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# Topological Anderson insulators induced by random binary disorders

Shu-Na Liu  Guangdong Provincial Key Laboratory of Quantum Engineering and
Quantum Materials, School of Physics and Telecommunication Engineering, South
China Normal University, Guangzhou 510006, China  Guo-Qing Zhang  Guangdong
Provincial Key Laboratory of Quantum Engineering and Quantum Materials, School
of Physics and Telecommunication Engineering, South China Normal University,
Guangzhou 510006, China  Guangdong-Hong Kong Joint Laboratory of Quantum
Matter, Frontier Research Institute for Physics, South China Normal
University, Guangzhou 510006, China  Ling-Zhi Tang  Guangdong Provincial Key
Laboratory of Quantum Engineering and Quantum Materials, School of Physics and
Telecommunication Engineering, South China Normal University, Guangzhou
510006, China  Dan-Wei Zhang  Guangdong Provincial Key Laboratory of Quantum
Engineering and Quantum Materials, School of Physics and Telecommunication
Engineering, South China Normal University, Guangzhou 510006, China
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Frontier Research
Institute for Physics, South China Normal University, Guangzhou 510006, China

###### Abstract

Different disorders lead to various localization and topological phenomena in
condensed matter and artificial systems. Here we study the topological and
localization properties in one-dimensional Su-Schrieffer-Heeger model with
spatially correlated random binary disorders. It is found that random binary
disorders can induce the topological Anderson insulating phase from the
trivial insulator in various parameter regions. The topological Anderson
insulators are characterized by the disorder-averaged winding number and
localized bulk states revealed by the inverse participation ratio in both real
and momentum spaces. We show that the topological phase boundaries are
consistent with the analytical results of the self-consistent Born approach
and the localization length of zero-energy modes, and discuss how the bimodal
probability affects the disorder-induced topological phases. The topological
characters can be detected from the mean chiral displacement in atomic or
photonic systems. Our work provides an extension of the topological Anderson
insulators to the case of correlated disorders.

##  I Introduction

In the past decades, topological states of matter have been widely
investigated in condensed-matter physics Hasan and Kane (2010); Qi and Zhang
(2011); Bansil  _et al._ (2016); Armitage  _et al._ (2018); Chiu  _et al._
(2016); Gao  _et al._ (2018); Zhang and Cao (2016); Qin and Guo (2016) and
artificial systems such as ultracold atoms Zhang  _et al._ (2018); Cooper  _et
al._ (2019); Goldman  _et al._ (2016), superconducting circuits Schroer  _et
al._ (2014); Roushan  _et al._ (2014); Tan  _et al._ (2018, 2019a, 2019b), and
photonic lattices Lu  _et al._ (2014); Ozawa  _et al._ (2019). Under strong
disorders, the topological states usually become trivial Anderson insulators
due to the fact that the eigenstates become localized and the band gap closes
Anderson (1958); Klitzing  _et al._ (1980); Evers and Mirlin (2008).
Surprisingly, it was found that topological insulators can be induced by
random disorders from trivial phases Li  _et al._ (2009), leading to the novel
interplay effect of topology and disorder. The disorder-induced topological
phases are dubbed topological Anderson insulators (TAIs) and have been widely
investigated Li  _et al._ (2009); Groth  _et al._ (2009); Jiang  _et al._
(2009); Sanchez-Palencia and Lewenstein (2010); Meier  _et al._ (2018); Li
_et al._ (2020); Mondragon-Shem  _et al._ (2014); Hsu and Chen (2020); Jiang
_et al._ (2019); Zhang  _et al._ (2020); Tang  _et al._ (2020); Titum  _et
al._ (2015); Yang  _et al._ (2021); Zhang  _et al._ (2021a); Stützer  _et al._
(2018); Peng  _et al._ (2021); Zhang and Shen (2013); Shi  _et al._ (2021);
Liu  _et al._ (2020a); Hsu and Chen (2020); Liu  _et al._ (2020b, 2021); Zhang
_et al._ (2012); Song  _et al._ (2012); Xu  _et al._ (2012); Zhang  _et al._
(2019); Yamakage  _et al._ (2013); Orth  _et al._ (2016); Xing  _et al._
(2011); Shen (2017); Prodan (2011); Wu  _et al._ (2019); Altland  _et al._
(2014); Longhi (2020). Experimental observations of TAIs were reported in
several artificial systems, such as 1D cold atomic wires Meier  _et al._
(2018), 2D photonic waveguide arrays Stützer  _et al._ (2018); Liu  _et al._
(2020a), electric circuits Zhang  _et al._ (2021a, 2019), coupled waveguides
Stützer  _et al._ (2015), and photonic quantum walks Lin  _et al._ (2021).
Notably, TAIs are induced by the uncorrelated disorders in these studies. For
instance, both uncorrelated random Meier  _et al._ (2018) and quasiperiodic
disorders Zhang  _et al._ (2021b); Tang  _et al._ (2022); Lu  _et al._ (2022)
can give rise to the TAI phase in 1D Su-Schrieffer-Heeger (SSH) chains. It was
shown that the spatial correlations in the disorder can entirely suppress the
emergence of the TAI phase on two-dimensional quantum spin Hall systems with
Gaussian correlated on-site potentials Girschik  _et al._ (2013). However, the
effects of correlated disorders on other topological phases are largely
unexplored. In particular, it remains unclear whether the TAI phase exhibits
in 1D SSH chiral chains with correlated disorders.

As a typical kind of spatially short-range correlated disorder, the random
binary disorder has been studied in localization phenomena. It was found that
the random binary disorder can induce unconventional localization-
delocalization transition Flores (1989); Hilke (1997); Dunlap  _et al._
(1990); Sedrakyan (2004); Schaff  _et al._ (2010), and the phase transitions
between Mott insulators and Bose glasses Krutitsky  _et al._ (2008) or metal
phases Semmler  _et al._ (2010). Moreover, the random site energies with a
discrete binary probability distribution can significantly influence the
electronic structure and transport properties Evangelou and Wang (1993);
Alvermann and Fehske (2005). The random binary disorder in random dimer models
can be used to explain the insulator-metal transition with new metallic states
in conducting polymers Phillips and Wu (1991) and a new transport mechanism Wu
and Phillips (1991). Realizations of the random-dimer model have been achieved
in GaAs-AlGaAs superlattices Bellani  _et al._ (1999), 1D waveguide array
fabricated infused silica Naether  _et al._ (2013), and a ultracold atomic
mixture in 1D optical lattices Schaff  _et al._ (2010). Except for these
localization effects, it is interesting to study the topological phases under
the random binary disorder. A nature question is whether the TAI phase can be
induced by this kind of correlated disorder.

In this paper, we study the topological and localization properties of the 1D
SSH model Su  _et al._ (1979); Heeger  _et al._ (1988) with random binary
disordered hoppings. We obtain the topological phase diagrams by numerically
calculating the real-space winding number, the energy gap, and the edge
states. It is found that the random binary disorder can induce the TAI phase
from the trivial insulator in a wide range of parameter regions. We also study
the influence of the bimodal probability on the topological phase. By
generalizing the existing self-consistent Born approximation (SCBA) approach
to our model, we reveal the trivial-topological phase transition driven by the
disorder-renormalization of the intracell hopping strength. An analytical
formula for the localization length of the zero-energy state is derived, whose
divergence indicates the topological phase transition. We further study the
localization properties of the system by computing the inverse participation
ratio (IPR) in both real and momentum space. We numerically show that the mean
chiral displacement of the bulk dynamics can be used to reveal the topological
phases, which can already be implemented in ultracold atomic and photonic
systems Cardano  _et al._ (2017); Maffei  _et al._ (2018); Meier  _et al._
(2018); Xiao  _et al._ (2021); Wang  _et al._ (2018); D’Errico  _et al._
(2020); Xie  _et al._ (2020, 2019); Wang  _et al._ (2021); Meng  _et al._
(2020). Finally, we discuss the TAI phase in a more general case with both
binary disordered inter- and intra-cell hoppings.

The rest of this paper is organized as follows. We first introduce the 1D SSH
model with random binary disordered hoppings in Sec. II. Section III is
denoted to present the results of the topological phase diagrams with
disorder-induced TAIs, the localization properties, and the influence of the
bimodal probability. In Sec. IV, we propose to detect the topological phases
by measuring the mean chiral displacement. A short discussion and a brief
summary are finally given in Sec. V.

##  II Model

We begin by considering the 1D SSH model with random binary disordered hopping
strengths. The tight-binding Hamiltonian consists of two sublattices (labeled
by A and B, respectively) per unit cell, which can be expressed as

${H}=\sum_{n}(m_{n}c_{A,n}^{\dagger}c_{B,n}+\mathrm{h.c.})+\sum_{n}(t_{n}c_{A,n+1}^{\dagger}c_{B,n}+\mathrm{h.c.}),$
(1)

where $c_{A(B),n}^{\dagger}$ and $c_{A(B),n}$ are the creation and
annihilation operators of electron on $A(B)$ sublattice in the $n$-th unit
cell, respectively. We consider a system has $N$ unit cells, and the total
number of sublattice sites is $L=2N$ with each sublattice site indexed by
$x\in[1,L]$.

For simplicity, we first consider the dilute case where the intercell hopping
strengths to be site-independent $t_{n}=t$, and the intracell hopping
strengths $m_{n}$ are binary disorders. In the last section, we will
generalize our results by considering $t_{n}$ also binary disordered. To do
this, $m_{n}$ is drawn from a bimodal probability distribution function Saha
_et al._ (1996); Mondal  _et al._ (2019)

$\mathcal{P}(m_{n})=P\delta(m_{n}-V_{0}-w_{a})+(1-P)\delta(m_{n}-V_{0}-w_{b}),$
(2)

which means $m_{n}$ takes bimodal value $V_{0}+w_{a}$ or $V_{0}+w_{b}$ with
probability $P$ or $1-P$ on $n$-th unit cell. $V_{0}=w_{a}=1$ is chosen as the
energy unit, and the disorder strength is defined as the difference between
$w_{a}$ and $w_{b}$: $W\equiv w_{a}-w_{b}$. For the special case that the
probability $P=0$ or $P=1$, it reduces to the clean SSH model Su  _et al._
(1979); Heeger  _et al._ (1988) of free fermions.

It is well-known that the SSH model has two topologically distinct phases and
exists two-fold degenerate zero-energy edge states Asbóth  _et al._ (2016) in
the topological insulating phase under open boundary conditions (OBCs). In
experiments, the SSH model has been realized in different systems, such as
cold atoms in 1D optical superlattices Atala  _et al._ (2013). In this clean
case, the 1D winding number of Bloch vectors in momentum space can be used to
characterize topologically trivial and nontrivial phases. In the presence of
disorders, the translational symmetry breaking and the wave vectors in
momentum space are no longer good quantum numbers, so the conventional winding
number defined in terms of Bloch vectors should be extended to the real space
Mondragon-Shem  _et al._ (2014). Here we use the real-space winding number to
characterize the topological property of our model. For a given disorder
configuration denoted by the label $s$, the corresponding real-space winding
number is given by Mondragon-Shem  _et al._ (2014):

$\nu_{s}=\frac{1}{L^{\prime}}\operatorname{Tr}^{\prime}(\Gamma
Q_{s}[Q_{s},X]),$ (3)

where $X=\mathrm{diag}(1,1,2,2,\cdots,N,N)$ denotes the unit-cell coordinate
operator,
$Q_{s}=\sum_{j}(\ket{\psi_{j}}_{ss}\bra{\psi_{j}}-\ket{\tilde{\psi}_{j}}_{ss}\bra{\tilde{\psi}_{j}})$
is the flat-band Hamiltonian with $\ket{\psi_{j}}_{s}$ being the $j$-th
eigenstate of $H$ and
$\ket{\tilde{\psi}_{j}}_{s}=\Gamma^{-1}\ket{\psi_{j}}_{s}$,
$\Gamma=\mathrm{diag}(1,-1,1,-1,\cdots,1,-1)$ is the chiral operator with
diagonal elements $1$ and $-1$ for A and B sublattices, respectively. Here
$L^{\prime}=L-2l$ is slightly less than the chain length $L$ to get rid of the
$l$ sites boundary effect at both ends, and ${\mathrm{Tr}}^{\prime}$ stands
for the trace within the central region of the length $L^{\prime}$ in the
chain. In numerical simulations, we choose $l=L/4$ to avoid boundary effects
and get quantized winding numbers. For the disordered SSH model, we use the
disorder-averaged (over $N_{s}$ configurations) winding number

$\nu=\frac{1}{N_{s}}\sum_{s=1}^{N_{s}}\nu_{s}$ (4)

as the topological invariant, where a modest configuration number $N_{s}$ is
usually sufficient in practice. Thus, $\nu=0$ and $1$ correspond to
topologically trivial and nontrivial in the limit of $L\rightarrow\infty$,
respectively.

##  III Topology and localization

In this section, we systematically investigate the topological and
localization properties of the 1D SSH model with random binary disordered
hoppings. First, we study the winding number, energy spectrum, and edged state
to consistently characterize the topological phase transition induced by
random binary hoppings. This phase transition can also be revealed by using
the SCBA method which accounts for the disorder effects into the
renormalization of the hopping strengths. We then analytically derive the
localization length of the zero-energy edge mode and numerically calculate the
real-space and momentum-space IPR. Finally, we study the influence of bimodal
probability on the topological and localization properties.

Figure 1: (Color online) (a) Disorder-averaged real-space winding number $\nu$
in the $W$-$t$ plane. The black dashed line denotes the topological phase
boundary determined by the SCBA method. The solid red line is the topological
phase boundary revealed by divergence of the localization length $\Lambda$ of
zero energy modes. (b) The winding number $\nu$, the energy gap $\Delta E$,
and the localization length $\Lambda$ as a function of $W$ for $t=1.5$. (c)
Four centered eigenenergies under OBCs as a function of $W$ for $t=1.5$. (d)
Density distribution of the $j=L/2$-th eigenstate for $t=1.5$ and $W=0,2,5.3$
(from top to bottom), respectively. Other parameters are $P=1/2$, and $L=200$.

###  III.1 Topological phase diagrams

We first investigate the topological properties of the 1D random binary
disordered SSH model with fixed $P=1/2$. By calculating the disorder-averaged
real-space winding number $\nu$ for the system of $L=200$ on the $W$-$t$
plane, we obtain the topological phase diagram shown in Fig. 1(a). In the
clean limit $W=0$, the system is in the trivial phase with $\nu=0$ for
intercell hopping $t<2$. Interestingly, there exists nontrivial $\nu=1$ phase
in the $t<2$ region with moderate disorder strength, indicating the existence
of TAIs induced by the random binary disorder. We also compute the bulk gap
under PBCs and the density distribution of the most centered two eigenstates
under OBCs. In this case, the disorder-averaged $j$-th eigenenergy $E_{j}$ and
the corresponding bulk gap $\Delta E$ are given by

$\displaystyle E_{j}$
$\displaystyle=\frac{1}{N_{s}}\sum_{s=1}^{N_{s}}E_{s,j},$ (5)
$\displaystyle\Delta E$
$\displaystyle=\frac{1}{N_{s}}\sum_{s=1}^{N_{s}}\left|E_{s,L/2+1}-E_{s,L/2}\right|.$
(6)

In Fig. 1 (b), we plot $\nu$ and $\Delta E$ as functions of $W$ for $t=1.5$.
Our results show that the topological phase transition points revealed by the
abrupt changes of real-space winding number $\nu$ under OBCs and closure
points of bulk gap $\Delta E$ under PBCs agree with each other. In addition,
there are two topological phase transitions including one from trivial phase
to nontrivial phases for $W\simeq 0.88$ and another from nontrivial to trivial
phases for $W\simeq 3.12$, respectively. In the TAI phase when $0.88\lesssim
W\lesssim 3.12$ under OBCs, two-fold degenerate zero-energy modes exhibit in
the energy spectrum shown in Fig. 1(c). The corresponding density
distributions of the $j=L/2$-th eigenstate for three typical disorder
strengths $W=0,2,5.3$ are plotted in Fig. 1(d). The system in the clean limit
is in the trivial band insulator with the extended bulk states at the center
of the energy spectrum. In the TAI phase under the moderate disorder strength,
the two centered eigenstates become zero-energy edge mode localized at two
ends. For sufficient disorder strength, the system becomes a trivial Anderson
insulator where the eigenstates are localized in the bulk.

Now we use the SCBA analysis to explain the disorder-induced trivial-
topological phase transition. The disorder effect can be renormalized to the
parameters of the clean Hamiltonian based on an effective medium theory and
the SCBA method Groth  _et al._ (2009); Chen  _et al._ (2015); Park  _et al._
(2017); Liu  _et al._ (2016). The key ingredient is to self-consistently
obtain the self-energy caused by the disorder, which acts as the
renormalization of the clean Hamiltonian. For our model, the self-energy term
$\sum(E_{F})$ satisfies the following self-consistent equation

$\frac{1}{E_{F}-H_{q}(k)-\Sigma\left(E_{F}\right)}=\left\langle\frac{1}{E_{F}-H_{\text{eff}}(k,W)}\right\rangle_{s}.$
(7)

Here $\braket{\cdots}_{s}$ means taking average over sufficient disorder
configurations, $H_{q}(k)$ is the clean Hamiltonian ($W=0$) in momentum space
with intracell hopping strengths $m=V_{0}+w_{a}$, $E_{F}\equiv 0$ is the Fermi
energy, and the effective binary-disordered Hamiltonian is given by

$\displaystyle H_{\mathrm{eff}}(k,W)=(m^{\prime}+t\cos k)\sigma_{x}+t\sin
k\sigma_{y},$ (8)

where $m^{\prime}=m$ with probability $P$, and $m^{\prime}=m-W$ with
probability $1-P$. The self-consistent self-energy equation under the first-
order SCBA can be written as Gonçalves  _et al._ (2018)

$\Sigma=\frac{\sigma^{2}}{2\pi}\int_{0}^{2\pi}\left\langle\frac{1}{G_{0}^{-1}-\Sigma}\right\rangle_{s}dk,$
(9)

where $G_{0}^{-1}=-H_{\mathrm{eff}}(k,W)$ is the Green’s function, and
$\sigma^{2}=W^{2}/4$ is the variance of the binary disorder distribution. The
self energy can be parameterized as
$\Sigma=\Sigma_{\mathrm{I}}I+\Sigma_{x}\sigma_{x}+\Sigma_{y}\sigma_{y}+\Sigma_{z}\sigma_{z}.$
As the inverse of Hamiltonians (8) has only $x$ and $y$ components, so the
self energy $\Sigma_{z}=0$ can satisfy Eq. (9). The $\Sigma_{y}$ component
from the off-diagonal entries of $G_{0}$ can be explicitly calculated as
$\Sigma_{y}=\int_{0}^{2\pi}\mathrm{Im}[\braket{G_{0}}_{s}]dk=0$. Finally, we
obtain the non-vanishing $\Sigma_{x}$ component, which renormalizes the
intracell hopping strength as

$\bar{m}=m+\Sigma_{x}(W).$ (10)

In the clean limit, $\bar{m}=2$ and the phase transition point occurs at
$\bar{m}=t=2$. For the weak disorder $W\leq 2$ and under the SCBA, the
renormalization of the hopping strength indicates the topological phase
transition happens at the critical value $W_{c}$ where $\bar{m}=t$. We
numerically obtain the self energy $\Sigma_{x}(W)$ and determine the
topological phase transition boundary for weak disorder strength. The SCBA
method fails at large disorders for $W>2$. We plot the left-hand-side phase
boundary revealed by this method as the black dashed line in Fig. 1(a), which
is consistent with the topological transition boundary revealed by the winding
number.

###  III.2 Localization properties

Now we show that the divergence of the localization length of zero-energy
states is related to the topological phase transition due to the appearance of
exponentially localized zero-energy edge modes. The Schrödinger equation for
the zero-energy eigenstate $\psi$ in this case reads $\hat{H}\psi=0$, which
leads to $m_{n}\psi_{n,A}+t_{n}\psi_{n+1,A}=0$ and
$t_{n}\psi_{n,B}+m_{n+1}\psi_{n+1,B}=0$. We can obtain the following solutions

$\displaystyle\psi_{N,A}$
$\displaystyle=(-1)^{N}\prod_{n=1}^{N}\frac{m_{n}}{t_{n}}\psi_{1,A},$ (11)
$\displaystyle\psi_{N,B}$
$\displaystyle=(-1)^{N}\prod_{n=1}^{N}\frac{t_{n}}{m_{n+1}}\psi_{1,B}.$ (12)

The inverse of the localization length is then given by (in the thermodynamic
limit $N\rightarrow\infty$) Mondragon-Shem  _et al._ (2014)

$\Lambda^{-1}=\max\\{\lim_{N\rightarrow\infty}\frac{1}{N}\ln{|\psi_{N,A}|},\lim_{N\rightarrow\infty}\frac{1}{N}\ln{|\psi_{N,B}|}\\}.$
(13)

Taking $\psi_{1,A}=\psi_{1,B}=1$, we can obtain

$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\ln{|\psi_{N,A}|}$
$\displaystyle=\lim_{N\rightarrow\infty}\frac{1}{N}\ln{|\psi_{N,B}|}$ (14)
$\displaystyle=|\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}(\ln{|t_{n}|}-\ln{|m_{n}})|.$

Inserting Eq. (14) into Eq. (13), we obtain inverse of the localization length
for the zero-energy mode as

$\Lambda^{-1}=|\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}(\ln{|t_{n}|}-\ln{|m_{n}|})|.$
(15)

For our random binary disordered SSH model with $t_{n}=t$ and
$m_{n}=V_{0}+w_{n}$, we can obtain the analytical result

$\Lambda^{-1}=|\ln|t|-P\ln(2)-(1-P)\ln|2-W||,$ (16)

when $V_{0}=1$, and $W=w_{a}-w_{b}$. We plot $\Lambda^{-1}=0$ as the red solid
line in Fig. 1 (a), which shows the topological phase boundary can be revealed
by the divergence of $\Lambda$. To be more clear, we also plot $\Lambda$ as
the blue solid line in Fig. 1(b). These results indicate that the topological
phase transition from the trivial band insulator to the TAI in this disordered
system can be revealed by the winding number, the energy gap, the
renormalization under the SCBA method, and the divergence of localization
length.

Figure 2: (Color online) The disorder-averages IPR of each eigenstate for
$L=200$ and $t=1.5$ under OBCs (a) and PBCs (b), respectively. (c) The
disorder- and eigenstate-averaged IPR $\bar{I}_{r}$ in the real space as
functions of $W$ and $t$ for $L=200$ under PBCs. (d) The momentum (real) space
IPR ${\bar{I}_{k}}$ ($\bar{I}_{r}$) as a function of $W$ for $t=1.5$ and
various $L$s.

We further investigate the bulk states localization induced by the random
binary disorder by computing the IPR in both real and momentum space. The
disorder-averaged real-space IPR of the $j$-th eigenstate reads

$\bar{I}_{j}=\frac{1}{N_{s}}\sum_{s=1}^{N_{s}}\sum_{x=1}^{L}\left|\psi_{j,x}^{(s)}\right|^{4}$
(17)

where $\psi_{j,x}^{(s)}$ denotes the probability amplitude of the $j$-th
normalized eigenstate at the $x$-th lattice site for the disorder
configuration $s$. We present $\bar{I}_{j}$ for $L=200$ and $t=1.5$ under OBCs
and PBCs in Figs. 2(a) and (b). The IPRs of the centered two eigenstates
($j=100,101$) show a large value in Fig. 2(a), indicating the existence of two
zero-energy edge states in the topological region under OBCs. The difference
of IPRs between OBCs and PBCs is merely at the centered two states in the
topological region. Therefore, we can use IPR under PBCs to characterize the
localization of bulk states without topological edge modes.

To characterize the bulk localization, the disorder- and eigenstate-average
IPR in the real space under PBCs is given by

$\bar{I}_{r}=\frac{1}{N_{s}}\frac{1}{L}\sum_{s=1}^{N_{s}}\sum_{j=1}^{L}\sum_{x=1}^{L}\left|\psi_{j,x}^{(s)}\right|^{4}$
(18)

For extended state, real-space IPR $\bar{I}_{r}\sim 1/L$ tends to $0$ in the
thermodynamic limit $L\rightarrow\infty$, while
$\bar{I}_{r}\sim\mathcal{O}(1)$ for localized states. The numerical result of
$\bar{I_{r}}$ in the $W$-$t$ plane for system size $L=200$ is shown in Fig.
2(c), where both extended and localized phases can be clearly seen. We also
consider the momentum-space IPR, and the disorder- and eigenstate-average IPR
in the momentum space is given by

$\bar{I}_{k}=\frac{1}{N_{s}}\frac{1}{L}\sum_{s=1}^{N_{s}}\sum_{j=1}^{L}\sum_{k_{d}}^{2\pi}\left|\psi_{j,k_{d}}^{(s)}\right|^{4}$
(19)

The probability amplitude $\psi_{j,k_{d}}^{(s)}$ is obtained by taking the
Fourier transformation
$\psi_{j,k_{d}}^{(s)}=\frac{1}{L}\sum_{x=1}^{L}e^{-ik_{d}\cdot
x}\psi_{j,x}^{(s)}$ where the wave vector is $k_{d}=2\pi d/L$ with integer
$d\in[1,L]$. A localized state in the real space becomes extended in the
momentum space, and thus $\bar{I}_{k}\rightarrow 0$ in the large $L$ limit
indicates localized phase. The results of $\bar{I}_{k}$ and $\bar{I}_{r}$ as
functions of $W$ for $t=1.5$ are shown in Fig. 2(d). Except those near
$W=0,4$, $\bar{I}_{k}$ can approach to zero by increasing the lattice length
$L$, which indicates the localized phase. For $W=0,4$, the vanishing of real-
space $\bar{I}_{r}$ and large value of momentum-space $\bar{I}_{k}$ indicate
the fully extended phase.

Figure 3: (Color online) Top panel: The topological phase diagram for $P=0$
(a) and $P=1$ (b) of $L=200$ systems revealed by the disorder-averaged winding
number $\nu$ in the $W$-$t$ plane. Bottom panel: The winding number $\nu$ (c)
and real-space IPR $\bar{I}_{r}$ (d) as functions of $P$ and $W$ for $t=1.5$
in $L=200$ systems. Solid red curves and dashed black lines indicate the
topological phase boundaries by the divergence of localization length
$\Lambda$s and by the SCBA method, respectively.

###  III.3  The influence of the bimodal probability

It has been shown that the bimodal probability has the influence on the
localization lengths in binary disordered systems Dunlap  _et al._ (1990);
Evangelou and Wang (1993). Now we consider how the phase diagram and
localization properties are affected by the bimodal probability $P$ in the SSH
model. We first consider the topological properties in two limits of $P=0$ and
$P=1$. The corresponding topological phase diagrams on the $W$-$t$ plane are
plotted in Figs. 3(a) and 3(b), respectively. For $P=0$ in Fig. 3(a), we can
see that there exist disorder-driven TAIs for $t$ varying from $0$ to $2$. The
parameter region of the TAI phase reduces when decreasing $t$. For $P=1$ in
Fig. 3(b), the phase diagram is divided into topological trivial and
nontrivial regions by the horizontal line $t=2$, which indicates the absence
of disorder-induced TAIs. In Fig. 3 (c), we show the topological phase diagram
on the $W$-$P$ plane for $t=1.5$. The parameter region of TAIs shrinks to zero
as $P$ increases to unity. The solid red curves in Figs. 3(a-c) correspond to
the critical case of $\Lambda^{-1}\rightarrow 0$. The dashed black lines in
Figs. 3(a) and 3(c) are phase boundaries revealed by the renormalized hopping
term $\bar{m}=t$ in the SCBA calculation. The corresponding real-space IPR
$\bar{I}_{r}$ as functions of $W$ and $P$ is plotted in Fig. 3(d), which shows
that the system is more localized for moderate $W$ and $P$. Therefore, by
increasing $P$, there exist the topological-trivial phase transition and the
localization-delocalization transition in the SSH model.

Figure 4: (Color online) The time evolution of $C$ for $P=1/2$ (a) and $W=1$
(b) with $t=1.5$ and $L=100$ under OBCs. The time- and disorder-averaged
$\bar{C}$ as a function of $W$ for $P=1/2$ (c) and as a function of $P$ for
$W=1$ (d) for $L=10,30,100,200$. The purple dashed line and red dot-dashed
lines in (a) and (b) correspond to the values of $\bar{C}\approx\nu=0,1$ for
topological and trivial phases, respectively.

##  IV Proposed detections

Finally, we propose that the TAI phase in this binary disordered model can be
detected from the mean chiral displacement, as recently demonstrated in atomic
and photonic systems Cardano  _et al._ (2017); Maffei  _et al._ (2018); Meier
_et al._ (2018); Xiao  _et al._ (2021); Wang  _et al._ (2018); D’Errico  _et
al._ (2020); Xie  _et al._ (2020, 2019); Wang  _et al._ (2021); Meng  _et al._
(2020); Xu  _et al._ (2018); St-Jean  _et al._ (2021). The time evolution of
the mean chiral displacement $C_{s}(\tau)$ for a single disorder configuration
$s$ is given by

$\displaystyle C_{s}(\tau)=2_{s}\langle\psi(\tau)|\Gamma
X|\psi(\tau)\rangle_{s},$ (20)

where $\tau$ denotes time, $\Gamma$ and $X$ are chiral and unit-cell
coordinate operators previously used in the real-space winding number. The
disorder average mean chiral displacement is then written as

$\displaystyle C(\tau)=\frac{1}{N_{s}}\sum_{s=1}^{N_{s}}C_{s}(\tau).$ (21)

After a long time evolution $\tau=T$, the mean chiral displacement averaged
over disorders and time is given by

$\displaystyle\bar{C}=\frac{1}{T}\int_{0}^{T}C(\tau)d\tau.$ (22)

We choose the initial state at the center of the lattice and simulate the
quantum dynamics for many disorder configurations, with typical results of
$C(\tau)$ and $\bar{C}$ shown in Fig. 4. As shown in Figs. 4(a) and 4(b), the
dynamics of $C(\tau)$ shows a transient and oscillatory behavior and its time-
and disorder-averaged $\bar{C}$ converges to the winding number $\nu$ in both
clean and disorder cases in the long time limit Cardano  _et al._ (2017);
Maffei  _et al._ (2018).

We further plot $\bar{C}$ as a function of $W$ for $P=1/2$ and $t=1.5$ in Fig.
4(c) for different lattice sizes. We can see that $\bar{C}$ converges to an
integer value which approaches to $0$ in the trivial phase and $1$ in the
topological phase by increasing the lattice size $L$, which agrees with the
result of $\nu$ shown in Fig. 1(b). We also numerically calculate $\bar{C}$ as
a function of $P$ for $W=1$ and $t=1.5$ in Fig. 4(d), which shows that
$\bar{C}\approx 1$ for small $P$ and drops to zero for $P\gtrsim 0.5$. The
decrement of $\bar{C}$ by increasing $P$ is in good agreement with the winding
number $\nu$ obtained in Fig. 3(c).

Figure 5: (Color online) (a) The topological phase diagram in the
$W$-$W^{\prime}$ plane for $t=1.5$, $P=P^{\prime}=0.5$. Solid red curves
indicate the topological phase boundaries revealed by the divergence of the
localization length $\Lambda$. (b) The disorder-averaged real-space IPR
$\bar{I}_{r}$ as a function of $W$ and $W^{\prime}$ for $t=1.5$,
$P=P^{\prime}=0.5$. The lattice size is $L=200$.

##  V discussion and conclusion

Before concluding, we discuss the 1D SSH model with overall random binary
disordered hoppings and reveal the TAI phase in this general case. The
intercell hoppings $t_{n}$ are drawn from the following bimodal probability
distribution function

$\mathcal{P}^{\prime}(t_{n})=P^{\prime}\delta(t_{n}-t)+(1-P^{\prime})\delta(t_{n}-t+W^{\prime}),$
(23)

where $W^{\prime}$ is the disorder strength of the intercell hopping. When
$W^{\prime}=0$, it returns to the previous case discussed in Sec. II. The
trivial-topological and topological-trivial phase transition points ($W\simeq
0.88$ and $W\simeq 3.12$) shown in Fig. 5(a) consist with results in Fig. 1(b)
when $t=1.5$. When increasing the disorder strength $W^{\prime}$ of intercell
hoppings, we find the TAI region shrinking and finally disappearing when
$W^{\prime}=1.5$. Further increasing $W^{\prime}$ can revive and broaden the
topological region. When $0<W^{\prime}<1.5$, the effective intercell hopping
$\bar{t}<t=1.5$, and from the SCBA renormalized relation Eq. (10) one can see
that the critical disorder $W_{c}$ from trivial to topology is larger than the
corresponding point for $W^{\prime}=0$ to satisfy $\bar{m}=\bar{t}$, and the
TAI region induced by $W$ is shrinking. When $W^{\prime}=1.5$, the probability
of $t_{n}=0$ leads to a breaking inter-cell hopping and vanishes the TAI
phase. Further increasing $W^{\prime}$ effectively enlarges the hopping
strength of $|t_{n}|$ and thus the topological phase revives. In this general
case, we obtain the inverse of the localization length as

$\displaystyle\Lambda^{-1}=$
$\displaystyle\left|P^{\prime}\ln\right|t\left|+\left(1-P^{\prime}\right)\ln\right|t-W^{\prime}|$
$\displaystyle-P\ln 2-(1-P)\ln|2-W||$ (24)

When $W^{\prime}=0$, Eq. (V) returns to Eq. (16). In Fig. 5(a), we plot the
solid red lines where $\Lambda^{-1}=0$ on the $W$-$W^{\prime}$ plane for
$t=1.5$ and $P^{\prime}=P=0.5$, which are consistent with the real-space
winding number. We also calculate the disorder-averaged real-space IPR
$\bar{I}_{r}$ in Fig. 5(b), which indicates the bulk states are localized in
the topological phase region.

In summary, we have explored the topological and localization properties of
the SSH model with the random binary disorder. We have shown that the random
binary disorder can induce TAIs in a wide range of parameter regions of
topological phase diagrams. We have analyzed trivial-topological phase
transitions by calculating the renormalization of hopping strengths with the
SCBA method and topological phase boundaries by calculating the localization
length of the zero-energy mode. The localization and the effect of the bimodal
probability have been studied. We have also numerically demonstrated that the
revealed topological phases can be detected from the mean chiral displacement,
which could be measured from the bulk dynamics in atomic or photonic systems.
This work provides an extension of the TAIs to the case of spatially
correlated disorders.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(No. 12174126, and No. 12104166), the Key-Area Research and Development
Program of Guangdong Province (Grant No. 2019B030330001), the Science and
Technology of Guangzhou (Grant No. 2019050001), and the Guangdong Basic and
Applied Basic Research Foundation (Grants No. 2020A1515110290 and No.
2021A1515010315).

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