance of the tiling has been used in renormalization group schemes for electronic states [27]and in the
calculation of ground state properties of an antiferromagnetic Heisenberg model [28]. Moreover, since the
AB tiling can be generated from a parent 4-dimensional cubic lattice, the Hamiltonian Eq. (1) can be related
to a four dimensional Quantum Hall system [29], and inherits certain topological properties which will not
however concern us here.
The DOS of the pure quasiperiodic tiling has a characteristic spiky shape (see Fig. 2 for W=0). The spiky
local maxima and minima of the DOS are consequences of the symmetries of the Hamiltonian Eq. (1), due to
the properties of the underlying structure such as scale invariance and repetitivity of environments.1Among
1The repetitivity property says that local configurations are guaranteed to repeat throughout the tiling with a maximal and a minimal
allowed spacing, and generalizes the notion of strict translational invariance present in periodic crystals.
5
the sharp minima or “pseudogaps”, the most prominent pair is located near E≈ ±1.95. The pseudogaps
are expected to be located at special values of the integrated density of states (IDOS) given by a gap labeling
scheme [26], with the main gaps corresponding to IDOS values that approach the values λ−2and its symmetric
1−λ−2as the system size increases.
In the pure model, additionally, one finds a delta-function peak of the DOS exactly at E=0, corresponding
to a macroscopic number of“confined” states. Similar E=0 states can be found for other bipartite tilings
including the well-known Penrose tiling [10, 11]. In the AB tiling, the smallest such E=0 state is a small ring,
with non-zero amplitudes on the 8 nearest neighbors of sites with a coordination number 8. The degeneracy
of this type of confined state is given by Nλ−4as N→ ∞. Other confined states can be similarly enumerated.
The spatial features of the set of confined states (which can overlap) and their degeneracies are discussed in
[30–32]. The support of these confined states is a finite fraction of sites of the tiling. They are unstable with
respect to most forms of disorder.
Adding disorder – by varying the parameters in (1), for example, or by modifying the geometry by ran-
dom phason flips – breaks the symmetries of H, and singularities of the DOS are progressively smoothed out.
Roughly speaking, disorder broadens the Bragg peaks of the structure, thus reducing quantum interference
due to backscattering.
A number of early studies considered the evolution of spectral properties under phason flip disorder via
the analysis of level statistics (see the review [33]). Grimm and Roemer have [19]returned recently to the
problem, performing a careful analysis of the r-value (ratio of the gaps of consecutive levels) statistics [34].
They reach the conclusion that states in both the pure and the phason-disordered tilings were described by
the GOE ensemble. In other words, phason disorder did not induce localization of states (in which case one
would have found not GOE but Poissonian statistics). From this, it seems clear that the phason disorder does
not suffice to drive the system into the strong localization regime, at least, for the system sizes considered.
In this work, instead, disorder is specifically introduced by assuming the onsite energies to be random vari-
ables taken from a box distribution of width W(i.e. εiare independent and identically distributed random
variables uniformly taken in the interval [−W/2, W/2]). We expect that the addition of quenched disorder
also in the ti j yields the same results. Differently from phason disorder, out model (1) allows one to tune
the disorder from very weak disorder (reproducing effects seen for the phason-flip disordered model) out to
arbitrarily high disorder, when all states are strongly localized.
Results for the DOS of pure and disordered tilings obtained from exact diagonalizations of Hare presented
in Fig. 2. Ensemble averages are performed over several independent realizations of the disorder, using, here
and throughout the rest of the paper, Mcopies of the system with M≈64 for N=47321, M≈256 for
N=8119, M≈8192 for N=1393, and M≈2·104for N=239.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 2 4 6 8 10
ρ(E)
E
W=0
W=0.4
W=1
W=2
W=4
W=8
W=16
FIG. 2. Density of states ρ(E)for several values of the disorder and for N=47321.
摘要：

Electronicstatesofadisordered2Dquasiperiodictiling:fromcriticalstatestoAndersonlocalizationAnuradhaJagannathan1andMarcoTarzia2,31LaboratoiredePhysiquedesSolides,Bât.510,UniversitéParis-Saclay,91405Orsay,France2LaboratoireThéoriquedelaMatièreCondensée,SorbonneUniversité,75252ParisCédex05,France3Insti...

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