††thanks: Corresponding author
<EMAIL_ADDRESS>Corresponding author
<EMAIL_ADDRESS>
# Disorder-induced topological phase transition in a 1D mechanical system

Xiaotian Shi  Aeronautics and Astronautics, University of Washington, Seattle,
Washington 98195, USA  Ioannis Kiorpelidis  LAUM, CNRS-UMR 6613, Le Mans
Université, Avenue Olivier Messiaen, 72085 Le Mans, France  Department of
Physics, University of Athens, 15784, Athens, Greece  Rajesh Chaunsali  LAUM,
CNRS-UMR 6613, Le Mans Université, Avenue Olivier Messiaen, 72085 Le Mans,
France  Vassos Achilleos  LAUM, CNRS-UMR 6613, Le Mans Université, Avenue
Olivier Messiaen, 72085 Le Mans, France  Georgios Theocharis  LAUM, CNRS-UMR
6613, Le Mans Université, Avenue Olivier Messiaen, 72085 Le Mans, France
Jinkyu Yang  Aeronautics and Astronautics, University of Washington, Seattle,
Washington 98195, USA

###### Abstract

We numerically investigate the topological phase transition induced purely by
disorder in a spring-mass chain. We employ two types of disorders – chiral and
random types – to explore the interplay between topology and disorder. By
tracking the evolution of real space topological invariants, we obtain the
topological phase diagrams and demonstrate the bilateral capacity of disorder
to drive topological transitions, from topologically nontrivial to trivial and
vice versa. The corresponding transition is accompanied by the realization of
a mechanical Topological Anderson Insulator. The findings from this study hint
that the combination of disorder and topology can serve as an efficient
control knob to manipulate the transfer of mechanical energy.

##  I Introduction

With the discovery of Topological Insulators in condensed matter [1, 2], there
have been tremendous efforts to explore topological phases in classical wave
systems, including photonic [3, 4, 5, 6], acoustic [7, 8, 9], and elastic [10,
11, 12, 13] systems. Based on the bulk-edge correspondence for topological
insulators, one can predict the existence of topological states on the
boundary of the system by characterizing the bulk of a material with an
invariant. Such states are protected by the internal symmetries and are immune
to certain types and levels of disorder [14]. As disorder inevitably destroys
the periodicity of the system, it is natural to think that the presence of
disorder tends to suppress topological properties. However, the discovery of
Topological Anderson Insulators (TAI) [15, 16, 17, 18, 19, 20, 21] suggests
that disorder can actually induce the abnormal transition from topologically
trivial to nontrivial states, which brings the studies on the interplay of
topology and disorder into a new era. Later, the concept of TAI has been
further illustrated and experimentally verified in atomic wires [22], photonic
systems [23], and recently in acoustic waveguides [24]. In all these cases,
the tight binding model of 1D Su-Schrieffer-Heeger (SSH) was used. However,
the interplay of both topology and disorder in lattices described by a second
order differential equations in time, capable of describing the dynamics of
not only a variety of mechanical systems (e.g., spring-mass model) but also
several electric structures (LC circuits), has not been explored.

Here, inspired by previous studies, we propose a dimer spring-mass system
mimicking the 1D Su-Schrieffer-Heeger (SSH) model [25] to realize the
topological phase transition. Despite some similarities with previous models,
this mechanical system has some important differences that can lead to
distinctive behaviors in the presence of disorder. First, the band gap region
is around a finite frequency as opposed to zero frequency in the SSH model.
Second, in the SSH model, hopping disorder only changes the energy spectrum
while preserving system’s chiral symmetry. On the contrary, disorder in the
spring stiffness of a spring-mass chain usually breaks the chiral symmetry,
since the perturbation is reflected in the diagonal entries of the dynamical
matrix as well. Third, as noted earlier, the equations of motions are governed
by the second-order differential equations in time, such that the topological
characterization of real space obtained from the transient response of the
system should be handled carefully. Therefore, to understand the effect of
disorder on the topological properties of this classical system
comprehensively, we introduce two types of disorders in our spring-mass chain:
i) a chiral (chiral-symmetry preserving) and ii) a random disorder. To probe
the topological properties for various disorder scenarios, we use three kinds
of topological invariants defined in the real space (displacements of the
masses). Moreover, we keep track of the localization length in the system to
capture the boundaries between topologically distinct phases. While the random
type of disorder is more common in mechanical settings, we demonstrate that in
the carefully designed setting of chiral disorder, we can achieve both kinds
of topological phase transitions, from topologically nontrivial to trivial and
vice versa. The later type leads to the mechanical analogue of TAI.

##  II Model

###  II.1 Equations of Motion

In this work, we consider a $1$D dimer spring-mass chain, as shown in Fig. 1,
which is composed of particles of uniform mass ($m=0.01$ throughout this
study) connected by alternating springs. All the particles are attached to the
ground with an onsite spring ($K_{0}$), which acts in the horizontal
direction. Each unit cell contains two particles. The springs located within
the unit cell are defined as intracell springs ($K_{a}$) while those
connecting neighboring unit cells are called intercell springs ($K_{e}$). We
assume the 1D dimer chain contains $n$ unit cells, that is $N=2n$ particles
and $N+1$ inter and intracell springs in total. By imposing fixed boundary
conditions on both ends, the equations of motion of the system can be written
as:

$\displaystyle{m}{\ddot{u}_{1}}$ $\displaystyle=$ $\displaystyle-
k_{1}u_{1}-k_{2}(u_{1}-u_{2})-k_{1}^{0}u_{1},$
$\displaystyle{m}{\ddot{u}_{j}}$ $\displaystyle=$ $\displaystyle
k_{j}(u_{j-1}-u_{j})-k_{j+1}(u_{j}-u_{j+1})-k_{j}^{0}u_{j},\quad j\in[2,N-1]$
$\displaystyle{m}{\ddot{u}_{N}}$ $\displaystyle=$ $\displaystyle
k_{N}(u_{N-1}-u_{N})-k_{N+1}u_{N}-k_{N}^{0}u_{N},$ (1)

where $u_{j}$ is the displacement of the $j$th particle, $k_{j}$ is the spring
constant between the $(j-1)$th and $j$th particles, and $k_{j}^{0}$ is the
$j$th onsite spring constant, such that $k_{j}=K_{e}$ for odd $j$,
$k_{j}=K_{a}$ for even $j$, and $k_{j}^{0}=K_{0}~{}\forall j$.

We can rearrange the equations of motion into a matrix form as follow:

$\ddot{\mathbf{U}}(t)+\mathbb{D}\mathbf{U}(t)=0,$ (2)

where $\mathbf{U}(t)$ and $\mathbb{D}$ are the displacement vector of length
$N$ and the dynamical matrix of dimension $N\times N$, respectively. The
dynamical matrix is real and symmetric and takes the following form:

$\mathbb{D}=\frac{1}{m}\begin{bmatrix}K_{a}+K_{e}+K_{0}&-K_{a}&...&0&0\\\
-K_{a}&K_{a}+K_{e}+K_{0}&-K_{e}&...&0\\\ ...&...&...&...&...\\\
0&...&-K_{e}&K_{a}+K_{e}+K_{0}&-K_{a}\\\
0&0&...&-K_{a}&K_{a}+K_{e}+K_{0}\end{bmatrix}.$ (3)

Figure 1: Schematic of the 1D dimer spring-mass chain with ground springs.
Unit cell is highlighted with the box.

To get the system’s time-history response, we can rewrite Eq. (2) as a first-
order state equation

$\dot{\mathbf{X}}=\mathbb{A}\mathbf{X},$ (4)

where $\mathbf{X}=\begin{bmatrix}\mathbf{U}(t)\\\ \mathbf{V}(t)\end{bmatrix}$
is the state vector consisting of displacement and velocity components of all
the particles, and $\mathbb{A}={\begin{bmatrix}0&{\mathbb{I}}\\\
{-\mathbb{D}}&0\end{bmatrix}}_{2N\times 2N}$ with $\mathbb{I}$ being the
identity matrix.

###  II.2 Spectrum

To perform the eigen analysis, we assume a harmonic system response and
substitute the ansatz $\mathbf{U}(t)=\mathbf{U}e^{i\omega t}$ into Eq. (2) and
get:

$\mathbb{D}\mathbf{U}_{j}=\omega_{j}^{2}\mathbf{U}_{j}$ (5)

where $\mathbf{U}_{j}$ is an eigenvector corresponding to the eigenfrequency
$\omega_{j}$. We note that after the removal of the diagonal term
$(K_{a}+K_{e}+K_{0})/m$ from the dynamical matrix of Eq. (3), which shifts the
spectrum to the frequency $\omega_{0}=\sqrt{\frac{K_{a}+K_{e}+K_{0}}{m}}$, the
remaining matrix obeys the anti-commutative relation:

$\Gamma[\mathbb{D}-\omega_{0}^{2}\mathbb{I}]+[\mathbb{D}-\omega_{0}^{2}\mathbb{I}]\Gamma=0,$
(6)

where $\mathbb{I}_{N\times N}$ denotes the identity matrix and $\Gamma$ is the
chiral operator (see Appendix A for details). The system in this case is said
to possess the chiral symmetry and in combination with the time reversal
symmetry, it results in a symmetric square of the spectrum ($\omega^{2}$)
around the mid-gap square frequency ($\omega_{0}^{2}$). In addition, all the
eigenvectors have their chiral partner, namely for every eigenvector
$\mathbf{U}_{k}$ with eigenfrequency $\omega_{k}$, there is an eigenvector
$\mathbf{U}_{l}$ with eigenfrequency $\omega_{l}$ such that
$\Gamma\mathbf{U}_{k}=\mathbf{U}_{l}$ and
$\omega_{k}^{2}-\omega_{0}^{2}=-(\omega_{l}^{2}-\omega_{0}^{2})$.

###  II.3 Disorder arrangement

Although in the clean limit (without disorder), the dynamic matrix of the
dimer spring-mass chain shares the same topological property with the
Hamiltonian matrix of a 1D SSH chain, these two systems become quite different
in the presence of disorder. Mathematically speaking, the diagonal terms of
the SSH Hamiltonian matrix depend only on the onsite energies. By introducing
disorder on the hoppings in the SSH tight-binding model, we will only vary the
off-diagonal terms of the SSH Hamiltonian matrix. In contrast, the diagonal
terms of the dynamic matrix of a spring-mass chain are the summation of two
neighboring springs and the onsite ground spring. In consideration of such
difference, we introduce two types of disorder to study their effects on
topological phase transitions in the 1D mechanical system. The disorder
strengths of intercell and intracell springs are noted as $W_{e}$ and $W_{a}$,
respectively. Then, the $j$th disordered spring stiffness can be written as:

$k_{j}=\begin{cases}K_{e}+\delta_{j}=K_{e}+W_{e}\epsilon_{j}&\quad\text{if
}j\text{ is odd}\\\ K_{a}+\delta_{j}=K_{a}+W_{a}\epsilon_{j}&\quad\text{if
}j\text{ is even}\end{cases}$ (7)

where $\epsilon_{j}$ are random, independent numbers chosen uniformly from the
range $[-1,1]$.

First, we discuss the case of chiral disorder, which we call Type I disorder
in this study. For this particular type, in order to keep the chiral symmetry
of the system, we choose the ground springs to take values
$k_{j}^{0}=K_{0}+\delta_{j}^{0}$ with
$\delta_{j}^{0}=-\delta_{j}-\delta_{j+1}$. By doing so, we balance out the
disorders of the coupling springs, and thus the diagonal terms of the spring
matrix are constant. Then, the dynamical matrix of the system with chiral
disorder can be expressed as:

$\mathbb{D}^{I}=\frac{1}{m}\begin{bmatrix}K_{a}+K_{e}+K_{0}&-{K_{a}}-\delta_{2}&...&0&0\\\
-{K_{a}}-\delta_{2}&K_{a}+K_{e}+K_{0}&-{K_{e}}-\delta_{3}&...&0\\\
...&...&...&...&...\\\
0&...&{-K_{e}}-\delta_{N-1}&K_{a}+K_{e}+K_{0}&{-{K_{a}}}-\delta_{N}\\\
0&0&...&{-{K_{a}}}-\delta_{N}&K_{a}+K_{e}+K_{0}\end{bmatrix}.$ (8)

Note that we have again
$\Gamma[{\mathbb{D}}^{I}-\omega_{0}^{2}\mathbb{I}]+[{\mathbb{D}}^{I}-\omega_{0}^{2}\mathbb{I}]\Gamma=0$.
Therefore, the disorder matrix $\mathbb{D}^{I}-\omega_{0}^{2}\mathbb{I}$ is
chiral symmetric, as in the clean case, and a symmetric spectrum is expected
to be formed around $\omega_{0}^{2}$.

We call the random disorder Type II disorder. Here, we only introduce
disorders independently on intercell ($K_{e}$) and intracell ($K_{a}$)
springs, while keeping the onsite springs unperturbed. In this case, the
dynamical matrix takes the following form

$\mathbb{D}^{II}=\frac{1}{m}\begin{bmatrix}K_{a}+K_{e}+K_{0}+\Delta_{1}&-{K_{a}}-\delta_{2}&...&0&0\\\
-{K_{a}}-\delta_{2}&K_{a}+K_{e}+K_{0}+\Delta_{2}&-{K_{e}}-\delta_{3}&...&0\\\
...&...&...&...&...\\\
0&...&{-{K_{e}}}-\delta_{N-1}&K_{a}+K_{e}+K_{0}-\Delta_{N-1}&{-{K_{a}}}-\delta_{N}\\\
0&0&...&{-{K_{a}}}-\delta_{N}&K_{a}+K_{e}+K_{0}-\Delta_{N}\end{bmatrix},$ (9)

where ${\Delta_{j}=\delta_{j}+\delta_{j+1}}$. As one can easily check, due to
the non-constant diagonal terms, the disorder matrix $\mathbb{D}^{II}$ is not
chiral, and the spectrum is not symmetric around the $\omega_{0}^{2}$ anymore.

##  III Topological characterization

For an infinitely-long clean dimer chain, its topological property could be
characterized by the winding number defined in the wave vector space [14]. It
is quantized and can take only integer values in a system possessing chiral
symmetry. However, such formula cannot be applied to the disorder system
directly, since the translational symmetry is broken. Therefore, we need to
handle the topological invariant in the real space. Following Ref. [22], we
introduce three types of topological invariants calculated by the real space
wave functions, specifically Local Topological Marker ($\nu$), mean chiral
displacement [$C(t)$] and infinite-time limit of mean chiral displacement
($C_{\infty}$).

Local Topological Marker (LTM).—This marker is based on the eigenfuctions of
the system and gives a local value for the topological invariant when this is
evaluated away from the boundaries [22], see the appendix of [22] for more
information. First, we construct a modal matrix $\mathbb{U}$ by arranging all
the normalized eigenvectors $\mathbf{U}_{j}$ with corresponding
eigenfrequencies in ascending order. Specifically,
$\mathbb{U}=[\mathbf{U}_{1},\mathbf{U}_{2},...,\mathbf{U}_{n},\mathbf{U}_{n+1},...,\mathbf{U}_{N}]$.
Let $\mathbb{U}_{-}=[\mathbf{U}_{1},\mathbf{U}_{2},...,\mathbf{U}_{n}]$, and
$\mathbb{U}_{+}=[\mathbf{U}_{n+1},\mathbf{U}_{n+2},...,\mathbf{U}_{N}]$. Then
the projectors of the negative (below the band gap) and positive (above the
band gap) energy spectrum are given as
$\mathbb{P}_{-}=\mathbb{U}_{-}\mathbb{U}_{-}^{T}$ and
$\mathbb{P}_{+}=\mathbb{U}_{+}\mathbb{U}_{+}^{T}$, respectively. We can then
define "flat band Hamiltonian" as $\mathbb{Q}=\mathbb{P}_{+}-\mathbb{P}_{-}$.
The $\mathbb{Q}$ matrix is decomposed as
$\mathbb{Q}=\mathbb{Q}_{AB}+\mathbb{Q}_{BA}=\Gamma_{A}\mathbb{Q}\Gamma_{B}+\Gamma_{B}\mathbb{Q}\Gamma_{A}$,
where $\Gamma_{A}$ and $\Gamma_{B}$ refer to the projectors onto the $A$ or
$B$ particles respectively, and $\Gamma=\Gamma_{A}-\Gamma_{B}$ is the chiral
operator (see Appendix A). Then, the LTM can be defined as:

$\nu(l)=\frac{1}{2}\sum_{a=A,B}\\{(\mathbb{Q}_{BA}[\mathbb{X},\mathbb{Q}_{AB}])_{la,la}+(\mathbb{Q}_{AB}[\mathbb{Q}_{BA},\mathbb{X}])_{la,la}\\},$
(10)

where $\mathbb{X}$ is the position operator (see Appendix A), $l$ is the unit
cell number, $lA$ and $lB$ indicate the entries of the matrix corresponding to
the $A$ or $B$ particle for the $l$th unit cell. This marker works as a local
topological invariant when evaluated in a region away from the boundary of the
system. To extract a value for the winding number in a disordered system, we
need to average $\nu(l)$ over a small region in the center of the chain for
single disorder realization ($\bar{\nu}$) then over multiple disorder
realizations ($\langle\bar{\nu}\rangle$).

Mean chiral displacement (MCD).—Another type of variable that can detect the
winding in a 1D chiral system is the mean chiral displacement [26, 27], given
as

$C(t)=2\langle\Psi(t)|\widehat{\Gamma}\widehat{\mathbb{X}}|\Psi(t)\rangle,$
(11)

where $\Psi(t)=e^{-iHt}|0_{a}\rangle$ represents the time evolution of an
initially localized state in a quantum mechanical system. Unlike the quantum
mechanical system, which is governed by the Schrödinger equation, a first-
order differential equation, our 1D spring-mass chain is governed by second-
order differential equations. Therefore, we cannot use the above definitions
of MCD directly. Instead, we first map our mechanical system to a quantum-
mechanical problem. Following the framework proposed by Süsstrunk and Huber
[28], we introduce

$\Psi(t)=\left[{\begin{array}[]{*{20}{c}}{\sqrt{\mathbf{\mathbb{D}}}}&{0}\\\
{0}&i\end{array}}\right]\left[{\begin{array}[]{*{20}{c}}{\mathbf{U}(t)}\\\
{\mathbf{V}(t)}\end{array}}\right],$ (12)

which transform the time evolution equation given in Eq. (4) into
$i\dot{\Psi}(t)=\mathbb{H}\Psi(t)$ with
$\mathbb{H}=\begin{bmatrix}0&\sqrt{\mathbb{D}}\\\
\sqrt{\mathbb{D}}&0\end{bmatrix}$ being a Hermitian matrix. This resembles the
Schrödinger equation. Moreover, we define the extended chiral operator and
displacement operator as
$\widehat{\Gamma}=\big{[}\begin{smallmatrix}\Gamma&0\\\
0&\Gamma\end{smallmatrix}\big{]}$ and
$\widehat{\mathbb{X}}=\big{[}\begin{smallmatrix}\mathbb{X}&0\\\
0&\mathbb{X}\end{smallmatrix}\big{]}$, respectively. With these extended
operators we can then calculate the MCD from Eq. (11).

Infinite-time-limit of mean chiral displacement (IMCD).—As
$t\rightarrow\infty$, the MCD converges to a time-independent variable given
as:

$C_{\infty}=2\sum_{j=1}^{2N}|\alpha_{aj}|^{2}\langle\widehat{\Psi}_{j}|\widehat{\Gamma}\widehat{\mathbb{X}}|\widehat{\Psi}_{j}\rangle$
(13)

where $\widehat{\Psi}_{j}$ is the $j$th normalized eigenvector of the
transformed Hamiltonian $\mathbb{H}$, and $\alpha_{aj}$ indicates the
projection of the initial state $\Psi(0)$ on the $j$th eigenvector.

### Capturing topological transition through localization length

Apart from the aforementioned topological invariants, we may also study the
system’s localization properties, since it is found that a topological
transition is accompanied with a divergence of the localization length
$\Lambda$ [19, 21] at the Fermi level, corresponding to the mid-gap frequency
$\omega_{0}$ in our case. Indeed, for the chiral disorder case, at
$\omega_{0}$ the following solutions can be obtained:

$\begin{split}u_{2n-1}=(-1)^{n-1}\prod_{j=1}^{n-1}\dfrac{k_{2j}}{k_{2j+1}}u_{1}\\\
u_{2n}=(-1)^{n-1}\prod_{j=1}^{n-1}\dfrac{k_{2j+1}}{k_{2j+2}}u_{2},\end{split}$
(14)

where the $k_{j}$’s are given by equation (7). Then, assuming an exponential
form for these solutions and applying the Birkhoff’s theorem, see [21], one
obtains the following expression for $\Lambda$:

$\Lambda^{-1}=\left|\dfrac{1}{2}\dfrac{\int_{-1}^{1}d\epsilon\int_{-1}^{1}d\epsilon^{\prime}\left(\ln|K_{e}+W_{e}\epsilon|-\ln|K_{a}+W_{a}\epsilon^{\prime}|\right)}{4}\right|,$
(15)

where an ensemble average has been used. Note that the normalization factor
$1/4$ appears since the random variables $\epsilon$ and $\epsilon^{\prime}$
are uniformly distributed in the interval $[-1,1]$ as mentioned before. After
performing the integration, we deduce

$\Lambda^{-1}=\frac{1}{4}\left|\ln\left[\dfrac{\left|K_{e}+W_{e}\right|^{(K_{e}/W_{e}+1)}}{\left|K_{e}-W_{e}\right|^{(K_{e}/W_{e}-1)}}\dfrac{\left|K_{a}-W_{a}\right|^{(K_{a}/W_{a}-1)}}{\left|K_{a}+W_{a}\right|^{(K_{a}/W_{a}+1)}}\right]\right|.$
(16)

The latter expression, which is valid for the case of chiral disorder, will be
used later to indicate the critical line in the ($K_{a},W$) parametric plane,
where $\Lambda$ diverges.

Complementarily, we also calculate the localization length at $\omega_{0}$
employing the transfer matrix approach [29, 30]. Specifically, we can rewrite
Eq. (II.1) as

$\begin{pmatrix}u_{j+2}\\\
u_{j+1}\end{pmatrix}=\mathbb{T}_{j}\begin{pmatrix}u_{j+1}\\\
u_{j}\end{pmatrix},$ (17)

and get the transfer matrix $\mathbb{T}_{j}$ given by

$\mathbb{T}_{j}=\begin{pmatrix}\dfrac{k_{j+2}+k_{j+1}+k_{j+1}^{0}-m\omega_{0}^{2}}{k_{j+2}}&&-\dfrac{k_{j+1}}{k_{j+2}}\\\
1&&0\end{pmatrix}.$ (18)

Then, the Lyapunov exponents $\gamma_{1}$ and $\gamma_{2}$ are calculated
numerically using the typical numerical schemes described in [31, 32, 30]. We
found that $\gamma_{1}\approx-\gamma_{2}=\gamma$. The localization length is
then determined through the relation $\Lambda=\dfrac{1}{\gamma}$.

##  IV Numerical Results

###  IV.1 No disorder

Figure 2: Eigenfrequencies for a finite (a) topologically nontrivial and (b)
trivial 1D spring-mass chain. Shapes of the topological edge modes are shown
in the inset of (a). The grey areas represent the band gaps. (c) LTM
calculated from the eigenmodes of the nontrivial (blue) and trivial (red)
system. (d) Time-dependent MCD for the nontrivial (blue) and trivial (red)
under initial displacement excitation. (e) Evolution of three kind of real
space topological invariants during a topological transition. (f) Localization
length of the eigenmode at the frequency $\omega_{0}$.

Before moving to the disorder study, we need to verify the aforementioned
three types of real space topological invariants in a finite clean system. We
consider a chain composed of $n=250$ unit cells, i.e., $500$ particles and
$501$ inter/intracell springs. In the following numerical simulations, we fix
$K_{0}=5$. First, let us look at a case with $K_{e}=1$ and $K_{a}=0.5$. For
$K_{e}>K_{a}$, the periodic system is topological, which is characterized by a
winding number $v=1$. In a finite chain, we expect to see two localized edge
modes exist in the topological band gap as shown in Fig. 2(a) [see also Fig.
7(a) in Appendix B for the whole spectrum]. The red and blue dots represent
the two states localized on the left, $\lvert L\rangle$, and right boundary,
$\lvert R\rangle$, respectively at the mid gap frequency $\omega_{0}$. In
fact, a hybrization of the edge states is expected, which leads to a frequency
splitting, meaning that the two eigenfrequencies are not exactly located at
$\omega_{0}$ but at $\omega_{0}^{+}$ and $\omega_{0}^{-}$ (the two
corresponding eigenstates are approximated as:
$\lvert\omega_{0}^{+}\rangle=\frac{e^{-i\phi/2}\lvert
L\rangle+e^{i\phi/2}\lvert R\rangle}{\sqrt{2}}$ and
$\lvert\omega_{0}^{-}\rangle=\frac{e^{-i\phi/2}\lvert
L\rangle-e^{i\phi/2}\lvert R\rangle}{\sqrt{2}}$ for some $\phi$) [33].
However, for large system sizes, as in our case, one can consider that the
energy splitting is sufficiently small, and thus we have a "degeneracy" of the
eigenstates $\lvert L\rangle$ and $\lvert R\rangle$ at $\omega_{0}$. The
shapes of the localized edge modes are shown in the insets of Fig. 2(a). Note
that the two localized edge modes have a characteristic profile with only
sites from one sub-lattice excited (either $j$ even or odd). This is a well-
known consequence of the chiral symmetry. As we increase $K_{a}$ to $1.5$, the
system becomes trivial with a winding number being zero, where the localized
edge modes do not exist anymore [see Fig. 2(b) and Fig. 7(b) in Appendix B].

Based on the eigenmodes obtained from Eq. (5), we can calculate the LTM of the
system as defined in Eq. (10). Figure 2(c) shows the calculated LTM as a
function of the unit cell number in the topologically nontrivial (red line)
and trivial cases (blue line), respectively. As we see, the LTM is well
quantized to either zero or one in the bulk of the chain, while approaching
the boundary, it deviates from the theoretical value. To overcome such
boundary effect, we take the average value over 100 unit cells in the middle
of the chain, noted as $\bar{v}$. We then get $\bar{v}=1$ for the nontrivial
and $\bar{v}=0$ for the trivial configurations. Similarly, by plugging the
eigenmodes into Eq. (13), we obtain the IMCD, which are $C_{\inf}=1$ and
$C_{\inf}=0$ for nontrivial and trivial configurations, respectively. To
derive the time-dependent MCD, we take the initial displacement
$u_{251}(0)=0.1$ and perform transient simulations. After normalizing the
time-history response at each time $t$ and plugging this in Eq. (11), we get
the MCD as a function of time. Figure 2(d) demonstrates its evolution in a
nontrivial (blue curve) or trivial (red curve) system. It is clear that the
MCD oscillates and tends to converge to 1 (0) in a topologically nontrivial
(trivial) system despite the oscillations. We conduct the time average from
$t=0$ to $t=60$ to eliminate the influence of fluctuations, which gives
$\bar{C}(t)=1$ [$\bar{C}(t)=0$]. The time window is selected to avoid possible
wave reflections at the system boundaries.

In these two cases, all three topological invariants are closely quantized to
$0$ or $1$, which shows a great agreement with the theoretical winding number
of an infinite system defined in the momentum space. Figure 2(e) summarizes
the evolution of the different topological invariants during an in situ
topological phase transition. During the process, we vary $K_{a}$ from $0.5$
to $1.5$ with a $0.025$ step while keeping $K_{e}=1$. Clearly, all three kinds
of real space topological invariants agree well with each other and can
accurately probe the topological phase transition in a clean 1D mechanical
system, which happens when the gap closes, namely at $K_{a}=K_{e}=1$. Finally,
in Fig. 2(f), we also plot the the localization length of the edge modes as we
vary the coupling $K_{a}$ from 0.5 to 1.5, which is given by the analytical
expression $\Lambda=\frac{2}{|\ln(K_{a}/K_{e})|}$. Notice that at the point
where the topological transition occurs, i.e., $K_{a}=K_{e}=1$, the
localization length diverges. This is expected since, as we mentioned, the
topological transition occurs when the gap closes and thus the edge mode
transforms to an extended mode.

###  IV.2 Chiral disorder

In this subsection, we explore the topological phase transition of a spring-
mass system induced by the chiral disorder with the dynamical matrix
$\mathbb{D}^{I}$. Figure 3(a) shows the numerically calculated LTM as a
function of the intracell spring stiffness $K_{a}$ and the disorder strength
$W=W_{a}=2W_{e}$ under chiral disorder. We again choose $K_{e}=1$. Each data
point is a disorder-averaged result over $250$ realizations. Evidently, the
surface map can be approximately divided into two regions separated by a
blurry white boundary. The region with a red (blue) tone has topological
marker close to $1$ ($0$) corresponding to a topologically nontrivial
(trivial) system. In most real mechanical systems, the spring stiffness tend
to be positive. The black dashed lines follow the relation given by
$\mathrm{min}(|K_{e}-W_{e}|,|K_{a}-W_{a}|)=0$ for sufficiently large $K_{0}$.
Therefore, only the area on the left of the dashed black line in the surface
plot could be realized without the necessity of negative-stiffness springs.
However, note that by tuning the disorder ratio ($W_{e}/W_{a}$), we can
further shift the topological phase diagram and change the critical boundary
(see Appendix C). Note also that effective negative-stiffness springs can be
designed utilizing the symmetry of modes and a sophisticated geometry of the
coupling of the neighbor particles, look for example refs. [34] and [35] for
an implementation of negative hoppings in elastic and acoustic systems,
respectively.

Figure 3: (a) Topological phase diagram under chiral disorder with disorder
ratio $W=W_{a}=2W_{e}$. Colorbar ranging from $0$ to $1$ stands for the LTM
averaged over $250$ disorder realizations. (b) The localization length at the
mid gap frequency, $\Lambda(\omega_{0})$ from the transfer matrix method
(colormap) and the analytical expression. (c) and (d) The evolution of
topological markers with the increase in disorder strength for $K_{a}=0.6$
[yellow line in (a)] and $1.05$ [green line in (a)], respectively. The lines
and shaded area represent the mean and standard deviation of the LTM. The
cross and sphere markers indicate the mean of MCD and IMCD.

Figure 3(b) shows the localization length at $\omega_{0}$, obtained
numerically using the transfer matrix method, as a function of the same
parameters ($K_{a}$ and $W$). We note here that the transfer matrix was
iterated $10^{6}$ times at each point of Fig. 3(b). The solid black curve
shown in Figs. 3(a) and 3(b) indicates the critical boundary obtained by the
analytical diverging line from Eq. (16). It is clear that the numerical
solutions [blurry white regions in Fig. 3(a) and highlighted area in Fig.
3(b)] match with the analytical results (solid black curve). This denotes that
the topological phase transition is related with a critical point at which the
localization length diverges.

Figure 3(c) shows the evolution of the topological marker with the increase of
the disorder strength for $K_{a}=0.6$, which corresponds to the yellow line in
Fig. 3(a). In the clean limit ($W=0$), this finite system is nontrivial with a
topological marker of $\bar{v}=1$. As expected, the topological marker remains
constant in the presence of weak and medium level disorder, as the topological
properties of this system are protected by the chiral symmetry and the
topological edge modes are immune to weak disorder. However, when the disorder
strength rises and reaches a sufficiently large amount, the topological marker
falls sharply and eventually stabilizes near zero under strong disorder, which
indicates a transition process from a topologically nontrivial system to a
trivial one. We also notice that the disorder-averaged results show larger
standard deviation in this region, which is reasonable in strong disorder. We
also show in the same figures disorder-averaged mean values of MCD and IMCD,
marked by the cross and sphere markers. See Appendix D for typical time-
history responses of the 1D spring mass chain under different levels of
disorder. All three kinds of topological invariants therefore show good
agreement in capturing the general trends of disorder-induced topological
transition. For the time-history response, note that negative stiffness of
intersite or onsite springs caused by the strong disorder does not necessarily
lead to dynamical instability of our system. Within the disorder ranges that
we are interested in, our system remains stable (See Appendix E for more
details).

For $K_{a}=1.05$ [green line in Fig. 3(a)], however, we observe a more
sophisticated topological phase transition process, as shown in Fig. 3(d). The
system is trivial with $\bar{v}=0$ when there is no disorder. For very weak
disorder $W<0.3$, the topological marker remains near zero. However, as we
introduce stronger disorder, the topological marker surprisingly surges up to
reach a plateau ($\bar{v}\approx 0.98$) which is very close to $1$ (not
exactly 1 due to finite size). If we keep increasing the disorder strength, it
drops again close to zero. This procedure is particularity interesting since
we demonstrate a two-way transition by purely increasing disorder in the
system. The spring-mass chain starts as a trivial system, then changes into a
nontrivial one and finally becomes a trivial Anderson insulator. The MCD and
IMCD results as marked with crosses and circles in Fig. 3(d) also corroborate
the LTM trend. Therefore, the intermediary nontrivial state is the realization
of a mechanical analogue to the TAI.

Figure 4: Eigen-frequencies and two modes in the center of frequency spectrum
of a finite spring-mass chain ($K_{a}=1.05$ and $K_{e}=1$) under chiral
disorder with disorder ratio $W=W_{a}=2W_{e}$ for (a) $W=0.25$, (b) $W=1.25$
(TAI phase), and (c) $W=2.5$. These correspond to three typical realizations
with different disorder strengths in Fig. 3(d). Chiral nature of the two edge
modes lying at the center of band gap is shown in the insets. (d)–(f) Spatial
profile of these two modes for one disorder realization. (g)-(h) Histogram of
the Center of Mode (COM) obtained by $1000$ disorder realizations, describing
the probability of COM lying in 50 particle interval throughout the chain.

So far, we have quantitatively demonstrated the topological phase transitions
by tracking the real-space topological invariants. Now we investigate their
characteristics in more details. We choose the configuration of the system
that supports the TAI phase, i.e., with $K_{a}=1.05$ and $K_{e}=1$ in Fig.
3(d) and analyze its eigenfrequencies and modes as a function of disorder
strength. Figures 4(a)-(c) show the eigenfrequencies of a finite chain under
one realization with disorder strengths $W=0.25$, $W=1.25$, and $W=2.5$. The
grey regions represent the topological band gaps in the clean ($W=0$) case.
After we introduce disorder in the system, the topological band gap shrinks in
the regime of weak disorder [Fig. 4(a) for $W=0.25$] and is eventually closed
under strong disorder [see Figs. 4(b) and 4(c) for $W=1.25$ and $W=2.5$,
respectively]. We then plot the shapes of the center modes ($250$th and
$251$st) in Figs. 4(d)-(f). We find that for weak disorder ($W=0.25$), the
modes are quite extended. Of course, as the disorder strength increases, the
modes become spatially strongly localized. The TAI phase ($W=1.25$) is
distinct in a way that the localized modes appear on the boundaries of the
system as in a nontrivial system, whereas the localized modes need not be at
the boundaries for an Anderson insulator ($W=2.5$) and reside at the same
location with only phase differences at some particles.

We further demonstrate the validity of this statement for all disorder
realizations by defining the Center of Mode (COM) as

$\mathrm{COM}=\frac{\displaystyle\sum_{i=1}^{N}x_{i}{{u}_{i}^{2}}}{\displaystyle\sum_{i=1}^{N}{u_{i}^{2}}}$
(19)

where $x_{i}$ and $u_{i}$ denote the position and displacement of the $i$th
mass. We study the statistical distribution of the COM for $1000$ disorder
realizations. Figures 4 (g)-(i) show the histograms of the COM of the 250th
and 251st modes under chiral disorder for varying disorder strengths. This
agrees with the topological phase diagram [see Fig. 3(d)] and the eigen
analysis [Figs. 4 (d)-(f)]. When the system is with weak ($W=0.25$) and strong
($W=2.5$) disorder, the 250th and 251st modes are likely to be all along the
chain; however, for the TAI phase ($W=1.25$), the modes are most likely
localized at the boundaries [Fig. 4 (h)]. A similar inference can be made for
the configuration with $K_{a}=0.6$ and $K_{e}=1$ and that has been included in
Appendix F.

###  IV.3 Random disorder

Figure 5: (a) Topological phase diagram under random disorder with disorder
ratio $W=W_{a}=2W_{e}$. Colorbar ranging from $0$ to $1$ stands for the LTM
averaged over $250$ disorder realizations. (b) Localization length
$\Lambda(\omega_{\text{0}})$ under random disorder. The transfer matrix was
again iterated $10^{6}$ times at each point. (c) The evolution of topological
marker with increase of disorder for $K_{a}=1.05$ (green) and $K_{a}=0.6$
(yellow). The shaded area shows the standard deviation.

We now discuss the effect of random disorder on topology by removing the
disorder on the onsite ground springs. As given in Eq. (9), the diagonal
entries of the random disordered stiffness matrix are not identical anymore,
which leads to the breaking of the system’s chiral symmetry. Since the
topology in the 1D SSH model is mainly protected by the chiral symmetry, we
expect to see a stronger influence of the random disorder over topology.
Figure 5(a) shows the topological phase diagram under random disorder with
$W=W_{a}=2W_{e}$. It is clear that the topological states can only survive
much weaker random disorder compared to the chiral one. Moreover, Fig. 5(b)
shows the localization length at $\omega_{\text{0}}$, and in contrast to the
case of chiral disorder there is no signature of the localization length
divergence at this frequency, which also contributes to the uncertainty of the
clear phase transition due to non-chiral disorder. Furthermore, even with very
small amount of disorder, we see that the topological marker deviates from the
quantized value as shown in Fig. 5(c). As we expected, breaking the chiral
symmetry leads to the variation of the real space topological marker, which is
not well quantized to an integer value but tends to saturate near a decimal
between $0$ and $1$. This leave us an open question whether it is appropriate
to discuss the concept of topology in a 1D mechanical system in absent of the
chiral symmetry. If so, what kind of quantities or phenomena should we use to
characterize the topological properties in these cases?

Figure 6: Eigen-frequencies and two modes in the center of frequency spectrum
of a finite spring-mass chain ($K_{a}=0.6$ and $K_{e}=1$) under random
disorder with disorder ratio $W=W_{a}=2W_{e}$ for (a) $W=0.25$, (b) $W=1.25$,
and (c) $W=2.5$. These correspond to three typical realizations with different
disorder strengths on the yellow curve in Fig. 5(c). Non-chiral nature of the
two modes (marked) is shown in the insets. (d)–(f) Spatial profile of these
two modes for one disorder realization. (g)-(h) Histogram of the Center of
Mode (COM) obtained by $1000$ disorder realizations, describing the
probability of COM lying in 50 particle interval throughout the chain.

Despite the lack of a theoretical framework, we try to extend the eigen
analysis and COM statistical study to examine the relation between random
disorder and the topological indicators in a 1D system. Figures 6(a)-(c) show
the eigenfrequencies around the center of the frequency spectrum of a finite
chain ($K_{a}=0.6$) under random disorder with strength $W=0.25$, $W=1.25$,
and $W=2.5$, respectively. The grey regions represent the topologically
nontrivial band gaps in the clean ($W=0$) case. It is clear that the random
disorder causes more drastic change of the frequency spectrum. Especially, the
two center modes no longer share a common frequency and may shift out of the
original bandgap region. Despite the frequency variations, the two center
modes of the system under weak ($W=0.25$) random disorder are still localized
modes existing at the boundary [Figs. 6(d)-(i)], which is the key character of
the topological edge states. This is consistent with the topological invariant
calculation, which is very close to $1$ in this configuration [see the yellow
line in Fig. 5(c)]. As we increase the disorder level, these center modes
start to move into the bulk. More interestingly, the two center modes tend to
localize at different positions in the chain, which is different from the
trivial chiral disordered system where the two modes exist at the same
location [Fig. 4(f)]. A similar trend can also be observed for the disordered
study in the case of $K_{a}=1.05$ (Appendix F). Finally, in Figs. 4 (g)-(i) we
show the histograms of the COM of the central modes under random disorder for
varying disorder strengths. For weak ($W=0.25$) disorder, the 250th and 251st
modes are likely to be localized at the boundaries, while for strong ($W=1.25$
and $W=2.5$) disorder (both in the trivial Anderson phase) they can be
localized all along the chain.

##  V Conclusion

In this work, we explore the disorder-induced topological phase transitions in
a spring-mass chain. To probe the topological property in a disordered 1D
mechanical system, we use three kinds of topological invariants obtained from
the real space wave functions to work as an effective winding number. We
introduce two types of disorder, chiral and random, to study their influences
on the topology in a mechanical system. Under the chiral disorder, we not only
demonstrate the transition from topological to trivial, but also the abnormal
reverse transformation leading to the realization of so-called Topological
Anderson Insulator (TAI) in a mechanical setup. However, for the case of
random disorder, we do not observe similar TAI phenomena. Instead, we find
that the random disorder suppresses the topological property, as it breaks the
internal system symmetry that protects the topology. Our findings can be
extended to higher dimensional mechanical systems to further study the
interactions between disorder and topology. In addition, despite the
difficulties in controlling stiffness in a disordered mechanical system, it
would be very interesting to implement and experimentally verify the proposed
framework in other highly tunable systems, such as electric circuits [36].

###### Acknowledgements.

We thank Professor Christopher Chong from Bowdoin College for fruitful
discussions. X. S., and J. Y. are grateful for the financial support from the
U.S. National Science Foundation (CAREER1553202 and EFRI-1741685). I.K.
acknowledges financial support from the Academy of Athens. G.T. acknowledges
financial support from the CS.MICRO project funded under the program Etoiles
Montantes of the Region Pays de la Loire.

## APPENDIX A: Definition of the chiral operator and position operator

For a 1D system with $n=250$ unit cells, that is $N=2n=500$ particles, we
first define the unit cell numbers as $l=[-125,-124,\cdots,0,\cdots,123,124]$.
Then the chiral operator and position operator are given as

$\displaystyle\Gamma={\left[{\begin{array}[]{*{20}{c}}{1}&{0}&0&0&\cdots\\\
{0}&{-1}&0&0&\cdots\\\ {0}&{0}&{1}&0&\cdots\\\ {0}&{0}&{0}&{-1}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}}\right]_{N\times N}},$ (25)

$\displaystyle\Gamma_{A}={\left[{\begin{array}[]{*{20}{c}}{1}&{0}&0&0&\cdots\\\
{0}&{0}&0&0&\cdots\\\ {0}&{0}&{1}&0&\cdots\\\ {0}&{0}&{0}&{0}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}}\right]_{N\times N}},$
$\displaystyle\Gamma_{B}={\left[{\begin{array}[]{*{20}{c}}{0}&{0}&0&0&\cdots\\\
{0}&{1}&0&0&\cdots\\\ {0}&{0}&{0}&0&\cdots\\\ {0}&{0}&{0}&{1}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}}\right]_{N\times N}},$ (36)

$\displaystyle\mathbb{X}={\left[{\begin{array}[]{*{20}{c}}{-125}&{0}&0&0&\cdots&0&0\\\
{0}&{-125}&0&0&\cdots&0&0\\\ {0}&{0}&{-124}&0&\cdots&0&0\\\
{0}&{0}&{0}&{-124}&\cdots&0&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
{0}&{0}&0&0&\cdots&124&0\\\ {0}&{0}&0&0&\cdots&0&124\\\
\end{array}}\right]_{N\times N}}.$ (44)

## APPENDIX B: Spectrum

Figure 7: The eigenfrequencies of a chain of $500$ particles for the
topological nontrivial phase (left panel, $K_{e}=1$, $K_{a}=0.5$) and trivial
case (right panel, $K_{e}=1$, $K_{a}=1.5$). For both cases, $K_{0}=5$. Band
gap is highlighted with the shaded area.

In Fig. 7, we show all the eigenfrequencies of a clean chain of $500$
particles for both topologically trivial and nontrivial phases. These are the
same cases as shown in Fig. 2a and 2b of the main text.

## APPENDIX C: Topological phase diagram with different combinations of
disorder strength

In Section IV-B of the main text, we discuss the disorder configuration with
specific ratio $W=W_{a}=2W_{e}$. Here, we briefly explore the effect of
disorder ratio ($W_{e}/W_{a}$) by studying the topological transitions in the
presence of chiral disorder with different combination of $W_{a}$ and $W_{e}$.
Figures 8(a)-(c) show the topological phase diagrams under chiral disorder
with (a) $W=W_{a}$ and $W_{e}=0$, (b) $W=W_{a}=4W_{e}$, and (c)
$W=W_{a}=0.5W_{e}$. It is clear that by changing the disorder ratio
($W_{e}/W_{a}$), we can shift the critical boundaries for the disorder-induced
topological transitions.

Figure 8: Topological phase diagram under chiral disorder with different
combination of $W_{a}$ and $W_{e}$. (a) $W=W_{a}$ and $W_{e}=0$, (b)
$W=W_{a}=4W_{e}$, and (c) $W=W_{a}=0.5W_{e}$. The solid black line indicates
the divergence of the localization length obtained analytically. The dashed
line separates the region of negative stiffness.

## APPENDIX D: Spatiotemporal diagrams in disordered systems

Figure 9: Spatio-temporal evolution of particle displacement ($K_{a}=0.6$ and
$K_{e}=1$). We use chiral-disorder strengths (a) $W=0$, (b) $W=0.25$, (c)
$W=1.25$, and (d) $W=2.5$.

Figures 9(a)-(d) show the typical time-history responses of the 1D spring mass
chain with chiral disorder under strength $W=0$, $W=0.25$, $W=1.25$, and
$W=2.5$, respectively. In a periodic system ($W=0$), the energy is evenly
spreading to both sides of the system after the initial disturbance in the
middle of the chain, as shown in Fig. 9(a). With the increase of disorder
strength, we start to see more and more localization of the energy in the
system due to Anderson localization.

Figure 10: Stability diagram of the 1D system as a function of $K_{a}$ and $W$
under the chiral disorder. Colorbar ranging from $0$ to $1$ stands for the
probability (over $1000$ disorder realizations) of the dynamical matrix being
positive-definite.

## APPENDIX E: Stability of disordered systems

The dynamics of our system is governed by Eq. (2). For the stability of this
system, the dynamical matrix must be positive definite, implying all the
eigenvalues ($\omega^{2}$) are positive. In Fig. 10, we show the stability
diagram under chiral disorder that correspond to the case shown in Fig. 3(a)
of the main text. This highlights the probability of the dynamical matrix
being positive-definite with varying system parameters. We can therefore
conclude that our system remains stable within the disorder range we are
interested in ($W<3$).

## APPENDIX F: Some other cases of chiral and random disorder

To have a comprehensive understanding of the topological transition process,
we study two extra cases with chiral and random disorders, respectively. In
Fig. 11, we study a spring-mass chain with configuration $K_{a}=0.6$ and
$K_{e}=1$ under chiral disorder, which corresponds to Fig. 3 (c). Similarly,
Fig. 12 shows the results of a system with configuration $K_{a}=1.05$ and
$K_{e}=1$ under random disorder corresponding to the green curve in Fig. 5
(c).

Figure 11: Eigen-frequencies and two modes in the center of frequency spectrum
of a finite spring-mass chain ($K_{a}=0.6$ and $K_{e}=1$) under chiral
disorder with disorder ratio $W=W_{a}=2W_{e}$ for (a) $W=0.25$, (b) $W=1.25$,
and (c) $W=2.5$. These correspond to three typical realizations with different
disorder strengths in Fig. 3(c). Chiral nature of the two edge modes lying at
the center of band gap is shown in the insets. (d)–(f) Spatial profile of
these two modes for one disorder realization. (g)-(h) Histogram of the Center
of Mode (COM) obtained by $1000$ disorder realizations, describing the
probability of COM lying in 50 particle interval throughout the chain. Figure
12: Eigen-frequencies and two modes in the center of frequency spectrum of a
finite spring-mass chain ($K_{a}=1.05$ and $K_{e}=1$) under random disorder
with disorder ratio $W=W_{a}=2W_{e}$ for (a) $W=0.25$, (b) $W=1.25$, and (c)
$W=2.5$. These correspond to three typical realizations with different
disorder strengths on the green curve in Fig. 5(c). Non-chiral nature of the
two modes (marked) is shown in the insets. (d)–(f) Spatial profile of these
two modes for one disorder realization. (g)-(h) Histogram of the Center of
Mode (COM) obtained by $1000$ disorder realizations, describing the
probability of COM lying in 50 particle interval throughout the chain.

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