# Anderson transitions in disordered two-dimensional lattices

Dayasindhu Dey<EMAIL_ADDRESS>S. N. Bose National Centre for
Basic Sciences, Block - JD, Sector - III, Salt Lake, Kolkata - 700098, India
Manoranjan Kumar<EMAIL_ADDRESS>S. N. Bose National Centre for
Basic Sciences, Block - JD, Sector - III, Salt Lake, Kolkata - 700098, India
Pragya Shukla  Department of Physics, Indian Institute of Technology
Kharagpur, Kharagpur - 721302, India.

###### Abstract

We numerically analyze the energy level statistics of the Anderson model with
Gaussian site disorder and constant hopping. The model is realized on
different two-dimensional lattices, namely, the honeycomb, the kagomé, the
square, and the triangular lattice. By calculating the well-known statistical
measures viz., nearest neighbor spacing distribution, number variance, the
partition number and the dc electrical conductivity from Kubo-Greenwood
formula, we show that there is clearly a delocalization to localization
transition with increasing disorder. Though the statistics in different
lattice systems differs when compared with respect to the change in the
disorder strength only, we find there exists a single complexity parameter, a
function of the disorder strength, coordination number, localization length,
and the local mean level spacing, in terms of which the statistics of the
fluctuations matches for all lattice systems at least when the Fermi energy is
selected from the bulk of the energy levels.

###### pacs:

71.23.An, 72.15.Rn, 72.80.Ng, 05.60.Gg

##  I Introduction

The effect of impurities in the low dimensional systems such as Graphene
Novoselov  _et al._ (2004); Geim and Novoselov (2007); Geim (2009); Geng  _et
al._ (2012); Bardarson  _et al._ (2007), nano flakes González-Santander  _et
al._ (2013), metallic thin films Biswas  _et al._ (2014); Liao  _et al._
(2015),arrays of quantum dots etc., has been an intense area of research in
the last decade. Some of these low dimensional structures have potential
applications in electronic devices, because of their finite size and the
performance under varying degree of disorder and dimensionality Liao  _et al._
(2015); Lu  _et al._ (2011). In disordered systems delocalization to
localization transition is one of the intersting phenomenon, therefore to
caracterize this transition, analysis of the electronic wavefunction is
necessary. The extended to localized transition, also known as the ‘Anderson
transition’, and its dependence on various system parameters such as,
disorder, dimensionality, lattice structure, system size etc., in the finite
systems are a frontier area of research. This motivates us to seek an
estimation of the critical disorder for the transition in finite size lattices
with different geometry.

In 1958, Anderson suggested that an electron inside a material can be fully
localized in the presence of a large disorder Anderson (1958), whereas Edwards
et al. showed that a transition from an extended to a localized states in the
square lattice can occur at a disorder strength of 5 or 6 times higher than
the band width of the Anderson model Hamiltonian Edwards and Thouless (1972).
In 1979, Abrahams et al. conjectured, based on a scaling hypothesis, that the
electronic states are localized in less than three dimensional (3D) systems in
the presence of any amount of disorder Abrahams  _et al._ (1979); Lee and
Ramakrishnan (1985). The scaling hypothesis was later on supported by many
studies of the Anderson model on square lattices MacKinnon and Kramer (1981);
*mackinnon83; *mackinnon85; *pichard81a; *pichard81b. Further, Altshuler et
al. showed that weak electron interaction enhances the localization in these
systems, whereas under strong electron interaction, 2D electrons behave like a
Wigner crystal and, as shown by Tanatar et al. Altshuler  _et al._ (1980),
even a small amount of disorder can make the system insulating at zero
temperature. The theoretical analysis Abrahams  _et al._ (2001) suggested that
under strong electron interactions and small disordered regime, 2D systems can
be conducting. Most of these studies are done for infinite systems and square
lattices, except some of the recent studies on the honeycomb lattices
Ostrovsky  _et al._ (2007); *ryu2007; *nomura2007; *san-jose2007;
*lewenkopf2008; *beenakker2007. A detailed review on the Anderson transition
is given in the Ref. Evers and Mirlin, 2008.

The study of Anderson transition is not only important for material sciences
but also relevant to understand the influence of the wavefunction dynamics on
the physical properties of the disordered systems. The presence of disorder
and/or interaction leads to a randomization of the Hamiltonian, resulting in a
random matrix representation in a physically suitable basis e.g., site basis.
The structure of the matrix e.g., degree of sparsity is sensitive to various
system conditions viz., dimensionality, shape, size, and boundary conditions.
The statistical behaviour of the system can therefore be analysed by an
ensemble of the disordered Hamiltonians. Such analysis has been a subject of
extensive study during the past decade. It is now well-known that, in the weak
disorder regime, the statistics can be well-modelled by the Wigner-Dyson
universality classes of random matrix ensembles which correspond to extended,
featureless eigenfunctions and a strong level-repulsion with statistics of the
eigenvalues and the eigenfunction independent from each other. Increasing the
disorder in finite size systems causes the statistics to crossover from
Wigner-Dyson to the Poisson universality class (with no level-repulsion and
fully localized eigenfunctions in strong disorder limit). The statistics in
the intermediate regime e.g., near critical disorder is sensitive to the
degree of eigenfunction localization which in turn is expected to depend on
the system conditions besides disorder. The study Mirlin (2000); Evers and
Mirlin (2008) however, showed that the statistics can be well-modeled by the
single parametric power-law random banded matrix (PRBM) ensembles. Another
theoretical study Shukla (2005a) later on indicated the application of a wide
range of random matrix ensembles (besides PRBM) as the model for the
intermediate statistics; this study was based on the common mathematical
formulation of the energy level statistics of a broad class of random matrix
ensembles (with varying degree of sparsity and disorder but same symmetry
class). The formulation is governed by a single parameter, referred as the
complexity parameter, a function of all system parameters including energy
range of interest and therefore different ensembles are expected to show
analogous statistics if their complexity parameters are same Shukla (2005a).
As the theoretical claim about the existence of a complexity parameter is in
clear agreement with the single parameter scaling conjecture of Ref. Abrahams
_et al._ , 1979, it is highly desirable to seek its numerical validity in
disordered systems.

Figure 1: (Color online) Two dimensional lattice structures considered in this
paper: (a) the honeycomb lattice with coordination number 3, (b) the kagomé
and (c) the square lattices both having 4 nearest neighbors and (d) the
triangular lattice with coordination number 6

In this paper, we study the transitions from extended to localized state of
Anderson Hamiltonian on finite 2D systems with different geometry in presence
of the disorder. As shown in Fig. 1, we consider four lattice geometries,
namely, square, triangular, honeycomb and kagomé lattice which differ from
each other in the coordination number and their bond connectivity. The
coordination numbers of honeycomb, square, kagomé and triangular, lattice are
three, four, four, and six respectively. We note that the square and the
kagomé both have four nearest neighbour but their spectrum is completely
different because of the bond connectivity. The paper is organized as follows.
In section II, we introduce the model disordered Hamiltonian; here the theory
leading to single parameter based formulation of the spectral statistics is
also briefly reviewed. The section III briefly describes the statistical
measures used in our numerical analysis. The section IV presents the details
of the numerical techniques as well as results; this section is divided into
four sub-sections one for each lattice system. The comparison of the
statistics of physical parameters with respect to the single complexity
parameter for various lattices is given in section V. In section VI, results
are discussed and compared with the existing literature.

##  II  The Model

The Hamiltonian: We consider the standard Anderson model Anderson (1958) under
tight binding approximation realized on four different 2D lattice systems. The
tight binding model Hamiltonian is well suited for a metallic system, where
electron-electron interactions are screened. The Hamiltonian can be written in
a single particle basis as

$H=\sum_{i}\epsilon_{i}|i\rangle\langle i|+\sum_{\langle
i,j\rangle}t_{ij}|i\rangle\langle j|,$ (1)

where $\epsilon_{i}$, and $t_{ij}$ are random site energies and the nearest
neighbour hopping energies. In this work, the sites energies are considered
Gaussian distributed for each lattice system and the hopping energies are kept
constant: $t_{ij}=-1.0$. The Gaussian type site disorder has been widely
studied for the square lattice system and has a close analogy with real thin
film materials.

The ensemble: The presence of the disorder in the system makes it inevitable
to consider an ensemble of the Hamiltonians. For the Hamiltonian in Eq.(1)
with on-site random energies, the probability density $\rho(H)$ of the
ensemble including all possibilities can be written in general as

$\displaystyle\rho(H)$ $\displaystyle=$ $\displaystyle C\,{\rm
exp}\left[{-\sum_{k}{1\over 2h_{kk}}(H_{kk}-b_{kk})^{2}}\right]$ (2)
$\displaystyle\times\prod_{k,l;k\not=l}\delta(H_{kl}-f_{kl}),$

where $C$ is the normalization constant, $f_{kl}=t$ if $\\{k,l\\}$ correspond
to nearest neighbour sites and otherwise $f_{kl}=0$. Writing the delta
function as a limiting Gaussian ($\delta(x)=\lim_{v\rightarrow
0}{1\over\sqrt{2\pi v^{2}}}\;{\rm e}^{-x^{2}/2v^{2}}$), reduces Eq.(2) to the
following form

$\rho(H,h,b)=C\exp\left[\sum_{k\leq
l}\frac{1}{h_{kl}}(H_{kl}-b_{kl})^{2}\right],$ (3)

where $h$ is the set of all variances $h_{kl}=\langle
H_{kl}^{2}\rangle-\langle H_{kl}\rangle^{2}$, and $b$ is the set of all mean
values $b_{kl}=\langle H_{kl}\rangle$. Here, for constant hopping between
nearest neighbor sites and an onsite Gaussian disorder $W$ in Eq.(1), one has

$\displaystyle h_{kk}$ $\displaystyle=$ $\displaystyle W^{2}/12,\qquad
b_{kk}=0,$ (4) $\displaystyle h_{kl}$ $\displaystyle=$ $\displaystyle 0,\qquad
b_{kl}=-f(kl)t,$ (5)

where $f(kl)=1$ for $\\{k,l\\}$ pairs representing nearest neighbors and zero
otherwise. The statistical behavior of the Hamiltonian $H$, Eq.(1), can now be
analyzed using the ensemble (3). An important aspect of Eq.(3) is that it can
represent the ensemble density for a wider class of lattice systems under
different conditions e.g., random, anisotropic, variable range hopping,
dimensionality and boundary conditions; the only constraint on these system is
to preserve the time-reversal symmetry which allows $H$ to be real-symmetric.
Its form is therefore appropriate for the verification of single parameter
scaling conjecture.

Single parameter formulation: Before proceeding to numerical analysis, it is
helpful to briefly review the complexity parametric formulation of the
spectral statistics. The variation of system conditions in general result in a
variation of the distribution parameters $h$ and $b$ and therefore an
evolution of the $H$-ensemble. As discussed in Ref. Shukla, 2005a, b, under a
change of parameters $h_{kl}\to h_{kl}+\delta h_{kl}$ and $b_{kl}\to
b_{kl}+\delta b_{kl}$, $\rho(H)$ undergoes a diffusion dynamics along with a
finite drift, and, using Gaussian nature of $\rho(H)$, it can exactly be shown
that

$T\rho=L\rho,$ (6)

where $T$ is a combination of parametric derivatives, and $L$ is a diffusion
operator in matrix space and are given by

$T=\sum_{k\leq l}\left[(g_{kl}-2h_{kl})\frac{\partial}{\partial
h_{kl}}-\,b_{kl}\,\frac{\partial}{\partial b_{kl}}\right]$ (7)

and

$L=\sum_{kl}\frac{\partial}{\partial
H_{kl}}\left[\frac{g_{kl}}{2}\frac{\partial}{\partial H_{kl}}+\,H_{kl}\right]$
(8)

with $g_{kl}=1+\delta_{kl}$. A suitable transformation of parametric space
maps $T$ to a single parametric derivative,
$T\rho=\frac{\partial\rho}{\partial Y}$, which in turn reduces Eq. (6) to a
single parametric diffusion equation Shukla (2005a, b)

$\frac{\partial\rho}{\partial Y}=L\rho.$ (9)

Here

$Y=\frac{-1}{N^{2}}\sum_{k\leq l}\left[\;{\rm
ln}\left|1-(2-\delta_{kl})h_{kl}\right|\;+\;{\rm
ln}\left|b_{kl}+\delta_{b0}\right|^{2}\right]+C_{y}$ (10)

with $C_{y}$ as an arbitrary constant and $\delta_{b0}=1$ if $b_{kl}=0$ else
$\delta_{b0}=0$. For the ensemble described by the set of parameters in
Eq.(5), this leads to

$\displaystyle Y=\frac{-1}{N}\;{\rm
ln}\left[\left|1-\frac{W^{2}}{12}\right|.\left|t+\delta_{t0}\right|^{z/2}\right]+C_{y},$
(11)

where $z$ is the coordination number of the lattices.

The joint probability distribution of the eigenvalues $P(\\{E_{n}\\})\equiv
P(E_{1},\ldots,E_{N})$ for a metal (fully extended eigenfunctions limit) is
given by the Wigner-Dyson distribution,
$P(\\{E_{n}\\})=\prod_{i<j}|E_{i}-E_{j}|\exp(-\frac{1}{2}\sum_{k}E_{k}^{2})$.
The eigenvalues are uncorrelated in insulator limit, thus implying
$P(\\{E_{n}\\})=\prod_{i<j}P_{n}(E_{n})$ Shklovskii  _et al._ (1993). The
distribution for the intermediate states of localization can be derived by
integrating $\rho$ over the associated eigenvector space. As shown in Ref.
Shukla, 2005a, an integration of Eq.(9) leads to a single parametric diffusion
of the eigenvalue distribution of the ensemble (3)

$\frac{\partial P}{\partial Y}=\sum_{n}\frac{\partial}{\partial
E_{n}}\left[\frac{\partial}{\partial E_{n}}+\sum_{m\neq
n}\frac{1}{E_{m}-E_{n}}+E_{n}\right]P.$ (12)

The above equation can be used to obtain the ensemble averaged level density
as well as its local fluctuations Shukla (2005a). As discussed in Ref. Pandey,
1995, while the diffusion of the average level density is governed by $Y$, the
diffusion of its fluctuations occurs at a scale determined by
$(Y-Y_{0})\sim\Delta_{l}^{2}$ where $Y_{0}$ is the value of $Y$ at the
beginning of evolution and $\Delta_{l}$ is the local mean level spacing:

$\displaystyle\Delta_{local}(E)\approx L^{d}\;\xi^{-d}\;\Delta(E)$ (13)

with $\xi$ as the localization length and $d$ as the dimension (here $d=2$).
The statistics other than mean level density is therefore governed by a
rescaled parameter $\Lambda(E)$:

$\Lambda(E)=\frac{|Y-Y_{0}|}{\Delta_{l}^{2}}.$ (14)

The solution of Eq.(12) at $\Lambda\to\infty$ corresponds to the Wigner-Dyson
statistics. The $\Lambda\to 0$ limit correspond to the distribution at the
initial state of the evolution; for an insulator initial state, the spectral
statistics reduces to the statistics of uncorrelated energy levels.

The transition parameter $\Lambda$ is in general a function of various
parameters e.g., disorder, system size, dimensionality, energy range of
interest, lattice topology. Although both $Y$ and $\Delta_{l}$ contribute to
the system dependence of $\Lambda$, the crucial influence comes from
$\Delta_{l}$ due to its dependence on the localization length $\xi$. For
finite system sizes $N$, a variation of system conditions e.g., disorder leads
to a smooth crossover of statistics between the stationary limits $\Lambda\to
0$ and $\Lambda\to\infty$. In infinite size limit, the statistics abruptly
changes from one stationary limit to the other. If however, the limit
$\Lambda^{*}=\lim_{N\rightarrow\infty}\Lambda(N)$ exists, the corresponding
statistics would belong to a universality class different from the two
stationary limits. The existence of $\Lambda^{*}$ therefore is a criteria for
the existence of critical spectral statistics Shukla (2005a).

##  III Fluctuation measures: the definitions

Our main objectives in this paper is to study the influence of the system
conditions on the statistical behavior and identify the critical regime. For
this purpose, we consider four different fluctuation measures namely, density
of states (DOS), reduced partition number ($P/L$), the peak position of the
NNSD and the dc electrical conductivity which can briefly be described as
follows.

Spectral measures: A well-known measure to analyse the short range
correlations among energy levels is the nearest neighbour spacing distribution
(NNSD) which describes the probability $P(s)$ of two nearest neighbour energy
levels to be found at a distance $s$ measured in the units of the mean level
spacing around the desired energy regime. For the weak disorder regime with
extended eigenfunctions, $P(s)$ is given by the Wigner surmise

$P_{W}(s)=\frac{\pi}{2}s\exp\left(-\frac{\pi}{4}\;s^{2}\right).$ (15)

For the opposite limit of strong disorder with fully localized wavefunctions,
$P(s)$ follows Poisson distribution

$P_{P}(s)=\exp\left(-s\right).$ (16)

To compare the level spacing distribution for the entire transition within an
arbitrary energy regime for different lattices, we use a traditional measure,
namely, the cumulative nearest neighbor spacing distribution $\eta_{i}$ which
depends on the tail behaviour of the nearest neighbour spacing distribution
and is defined as

$\eta_{i}=\frac{\int_{s_{i}}^{\infty}P(s,\Lambda)ds-\int_{s_{i}}^{\infty}P_{W}(s)ds}{\int_{s_{i}}^{\infty}P_{P}(s)ds-\int_{s_{i}}^{\infty}P_{W}(s)ds},$
(17)

where $s_{i}$, $i=1,2$ refer to the two crossing points of $P_{W}(s)$ and
$P_{P}(s)$: $s_{1}=0.473$ and $s_{2}=2.002$ Shklovskii  _et al._ (1993);
Berkovits and Avishai (1998); Cuevas (1999). As the system makes a transition
from delocalized to localized state, $\eta_{i}$ changes from 0 to 1. While
NNSD gives the short range correlations of the energy levels, there is another
measure which gives the long range correlations, namely, the number variance.
It is defined as the variance of the number $n$ of unfolded energy levels in
an energy interval $r$ centered at the energy regime of interest i.e., the
number variance $\Sigma^{2}(E,r)=\langle(n(E,r)-\langle
n(E,r)\rangle)^{2}\rangle$.

Participation number: The dependence of $\Lambda$ on the localization length
$\xi$ through $\Delta_{l}$ results in sensitivity of the energy level
statistics to eigenfunction behaviour. This motivates to analyse a standard
measure for the eigenfunction localization, namely, the reduced participation
number, referred here as $P/L$, which characterize the spread of
eigenfunctions in the site basis. The partition number $P$ for a wavefunction
$\psi_{n}$ is defined as $P^{-1}=\sum_{i}^{N}|\psi_{in}|^{4}$ and is
proportional to the localization length.

DC conductivity: The localization of electronic wave-function can be
characterized by the DC electrical conductivity $\sigma$ which can be measured
also experimentally for real systems. Here we numerically calculate $\sigma$
using Kubo-Greenwood formula (see Ref. Niizeki, 1979 for details)

$\sigma(E_{F})=\frac{2\pi\hbar}{\Omega}{\rm{Tr}}\left[\bm{J}\delta(E_{F}\bm{I}-\bm{H})\bm{J}\delta(E_{F}\bm{I}-\bm{H})\right]$
(18)

where $E_{F}$ is the Fermi energy, $\bm{H}$ is the Hamiltonian, $\Omega$ is
the volume of the system and $\bm{J}$ is the one electron current operator

$\vec{\bm{J}}=\frac{ieV}{\hbar}\sum_{\langle
i,j\rangle}(\bm{R}_{i}-\bm{R}_{j})(|i\rangle\langle j|-|j\rangle\langle i|)$
(19)

with $\bm{R}_{i}$ as the position vector of site $i$.

##  IV  Numerical analysis and Results

To study the influence of system conditions on the statistical behavior, we
apply exact diagonalization technique to numerically obtain the eigenvalues
and the eigenfunctions of the Hamiltonian (1) for four different lattice
types. For each lattice type, an ensemble of approximately $1000$ realizations
is considered to attain statistical accuracy. To explore the size-dependence
of the statistics and its sensitivity to the energy range, many system sizes,
varying from $L=40$ to $L=100$, are analysed for each lattice type and for two
energy ranges. The latter correspond to two filling of the Hamiltonian:
firstly, the half-filling which is a more natural choice of the systems like
graphene, gold etc., and secondly, a filling where the density of states is
significantly high, making a rigorous statistical analysis of energy levels
more feasible. For the local fluctuations analysis, it is necessary to first
rescale the energy eigenvalues by the local mean level spacing Haake (2010)
(also known as unfolding of the levels). The average density of states for
this purpose is calculated using the binning method. To analyse the local
spectral fluctuations in the desired energy-range, approximately $\sim 3\%$ of
the rescaled energy levels are taken around the chosen Fermi energy. The
rescaled energy levels and normalized eigenfunctions are then used to
calculate the density of states (DOS), reduced partition number ($P/L$), the
peak position of the NNSD and the dc electrical conductivity. The results are
arranged in order of increase in the coordination numbers in the subsections
A, B, C and D containing honeycomb, kagomé, square and triangular lattices
respectively. Here we describe three different methods of calculation in
detail for the honeycomb lattice only; the calculations for other lattice
systems are carried out following similar methods.

###  IV.1 Honeycomb lattice

The 2D honeycomb structure is one of the most interesting lattice types which
occurs in many systems of industrial importance e.g., graphene. In this case,
the non-interacting electronic models like uniform Anderson model or tight
binding model have two unique Dirac cones in a brillouin zone (BZ). The DOS
$\rho(\epsilon)$ for a 2D honeycomb lattice in the clean limit (i.e. absence
of disorder) is given by

$\rho(\epsilon)=-\frac{1}{\pi}\text{Im}\left[\lim_{\Delta\epsilon\to
0^{+}}\int_{1^{\text{st}}\text{BZ}}d{\bf{k}}\frac{V}{V^{2}-|\mu|^{2}t^{2}}\right],$
(20)

where $V=\epsilon+i\Delta\epsilon$ and
$\mu=\exp(ik_{x}a)+\exp[i(-k_{x}a/2+k_{y}\sqrt{3}a/2)]+\exp[i(-k_{x}a/2-k_{y}\sqrt{3}a/2)]$
with $k_{x}$ and $k_{y}$ as the components of the wave vector McKinnon and
Choy (1993); $\rho(\epsilon)$ in this case vanishes at $\epsilon=\pm 3t$. In
half-filled limit, the system shows a gapless state and a linear dispersion
relation which can be mapped to that of a massless Dirac state. In real
materials, the vacancies can lead to coupling of cones but coupling strength
remains very small, the two cones being separated by a large momentum vector
Suzuura and Ando (2002). The microscopic calculation for the two valley
Hamiltonian indicates a crossover from anti-localization to weak localization
Suzuura and Ando (2002). A recent experiment also confirms the anti-
localization and weak localization of states at low and high carrier
concentration respectively (for defects like charge impurity etc.).

To observe the effect of disorder, the DOS $\rho(\epsilon)$ is calculated
numerically for different disorders; the results are shown in Fig. 2. In the
clean system ($W=0$), $\rho(\epsilon)$ is vanishingly small near energy
$\epsilon=0$ (the Fermi-energy at half filling) and has van-Hove singularities
at $\epsilon=\pm t$ . The effect of varying disorder on $\rho(\epsilon)$ at
$\epsilon=-1.5t$ (i.e., at the bulk of DOS) and at $\epsilon=0$ (i.e., at the
half filling) is shown in the inset of Fig. 2. At the half filled energy
regime, $\rho(\epsilon)$ increases till $W=7.0t$ and then decrease
exponentially as shown in the inset of the Fig. 2. In large $W$ limit, DOS
becomes flat as shown in the Fig. 2. As clear from the Fig. 2, the disorder
has a significant impact on the DOS of this lattice type in the energy regime
near $\epsilon=0$ (with DOS showing a prominent dip for small disorder) and
$\epsilon=\pm 1.5t$. For the fluctuations analysis we therefore, choose Fermi
energy from these two regimes, namely, $\epsilon=0$ (half filling) and
$\epsilon=-1.5t$ (bulk).

Figure 2: (Color online) The DOS of the honeycomb lattice. The solid line
shows the DOS in the absence of disorder. Three disorders strength are chosen
around the critical disorder (see table 1). The DOS at very large disorder is
also shown. The inset shows the variation of DOS with disorder at two energy
regimes. Figure 3: (Color online) The partition number per unit length with
disorder for different system sizes are shown at (a) $E_{F}=-1.5t$ and (b)
$E_{F}=0$. The linear fits of the curves cut the disorder axis at $W_{x}$. The
insets of both (a) and (b) show the critical disorder $W_{x}$ with $1/L$ which
are fitted with the lines: $W_{x}=3.13+68.6/L$ for inset (a) and
$W_{x}=2.64+72.6/L$ for inset (b). Figure 4: (Color online) The peak-position
of the NNSD with the inverse of the disorder (a) at $E_{F}=-1.5t$ and (b) at
$E_{F}=0$. The lines in the main part of (a) and (b) are the fitted curves
with the function $0.77\tanh((W^{*}/W)^{2})$. The critical disorder for
various system sizes are depicted in insets (i) and (iii). The insets (ii) and
(iv) show the NNSD at different disorders for the system size $L=100$ Figure
5: (Color online) The dc conductivity with disorder at (a) $\epsilon=-1.5t$
and (b) $\epsilon=0$. The upper part in (b) is fitted with:
$\ln\sigma=0.99-1.84\ln W$ and the lower part is fitted with
$\ln\sigma=1.577-2.265\ln W$ which gives $W_{c}=3.9t$.

Our next step is to seek the critical disorder for the extended to localized
state transition in the honeycomb lattice by three different routes. While the
transition in principle takes place in the thermodynamics limit
$L\rightarrow\infty$ at a critical disorder (or critical energy), the finite
systems undergo a smooth crossover within a critical regime (around critical
disorder or energy). It is therefore imperative to analyze the critical
disorder $W_{crit}(L)$ for many finite system sizes, with exact critical
disorder given by $\lim_{L\rightarrow\infty}W_{crit}(L)=W_{crit}$. For this
purpose, we first analyse the disorder dependence of the reduced participation
number $P/L$. Fig. 3 elucidates the behavior of the reduced partition number
at two Fermi energies at (a) the bulk and (b) at the half filling
respectively. For each case $P/L$ varies linearly with the disorder $W$ in the
small disorder regime while it varies exponentially for strong disorders. The
linear portion of the curves are fitted with lines ($P/L=a-bW$) which cut the
disorder axis at $W_{x}=a/b$. For $W>W_{x}$, the localization length of the
systems is less than the system size. At $W=W_{x}$, the localization length
become equal to the system size $L$, thus suggesting $W_{x}$ as the finite
size critical disorder (more clearly $W^{pn}_{crit}(L)=W_{x}$ with superscript
referring to the method applied) . (Note, for later reference, here we use
different notations for the critical disorder obtained by different methods).
The critical disorder measured from this method for different system sizes of
the honeycomb lattice are given in the table 1. As shown in the insets of Fig.
3(a) and (b), $W_{x}$ for different system sizes vary linearly with the
inverse of the system size. The numerics reveals the sensitivity of $W_{x}$ to
Fermi energy too: for $L=100$, the value of $W_{x}$ is 3.4 at the half filling
and 3.8 at the bulk of the DOS.

Continuing with our quest for critical disorder, our next step is to analyze
the Nearest neighbor level spacing distribution (NNSD) defined in section III.
The decrease of disorder in the finite systems causes the NNSD to crossover
from the Poisson distribution (eq.(16)) to the Wigner surmise (eq.(15)). To
analyse the NNSD for different disorder strengths, the energy eigenvalues are
apriori unfolded using the local mean level spacing. The numerically obtained
NNSD for a given disorder and system size is compared with the Brody
distribution Brody  _et al._ (1981):

$P_{B}(S)=\alpha(1+\omega)S^{\omega}\exp(-bS^{1+\omega})$ (21)

which gives the Brody parameters $\omega$ and $b$ as a function of $W$ and
$L$. The fitted Brody distribution is then used to calculate the peak position
dependence on the disorder strengths $W$ for a fixed size $L$. The dependence
is plotted with respect to inverse of the disorder in Fig. 4 for many system
sizes and at two Fermi energies (the bulk and the half filling). The curve
depicting peak positions with respect to disorder for each $L$ is now fitted
with the function

$\displaystyle F(W)=0.77\tanh((W^{*}/W)^{n})$ (22)

where $n=2$ and $W^{*}$ is the critical disorder strength (in this method) at
which localization occurs for a finite system size $L$ (more clearly
$W^{nnsd}_{crit}(L)=W^{*})$. This type of functional dependence of the peak
position of NNSD was suggested by A J Millis et al. in context of the energy
level statistics of one dimensional spin-1/2 chain to find the critical value
$J_{c}$ indicating integrable and non-integrable boundary Rabson  _et al._
(2004). As shown in the main Fig. 4 (b), there is large deviation in the peak
position-disorder curve from the function $F(w)$ in the small disorder regime
for the half filled case. $F(W)$ with $n=2$ however fits well in small $W$
limit for the case where the Fermi energy is chosen away from the half
filling. As shown in the insets (i) and (iii) of Fig. 4 (a) and (b)
respectively, the crossover disorder $W^{*}$ decreases with system size $L$.
Again $W^{*}$ shows an energy dependence for a fixed $L$: for $L=100$,
$W^{*}=3.9$ and $3.6$ for $\epsilon=0$ and $\epsilon=-1.5t$ respectively (see
table 1). The insets (ii) and (iv) of Fig. 4 (a) and (b) also display the NNSD
behavior for $\epsilon=0$ and $\epsilon=-1.5t$ respectively. We notice that
for $\epsilon=0$ or half filled case the NNSD vary significantly in small
impurity limit but the variation in the bulk limit $\epsilon=-1.5t$ is weak
below $W=W^{*}$. The unusual behaviour can be explained from the rapid change
in the DOS with disorder at the half filling. The half filling case suggests
that the system is still in the ballistic limit for small disorder.

The electronic conductivity is an important characteristic of the localization
to delocalization transition, with high conductivity an indicator of the large
localization length. The dc electrical conductivity for various system sizes
for the two energy ranges are calculated using the Kubo-Greenwood formula. The
minimum conductivity $\sigma_{m}$ of a clean sample at $T=0$ and half-filled
limit is analytically predicted to be $4e^{2}/\pi h$; this is also confirmed
by our analysis. The effect of changing disorder on the behavior of
conductivity ($\sigma$) in honeycomb lattice is displayed in the figure 5 (a)
and (b) at the bulk of DOS and at the half filling respectively. In the bulk
limit, the conductivity decreases with increasing disorder following a power
law dependence with exponents $-1.84$ in the weak disorder regime and $-2.265$
in the strong disorder regime. The intersection of these two lines gives the
crossover disorder $W^{cond.}_{crit}(L)=W_{c}=3.9t$ which is almost in
agreement with the values calculated from the other two methods. The disorder-
dependence of conductivity at the half filling however shows some atypical
behavior: it remains almost constant at very small disorders up to $W=0.4t$,
decreases thereafter up to $W<2.8t$, then increases till $W=5.0$ and decrease
afterwards for further increase in $W$. A possible explanation of this flip-
flop behavior may lie in poor statistics due to small number of states
available at this filling or due to dominant finite size effect. We intend to
probe this behavior with more numerical rigor separately.

$\epsilon$ | $L$ | $W_{x}$ | $W*$ | $W_{c}$
---|---|---|---|---
$-1.5t$ | 40 | 4.9 | 5.2 |
50 | 4.4 | 4.8 |
70 | 4.2 | 4.3 |
100 | 3.8 | 3.9 | 3.9
Table 1: Critical disorders calculated from all three methods for the
honeycomb lattice at two different Fermi energies for four system sizes

Next we compare the crossover values of $W_{crit}(L)$ calculated from three
different methods (i.e $W_{x},W^{*},W_{c}$) for four different $L$. Due to
statistical reliability, the comparison is shown in table 1 (only for the bulk
energy limit). As given in the table 1, $W_{x}$ and $W^{*}$ decrease with an
increase in the system size $L$, varying as $L^{-1}$ (As the size dependence
on the conductivity is negligibly small for this case, we present the data for
$L=100$ case only). This clearly indicates conductivity as a better measure to
find the critical disorder for the transition (due to faster convergence with
$L$). The analysis suggests the critical disorder $W_{crit}\approx 3.8$.

###  IV.2  Kagomé lattice

The DOS $\rho(\epsilon)$ of the Anderson model with uniform site energy for a
kagomé lattice is exactly solvable in the clean limit, and is composed of the
two bands of the honeycomb lattice shifted in energy by $t$ with amplitude
reduced by a factor of 2/3 and a delta peak at $\epsilon=2t$ with weight 1/3.
The DOS diverges at three values of energy (at $\epsilon=0$ and $\pm 2t$) and
is zero for $\epsilon=-t$ for the clean system Hanisch  _et al._ (1995). The
band width in the clean limit of this lattice is spanned from $\epsilon=-4t$
to $\epsilon=2t$. Effect of the onsite disorder on the DOS of the kagomé
lattice Hamiltonian of system size $L^{2}$ with $L\sim 100$ is shown in the
Fig. 6 for three disorder strengths alongwith the clean limit. As clear from
the figure, the singularities in the DOS vanish with the increase in disorder,
with DOS approaching a Gaussian distribution in energy at very large disorder.
The inset in Fig 6 displays the behavior of the DOS at the Fermi energy for
varying disorders which turns out to be an exponential decay in large disorder
limit.

Figure 6: (Color online) The DOS of the kagomé lattice for different
disorders. The solid line is the DOS for the clean system. The inset shows the
DOS at half filling with the disorder. Figure 7: (Color online) The reduced
partition number for different system sizes of the kagomé lattice. The linear
portion of the curves are fitted with lines which cut the disorder axis at
$W_{x}$. The inset shows the critical disorder $W_{x}$ with the inverse of the
system size and which follows the fitted line: $W_{x}=4.0+91.3/L$. Figure 8:
(Color online) The peak-position of the NNSD vs the inverse of the disorder.
The lines in the main figure are fitted curves with
$0.77\tanh((W^{*}/W)^{2})$. The top left inset shows the critical disorder
with the system size which follows: $W^{*}=3.67+108.5/L$. The bottom right
inset shows the NNSD of the kagomé lattice of system size $L\sim 100$ for
different disorders Figure 9: (Color online) DC conductivity of the kagomé
lattice of linear size $\approx$ 100 with disorder. The upper part of the plot
follows: $\ln\sigma=0.938-1.9\ln W$ and the lower part follows:
$\ln\sigma=3.514-3.2\ln W$ and the critical disorder $W_{c}=5.9$.

To determine the critical disorder $W_{crit}$ in this case, we again apply the
three methods mentioned in detail for the honeycomb lattice. The results for
reduced participation number $P/L$ are displayed in Fig. 7 for different
system sizes in the kagomé lattice. As shown in the inset of Fig. 7, the
finite size critical disorder $W_{x}$ calculated from this method varies
linearly with the inverse of the system size. The results for the search of
critical disorder through the NNSD peak position analysis are displayed in
Fig. 8, for four different system sizes of kogomé lattice. Here again the
curve describing peak position-disorder dependence is fitted with the function
$F(W)$ given by Eq.(22) with $n=2$. As shown in the inset of Fig. 8, $W^{*}$
decrease with system size $L$. The behavior of NNSD shown in the inset of Fig.
8 indicates a very small variance in NNSD for $W<W^{*}$.

The disorder dependence of the electronic conductivity for kagomé lattice is
displayed in Fig. 9 (shown here only for $L=100$); it reveals a decreasing
conductivity with increase of disorder, along with two different power law
behaviors in small and large disorder ranges (indicated by two linear regimes
with different slopes in the log $\sigma$-log $W$ plot). For a system size
$L\approx 100$, the crossing point of the two lines lies at $W_{c}\sim 5.9t$.

Table 2 displays the critical disorder $W_{crit}(L)$ calculated from three
different methods for four different system sizes. Similar to the honeycomb
lattice case; the size dependence of $W_{c}$ is not presented as it is
negligibly small for this system also. whereas $W_{x}$ and $W^{*}$ calculated
from participation number and shift in peak height of NNSD is linear with
$L^{-1}$. Similar to the case of honeycomb lattice, here again the
calculations suggest a convergence of $W_{c}$ to its critical value lattice,
once again confirming a weaker sensitivity of the conductivity approach to
finiteness of the kagomé lattice.

$L$ | $W_{x}$ | $W*$ | $W_{c}$
---|---|---|---
40 | 6.2 | 6.4 | 5.9
50 | 5.9 | 5.9
70 | 5.3 | 5.2
100 | 4.9 | 4.8
Table 2: Critical disorders calculated from all three methods for the kagomé
lattice for four system sizes

###  IV.3 Square lattice

The DOS $\rho(\epsilon)$ of an Anderson model with uniform site energy for a
square lattice at the clean limit can be calculated from the band dispersion
relation

$\epsilon({\mathbf{k}})=2t(\cos k_{x}+\cos k_{y}).$ (23)

Figure 10: (Color online) The DOS of the square lattice at different
disorders. The solid line is the DOS for a clean system. The inset shows the
DOS at the half filling with the disorder.

In the clean limit, the DOS has a van Hove singularity at $\epsilon=0$ and
vanishes for $|\epsilon|>4t$. Effect of on-site disorder on the DOS for four
system sizes of square lattices is shown in the Fig. 10. As shown in the inset
of Fig 10 the DOS at half filling decays exponentially with disorder which is
consistent with the Ref. Yu  _et al._ , 2005.

Following the same procedure as mentioned in the case of honeycomb lattice,
here again we analyze the disorder dependence of the reduced partition number
$P/L$, NNSD and DC conductivity for different system sizes but only in bulk
energy regime; the results are shown in Fig. 11, Fig. 12 and Fig. 13
respectively. As clear from the figures, the qualitative behavior in this case
is same as in the bulk energy regimes of honeycomb and kagome lattices; a
quantitative difference however shows up in the fitted line
$W_{x}=5.39+106.8/L$ for $P/L$, with power $n$ for fit $F(W)$ as $3$ (instead
of $2$ as in previous two cases), with $W^{*}=5.38+118.4/L$ and $W_{c}=6.5t$.
The critical disorders $W_{x},W^{*}$ and $W_{c}$ from the three methods are
given in table 3. $W_{x}$, $W^{*}$ and $W_{c}$ for this case show an inverse
linear dependence on the system size.

Figure 11: (Color online) The reduced partition number of the square lattice of different system sizes. The upper part of the curves are fitted with lines which cut the disorder axis at $W_{x}$. The inset shows the critical disorder $W_{x}$ with the inverse of the system size and the fitted line: $W_{x}=5.39+106.8/L$. Figure 12: (Color online) The peak position of the NNSD of the square lattices of various system sizes. The curves in the main plot are fitted with: $0.77\tanh((W^{*}/W)^{3})$. The top left inset $W^{*}$ are plotted with inverse of the system size which follows the line: $W^{*}=5.38+118.4/L$. The bottom right inset depicts the NNSD at different disorder for the square lattice of size $L=100$. Figure 13: (Color online) DC conductivity of the disordered square lattice at the half filling. The upper part is fitted with: $\ln\sigma=1.807-1.723\ln W$ and the lower part is fitted with: $\ln\sigma=5.845-3.879\ln W$ which gives $W_{c}=6.5t$. $L$ | $W_{x}$ | $W^{*}$ | $W_{c}$
---|---|---|---
40 | 8.0 | 8.3 |
50 | 7.6 | 7.8 | 7.6
70 | 6.9 | 7.0 | 7.0
100 | 6.5 | 6.6 | 6.5
Table 3: Critical disorders calculated from all three methods for the square
lattice at half filling for four system sizes

###  IV.4 Triangular lattice

In the clean limit, the DOS $\rho(\epsilon)$ for a triangular lattice can
exactly be obtained from the band dispersion relation

$\epsilon({\mathbf{k}})=-t\left[2\cos(k_{x})+4\cos\left(\frac{k_{x}}{2}\right)\cos\left(\frac{\sqrt{3}}{2}k_{y}\right)\right],$
(24)

where ${\mathbf{k}}$ is confined to the first Brillouin Zone Hanisch  _et al._
(1995). The DOS obtained numerically in the clean limit along with the
presence of disorder is shown in the Fig 14. The analysis indicates a van Hove
singularity in the DOS at energy $E=2t$ and it approaches zero for
$\epsilon<-6t$ and $\epsilon>3t$. The energy dependence of the DOS for four
disorders is shown in the Fig. 14. In the inset of Fig 14, the disorder
dependence of the DOS at a fixed energy (half filled energy) is displayed
which indicates an exponential decrease in DOS with increasing disorder.

Figure 14: (Color online) The DOS of the triangular lattice in presence of
different disorder strengths. The solid line is the DOS for the system without
any disorder. The inset shows the variation of DOS at the half filling with
disorder. Figure 15: (Color online) The reduced partition number of triangular
lattice of different system sizes with disorder. The are the linear fit for
the reduced partition number curve which cut the $W$ axis at $W_{x}$. The
finite size dependence of the critical disorder $W_{x}$ is shown in the inset
which follows the line: $W_{x}=6.88+169.3/L$. Figure 16: (Color online) The
peak position of the NNSD with the inverse disorder for the triangular lattice
of different system sizes. The data is fitted with the function
$0.77\tanh((W^{*}/W)^{2})$. The top left inset shows the variation of $W^{*}$
with system size which follows the fitted line: $W^{*}=7.10+165.8/L$. The
bottom right inset shows the NNSD of the triangular lattice of system size
$L=100$ calculated at the half filling.

The results for the search of critical disorder in triangular lattice by the
three methods (same as mentioned above for previous lattices) are shown in
Fig. 15, Fig. 16 and Fig. 17 respectively. The figures again indicate the same
qualitative behavior as in the bulk energy regimes of honeycomb, kagomé and
square lattices but the quantitative difference shows up in the fitted line
$W_{x}=6.88+169.3/L$ for $P/L$, with power $n$ for fit $F(W)$ as $3$ (same as
in square lattice but different from honeycomb and kagomé), with
$W^{*}=7.10+165.8/L$ and $W_{c}=8.7t$. A comparison of critical disorders
$W_{x},W^{*}$ and $W_{c}$ (for Fermi energy in bulk) from the three methods is
displayed in table 4 which confirms, as for previous three lattice types, an
inverse linear dependence for $W_{x}$, $W^{*}$ and $W_{c}$ in this case too.

Figure 17: (Color online) DC conductivity of the disordered triangular lattice at the half filling. The upper part is fitted with: $\ln\sigma=2.126-1.902\ln W$ and the lower part is fitted with: $\ln\sigma=5.812-3.606\ln W$. The crossing point of these two fitted curves gives $W_{c}=8.7$. $L$ | $W_{x}$ | $W^{*}$ | $W_{c}$
---|---|---|---
40 | 11.1 | 11.2 |
50 | 10.3 | 10.4 | 10.4
70 | 9.2 | 9.5 | 9.4
100 | 8.6 | 8.7 | 8.7
Table 4: Critical disorders calculated from all three methods for the
triangular lattice at half filling for four system sizes

##  V complexity parameter formulation of the transition

As discussed in previous section, the numerical analysis reveals the
qualitative insensitivity of the local fluctuations in physical properties to
system parameters (within a fixed energy range) although a quantitative
dependence is indicated. More clearly, for each lattice type with Fermi energy
in the bulk we observe the following behavior: (i) the reduced particiapation
ratio has a linear/exponential dependence on the disorder in weak/strong
disorder regime, respectively, (ii) the disorder dependence of the peak
positions of NNSD can be described by the function $F(w)$ (with different $n$
values), (iii) the conductivity has a power law dependence on the disorder
with different exponents in weak and strong disorder regime.

Figure 18: (Color online) The variation of $\eta_{1}$ with (a) the rescaled
complexity parameter $\Lambda$ for four different two dimensional lattices and
(b) the disorder $W$. As obvious, the behaviour for different lattices
coincide in terms of $\Lambda$ but not in terms of disorder.

The observed behavior therefore strongly suggests the possibility of a common
mathematical formulation of the statistical properties where system dependence
enters through a single function of all system parameters. The theoretical
steps for such a formulation are briefly reviewed in section II, with the
function referred as the complexity parameter. As defined in Eq. 11, the
function $Y$ is a combination of various system parameters, with explicit
dependence on the disorder strength, hopping and system size. The information
about dimensionality and boundary conditions is implicitly contained in the
sparsity of the matrix $H$ as well matrix element identities and therefore in
the summation in Eq. 10. The necessary rescaling of energy levels for
comparison of the fluctuations however leads to $\Lambda$ (Eq. 14) as the
relevant transition parameter; the rescaling therefore introduces the crucial
dependence on the dimensionality as well as on the Fermi energy. The obvious
relevance of the theoretically obtained single parameter governing the
transition renders its numerical/ experimental verification very desirable.
For this purpose, we consider here three well-known spectral fluctuation
measures namely, the cumulative NNSD $\eta_{1},\eta_{2}$ and the number
variance $\Sigma^{2}(r)$ (defined in section III) of the four lattices and
analyze their evolution in terms of the complexity parameter $\Lambda$ instead
of disorder $W$. The verification in context of the eigenfunction fluctuations
is yet to be carried out and will be reported elsewhere.

Figure 19: (Color online) The variation of $\eta_{2}$ with (a) the rescaled
complexity parameter $\Lambda$ for four different two dimensional lattices and
(b) the disorder $W$. As obvious, the behaviour for different lattices
coincide in terms of $\Lambda$ but not in terms of disorder. Figure 20: (Color
online) The number variance $\Sigma^{2}(r)$ at $r=10$ with (a) the rescaled
complexity parameter $\Lambda$ for four different two dimensional lattices and
(b) the disorder $W$. As obvious, the behaviour for different lattices
coincide in terms of $\Lambda$ but not in terms of disorder.

To calculate $\Lambda$ for our analysis, we use the fact that the localization
length $\xi$ is proportional to the average participation number $P$. Fig. 18,
19 and 20 show the results for $\eta_{1},\eta_{2}$ and $\Sigma^{2}(r)$,
respectively, for four lattices for Fermi energy at $e=0$. As clear from the
part (a) of these figures, $\Lambda$-governed evolution of each of these
measure for all four lattices falls almost on the same curve for entire
crossover from the localization to delocalization; the difference of
connectivity of the lattices does not affect their behavior. Note a disorder
($W$)-dependent evolution of $\eta_{1}$ and $\eta_{2}$ for honeycomb and
kagome lattices is expected to differ from that of square and triangular ones
(as suggested by the observed difference in disorder governed evolution of the
NNSD-peaks of the lattices, with $\eta_{1}$ and $\eta_{2}$ being cumulative
NNSDs); the deviation of the results for four lattices is clearly visible from
part (b) of Fig. 18, 19 and 20). This clearly reveals $\Lambda$ as the
parameter in terms of which the transition in spectral statistics in Anderson
lattices follows a universal route.

##  VI Discussion

From our results it is clear that the statistical behavior of two dimensional
finite size lattices depends on the coordination number, the lattice
connectivity and the system size. The statistics is different for the lattices
with same coordination number with different connectivity which is evident
from the spectral statistics of the square lattice and the kagomé lattice
(both having same coordination number). The spectral averaged density of
states for all the lattices, considered in this paper, show strong disorder
sensitivity in weak disorder regime, and, at least one van-Hove singularity in
clean limit. The position of the singularity is sensitive to the lattice type;
it occurs at energy $\epsilon=0$ for the square lattice, at $\epsilon=2.0t$
for the triangular lattice, at $\epsilon=\pm t$ for the honeycomb lattice and
at $\epsilon=0$, $-2t$ for the kagomé lattice. The density of states for the
honeycomb lattice and the kagomé lattice are related to that of the triangular
lattice Hanisch  _et al._ (1995) and can be written in terms of the DOS of the
triangular lattice in the clean limit. The DOS at $\epsilon=\epsilon_{F}$
varies differently for different lattice systems at small disorder region
whereas they all decay exponentially for strong disorder region. At very large
disorder, the DOS for all the lattice system lead to a Gaussian distribution.

Three methods are used to estimate the critical disorder for delocalization to
localization transition in two dimensional finite size lattices. First, the
reduced partition number ($P/L$) are used to find the critical disorder
$W_{x}$. $P/L$ varies linearly in the weak disorder regime whereas, it varies
exponential in the strong disorder regime. Next, we study the peak position of
the NNSD of each lattice type for four system sizes; which fits with function
$0.77\tanh((W^{*}/W)^{\alpha})$ where $\alpha=3$ for the square and the
triangular lattices whereas, $\alpha=2$ for the honeycomb and the kagomé
lattices. Last, the Kubo-Greenwood dc conductivity is to find the critical
disorder ($W_{c}$) as the conductivity follows two power law decay in the weak
and strong disorder regimes. The critical disorders calculated from all the
methods are in agreement with each other.

We also analyze dependence of the critical disorder on finite size and lattice
structure considering four system sizes for all the cases. Our results
indicate (i) a linear varaition of critcal disorder with $1/L$, (ii) it is
smallest for the honeycomb lattice and largest for the triangular lattice and
increases with the coordination number for a finite lattice of size $L$.
Although the coordination number is same for the kagomé lattice and the square
lattice the critical disorders are different for the two cases. Therefore, the
critical disorder depends not only on the coordination number but also on the
lattice connectivity.

Finally, we compare three fluctuation measures namely, the cumulative NNSD
measures $\eta_{1}$, $\eta_{2}$ and the number variance $\Sigma^{2}(r)$ for
the four lattices and study their evolution with the single complexity
parameter $\Lambda$. The reults confirm the single parameter dependence of the
localization to delocalization transition in Anderson Hamiltonian in context
of the spectral statistics.

##  VII Acknowledgments

M. K. thanks DST for Ramanujan fellowship grant vide No. SERB/F/3290/2013-2014
and DST Nanomission for the CRAY computational facility.

## References

  * Novoselov  _et al._ (2004) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
  * Geim and Novoselov (2007) A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007).
  * Geim (2009) A. K. Geim, Science 324, 1530 (2009).
  * Geng  _et al._ (2012) D. Geng, B. Wu, Y. Guo, L. Huang, Y. Xue, J. Chen, G. Yu, L. Jiang, W. Hu, and Y. Liu, PNAS 109, 7992 (2012).
  * Bardarson  _et al._ (2007) J. H. Bardarson, J. Tworzydło, P. W. Brouwer, and C. W. J. Beenakker, Phys. Rev. Lett. 99, 106801 (2007).
  * González-Santander  _et al._ (2013) C. González-Santander, F. Domínguez-Adame, M. Hilke, and R. A. Römer, EPL (Europhysics Letters) 104, 17012 (2013).
  * Biswas  _et al._ (2014) A. Biswas, K.-S. Kim, and Y. H. Jeong, Journal of Applied Physics 116, 213704 (2014).
  * Liao  _et al._ (2015) J. Liao, Y. Ou, X. Feng, S. Yang, C. Lin, W. Yang, K. Wu, K. He, X. Ma, Q.-K. Xue, and Y. Li, ArXiv e-prints (2015), arXiv:1504.01847 [cond-mat.mes-hall] .
  * Lu  _et al._ (2011) H.-Z. Lu, J. Shi, and S.-Q. Shen, Phys. Rev. Lett. 107, 076801 (2011).
  * Anderson (1958) P. W. Anderson, Phys. Rev. 109, 1492 (1958).
  * Edwards and Thouless (1972) J. T. Edwards and D. J. Thouless, J. Phys. C: Solid State Phys. 5, 807 (1972).
  * Abrahams  _et al._ (1979) E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
  * Lee and Ramakrishnan (1985) P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
  * MacKinnon and Kramer (1981) A. MacKinnon and B. Kramer, Phys. Rev. Lett. 47, 1546 (1981).
  * MacKinnon and Kramer (1983) A. MacKinnon and B. Kramer, Zeitschrift für Physik B Condensed Matter 53, 1 (1983).
  * MacKinnon (1985) A. MacKinnon, Zeitschrift für Physik B Condensed Matter 59, 385 (1985).
  * Pichard and Sarma (1981a) J. L. Pichard and G. Sarma, Journal of Physics C: Solid State Physics 14, L127 (1981a).
  * Pichard and Sarma (1981b) J. L. Pichard and G. Sarma, Journal of Physics C: Solid State Physics 14, L617 (1981b).
  * Altshuler  _et al._ (1980) B. L. Altshuler, A. G. Aronov, and P. A. Lee, Phys. Rev. Lett. 44, 1288 (1980).
  * Abrahams  _et al._ (2001) E. Abrahams, S. V. Kravchenko, and M. P. Sarachik, Rev. Mod. Phys. 73, 251 (2001).
  * Ostrovsky  _et al._ (2007) P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. Lett. 98, 256801 (2007).
  * Ryu  _et al._ (2007) S. Ryu, C. Mudry, H. Obuse, and A. Furusaki, Phys. Rev. Lett. 99, 116601 (2007).
  * Nomura  _et al._ (2007) K. Nomura, M. Koshino, and S. Ryu, Phys. Rev. Lett. 99, 146806 (2007).
  * San-Jose  _et al._ (2007) P. San-Jose, E. Prada, and D. S. Golubev, Phys. Rev. B 76, 195445 (2007).
  * Lewenkopf  _et al._ (2008) C. H. Lewenkopf, E. R. Mucciolo, and A. H. Castro Neto, Phys. Rev. B 77, 081410 (2008).
  * Evers and Mirlin (2008) F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
  * Mirlin (2000) A. D. Mirlin, Phys. Rep. 326, 259 (2000).
  * Shukla (2005a) P. Shukla, J. Phys.: Condens. Matter 17, 1653 (2005a).
  * Shukla (2005b) P. Shukla, Phys. Rev. E 71, 026226 (2005b).
  * Shklovskii  _et al._ (1993) B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianides, and H. B. Shore, Phys. Rev. B 47, 11487 (1993).
  * Pandey (1995) A. Pandey, Chaos, Solitons & Fractals 5, 1275 (1995), quantum Chaos: Present and Future.
  * Berkovits and Avishai (1998) R. Berkovits and Y. Avishai, Phys. Rev. Lett. 80, 568 (1998).
  * Cuevas (1999) E. Cuevas, Phys. Rev. Lett. 83, 140 (1999).
  * Niizeki (1979) K. Niizeki, Prog. Theor. Phys. 62, 1 (1979).
  * Haake (2010) F. Haake,  _Quantum Signatures of Chaos_ , 3rd ed. (Springer, 2010).
  * McKinnon and Choy (1993) B. A. McKinnon and T. C. Choy, Aust. J. Phys. 46, 601 (1993).
  * Suzuura and Ando (2002) H. Suzuura and T. Ando, Phys. Rev. Lett. 89, 266603 (2002).
  * Brody  _et al._ (1981) T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. 53, 385 (1981).
  * Rabson  _et al._ (2004) D. A. Rabson, B. N. Narozhny, and A. J. Millis, Phys. Rev. B 69, 054403 (2004).
  * Hanisch  _et al._ (1995) T. Hanisch, B. Kleine, A. Ritzl, and E. Müller-Hartmann, Annalen der Physik 507, 303 (1995), cond-mat/9501116 .
  * Yu  _et al._ (2005) G. Yu, Y. Yong-Hong, W. Yong-Gang, and Z. Qun, Commun. Theor. Phys. 43, 743 (2005).