# Self-duality of One-dimensional Quasicrystals with Spin-Orbit Interaction

Deepak Kumar Sahu  Aruna Prasad Acharya  Department of Physics and Astronomy,
National Institute of Technology, Rourkela, Odisha, India  Debajyoti Choudhuri
Department of Mathematics, National Institute of Technology, Rourkela, Odisha,
India  Sanjoy Datta<EMAIL_ADDRESS>Department of Physics and Astronomy,
National Institute of Technology, Rourkela, Odisha, India  Center for
Nanomaterials, National Institute of Technology, Rourkela, Odisha, India

(August 27, 2024)

###### Abstract

Non-interacting spinless electrons in one-dimensional quasicrystals, described
by the Aubry-André-Harper (AAH) Hamiltonian with nearest neighbour hopping,
undergoes metal to insulator transition (MIT) at a critical strength of the
quasi-periodic potential. This transition is related to the self-duality of
the AAH Hamiltonian. Interestingly, at the critical point, which is also known
as the self-dual point, all the single particle wave functions are
multifractal or non-ergodic in nature, while they are ergodic and delocalized
(localized) below (above) the critical point. In this work, we have studied
the one dimensional quasi-periodic AAH Hamiltonian in the presence of spin-
orbit (SO) coupling of Rashba type, which introduces an additional spin
conserving complex hopping and a spin-flip hopping. We have found that,
although the self-dual nature of AAH Hamiltonian remains unaltered, the self-
dual point gets affected significantly. Moreover, the effect of the complex
and spin-flip hoppings are identical in nature. We have extended the idea of
Kohn’s localization tensor calculations for quasi-particles and detected the
critical point very accurately. These calculations are followed by detailed
multifractal analysis along with the computation of inverse participation
ratio and von Neumann entropy, which clearly demonstrate that the quasi-
particle eigenstates are indeed multifractal and non-ergodic at the critical
point. Finally, we mapped out the phase diagram in the parameter space of
quasi-periodic potential and SO coupling strength.

##  I Introduction

In one dimensional (1D) lattice with random disorders, all the electronic
single-particle states (SPS) of the non-interacting Anderson Hamiltonian
localize exponentially even if the strength of the disorder is arbitrarily
small Anderson ; Abrahams . In pure Anderson Hamiltonian, metal (extended SPS)
to insulator (localized SPS) transition exists only in three dimension (3D),
while in two dimension (2D) there are no extended states, but for weak
disorder SPS are marginally localizedAbrahams . The extended states are also
ergodic, that is, in the thermodynamic limit the real space average of the
$q$-th moment of $\left|\psi(i)\right|^{2}$ converges to its ensemble average
value Deng-Sinha . In contrast, a quasiperiodic lattice, described by the
Aubry-André-Harper (AAH) Hamiltonian Harper ; Azbel ; Aubry-Andre  with
nearest-neighbour hopping, undergoes metal to insulator (MIT) transition at a
critical disorder strength, even in 1D. The fundamental, spinless AAH
Hamiltonian referred above can be represented by the eigenvalue equation,
$t(\psi_{n+1}+\psi_{n-1})+W\text{cos}(2\pi bn+\phi)\psi_{n}=E\psi_{n}$, where
$W$ is the strength of the disorder, $t$ is the nearest neighbour hopping
amplitude, and $\psi_{n}$ is the amplitude of the electronic wave function at
the lattice site $n$. When $b$ is irrational we have the quasi-periodic
lattice. The AAH Hamiltonian in reciprocal space is given by,
$\frac{W}{2}(\psi_{k+1}+\psi_{k-1})+2t\text{cos}(2\pi
bn+\phi)\psi_{k}=E\psi_{k}$. The Fourier transformed AAH Hamiltonian becomes
same as the original Hamiltonian when $W/t=2$ and their roles are
interchanged. This unique feature of AAH Hamiltonian makes it self-dual, as
states below and above this critical value $W_{c}/t=2$ are related by Fourier
transformation. All the SPS are extended for $W_{c}/t<2$ while they are
localized for $W_{c}/t>2$. The self-dual point $W_{c}/t=2$ is particularly
special, as all the states show multifractal character, that is, they are
extended but non-ergodic.

Disordered electronic systems have continued to receive wide attention, even
after more than five decades of the publication of the seminal paper by
Anderson Anderson . A major driving force behind this is due to the importance
of disorders, which are unavoidable in real materials. Also, the debate about
whether MIT exists in dimension lower than three in such lattices is not
settled completely. The search for MIT in lower dimensional disordered systems
is not just an academic interest, but it is an extremely important problem
from the point of view of device applications as well. Moreover, successful
experimental simulations of these disordered Hamiltonians in optical lattice
set-up roati ; Lahini ; Kraus ; madsen  have opened the possibility of
verification of the theoretical predictions. More recently, the critical
behaviour of the spin-less version of the AAH Hamiltonian has been
experimentally observed in polaritonic 1D wires, created with the help of
cavity-polariton devicesZilberberg . Consistent effort towards finding MIT in
lower dimension have led researchers to consider Anderson Hamiltonian with SO
coupling quite extensively Hikami ; Evangelou-Ziman ; Evangelou-PRL ; Ando-
PRB89 ; Janssen-Huckestein ; Minakuchi ; Yakubo ; Asada-Slevin ; Su-Wang .
However, the effect of SO coupling in quasi-periodic systems has not received
much attention.

In this present work, we have explored the effect of SO coupling on MIT in
quasi-periodic 1D lattice. More specifically, we have studied the AAH model
with nearest neighbour hopping in the presence of Rashba type spin-orbit (RSO)
coupling Rashba . RSO coupled electronic systems receive wide attention
because of its potential in various device applications, especially in
spintronic devices Zutic . The main advantage of RSO coupled systems is the
possibility of easier external control of the coupling strength. In order to
find a comprehensive and clear answer, we have systematically used three
different approaches. First, to determine the effect of RSO on the nature of
MIT in 1D quasi-periodic lattice, we have used the idea of many-body
localization tensor Resta-PRL98 ; Resta-Sorella-PRL99 , which is based on
Kohn’s idea of localization in disordered systems kohn . In the process, we
have demonstrated that this many-body localization tensor, also known as
Kohn’s localization tensor(KLT), is a truly powerful method to study MIT for
spinfull electronic systems, where spin states mix. After obtaining precise
answer on the effect of RSO on MIT, we studied the nature of the eigenstates
across the entire energy spectrum for different disorder strengths. For this
we have used the inverse participation ratio (IPR) and von Neumann entropy
(vNE) to get a quick overall idea about the nature of the quasi-particle
eigenstates. Finally, we have carried out detailed multifractal analysis to
prove that the self-duality of AAH Hamiltonian remains preserved in the
presence of RSO, although the self-dual point shifts towards higher disorder
strength with the increase of RSO coupling. While carrying out the
multifractral analysis, we have used both periodic and open boundary
conditions (OBC) to show that OBC can be used for the quasi-periodic lattices
as well, effectively removing the constraint on the system sizes for numerical
computation of the multifractal spectrum. To summarize our conclusions, at the
end, we present the phase diagram in the parameter space spanned by the
disorder strength and RSO couplings. It is important to note that in an
earlier study Kohomoto , a 1D quasi-periodic AAH Hamiltonian with RSO coupling
was obtained starting from a tight-binding square lattice in presence of
uniform magnetic field. However, the RSO Hamiltonian used in the above
mentioned reference is not same as ours. Although, it had also been concluded
that the self-dual nature remains intact, but it was reported that the self-
dual point remains unaffected by RSO coupling strength, while there exists two
new phases, self-dual of each other, in the parameter space of disorder
strength and SO coupling. These phases are characterized by coexistence of
delocalized and localized states. Naturally, we have not found any evidence of
these phases.

The paper is organized as follows: in Sec.III, we introduce briefly the idea
of Kohn’s localization tensor (KLT) for non-interacting electrons. From the
KLT calculations, we can immediately identify the critical point for MIT. In
Sec. III.3, we have explored the evolution of the critical point with the RSO
coupling strengths. However, these results do not provide clear idea about the
nature of the SPS at the critical point. To study the nature of the SPS, we
have computed the single particle IPR spectrum in Sec.IV and the von Neumann
entropy (vNE) in Sec.V. To demonstrate that all the SPS are non-ergodic at the
critical point, in Sec.VI.1, we have calculated the multifractal spectrum for
the three regions: below, above, and at the critical point. Finally, in
Sec.VII, we present the phase diagram with respect to the parameters that
control the disorder strength and the strengths of RSO coupling.

##  II Aubry-Andrè Model With RSO coupling

The Hamiltonian considered in this work consists of two parts,

$H^{\prime}=H+H_{R},$ (1)

where $H$ is the usual AAH Hamiltonian given by,

$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\displaystyle\sum_{{i=1},\sigma}^{L-1}(c^{\dagger}_{{i+1},{\sigma}}c_{{i},{\sigma}}+h.c)+$
(2) $\displaystyle W\displaystyle\sum_{{i=1},{\sigma}}^{L}cos(2\pi
bi+\phi)c^{\dagger}_{{i},{\sigma}}c_{{i}.{\sigma}}$

Here, $t$ is the hopping amplitude from site $i$ to site $i+1$ and $L=Na$ is
the length of the lattice, where $N$ is the number of lattice sites and
$a=1$(arbitrary unit) is the lattice spacing. $c^{\dagger}_{{i},{\sigma}}$ and
$c_{{i},{\sigma}}$ are the fermionic creation and annihilation operators
respectively for spin operator $\sigma=\uparrow$,$\downarrow$ particle at site
$i$. $W$ is the strength of quasi-periodic potential. $\phi$ is an arbitrary
phase varying from $(0,2\pi)$. The choice of the phase $\phi$ does not affect
our conclusion, and henceforth we shall set it to zero. We have used
$b=(\sqrt{5}+1)/2$. Please note that, sometimes in the literature
$b=(\sqrt{5}-1)/2$ is also used, but our conclusions are independent of the
particular choice of $b$.

The RSO Hamiltonian $H_{R}$ is given by Birkholz ,

$\displaystyle H_{R}$ $\displaystyle=$
$\displaystyle-\alpha_{z}\displaystyle\sum_{{i=1},{\sigma},{\sigma^{\prime}}}^{L-1}(c^{\dagger}_{i+1,\sigma}(i\sigma_{y})_{\sigma,\sigma^{\prime}}c_{i,\sigma^{\prime}}+h.c.)$
(3)
$\displaystyle-\alpha_{y}\displaystyle\sum_{{i=1},{\sigma},{\sigma^{\prime}}}^{L-1}(c^{\dagger}_{i+1,\sigma}(i\sigma_{z})_{\sigma,\sigma^{\prime}}c_{i,\sigma^{\prime}}+h.c.),$

where $\sigma_{y}$ and $\sigma_{z}$ are Pauli spin matrices in $y$\- and
$z$-direction respectively. $\alpha_{y}$ is a complex spin-conserving hopping
due to the confinement in $y$-direction and $\alpha_{z}$ is a spin-flip
hopping due to the confinement in $z$-directions. The hopping amplitude
$\alpha_{y}$ and $\alpha_{z}$ could be different in general and they could
also be site dependent. The pure RSO Hamiltonian $H_{R}$, that is studied in
this work, has also been studied in the context of transport properties in
quantum nanowires Ando-Tamura ; Mireles . Also, recently localization
properties of attractive fermions have been studied in presence of the spin-
flip component of the RSO HamiltonianXianlong-PRA16 . Since $\alpha_{y}$ is a
spin preserving hopping process, it is expected that it will not change the
self-dual nature of AAH Hamiltonian. However, it is not clear how $W_{c}/t$ is
going to be affected by $\alpha_{y}$ alone. On the other hand, the effect of
spin-flip hopping, $\alpha_{z}$, on the self-duality as well as on $W_{c}/t$
is not evident. Hence, we are going to focus mainly on the problem where
$\alpha_{z}\neq 0$ and $\alpha_{y}=0$. However, for completeness, we also
present KLT results for $\alpha_{y}\neq 0$ and $\alpha_{z}=0$.

Figure 1: (a) Squared localization length $\lambda^{2}$ for half-filled 1D
pure AA model with respect to disorder strength $W/t$ for some selected
lattices. (b) Scaling of inverse localization length $1/\lambda^{2}$ with
inverse chain length $(1/L)$. For finite size scaling we have used
$1/\lambda^{2}=a_{0}+b_{0}/L$. (c) Plot of $1/\lambda^{2}$ with $1/L$ at the
critical point. It is evident that at the critical point, $1/\lambda^{2}$ does
not follow the scaling pattern of either metallic or insulating states. All
the results for open boundary condition.

Figure 2: Localization tensor $\lambda^{2}$ for half-filled 1D pure AAH model
with PBC. (a) $\lambda^{2}$ vs. $W/t$ for a few selected chain lengths. (b)
Finite size scaling of $1/\lambda^{2}$ vs. $(1/L)$. (c) Once again, similar to
the OBC case, $1/\lambda^{2}$ lacks simple scaling behaviour with $1/L$ at the
critical point.

##  III Kohn’s Localization Tensor and Metal to Insulator Transition

For a comprehensive analysis of the interplay of disorder and RSO coupling we
have systematically used different approaches to capture the full picture. In
this section, we discuss the idea of Kohn’s localization tensor. It is a
reliable way to characterize metallic and insulating state. Based on the idea
first proposed by W. Kohnkohn , Resta and Sorella Resta-PRL98 ; Resta-Sorella-
PRL99  formulated a localization tensor, alternatively called as Kohn’s
localization tensor (KLT). In the thermodynamic limit, it is independent of
system size for insulating states, while for metallic states it diverges. For
1D AAH model with nearest-neighbour hopping and no RSO coupling, a metal to
insulator transition takes place at the critical disorder strength
$W_{c}/t=2.0$. Recently, it has been shown varma  that the localization tensor
$\lambda_{\alpha\beta}$, where $\alpha$ and $\beta$ are the space co-
ordinates, can capture this transition accurately. Furthermore, it has been
shown that computation of $\lambda_{\alpha\beta}$ becomes particularly simpler
for non-interacting electrons as well as for interacting electrons, if
interaction is treated within mean-field approximation. However, in these two
above mentioned scenarios spin states were completely decoupled. As mentioned
earlier, in the presence of spin-flip hopping induced by RSO coupling, spin
states mix to giving rise to quasi-particle eigenstates. Here, in this work we
extended the computational approach of $\lambda_{\alpha\beta}$ for quasi-
particle states and simultaneously capture the MIT, if it exists, accurately.
We have presented our results for periodic and open boundary conditions.
However, as we are going to see, we have found that it is not always possible
to calculate $\lambda_{\alpha\beta}$ with open boundary condition.

Here, we briefly discuss the key aspects of KLT and methods to calculate it
for different boundary conditions Resta-JPC-02-review ; varma ; Resta-
EurPhys2011 . A more elaborate discussion on KLT and ways to compute it can be
found in Ref.Resta-JPC-02-review  and in Ref.Resta-JCP06 . For periodic
boundary condition (PBC), the localization tensor,
$\lambda_{\alpha\beta}(\text{here\leavevmode\nobreak\ }\alpha=x,\beta=x)$ can
be expressed as,

$\lambda_{xx}^{2}=-\frac{L^{2}}{4\pi^{2}N}ln\frac{|z_{N}^{x}||z_{N}^{x}|}{|z_{N}^{xx}|},$
(4)

where the quantity $z_{N}^{x}$ is given by,

$z_{N}^{x}=\langle\Psi\rvert
e^{i\frac{2\pi}{L}\hat{\mathbf{R}}_{x}}\lvert\Psi\rangle,$ (5)

whereas, $z_{N}^{\alpha\beta}$ can be obtained by replacing $\hat{R}_{\alpha}$
with $\hat{R}_{\alpha}-\hat{R}_{\beta}$. Hence, in our case $|z_{N}^{xx}|=1$.
In the above equation, $\ket{\Psi}$ is the many-body ground state wave
function, $\hat{\mathbf{R}}=\sum_{i=1}^{N}\hat{r}_{i}$ is the the many-body
position operator with $\hat{\mathbf{R}}_{x}$ being the $x$ component. $N$ is
the number of lattice sites, while $L=N\leavevmode\nobreak\ a$, with $a=1$(in
arbitrary unit) being the lattice constant, is the size of the system. For a
half-filled system ($L=N$) in 1D, the localization tensor reduces to

$\lambda_{xx}^{2}=\lambda^{2}=-\frac{L}{2\pi^{2}}\text{ln}\left\\{|z_{N}|\right\\}.$
(6)

In absence of electron-electron interaction, $z_{N}^{(x)}$ can be simplified
further Resta-JPC-02-review ; Resta-EurPhys2011  and represented as
$z_{N}^{(x)}={det}^{2}[S_{jj^{\prime}}^{x}]$, where $S_{jj^{\prime}}^{x}$ is a
matrix whose elements are given by,

$S_{j,j^{\prime}}^{x}=\int dr\leavevmode\nobreak\
\psi_{j}^{*}(r)e^{i\frac{2\pi}{L}\hat{r}_{x}}\psi_{j^{\prime}}(r).$ (7)

In the above equation $\psi_{j}(r)$ represents the amplitude of single-
particle wave function at position $r$ for a spin-up or spin-down electron
arranged in ascending order in energies. The indices $j$ and $j^{\prime}$
indicate the energy level. Since, in absence of RSO spin-up and spin-down
electrons are completely decoupled, it is sufficient to consider only one type
of spin and $j,j^{\prime}=1,2,\cdots,N/2$ to compute $\lambda_{xx}^{2}$ at
half-filling. But, in presence of RSO, more specifically because of the spin-
flip hopping process, in Eq. 7 $\phi_{j}(r)$ now represents the amplitude of a
single quasi-particle wave function at position $r$ corresponding to the
$j^{th}$ eigen-energy. Hence, in our case $j,j^{\prime}=1,2,\cdots,N.$

In case of open boundary conditions (OBC), squared localization length
$(\lambda^{2})$, in units of the nearest-neighbour distance, can be expressed
as follows Resta-JCP06 ; Bendazzoli-Resta-JCP2010 :

$\lambda^{2}=\frac{1}{\nu
N}\displaystyle\sum_{i,i^{\prime}=1}^{N}\rho_{ii^{\prime}}^{2}(\nu)(i-i^{\prime})^{2},$
(8)

where $i,i^{\prime}=1,\cdots,N$ represent the lattice site, $\nu$ is the
filling factor and $\rho_{ii^{\prime}}(\nu)$ is the one-body density matrix,
defined as

$\rho_{ii^{\prime}}=\displaystyle\sum_{j=1}^{N}\phi_{j}(i)\phi_{j}^{*}(i^{\prime}),$
(9)

where $\phi_{j}(i)$ is the amplitude of single quasi-particle wave function at
lattice site $i$ corresponding to $j^{th}$ eigenvalue. Since we are interested
to compute $\lambda^{2}$ at half-filling, we set $\nu=1$ in Eq. 8 .

Figure 3: Localization tensor $\lambda^{2}$ for half-filled 1D AAH model with
RSO coupling. (a)$\lambda^{2}$ vs. $W/t$ for selected lattices with OBC. (b)
Scaling of inverse localization length $1/\lambda^{2}$ vs. $1/L$. Here
$1/\lambda^{2}$ scales linearly with $1/L$ similar to Fig. 1(b). (c) Plot of
$1/\lambda^{2}$ vs. $1/L$ at the critical point $W_{c}/t=2.55$. Lack of proper
scaling of $1/\lambda^{2}$ indicates that our estimation of critical point is
quite correct.

Figure 4: Localization tensor $\lambda^{2}$ for half-filled 1D AAH model with
RSO coupling. (a) $\lambda^{2}$ vs. $W/t$ for selected lattices with PBC. (b)
Finite size scaling of $1/\lambda^{2}$ with inverse chain length $1/L$ below
and above the critical point. We have used
$1/\lambda^{2}=a_{0}+b_{0}/L+c_{0}/L^{2}$ for the purpose of scaling in the
metallic phase. (c) With PBC as well we find anomalous behaviour of
$1/\lambda^{2}$ with increasing system size at $W/t=2.55$.

###  III.1 Localization tensor without RSO

Before presenting the localization tensor results for our Hamiltonian, in this
section we first benchmark our $\lambda^{2}$ calculations for pure AA model
without the RSO coupling at half-filling. We also highlight the behaviour of
$\lambda^{2}$ with system sizes at the critical point. In Fig.1(a) and
Fig.2(b), we have shown these calculation for open and periodic boundary
conditions respectively. In case of PBC, system sizes are restricted to
lattice sizes given by the Fibonacci sequence, while for OBC there is no such
restriction. One of the main objectives to compare results of different
boundary conditions is to show that the localization tensor is a robust
indicator to differentiate metallic and insulating state irrespective of the
boundary conditions. This observation is going to be useful, as we are going
to show in Sec.VI, while computing the multifractal spectrum for quasi-
periodic lattices, as we do not need to restrict ourselves to very specific
system sizes dictated by the Fibonacci sequence in order to extrapolate the
results to thermodynamic limit.

In Fig.1(a)(OBC) and Fig.2(a)(PBC), we have shown the variation of squared
localization length $\lambda^{2}$ with increasing disorder strength $(W/t)$
for different 1D lattices at half-filling. As expected, the transition from
delocalized to localized phase occurs at $W_{c}/t\simeq 2$ for both the
boundary conditions. $\lambda^{2}$ values are finite and independent of system
sizes above $W/t>2$ whereas for $W/t<2$, it increases with the increase of
system size. These observation are similar to Ref. varma . In Sec.III it has
been mentioned that in case of metallic phase $\lambda^{2}$ diverges in the
thermodynamic limit, while it is independent of system sizes in the insulating
phase. From the plot of $1/\lambda^{2}$ vs. $1/L$ of Fig. 1(b) and Fig.2(b),
it is clear that, irrespective of the boundary condition, $\lambda^{2}$
diverges for $L\rightarrow\infty$ in the metallic phase, while it is nearly
constant and converges to a finite value in the insulating phase. For pure AA
Hamiltonian, we have found that $1/\lambda^{2}$ scales linearly with the
inverse of the system size. For finite size scaling, in this case, we have
used $1/\lambda^{2}=a_{0}+b_{0}/L$, where $a_{0}$ and $b_{0}$ are two
adjustable parameters.

At the critical point $W_{c}/t=2$, scaling behaviour of $\lambda^{2}$ with
increasing system size is expected to be erratic as all the single particle
eigenstates are mutifractral in case of pure AA model. However, it can be an
useful and quick indicator to detect the existence of multifractal eigenstates
within a spectrum. Since there is no characteristic length-scale for
multifractal states, we can expect an anomalous behaviour of $1/\lambda^{2}$
with respect to $1/L$ compared to pure metallic and insulating phases. In Fig.
1(c) and in Fig. 2(c) we have plotted $1/\lambda^{2}$ versus $1/L$ for
$W_{c}/t=2$. In case of OBC, it is clear that $\lambda^{2}$ neither converges
to a finite value nor does it go to zero in the thermodynamic limit. With PBC
as well, $\lambda^{2}$ oscillates with increasing $L$ without any indication
of convergence. In this work, however, we have used a well established and
computationally less costly approach for the multifractal analysis of the
eigenstates.

###  III.2 Localization tensor in presence RSO

We now present our localization tensor results for one-dimensional AA model in
the presence of RSO coupling. Since the effect of the spin-flip hopping
process, induced by RSO, on the self-duality is much harder to anticipate, we
primarily focus on the $\alpha_{y}=0$ case. We have chosen $\alpha_{z}/t=0.8$.
There is nothing special about this value, as all our conclusions are
independent of it, except the location of self-dual point. In Fig.3(a)(OBC)
and 4(a)(PBC) the variation of squared localization length, $\lambda^{2}$,
with increasing value of disorder strength $W/t$ have been shown. It is
evident that inclusion of RSO coupling enhances the critical point towards
larger disorder strength, in this case $W_{c}/t\simeq 2.5$, as compared to
pure AA model ($W_{c}/t\simeq 2$). Furthermore, this conclusion is independent
of the applied boundary conditions on the system.

To confirm our observation in the thermodynamic limit we have done finite size
scaling of $1/\lambda^{2}$ with different system sizes below and above the
critical point. These results are presented in Fig. 3(b) and in Fig. 4(b). In
the case of OBC, it is evident that $1/\lambda^{2}$ converges to zero as
$L\rightarrow\infty$ for $W/t<2.5$, that is, for delocalized phase, while it
is independent of system size for $W/t>2.5$, indicating an insulating phase.
In case of PBC also, these fundamental conclusions remain same. Interestingly,
however, with PBC the dependence of $1/\lambda^{2}$ on $1/L$ deviates
significantly in the presence of RSO in the metallic phase. For finite size
scaling, specially for the metallic phase, we have used
$1/\lambda^{2}=a_{0}+b_{0}/L+c_{0}/L^{2}$, where $a_{0},b_{0}$ and $c_{0}$ are
adjustable parameters.

It is easy to pinpoint the critical disorder strength from the $\lambda^{2}$
vs $W/t$ data almost exactly. We are going to see that this information can
reliably be extracted from the multifractal analysis (Sec. VI.1) as well. From
the localization tensor result we estimate $W_{c}/t\simeq 2.55$ for
$\alpha_{z}/t=0.8$. To cross-check this estimation, we have plotted
$1/\lambda^{2}$ versus $1/L$ separately at $W_{c}/t\simeq 2.55$ in Fig. 3(c)
and in Fig. 4(c). It is quite clear that $1/\lambda^{2}$ does not follow any
discernible pattern with $1/L$. This, as we have pointed out in Sec. III.1, is
the hallmark of critical/multifractal eigenstates.

Figure 5: (a) Evolution of the self-dual point in AAH with only spin-flip
hopping induced by RSO coupling. The results are for a lattice $L=1597$ with
OBC. (b) The self dual point moves towards higher strength of the quasi-
periodic potential when pure AAH Hamiltonian is considered with only the
complex hopping induced by the RSO coupling. (c) Evolution of the self-dual
point when the AAH Hamiltonian is considered with the full RSO Hamiltonian.
Here we have only shown the change with the spin-flip hopping while the
complex hopping has been fixed at a value $0.6$.

###  III.3 Evolution of the critical point with RSO interaction

In this section, we study the evolution of the critical point for three
different cases, (i) $\alpha_{y}=0$, (ii) $\alpha_{z}=0$ and (iii)
$\alpha_{z}\neq 0,\alpha_{y}=\rm{fixed}.$ For case (i) and (iii) a lattice
with $N=1597$ and OBC have been used, while for case (ii) we have used PBC and
$N=987$. It is important to note that $\lambda^{2}$ could not be computed with
OBC for case (ii). In Fig. 5(a), we have considered only the spin-flip hopping
$\alpha_{z}$ in the RSO Hamiltonian. It is clear, even from the results for a
single finite lattice, that the critical point moves to higher strength of the
quasi-periodic potential. Interestingly, when only the complex hopping of the
RSO Hamiltonian is considered along with pure AA Hamiltonian, similar
variation of the critical point is observed in Fig. 5(b). Fig 5(c), represents
the general nature of the evolution of the critical point when the full RSO
Hamiltonian is considered along with pure AA Hamiltonian. In these results,
variation of the critical point with spin-hopping has been studied for a fixed
strength of the spin conserving complex hopping having the amplitude
$\alpha_{y}/t=0.6$. For $\alpha_{z}=0.2$, $\lambda^{2}$ appears to indicate a
cross-over rather than a clear transition. However, this is an artifact of the
finite system being used for these calculations. We have checked that as the
system size is increased, a clear transition from delocalized to localized
transition emerges.

##  IV Inverse Participation Ratio

Existence of MIT in presence of RSO is evident from the results of the
localization tensor calculations. Although at the critical point, anomalous
behaviour of the localization tensor with the inverse of the system size do
indicate the absence of a characteristic length scale, the true nature of the
quasi-particle eigenstates across the entire energy spectrum is not clear. For
example, if an energy spectrum contains predominantly delocalized states then
the localization tensor can suppress the contribution of localized states
while calculating it upto certain filling fraction. To distinguish localized
and delocalized states, Inverse Participation Ratio (IPR) is regularly used as
a first indicator. IPR can also provide some hint, although only
qualitatively, of multifractal eigenstates, if it exists. Generally, IPR is
used to quickly identify the existence of mobility-edge. Typically the
mobility edge is defined as the energy which separates localized and
delocalized eigenstates in the energy spectrum. If a mobility edge exists, IPR
jumps from system size independent (insulating/localized states) higher value
to a lower value (ergodic metallic/extended states) that scales inversely with
the system size. In our case, we have observed that away from the critical
point, the states are either delocalized $(W<W_{c})$ or localized $(W>W_{c})$.
The other important question that remains to be answered is, what are the
nature of the eigenstates at the critical point? In AA model without RSO, all
the eigenstates are extended but non-ergodic at the critical point. IPR data
qualitatively indicates that RSO does not alter this behaviour. At this point
we would like to mention that, studying the single particle energy spectra
$E_{n}$ and the distribution of the level-spacing $\delta_{n}=E_{n+1}-E_{n}$
can also reveal a great deal about the nature of the eigenstates Evangelou-
Pichard ; Takada ; Machida ; for example, in AAH model the energy level-
spacing follows Poissonian distribution, while at the critical point this
distribution follows an inverse power law. Typically, the energy spectrum of
AAH Hamiltonian consists of subbands and many gaps between these subbands.
Recently, it has been reported that there are some special states inside these
subband gaps Roy-Sharma , which are localized even in the metallic phase of
pure AAH Hamiltonian. We have checked that these states are mostly
concentrated at the lattice edges. Appearance of these states depends on many
factors; for example, they can appear for system with OBC, with PBC, but the
system size is not same as certain (not all) Fibonacci number. These states do
not influence the critical behaviour in any way, as can be seen from the
results of the previous section.

Figure 6: Evolution of IPR ${I_{n}}$ with increasing disorder strength for a
fixed RSO coupling $\alpha_{z}/t=0.8$ and $\alpha_{y}=0.$
(a)$W/t=2.5$,(b)$W/t=2.55$, and (c) $W/t=2.6$. Here index represents the ratio
of serial number of eigenstate to total number of eigenstates
($\mathrm{i}/(2L)$), where $i=1,2,\cdots,2L$.

Figure 7: Evolution of von Neumann Entropy ${S_{n}}$ with increasing disorder
strength for a fixed RSO coupling $\alpha_{z}/t=0.8$ and $\alpha_{y}=0.$
(a)$W/t=2.5$,(b)$W/t=2.55$, and (c) $W/t=2.6$.

Formally, for $\nu$-th quasiparticle eigenstate the usual definition of IPR
Evers-Mirlin  can be generalized as follows;

$P_{\nu}(q)=\sum_{\sigma}\sum_{i=1}^{N}\left|\psi_{\nu,\sigma}(i)\right|^{2q},\leavevmode\nobreak\
\leavevmode\nobreak\ q=2.$ (10)

Here $\nu$th single quasiparticle eigenstate is given by
$\ket{\Psi_{\nu}}=\sum_{\sigma}\sum_{i=1}^{N}\psi_{\nu,\sigma}(i)\ket{1,\sigma}_{i},$
where $\ket{1,\sigma}_{i}=\ket{0,0,\cdots,\sigma_{i},\cdots,0,0}$ represents
the localized basis state having one particle with spin $\sigma$ at site $i$.
Since, this is a non-interacting problem, usual scaling properties with
respect to system size is also expected to hold for the quasiparticle states,
i.e. for a perfectly extended metallic state $P_{\nu}(q)=1/N$, and for a
completely localized state $P_{\nu}(q)=1$. For a localized state, the IPR
value is supposed to be system size independent. These distinct scaling
properties of IPR enables it to identify delocalized and localized states
quickly. We have numerically verified that the spin-up and spin-down
contributions are identical towards the IPR of the quasiparticle they
constitute.

In Fig. 6 we have plotted the IPR spectrum for (a) $W/t=2.5$, (b) $W/t=2.55$,
and (c)$W/t=2.6$. These results are for $\alpha_{z}/t=0.8,\alpha_{y}/t=0.$ It
is evident that for $W/t<2.55$, all the eigenstates are delocalized as the IPR
value in this region depends inversely on the system size across the entire
energy spectrum, while all the states are localized for $W/t>2.55.$ On the
other hand, at the critical point the IPR spectrum behaves differently, it is
neither independent of system size nor does it scale inversely with $L$ like
the extended states. This behaviour is similar to the IPR spectrum of AA model
without RSO at the critical point, i.e. these are multifractal or in other
words extended yet non-ergodic. Furthermore, the correlation between the
energy level-spacing spectra and the scaling behaviour of IPR with L for all
three different types of electronic states are similar to the AA model without
RSO Roy-Sharma . In the delocalized phase, IPR typically behaves inversely
with $L$ across the whole energy-spectrum except at the special positions
where the level-spacing jumps abruptly. At the critical point, both of them
show anomalous behaviour with the system size, while in the localized phase
IPR spectrum behaves opposite compared to the delocalized phase at these
special point of level-spacing spectra apart from being system size
independent. All of these results are presented for PBC. OBC does not change
the fundamental conclusions.

Figure 8: Multifractal spectrum of AAH Hamiltonian with RSO coupling for PBC
(top row) and OBC (bottom row). Here $\alpha_{y}/t=0$ and $\alpha_{z}/t=0.8$.
From left to right: (a, d)$W/t=2.5$, (b, e)$W/t=2.55$ and (c, f) $W/t=2.6$.
$D(2)$ is the generalized dimension $D(q)$ for $q=2$. For extended ergodic
states $D(2)=1$ and $D(2)=0$ for insulating states, while for multifractal
/non-ergodic extended states $0<D(2)<1.$

##  V von Neumann entropy

From the results of the previous sections, it is evident that at half-filling
there is a transition from metallic phase to an insulating phase at a critical
disorder strength $W_{c}/t>2.0$, which increases as the strength of RSO is
increased. Furthermore, these results also hint that at the critical point the
states are multifractal, a preliminary observation that we are going to
establish firmly in Sec. VI. In this section, we present the results of von-
Neumann entropy (vNE), an alternative indicator of single particle properties,
which can also be used to qualititavely identify the nature of the
eigenstates.

In case of non-interacting spin-1/2 fermions, and in presence of RSO coupling,
individual eigenstates are occupied by quasiparticles. The quasiparticle
eigenstate having energy $E_{\nu}$ can be written as,

$\ket{\Psi_{\nu}}=\sum_{i=1}^{N}\left[\psi^{\nu}_{i,\uparrow}\ket{1,\uparrow}_{i}+\psi^{\nu}_{i,\downarrow}\ket{1,\downarrow}_{i}\right],$
(11)

where $\ket{1,\uparrow}_{i}=\ket{1}_{i}\otimes\ket{\uparrow}$ and
$\ket{1,\downarrow}_{i}=\ket{1}_{i}\otimes\ket{\downarrow}$.
$\ket{1,\uparrow}_{i}=c^{\dagger}_{i,\uparrow}\ket{0}$ and
$\ket{1,\downarrow}_{i}=c^{\dagger}_{i,\downarrow}\ket{0}$. Here $\ket{0}$
represents the vacuum state for the lattice in real space basis.
$c^{\dagger}_{i,\uparrow},\leavevmode\nobreak\ c^{\dagger}_{i,\downarrow}$ are
the creation operators for spin up and spin down particles respectively at the
lattice site $i$. The average number of spin up and down particles at site $i$
are given by
$|\psi^{\nu}_{i,\uparrow}|^{2}=\bra{\Psi_{\nu}}c^{\dagger}_{i,\uparrow}c_{i,\uparrow}\ket{\Psi_{\nu}}$
and
$|\psi^{\nu}_{i,\downarrow}|^{2}=\bra{\Psi_{\nu}}c^{\dagger}_{i,\downarrow}c_{i,\downarrow}\ket{\Psi_{\nu}}$
respectively. Then, the local density matrix $\rho^{\nu}_{j}$ can be obtained
from the total density matrix $\rho^{\nu}$ by tracing over all the lattice
sites except site $j$ and can then be written as,

$\displaystyle\rho^{\nu}_{j}$ $\displaystyle=$
$\displaystyle\left|\psi^{\nu}_{j,\uparrow}\right|^{2}\ket{1,\uparrow}_{j}\bra{1,\uparrow}_{j}+(1-|\psi^{\nu}_{j,\uparrow}|^{2})\ket{0,\uparrow}_{j}\bra{0,\uparrow}_{j}+$
(12)
$\displaystyle\left|\psi^{\nu}_{j,\downarrow}\right|^{2}\ket{1,\downarrow}_{j}\bra{1,\downarrow}_{j}+(1-|\psi^{\nu}_{j,\downarrow}|^{2})\ket{0,\downarrow}_{j}\bra{0,\downarrow}_{j}$

It is important to note that $\ket{0,\uparrow}_{j}$ and $\ket{0,\uparrow}_{j}$
represent local vaccum states for $j$-th site. A similar interpretation
applies to the states $\ket{1,\uparrow}_{j}$ and $\ket{1,\downarrow}_{j}$. The
von Neumann entropy for spin-1/2 quasiparticles follows easily from Eq. 12 as,

$\displaystyle S^{nis}_{j,\nu}$ $\displaystyle=$
$\displaystyle-\left(|\psi^{\nu}_{j,\uparrow}|^{2}\mathrm{log}_{2}|\psi^{\nu}_{j,\uparrow}|^{2}+(1-|\psi^{\nu}_{j,\uparrow}|^{2})\mathrm{ln}(1-|\psi^{\nu}_{j,\uparrow}|^{2})\right)+$
$\displaystyle-\left(|\psi^{\nu}_{j,\downarrow}|^{2}\mathrm{ln}|\psi^{\nu}_{j,\downarrow}|^{2}+(1-|\psi^{\nu}_{j,\downarrow}|^{2})\mathrm{ln}(1-|\psi^{\nu}_{j,\downarrow}|^{2})\right)$

Finally summing over all the lattice sites, the von-Neumann entropy for a
quasiparticle eigenstate is defined as,

$S^{nis}_{\nu}=\sum_{j=1}^{N}S^{nis}_{j,\nu}.$ (14)

Similar to the spinless AA model, for a purely extended quasi-particle state
$S^{nis}_{\nu}\approx(\mathrm{log}_{2}N+1)$ and for completely localized state
it is $\approx 0$.

In Fig. 7(a)-(c) results of our von Neumann entropy calculations are presented
for $\alpha_{z}/t=0.8$ and $\alpha_{y}/t=0.0$. To show the dramatic change in
vNE as we move slightly away from the critical point ($W_{c}/t\simeq 2.55$),
in Fig. 7(a) and in Fig. 7(c) we have plotted our results for $W/t=2.5$ and
$W/t=2.6$ respectively. It is clear that for $W/t=2.5$ the von Neumann entropy
increases with system size and roughly scales as expected, while it is
independent of the system sizes and the value is close to zero for $W/t=2.6$.
It is clear from the results of Fig. 7(b), at the critical point these results
do not completely follow the expected pattern of purely extended or localized
states. These results along with the localization tensor calculations and IPR
data qualitatively capture the nature of the eigenstates at the critical
point. However, it is necessary to have a more rigorous analysis to quantify
the degree of multifractality of the eigenstates at the critical point. To
address this, in the next section, we present a detail and careful analysis of
the multifractal spectrum around the critical point.

Figure 9: Top row: finite size scaling of $\alpha(q)$ with $1/L$ and bottom
row: finite size scaling of $f(\alpha(q))$ with $1/L$ for some limiting values
of the moment q. The scaling functions are given in Eq.20. The complex hopping
amplitude $\alpha_{y}/t=0.$ From left to right, $W/t=2.5,2.55$, and $2.6$. All
of these results are presented for open boundary condition.

##  VI Multifractal Spectrum and the Quasi-particle Eigenstates

“Absence of length scale” at the critical point of a phase transition has led
to the understanding that localization-delocalization (LD) transition can also
be viewed as a class of a much broader set of critical phenomena. Typically,
the critical phenomena are characterized by critical exponents. In case of LD
transition, multifractal analysis of the eigenstates plays a similar role like
the critical exponents. Following the arguments of multifractal analysis De-
luca  in our case, we start by identifying the q-th moment of the probability
of finding a quasiparticle within a linear box of length
$L\leavevmode\nobreak\ (L=Na,a=1\textrm{\leavevmode\nobreak\ in arb. unit})$
is $P(q)=\sum_{i=1}^{N}\left|\psi_{n}(i)\right|^{2q}\propto N^{-\tau(q)}$,
where $\psi_{n}$ is the quasi-particle wave function corresponding to n-th
eigenvalue and $i=1,2,\cdots,N$. The exponent $\tau(q)$ is alternatively
expressed in terms of $D(q)$ as

$\tau(q)=D(q)(q-1),$ (15)

where $D(q)$ is called the generalized dimension. In case of ergodic extended
(EE) eigenstates De-luca  $\tau(q)=q-1$ This conclusion follows from the
argument that the real space average $P(q)/N$ converges to the ensemble
average $\left\langle P_{q}\right\rangle/N=\bra{}\psi_{n}(i)\ket{{}^{2q}}$ in
the limit $N\rightarrow\infty$. Effectively it means that for EE states,
$D(q)=1$, whereas for a completely localized eigenstate $D(q)=0.$ For
multifractal states, $\tau(q)$ deviates from these two limiting cases leading
to $q$ dependence of the generalized dimension $D(q)$. These states are
extended yet non-ergodic. Out of the possible set of generalized dimesnions,
$D(2)$ is frequently used to characterize different states. In practice,
however, instead of computing $\tau(q)$ directly, an equivalent quantity
$f(\alpha)$ is evaluated. It characterizes the multifractal property of the
eigenstates. $f(\alpha)$ and $\tau(q)$ are connected by the Legendre
transformation,

$f(\alpha(q))=q\alpha(q)-\tau(q),$ (16)

where $\alpha(q)=d\tau(q)/dq$. In general, $f(\alpha)$ is a smooth non-
monotonic positive valued function having negative curvature and a global
maxima but no local minima or maxima. In fact, $f_{max}=f(\alpha(q=0))=d$,
where $d$ is the Euclidean dimension of the system Martin Janssen . From the
analysis of the function $f(\alpha)$, one can easily identify the nature of
the eigenstates. For EE states, in the thermodynamic limit $f(\alpha=1)=1$,
while $f(\alpha\neq 1)=-\infty$. For non-ergodic (NE) eigenstates
$f(\alpha(q))\rightarrow,0$ for $0<\alpha_{min}<\alpha(q)<\alpha_{max}$, while
$f_{\alpha(q=0)}=f_{max}$ appears for $\alpha(q=0)>1$. In contrast to EE and
NE states, for insulating states $f(\alpha(q))\rightarrow 0$ as
$\alpha(q)\rightarrow 0$, while $\alpha(q=0)$, i.e., the position of the
maxima of $f(\alpha)$ spectrum shifts towards larger value than 1 as the
disorder strength is increased. It is quite evident that to identify the
nature of the quasi-particle eigenstates it is sufficient to have an
estimation of $\alpha_{min}$ $(f(\alpha_{min})\rightarrow 0)$ and
$\alpha(q=0)$ ($f_{max}=f(\alpha(q=0))=d$). This allows us to use a well-
established method Martin Janssen ; Cuevas  to compute the multifractal
spectrum for our Hamiltonian. This spectrum is computed and compared for PBC
and OBC. Both of these boundary conditions lead to identical conclusions.

###  VI.1 Calulcation of Multifractal Spectrum

Before discussing the results, we briefly summarize the key steps of to
compute the multifractal spectrum. Initially the lattice is divided into small
boxes of linear size $l<L$ The first step is to find the normalized box-
probability,

$\mathcal{P}_{k}(l,q)=\frac{\mathcal{P}_{k}^{q}(l)}{\sum_{j=1}^{N_{b}}\mathcal{P}_{j}^{q}(l)},$
(17)

where $1\leq k\leq N_{b}=L/l$ represents the $k$-th box and
$\mathcal{P}_{k}(l,q)=\sum_{i\in
l_{k}}\left|\psi_{n}(i)\right|^{2q},l_{k}=l\leavevmode\nobreak\
\forall\leavevmode\nobreak\ k,$ is the probability of the $n$-th eigenstate.
Then $\alpha(q,L)$ and $f(\alpha(q,L))$ are obtained from the following
relations;

$\displaystyle\alpha(q,L)$ $\displaystyle=$
$\displaystyle\underset{\delta\rightarrow
0}{\mathrm{lim}}\frac{\sum_{k=1}^{N_{b}}\mathcal{P}_{k}(l,q)\textrm{ln}(\mathcal{P}_{k}(l,1))}{\textrm{ln}\delta}$
(18) $\displaystyle f(\alpha(q,L))$ $\displaystyle=$
$\displaystyle\underset{\delta\rightarrow
0}{\mathrm{lim}}\frac{\sum_{k=1}^{N_{b}}\mathcal{P}_{k}(l,q)\textrm{ln}(\mathcal{P}_{k}(l,q))}{\textrm{ln}\delta},$
(19)

where $\delta=l/L$. It is important to note that this method of computing the
multifractal spectrum is valid for $a\ll l<L$. For different system sizes $L$,
we have chosen $l$ in a way that the above condition is satisfied and
$0.1\leq\delta\leq 0.5.$ $\alpha(q,L)$ and $f(\alpha(q,L))$ have been computed
for system size upto $L=8\times 10^{3}$ and averaged over the entire energy
window till half filling. Finally, the thermodynamic limit value of
$\alpha(q)=\underset{L\rightarrow\infty}{\textrm{lim}}\alpha(q,L)$ and
$f(\alpha(q))=\underset{L\rightarrow\infty}{\textrm{lim}}f(\alpha(q,L))$ have
been estimated using finite size scaling. For finite size scaling, we propose
the following set of functions for $\alpha_{q}(L)$ and $f(\alpha_{q}(L))$,

$\displaystyle\alpha_{q}(L)$ $\displaystyle=$
$\displaystyle\alpha_{q}+a_{q}L^{-1}+b_{q}L^{-2},$ $\displaystyle
f(\alpha_{q}(L))$ $\displaystyle=$ $\displaystyle
f(\alpha_{q})+c_{q}L^{-1}+e_{q}L^{-2},$ (20)

where $\alpha_{q},f(\alpha_{q}),a_{q},b_{q},c_{q},$ and $e_{q}$ are adjustable
parameters. In Fig. 9 we present results of these scaling of $\alpha_{q}(L)$
and $f(\alpha_{q}(L))$ for three different region, at the critical point and
below and above it. Before discussing the scaling results, we first discuss
the $f(\alpha(q))$ vs $\alpha(q)$ spectrum, presented in Fig. 8. In Fig.
8(a)-(c) we present the results for PBC, while the results with OBC are
presented in Fig. 8(d)-(f). All these results are for $\alpha_{z}/t=0.8$. As
mentioned in earlier, one of the main purpose of presenting the results with
two different boundary conditions is to demonstrate that it does not affect
our fundamental conclusion about the nature of the states. Furthermore, these
results indicate that one can use OBC to calculate the multifractal spectrum
quite accurately using reasonably large system sizes, at least in 1D.

In Fig 8(a) and (d), $\alpha_{z}/t=0.8$ and $W/t=2.5$. From the results of
localization tensor, the quasi-particle states are expected to be extended and
ergodic. It is clear from the results of multifractal spectrum that, in the
thermodynamic limit we have $f(\alpha(q=0))\rightarrow 1$, while
$\alpha(q)\rightarrow 1$. These two values are not exactly $1$ as one would
expect for ideal extended ergodic states, but they are very close to the ideal
value. Using Eq. 15 and Eq. 16 we have computed $D(2)$. As expected for
metallic states, our estimated value of $D(2)$ is $0.98$ $(0.99)$ for periodic
(open) boundary conditions respectively. As we increase $W/t$ to $2.55$ (Fig.
8 (b) and (e)), we can see the dramatic change in the multifractal spectrum.
In the thermodynamic limit, $f(\alpha_{q})\rightarrow 0$ for $\alpha(q)<1$,
while $f_{max}=f(\alpha(q=0))=1$ for $\alpha(q=0)>1$. It clearly indicates
that all the states are extended, yet non-ergodic. This observation is also
supported by our estimation of $D(2)$. At the critical point $D(2)=0.89$ for
PBC, while $D(2)=0.93$ for OBC according to our estimate.

It is interesting to have a closer look at the behaviour of $f(\alpha_{q}(L))$
vs. $\alpha_{q}(L)$ spectrum with system size $L$. For different system sizes
the spectrum cross each other at some point. How these spectrum move with
increasing system size on either side of the crossing indicates the nature of
the eigenstates. For EE and NE states, the spectrum move towards $\alpha=1$
with increasing system size. This evolution of the multifractal spectrum with
system size gets reversed quite dramatically as we increase the disorder
strength just a little to the value $W/t=2.6$. In this case, with increasing
system size the spectrum moves towards $\alpha=0$ on the the left of the
crossing point, while on the right of the crossing point it moves further away
from $\alpha=1$. The results are presented in Fig. 8(c) and (f). It is evident
that in the thermodynamic limit, $f(\alpha_{q})\rightarrow 0$ for
$\alpha_{q}\simeq 0.0$, while $f_{max}=f(\alpha(q=0))=1$ for $\alpha(q=0)>>1$,
indicating that all the states are localized. From the numerical data, we find
that $D(2)=0.02(0.01)$ for PBC (OBC), as expected. These results are also
consistent with our estimation of the critical point from the localization
tensor calculations, as well as with the IPR and vNE results.

In Fig. 9, we have presented the scaling data for for
$\alpha_{y}/t=0,\alpha_{z}/t=0.8$ and few limiting values of $q$. Here results
are presented for OBC only. For PBC the results are nearly identical. To
demonstrate the dramatic change in the multifractal spectrum we have chosen
the self dual point $W_{c}/t=2.55$ and two different potential strengths just
below and above it. In Fig. 9(a)-(c), $\alpha_{q}(L)$ has been plotted with
$1/L$, while Fig. 9(d)-(f) are for $f(\alpha_{q}(L))$. From Fig. 9(a) it is
clear that just below the critical point $\alpha_{q}(L)\rightarrow 1$ with
increasing system size for all $q$. Same pattern can be observed for
$f(\alpha_{q}(L))$ as well, although for higher $q$ the convergence is not
perfect. The convergence of $f(\alpha_{q}(L))$ to 1 becomes perfect as
disorder strength is lowered slightly from $W/t=2.5$. From Fig. 9(b) we can
see that at the critical point $\alpha_{q}(L)$ and $f(\alpha_{q}(L))$ do not
converge to a single value for different $q$ in the thermodynamic limit.
$\alpha_{q=0}(L)$ converges to a value greater than 1, while it converges to a
single value much less than 1 as q increases beyond 2.0. At the same time,
$\underset{L\rightarrow\infty}{\textrm{lim}}f(\alpha_{q}(L))\rightarrow 0$ as
$q$ is increased. This indicates that the eigenstates are extended but non-
ergodic. Quite dramatically, as $W/t$ is increased just by a small amount to
$2.6$, $\underset{L\rightarrow\infty}{\textrm{lim}}{\alpha_{q=0}(L)}$
converges to a value much larger than 1, while in the thermodynamic limit
$\alpha_{q\neq 0}(L)$ and $f(\alpha_{q}(L))$ converge rapidly to the origin at
the same time with higher moment q. This indicates that the states are
localized across the entire spectrum.

Figure 10: Phase diagram in the parameter space spanned by the strength of
quasi-periodic potential $W/t$ and the spin-flip hopping amplitude
$\alpha_{z}/t$. $\alpha_{y}/t$ represents the spin conserving complex hopping
amplitude induced by RSO coupling. The dotted lines indicate the phase
boundary.

##  VII Phase Diagram

Finally, we present the phase diagram in the parameter space spanned by $W/t$
and $\alpha_{z}/t$. The phase boundaries has been obtained for increasing
strength of the complex hopping of the RSO Hamiltonian. Each of these phase
boundaries are indicated schematically by dotted line in Fig. 10. The phase is
metallic below it, while it is an insulating phase above the boundary. On each
of these boundaries, all the states are extended and non-ergodic.

##  VIII Conclusions

In conclusion, we have studied the effect of RSO coupling on the critical
behaviour of one dimensional quasi-periodic lattice, described by AAH
Hamiltonian. We have found that RSO coupling does not affect the self-dual
nature of AAH model, however the self-dual point moves towards higher strength
of the quasi-periodic potential. Moreover, individual influences of the two
different kinds of hopping induced by Rasha spin-orbit coupling, that is the
spin conserving complex hopping and spin-flip real hopping, are identical in
nature. We have found no evidence of coexistence of extended and localized
states in the entire parameter space. Furthermore, the phases are insensitive
towards the existence of any sub-gap states that might exist in the energy
spectrum depending on the lattice size and boundary conditions. In the process
of studying the MIT in this system, we have also demonstrated that can be also
used to study the transition from delocalized to localized phase in system
where the spin states mix and eigenstates are quasi-particles rather pure spin
states. For a rigorous analysis of the eigenstates, we have performed the
multifractal analysis, and it has been demonstrated that the algorithm works
equally well for PBC as well OBC. This effectively removes the constraint on
the system sizes that can be used to obtain the results in the thermodynamic
limit for quasi-periodic systems.

##  IX Acknowledgement

This work is supported by SERB (DST), India (File No. EMR/2015/001227, Diary
No. SERB/F/6197/2016-17). S.D would like to thank Aditi Chakrabarty for
helpful feedback on the manuscript.

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