# Out-of-Time-Order Correlation as a Witness for Topological Phase Transitions

Qian Bin  School of Physics and Institute for Quantum Science and Engineering,
Huazhong University of Science and Technology, Wuhan, 430074, China  Liang-
Liang Wan  School of Physics and Institute for Quantum Science and
Engineering, Huazhong University of Science and Technology, Wuhan, 430074,
China  Franco Nori  Theoretical Quantum Physics Laboratory, RIKEN Cluster for
Pioneering Research, Wako-shi, Saitama 351-0198, Japan  RIKEN Center for
Quantum Computing (RQC), 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan
Physics Department, The University of Michigan, Ann Arbor, Michigan
48109-1040, USA  Ying Wu  School of Physics and Institute for Quantum Science
and Engineering, Huazhong University of Science and Technology, Wuhan, 430074,
China  Xin-You Lü<EMAIL_ADDRESS>School of Physics and Institute for
Quantum Science and Engineering, Huazhong University of Science and
Technology, Wuhan, 430074, China  School of Physics and Institute for Quantum
Science and Engineering, Huazhong University of Science and Technology, Wuhan,
430074, China  School of Physics and Institute for Quantum Science and
Engineering, Huazhong University of Science and Technology, Wuhan, 430074,
China  Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering
Research, Wako-shi, Saitama 351-0198, Japan  RIKEN Center for Quantum
Computing (RQC), 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan  Physics
Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
School of Physics and Institute for Quantum Science and Engineering, Huazhong
University of Science and Technology, Wuhan, 430074, China
<EMAIL_ADDRESS>School of Physics and Institute for Quantum Science and
Engineering, Huazhong University of Science and Technology, Wuhan, 430074,
China

###### Abstract

We propose a physical witness for dynamically detecting topological phase
transitions (TPTs) via an experimentally observable out-of-time-order
correlation (OTOC). The distinguishable OTOC dynamics appears in the
topological trivial and non-trivial phases due to the topological locality. In
the long-time limit, the OTOC undergoes a zero-to-finite-value transition at
the critical point of the TPTs. This transition is robust to the choices of
the initial state of the system and the used operators in OTOC. The proposed
OTOC witness can be applied into the systems with and without chiral symmetry,
e.g., the lattices described by the SSH model, Creutz model, and Haldane
model. Moreover, our proposal, as a physical witness in real space, is still
valid even in the presence of disorder. Our work fundamentally offers a new
prospect of exploring topological physics with quantum correlations.

Topological phase transitions (TPTs) are fundamentally interesting in modern
physics because these go beyond the paradigm of traditional phase transitions
associated with symmetry breaking Bansil2016LD . It offers a non-trivial
paradigm for the classification of matter phases, and thus is attracting
enormous attention in condensed matter physics Hasan2010Kane ; Qi2011Zhang ;
Bernevig2013Hughes ; Chiu2016TS , optics Ozawa2019PA , and non-Hermition
physics Bergholtz2021BK . The occurrence of TPTs involve the gap-closing-and-
opening of band (the change of system topology) with symmetry preserving.
According to the extended bulk-boundary correspondence, the $n$th-order TPT in
a $d$-dimensional (dD) system leads to the appearance of a $(d-n)$-dimensional
gapless boundary state in the topological non-trivial phase Langbehn2017PT ;
Schindler2018CV ; Liu2019ZA ; Kunst2018MB ; Imhof2018BB ; Ezawa2018 ;
Schindler2018WV ; Noh2018BH ; Geier2018TH ; Chen2018CG ; Zhu2021HZ ; Che2020GL
. This symmetry-protected boundary state has strong robustness to disorder
Mittal2014FF ; MondragonShem2014HS ; Meier2018AD  and defects Bandres2018WH .
It can be used to realize topological lasers exhibiting robust transports
StJean2017GG ; Zhao2018MT ; Parto2018WH ; Harari2018BL ; Bandres2018WH ,
topological protected quantum coherence Bahri2015VA ; Nie2020PN , and quantum
state transfer Yao2013LG . Thus, the detection of TPTs is a key for exploring
topological physics. To quantitatively distinguish the topological trivial and
non-trivial phases, normally one calculates topological invariants (e.g.,
winding number and Chern number) in momentum space Asboth2016OP . However,
identifying TPTs with those commonly used topological invariants is not suited
for disorder systems where it is difficult to give the Hamiltonian in momentum
space. Then, it becomes a significant task to identify TPTs via an alternative
physical witness in real space that is robust to disorder.

The OTOC, defined as $\mathcal{O}(t)=\langle
W^{\dagger}(t)V^{\dagger}W(t)V\rangle$ with $W(t)=e^{iHt}We^{-iHt}$, was
proposed in investigating the holographic duality between a strongly
interacting quantum system and a gravitational system Shenker2014Stanford ;
Shenker2015Stanford ; Maldacena2016SS ; Jensen2016 ; Roberts2016Swingle ;
Garcia-Garcia2018LRB . Here $W$ and $V$ are initially commuting operators
Swingle2018 . Different from the normal time-order correlation function
characterizing classical and quantum statistics Glauber1963 ; Scully1997 ;
Agarwal2013 ; Salmon2022Gustin ; Mercurio2022Macri ; Lin2021LK , the OTOC can
quantify the temporal and spatial correlations throughout many-body quantum
systems, which is closely related to information scrambling. Thus, it is a
widely used tool for diagnosing chaotic behavior Roberts2015Stanford ;
Gu2016Qi ; Zhu2016HG ; Garcia-Mata2018SJ ; Aleinera2016FI ; Rozenbaum2017GG ;
Nahum2018VH ; Khemani2018VH ; Keyserlingk2018RP ; Chan2018LC ; Rozenbaum2019GG
; Cao2020ZS ; Liao2018Galitski ; Syzranov2018GG ; Lin2018Motrunich ;
Akutagawa2020HS ; Xu2020SC ; Alonso2022SA ; Blocher2020AM , many-body
localization Huang2017ZC ; Chen2017ZH ; Fan2017ZS ; He2017Lu ; Gu2017 ;
Lee2019KK ; Swingle2017Chowdhury ; Smit2019KM , entanglement McGinley2019NK ;
Riddell2019Sorensen ; Luitz2017Lev ; Garttner2018HR ; Hosur2016QR , and
quantum phase transitions Shen2017ZF ; Heyl2018PD ; Dag2019SD ; Chen2020HZ ;
Sun2020CT ; Wang2019PerezBernal ; LewisSwan2020MR ; Dag2020DS . Here, many-
body localization is a kind of many-body phenomenon in the nonequilibrium
system caused by many-body interactions. This is essentially different from
TPTs that describe the change of topological structure of systems. Under the
frame of band topology theory, normally the TPTs occurs in the system without
the many-body interactions. Recently, the dynamical detection of TPTs with
OTOCs has been proposed Dag2020DS , where the infinite-temperature OTOCs can
directly probe zero-temperature quantum phases via detecting the presence of
Majorana zero modes. This construct a relation between infinite-temperature
information scrambling and zero-temperature $\mathbb{Z}_{2}$ topological
order. Moreover, the OTOC can also be implemented experimentally
Garttner2017BSN ; Li2017FW ; Landsman2019FS ; Nie2020WC ; Wei2019PS  by
connecting the time reversal to the Loschmidt echo technique Hahn1950SE ;
Swingle2016BSS ; Sanchez2020SC . Further exploiting OTOC dynamics in
topological systems may open a door for completing the challenging problem of
identifying TPTs in the presence of disorder. Until now, the relation between
zero-temperature OTOC and TPTs remains largely unexplored, which may
substantially advance the fields of quantum correlation and topological
physics.

Here we propose a zero-temperature OTOC witness for dynamical detecting
$\mathbb{Z}$-type TPTs in lattice systems. As shown in Fig. 1(a), the
constructed OTOC becomes an experimentally observable fidelity Garttner2017BSN
of a final state $\rho_{f}$ projected onto an initial state $\rho_{0}$ by
defining $V=V\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$, i.e.,

$\displaystyle\mathcal{O}(t)={\rm
tr}[\rho_{0}e^{iHt}W^{\dagger}e^{-iHt}\rho_{0}e^{iHt}We^{-iHt}]=F(t).$ (1)

Due to the topological locality, the long-time limit of the OTOC
$\mathcal{O}(t\rightarrow\infty)$ undergoes a zero-to-finite-value transition
along with the system entering into the non-trivial phase from the trivial
phase. This sudden change is not limited by the choices of the operators $V$
(corresponding to the initial state of system) and $W$. In comparison with
previous methods of detecting TPTs Chiu2016TS , the proposed OTOC, as a
witness in real space, can be applied in disordered systems. Moreover, it is
not only suitable for the systems with chiral symmetry described by the
nearest-neighbor (NN) Su-Schrieffer-Heeger (SSH) model, next-next-nearest-
neighbor (NNNN) SSH model and Creutz model, but also can be used to the
systems without chiral symmetry, such as 2D lattices described by the Haldane
model and Qi-Wu-Zhang model. We also demonstrate the validity of the OTOC
witness for detecting second-order TPTs.

Figure 1: (a) A schematic illustration of implementing the OTOC, which is
equal to the fidelity $F(t)={\rm tr}[\rho_{0}\rho_{f}]$ Garttner2018HR ;
Garttner2017BSN . First, the initial state $\rho_{0}$ evolves to the state
$\rho_{1}(t)$ under $T_{-}=e^{-iHt}$. Second, the system changes from
$\rho_{1}(t)$ to $\rho_{2}(t)$ after the operation of $W$. Lastly, the system
evolves backward to get the final state $\rho_{f}$ under $T_{+}=e^{iHt}$. (b,
c) Schemes of the 1D SSH model and Creutz model, which describe the lattice
systems with chiral symmetry. (d, e) Phase diagrams of the NN SSH model: the
OTOC versus $\epsilon t$ and $\nu$ for (d) $W=a_{1,A}^{\dagger}a_{1,A}$ and
(e) $W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{3}a_{n}$, where $N=200$,
$|\psi_{0}\rangle=|1,A\rangle$, and $d_{1}=d_{2}=0$. The topological non-
trivial and trivial phases are denoted as TNP and TTP, respectively.

_Detecting TPTs in the systems with chiral symmetry._ —Without loss of
generality, we choose the 1D SSH model and Creutz model depicted in Figs.
1(b,c) as examples for demonstrating the validity of detecting TPTs with OTOC
in the systems with chiral symmetry. The corresponding system Hamiltonians can
be written as Asboth2016OP ; Su1979SH ; Rufo2019LC ; Creutz1999

$\displaystyle\\!\\!H_{\rm
s}\\!=\\!\\!\\!\sum_{n}\\!\\{\nu_{n}a_{n}^{\dagger}\sigma_{1}a_{n}\\!\\!+\\!\\![(\omega_{n}a_{n+1}^{\dagger}\\!\\!+\\!\epsilon\eta
a_{n+2}^{\dagger})\frac{\sigma_{1}\\!\\!+\\!\\!i\sigma_{2}}{2}a_{n}\\!\\!+\\!{\rm
h.c.}]\\},$ (2a) $\displaystyle\\!\\!H_{\rm
cr}\\!=\\!\\!\sum_{n}\\{\eta_{0}a_{n}^{\dagger}\sigma_{1}a_{n}+\eta_{0}^{\prime}[a_{n+1}^{\dagger}\frac{\sigma_{1}\\!-\\!i\sigma_{3}}{2}a_{n}+{\rm
h.c.}]\\},$ (2b)

where the number of cells is $N$, $\sigma_{j}$ ($j=0,1,2,3$) is Pauli
operator, and $a_{n}^{\dagger}=(a_{n,A}^{\dagger},a_{n,B}^{\dagger})$ is the
annihilation operator of the unit cell $n$ with sublattices $A$, $B$. For the
SSH model with Hamiltonian $H_{\rm s}$, $\omega_{n}=\epsilon(1+d_{1}r_{n})$
[or $\nu_{n}=\epsilon(\nu+d_{2}r_{n}^{\prime})$] is the intercell (or
intracell) hopping strength. Disorder with the dimensionless strengths
$d_{1}$, $d_{2}$ has been included here, and $r_{n}$, $r_{n}^{\prime}$ are the
independent random real numbers chosen from the uniform distribution
$[-0.5,0.5]$. Physically, $\epsilon$ is the characteristic intercell strength,
$\nu$ is the ratio of intra- to inter-cell hopping in the clean system, and
$\epsilon\eta$ is the NNNN hopping strength. Here, $H_{\rm s}$ is reduced to a
standard Hamiltonian of the NN SSH model when $\eta=0$. For the Creutz model
with Hamiltonian $H_{\rm cr}$, the arrows indicate the sign of the hopping
phase, and $\eta_{0}$ ($\eta_{0}^{\prime}$) is the vertical (horizontal and
diagonal) hopping strength. The above models possess a chiral symmetry with a
well-defined chiral operator $\mathcal{C}_{\rm 1d}$, which can reverse the
energy of the system, i.e., $\mathcal{C}_{\rm 1d}H\mathcal{C}_{\rm
1d}^{-1}=-H$ ($H=H_{\rm s},H_{\rm cr}$), where $\mathcal{C}_{\rm
1d}=\sum_{n=1}^{N}a_{n}^{\dagger}\sigma_{3}a_{n}$ for the SSH model and
$\mathcal{C}_{\rm 1d}=\sum_{n=1}^{N}a_{n}^{\dagger}\sigma_{2}a_{n}$ for the
Creutz model.

Let’s first consider the case of no disorder, i.e., $d_{1}=d_{2}=0$, the NN
(and NNNN) SSH model and Creutz model feature the TPTs at $\nu=1$ (and
$\eta=0,1$) and $\eta_{0}=\eta^{\prime}_{0}$, respectively Asboth2016OP ;
Su1979SH ; Rufo2019LC ; Creutz1999 . To identify the topological non-trivial
and trivial phases in real space, in Fig. 2, we numerically calculate the OTOC
dynamics with Eq. (1), which involves the backward evolution. Note that, Fig.
2 includes the results for choosing different OTOC operators $V$ and $W$. It
clearly shows that, both for the SSH model and Creutz model, the
distinguishable OTOC dynamics appears in the non-trivial and trivial phases.
Specifically, the OTOC evolves to a finite value and almost zero in the
topological non-trivial and trivial phases, respectively [see the insets of
Figs. 2(b,d,f)]. This relates to the physical mechanism that the information
does scramble in the trivial phase, while this scrambling is suppressed
immensely in the nontrivial phase. There exists a zero-to-finite-value
transition in the long-time limit of the OTOC, when the system enters into the
non-trivial phase from the trivial phase. This distinguishable OTOC dynamics
is robust to the initial state of the system (i.e., the operator $V$), which
could be a single-site occupation or multi-site occupation state. Moreover,
the averaged OTOC becomes discrete at the critical point, when the initial
state is the eigenstate of the system whose eigenvalue has the lowest absolute
value supp . Figure 2 also shows that the OTOC witness is not limited by the
choice of the operator $W$. In our proposal, the operator $W$ can either a
few-site (including single-site) operation on sublattice $A$ (e.g.,
$W=\sum_{l=1}^{L}a_{l,A}^{\dagger}a_{l,A}$, $L=1,2,3$) or a multi-site
operation on sublattices $A$ and $B$ (e.g.,
$W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{j}a_{n}$, $j=2,3$), and the chosen
operators $W$ neither commute nor anti-commute with the system Hamiltonian,
i.e., $[W,H]_{\pm}\neq 0$.

Figure 2: The dependence of $\mathcal{O}(t\rightarrow\infty)$ on $\nu$,
$\eta$, and $\eta_{0}/\eta_{0}^{\prime}$ for (a,c,e)
$W=\sum_{l=1}^{L}a_{l,A}^{\dagger}a_{l,A}$ and (b,d,f)
$W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{j}a_{n}$ [$j=3$ for (b,d) and $j=2$
for (f)]. Panels (a,b), (c,d), and (e,f) correspond to the systems described
by the NN SSH model, NNNN SSH model, and Creutz model, respectively. The
initial states are set as (a,c,e) $|\psi_{0}\rangle=|1,A\rangle$, (b,d)
$|\psi_{0}\rangle=\sum_{m=1}^{M}(-1)^{m-1}|m,A\rangle/\sqrt{M}$ and (f)
$|\psi_{0}\rangle=\sum_{m=1}^{M}(-1)^{m-1}(|m,A\rangle+i|m,B\rangle)/\sqrt{2M}$.
Insets: the evolution of the OTOC for different values of $\nu$, $\eta$ and
$\eta_{0}/\eta_{0}^{\prime}$ when $M=1$. The lines and dots correspond to the
fully numerical simulations obtained by Eq. (1) and the analytical results
obtained by Eqs. (3,4), respectively. Other system parameters are $N=200$,
$d_{1}=d_{2}=0$, (a,b) $\eta=0$, (c,d) $\nu=1$. The TNPs and TTPs are
indicated by the gray shadings and write areas, respectively.

To fully show the dependence of the OTOC witness on system parameters, we also
calculate the analytical solution of $\mathcal{O}(t)$ under the condition of
$N\gg 1$. Let’s consider the NN SSH model as an example, and choose
$|\psi_{0}\rangle=\sum_{m=1}^{M}\frac{(-1)^{m-1}}{\sqrt{M}}|m,A\rangle$, where
$M=1$ corresponds to the case of single-site occupation state, i.e.,
$|\psi_{0}\rangle=|1,A\rangle$. Here, $m$ and $A/B$ in state $|m,A/B\rangle$
represent the $m$th cell and sublattice $A/B$, respectively. Corresponding to
$W=\sum_{l=1}^{L}a_{l,A}^{\dagger}a_{l,A}$ and
$W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{3}a_{n}$, we respectively obtain
supp

$\displaystyle\\!\\!\mathcal{O}(t)\\!\approx\\![1/\\!\sum_{n=0}^{N}\nu^{2n}\\!+\\!\sum_{k=1}^{N}\\!\frac{2\epsilon^{2}\nu^{2}\cos(\lambda_{+}^{(k)}t)}{(N+1)(\lambda_{\pm}^{(k)})^{2}}\sin^{2}(\\!\frac{k\pi}{N+1}\\!)]^{4}$
(3)

and

$\displaystyle\\!\\!\mathcal{O}(t)\\!\approx\\![1/\\!\sum_{n=0}^{N}\\!\nu^{2n}\\!\\!+\\!\\!\sum_{k=1}^{N}\\!\frac{2\epsilon^{2}\nu^{2}\cos(2\lambda_{+}^{(k)}t)}{(N+1)(\lambda_{\pm}^{(k)})^{2}}\sin^{2}(\\!\frac{k\pi}{N+1}\\!)]^{2}$
(4)

for $L,M=1$. Here
$\lambda_{\pm}^{(k)}=\pm\epsilon[1+\nu^{2}+2\nu\cos(\frac{k\pi}{N+1})]^{1/2}$
and $k=1,2,\dots,N$. Note that the above equations require $\nu\neq 0$, and
$\nu=0$ means that the hopping cannot occur in the intracells, corresponding
to $\mathcal{O}(t)=1$. The similar analytical results for $L,M>1$ are shown in
the supplementary material supp . As shown in Figs. 1(a,b), the analytical
solutions also present a zero-to-finite-value transition of OTOC at the
critical point of TPTs. This conclusion is valid for both the cases of
choosing $W$ as a single-site operation and a multi-site operation. Figures
2(a,b) show a very good agreement between the analytical solutions and the
fully numerical simulations, which demonstrates the validity of our solutions.

Now let’s discuss the influence of disorder on our proposal by choosing the NN
SSH model as an example. The proposed OTOC witness for identifying the TPTs is
also suitable for disordered systems. As shown in Figs. 3(a,b),
$\mathcal{O}(t\rightarrow\infty)$ still undergoes the zero-to-finite-value
transition along with the occurrence of the TPTs, even when weak disorder is
introduced into the system. In terms of information, this transition
originally comes from the topological locality in the non-trivial phase.
Specifically, the information scrambling occurs in the trivial phase, and is
suppressed immensely in the non-trivial phase. Similar as the case of no
disorder, this result is robust to the choices of the operator $W$. Figures
3(a,b) also show that the above distinguishability of the OTOC dynamics
disappears in the strong disorder regime (e.g., $d>4$). Physically, this is
because the TPTs, together with the symmetry-protected boundary state, will
disappear as the disorder is too large. Figures 3(c,d) further demonstrate the
vanishing of the topological non-trivial phase induced by strong disorder.
Moreover, the proposed OTOC witness can also be considered as an order
parameter of the topological phase diagram, and predict topological Anderson
insulator physics supp . It is consistent with previous works in Refs.
MondragonShem2014HS ; Meier2018AD , which further verify the validity of our
OTOC witness.

Figure 3:  (a,b) The dependence of $\mathcal{O}(t\rightarrow\infty)$ on $\nu$
for different disorder strengths $d$ when (a) $W=a_{1,A}^{\dagger}a_{1,A}$ and
(b) $W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{3}a_{n}$. (c) The value of
$\mathcal{O}(t\rightarrow\infty)$ versus $d$ for different choices of the
operator $W$ when $\nu=0.2$. (d) The evolution of the OTOC for different $d$
indicated by the circles in (c). Here all data are averaged over 30
independent disorder configurations, and we have chosen $N=200$,
$d_{2}=2d_{1}=d$, and $|\psi_{0}\rangle=|1,A\rangle$. The TNPs and TTPs are
indicated by the gray shadings and write areas, respectively.

_Detecting TPTs in the systems without chiral symmetry._ —The proposed OTOC
witness for identifying the TPT is not limited to the above systems with
chiral symmetry, but is applicable for the systems without chiral symmetry,
such as 2D lattice systems described by the Haldane model and Qi-Wu-Zhang
model. As shown in Fig. 4(a), the Haldane model on the honeycomb lattice has
Hamiltonian Haldane1988 ; Jotzu2014MD

$\displaystyle H_{\rm ha}\\!=\\!\eta_{1}\\!\sum_{\langle
j,j^{\prime}\rangle}c_{j}^{\dagger}c_{j^{\prime}}\\!+\\!\eta_{2}\\!\\!\sum_{\langle\langle
j,j^{\prime}\rangle\rangle}\\!e^{is_{jj^{\prime}}\phi}c_{j}^{\dagger}c_{j^{\prime}}\\!+\\!\mu
s^{\prime}\\!\sum_{j}c_{j}^{\dagger}c_{j},$ (5)

where $c_{j}^{\dagger}$ ($c_{j}$) is the creation (annihilation) operator of
the $j$th site, and the summation indexes cover all sites. The symbol $\mu$ in
last term denotes the sublattice potential, where $s^{\prime}=+1$ and
$s^{\prime}=-1$ correspond to sublattices $A$ and $B$, respectively. Here,
$\eta_{1}$ and $\eta_{2}$ are the real-valued nearest- and next-nearest-
neighbor hopping amplitudes, respectively. The next-nearest-neighbor hopping
contains the phases $s_{jj^{\prime}}\phi$ with $s_{jj^{\prime}}=\pm 1$, which
can break the time-reversal symmetry. The system has no chiral symmetry and is
a paradigmatic example of 2D lattice featuring TPTs. For example, the
parameter ranges $|\mu/\eta_{2}|<3\sqrt{3}$ and $\mu/\eta_{2}=other$
correspond respectively to the topological non-trivial and trivial phases when
$\phi=\pi/2$. Similar as the procedure used in 1D systems with chiral
symmetry, we numerically calculate the OTOC dynamics with Eq. (1) to identify
the occurrence of TPTs in real space. As shown in Fig. 4(b), the zero-to-
finite-value transition of $\mathcal{O}(t\rightarrow\infty)$ can still be
observed when the system enters into the topological nontrivial phase from the
trivial phase. The similar results can also be obtained in the system
described by the Qi-Wu-Zhang model supp .

Figure 4: (a) Scheme of the Haldane model, where the unit cell consists of
sublattices $A$ and $B$. (b) The dependence of
$\mathcal{O}(t\rightarrow\infty)$ on $\mu/\eta_{1}$ for different cell numbers
when $|\psi_{0}\rangle=|1,A\rangle$ and $W=\sum_{j}c_{j}^{\dagger}c_{j}$ (the
summation index $j$ only cover all sublattice $B$). Here we have chosen
$\eta_{1}=\eta_{2}$ and $\phi=\pi/2$. The red and blue lines correspond to the
cell numbers of $4\times 4$ and $20\times 20$, respectively. (c) Scheme of the
2D SSH model with gauge flux $\pi$ penetrating any plaquette. (d) The
dependence of $\mathcal{O}(t\rightarrow\infty)$ on $\nu^{\prime}/w$ for
$W=a_{1,1}^{\dagger}a_{1,1}$ (black curve) and
$W=a_{1,1}^{\dagger}a_{1,1}+a_{1,3}^{\dagger}a_{1,3}+a_{3,1}^{\dagger}a_{3,1}$
(blue curve) when $|\psi_{0}\rangle=|1,1\rangle$. The TNPs and TTPs are
indicated by the gray shadings and write areas, respectively.

_Application to the second-order TPTs_.—Higher-order topological insulators,
as an extension of the topological insulators, have recently attracted
extensive attention Langbehn2017PT ; Schindler2018CV ; Liu2019ZA ; Kunst2018MB
; Imhof2018BB ; Ezawa2018 ; Schindler2018WV ; Noh2018BH ; Geier2018TH ;
Chen2018CG ; Zhu2021HZ ; Che2020GL . High-order TPTs usually can be identified
by detecting the boundary states in real space. For example, the topological
protected corner states have been used to identify the second-order TPT in a
2D system Benalcazar2017BH ; SerraGarcia2018PS ; Peterson2018BH ; Mittal2019OZ
. Here, our proposed OTOC witness is also applicable for detecting second-
order TPTs. As shown in Fig. 4(c), we take the extended 2D SSH model with non-
zero gauge flux as an example, and its Hamiltonian reads Mittal2019OZ

$\displaystyle\\!\\!H_{\rm 2s}({\rm{\bf k}})=$
$\displaystyle(\nu^{\prime}+w\cos k_{y})\tau_{0}\otimes\sigma_{1}\\!-\\!w\sin
k_{y}\tau_{3}\otimes\sigma_{2}$ $\displaystyle-(\nu^{\prime}+w\cos
k_{x})\tau_{2}\otimes\sigma_{2}\\!-\\!w\sin k_{x}\tau_{1}\otimes\sigma_{2},$
(6)

where ${\rm{\bf k}}=\\{k_{x},k_{y}\\}$ are the wave number, and
$\pm\nu^{\prime}$ ($\pm w$) is the intracell (intercell) hopping strength.
This system features a second-order TPT when increasing the value of
$\nu^{\prime}/w$, i.e., $\nu^{\prime}\\!<\\!w$ and $\nu^{\prime}\\!>\\!w$
corresponding to the topological non-trivial and trivial phases, respectively.
To identify the occurrence of second-order TPTs, in Fig. 4(d), we numerically
calculate the OTOC in the lattice system with $20\times 20$ cells when the
different OTOC operators $W$ are considered. Figure 4(d) clearly shows the
distinguishable OTOC dynamics in the topological non-trivial and trivial
phases. Both for $W=a_{1,1}^{\dagger}a_{1,1}$ and
$W=a_{1,1}^{\dagger}a_{1,1}+a_{1,3}^{\dagger}a_{1,3}+a_{3,1}^{\dagger}a_{3,1}$,
the zero-to-finite-value transition of $\mathcal{O}(t\rightarrow\infty)$
appears at the critical point of the second-order TPT. Moreover, the system is
initially in the corner site $(1,1)$ (i.e., $|\psi_{0}\rangle=|1,1\rangle$),
which is experimentally feasible. Here $(x,y)$ represents a lattice point in
the square lattice, and $|x,y\rangle$ denotes the state occupying in the site
$(x,y)$. The creation (annihilation) operator of the site $(x,y)$ is denoted
by $a_{x,y}^{\dagger}$ ($a_{x,y}$).

_Experimental implementation and conclusions._ —Regarding experimental
implementations, the trapped ion Nevado2017FLP ; Garttner2017BSN ;
Haffner2008RB ; Blatt2012Roos ; Monroe2021CD  is an ideal candidate for our
proposal. We consider a set of $2N$ trapped ions with excited and ground
states arranged along a 1D chain as the SSH model. First, the system is
initialized in $\rho_{0}=|1,A\rangle\langle 1,A|$ by applying a $\pi$ pulse to
excite the first ion in the chain into its excited state Haffner2008RB ;
Blatt2012Roos ; Monroe2021CD . Then, one should make the system evolve under
the Hamiltonian for a time $t$ to the state
$\rho_{1}(t)=e^{-iHt}\rho_{0}e^{iHt}$. Subsequently, applying the operator $W$
to get $\rho_{2}(t)=W^{\dagger}\rho_{1}(t)W$. When the operator $W$ is a
single-site operator on sublattice $A$, it can be achieved by removing the
polarizations of the ions except for that of the first ions by using selective
pulses Haffner2008RB ; Blatt2012Roos ; Monroe2021CD ; Garttner2017BSN . Next,
inverting the sign of $H$ by the spin echo technique (i.e., applying a $\pi$
pulse to reverse the polarization of one of the ions) Hahn1950SE  and making
the system evolve again for $t$ to obtain the final state
$\rho_{f}=e^{iHt}\rho_{2}(t)e^{-iHt}$ Swingle2016BSS ; Sanchez2020SC .
Finally, the OTOC can be obtained by measuring the overlap of the final state
with respect to the initial state via a fluorescence detection Garttner2017BSN
; Monroe2021CD , similar as the many-body Loschmidt echo technique. For 2D
lattice systems, the OTOC measurement is similar to that of the 1D lattice
systems except for the construction of the model. Note that our proposal is
not limited to this particular architecture, and could be implemented or
adapted in a variety of platforms that have full local quantum control
Wang2018ZX ; Cai2019HM ; Li2017FW ; Nie2020WC ; Atala2013AB ; Lohse2016SZ ;
Leseleuc2019LS ; Wei2019PS ; Zhao2022Ge , such as a nuclear magnetic resonance
quantum simulator Li2017FW ; Nie2020WC ; Wei2019PS  and superconducting qubit
Wang2018ZX ; Cai2019HM ; Zhao2022Ge .

In conclusion, we have proposed a zero-temperature OTOC witness in real space
for identifying $\mathbb{Z}$-type TPTs in general lattice systems with or
without chiral symmetry. Our proposal is robust to the choices of the initial
state of the system and the used operators in OTOC. It is also suitable for
disordered systems, and can predict topological Anderson insulator physics in
the strong disorder regime. Moreover, the proposed OTOC witness can be used to
detect not only first-order TPTs, but also second-order TPTs. Applying it into
non-Hermitian systems supp , the TPTs can be identified without implementing
the transition from non-Bloch to Bloch theory. The generality of our proposal
leads to that the proposed OTOC witness has predictive power in detecting
TPTs. For example, we could construct the OTOC witness by preparing the system
initially being in the first site and choosing a single-site operation as the
$W$ operator, even in a situation where we don’t already understand the
structure of a 1D lattice.

We thank Prof. T. Liu and Prof. J.-H. Gao for helpful discussions. This work
is supported by the National Key Research and Development Program of China
grant 2021YFA1400700, the National Science Foundation of China (Grants No.
11974125, No. 12205109, No. 12147143), and the China Postdoctoral Science
Foundation No. 2021M701323. F.N. is supported in part by: Nippon Telegraph and
Telephone Corporation (NTT) Research, the Japan Science and Technology Agency
(JST) [via the Quantum Leap Flagship Program (Q-LEAP), and the Moonshot R&D
Grant Number JPMJMS2061], the Japan Society for the Promotion of Science
(JSPS) [via the Grants-in-Aid for Scientific Research (KAKENHI) Grant No.
JP20H00134], the Army Research Office (ARO) (Grant No. W911NF-18-1-0358), the
Asian Office of Aerospace Research and Development (AOARD) (via Grant No.
FA2386-20-1-4069), and the Foundational Questions Institute Fund (FQXi) via
Grant No. FQXi-IAF19-06.

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Supplemental Material for “Out-of-Time-Order Correlation as a Witness for
Topological Phase Transitions”

Supplemental Material for “Nonreciprocal photon-phonon and photon-magnon pairs
emission”

Supplemental Material for “Out-of-Time-Order Correlation as a Witness for
Topological Phase Transitions” Qian Bin

Liang-Liang Wan

Franco Nori

Ying Wu

Xin-You Lü This supplemental material contains five parts: I. A detailed
derivation of the analytical OTOC dynamics for the nearest-neighbor SSH model.
II. Discussion of OTOC witness when the initial state is the eigenstate of the
system. III. Additional discussion of the application of OTOC witness in the
disordered systems. IV. Discussion of the application of OTOC witness in two-
dimensional lattice described by the Qi-Wu-Zhang model. V. Discussion of the
application of OTOC witness in the non-Hermitian systems.

##  I Derivation of the analytical OTOC dynamics

In this section, we present the analytical derivation of the OTOC dynamics in
the system described by the 1D nearest-neighbor Su-Schrieffer-Heeger (SSH)
model. We remind the Hamiltonian of the nearest-neighbor SSH model in the
absence of disorder

$\displaystyle H_{\rm s}=\sum_{n}\left\\{\epsilon\nu
a_{n}^{\dagger}\sigma_{1}a_{n}+\frac{\epsilon}{2}\left[a_{n+1}^{\dagger}(\sigma_{1}+i\sigma_{2})a_{n}+{\rm
h.c.}\right]\right\\},$ (S1)

where $\sigma_{j}$ ($j=0,1,2,3$) are the Pauli matrices, corresponding to the
identity matrix $I$, $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$,
respectively. Here $a_{n}^{\dagger}=(a_{n,A}^{\dagger},a_{n,B}^{\dagger})$ is
the annihilation operator of the unit cell $n$ with sublattices $A$, $B$, and
$\epsilon$ ($\epsilon\nu$) is the intercell (intracell) hopping strength. The
parameter regimes $\nu<1$ and $\nu>1$ correspond to the topological non-
trivial and trivial phases, respectively. This model has a chiral symmetry
defined by a chiral operator $\mathcal{C}_{\rm
1d}=\sum_{n}a_{n}^{\dagger}\sigma_{3}a_{n}$ satisfying $[H_{\rm
s},\mathcal{C}_{\rm 1d}]_{+}=0$. By defining
$V=V\rho_{0}=|\psi_{0}\rangle\langle\psi_{0}|$, the constructed OTOC becomes
an experimentally feasible fidelity of the final state $\rho_{f}$ projected
onto an initial state $\rho_{0}$, with

$\displaystyle\mathcal{O}(t)={\rm tr}[\rho_{0}e^{iH_{\rm
s}t}W^{\dagger}e^{-iH_{\rm s}t}\rho_{0}e^{iH_{\rm s}t}We^{-iH_{\rm s}t}].$
(S2)

In the following, we address in turn the analytical solutions of the OTOC
dynamics for different choices of the operator $W$ and the initial state
$|\psi_{0}\rangle$.

We consider a general initial state

$\displaystyle|\psi_{0}\rangle=\frac{1}{\sqrt{M}}\sum_{m=1}^{M}(-1)^{m-1}|m,A\rangle,$
(S3)

where $M=1$ corresponds to $|\psi_{0}\rangle=|1,A\rangle$. The matrix of the
SSH Hamiltonian $H_{\rm s}$ including $2N$ sites in single-particle space has
an implicit formula for the eigenpairs when $\nu\neq 1$S_Shin1997 . Then it is
very difficult to analytically calculate the OTOC dynamics under this
Hamiltonian. Comparing the cases of including $2N$ sites and $2N+1$ sites, the
numerical energy spectrum of the latter is hardly changed except for one zero
energy level is added, when the value of $N$ is large enough. The OTOC
dynamics is almost not influenced by this extending of the Hamiltonian size.
Thus, to analytically calculate the OTOC dynamics, we extend the size of the
SSH model to $2N+1$ sites, and the Hamiltonian $H_{\rm s}$ becomes

$\displaystyle H_{\rm s}=\epsilon\left(\begin{array}[]{cccccc}0&\nu&&&&\\\
\nu&0&1&&&\\\ &1&0&\nu&&\\\ &&\ddots&\ddots&\ddots&\\\ &&&\nu&0&1\\\
&&&&1&0\end{array}\right)_{(2N+1)\times(2N+1)}.\quad$ (S10)

The eigenvalues of Eq. (S10) are

$\displaystyle\lambda_{0}=0,~{}~{}~{}~{}~{}\lambda_{\pm}^{(k)}=\pm\epsilon\sqrt{1+\nu^{2}+2\nu\cos\left(\frac{k\pi}{N+1}\right)},~{}~{}~{}~{}~{}~{}~{}~{}(k=1,2,\dots
N)$ (S11)

and the corresponding eigenstates are

$\displaystyle
V^{(0)}=\frac{1}{\sqrt{A_{0}}}\left[(-\nu)^{0},0,(-\nu)^{1},0,(-\nu)^{2},0,\dots,0,(-\nu)^{N-1},0,(-\nu)^{N}\right]^{T}$
(S12)

and

$\displaystyle V^{(k)}_{\pm}=$
$\displaystyle\frac{1}{\sqrt{A_{\pm}^{(k)}}}\left[\frac{1}{\nu}\sin\left(\frac{0k\pi}{N+1}\right)+\sin\left(\frac{k\pi}{N+1}\right),\frac{\lambda^{(k)}_{\pm}}{\epsilon\nu}\sin\left(\frac{k\pi}{N+1}\right),\frac{1}{\nu}\sin\left(\frac{k\pi}{N+1}\right)+\sin\left(\frac{2k\pi}{N+1}\right),\right.$
$\displaystyle\left.\frac{\lambda^{(k)}_{\pm}}{\epsilon\nu}\sin\left(\frac{2k\pi}{N+1}\right),\frac{1}{\nu}\sin\left(\frac{2k\pi}{N+1}\right)+\sin\left(\frac{3k\pi}{N+1}\right),\frac{\lambda^{(k)}_{\pm}}{\epsilon\nu}\sin\left(\frac{3k\pi}{N+1}\right),\dots,\right.$
$\displaystyle\left.\frac{1}{\nu}\sin\left(\frac{(N-1)k\pi}{N+1}\right)+\sin\left(\frac{Nk\pi}{N+1}\right),\frac{\lambda^{(k)}_{\pm}}{\epsilon\nu}\sin\left(\frac{Nk\pi}{N+1}\right),\frac{1}{\nu}\sin\left(\frac{Nk\pi}{N+1}\right)+\sin\left(\frac{(N+1)k\pi}{N+1}\right)\right]^{T},$
(S13)

respectively, where

$\displaystyle
A_{0}=\sum_{n^{\prime}=0}^{N}\nu^{2n^{\prime}}=\begin{cases}(1-\nu^{2N+2})/(1-\nu^{2})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\nu\neq
1)\\\
N+1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\nu=1)\end{cases},~{}~{}~{}~{}~{}~{}~{}~{}$
(S14) $\displaystyle
A_{\pm}^{(k)}=\frac{(N+1)(\lambda^{(k)}_{\pm})^{2}}{\epsilon^{2}\nu^{2}}.$
(S15)

In the space spanned by the eigenstates $\\{V^{(0)},V^{(k)}_{\pm}\\}$, the
Hamiltonian $H_{\rm s}$ is given by

$\displaystyle H_{\rm s}=\lambda^{(0)}|V^{(0)}\rangle\langle
V^{(0)}|+\sum_{k=1}^{N}\left(\lambda_{+}^{(k)}|V_{+}^{(k)}\rangle\langle
V_{+}^{(k)}|+\lambda_{-}^{(k)}|V_{-}^{(k)}\rangle\langle V_{-}^{(k)}|\right).$
(S16)

Then we obtain the instantaneous state dominated by the forward evolution of
$H_{\rm s}$, with

$\displaystyle e^{-iH_{\rm
s}t}|\psi_{0}\rangle=\exp\\{-i[\lambda^{(0)}|V^{(0)}\rangle\langle
V^{(0)}|+{\sum}_{k=1}^{N}(\lambda_{+}^{(k)}|V_{+}^{(k)}\rangle\langle
V_{+}^{(k)}|+\lambda_{-}^{(k)}|V_{-}^{(k)}\rangle\langle
V_{-}^{(k)}|)]t\\}|\psi_{0}\rangle$
$\displaystyle=\frac{1}{\sqrt{MA_{0}}}\sum_{m=1}^{M}\nu^{m-1}|V^{(0)}\rangle+\sum_{k=1}^{N}\sum_{m=1}^{M}\frac{(-1)^{m-1}}{\sqrt{M}}\left[\frac{1}{\nu}\sin(\frac{m-1}{N+1}k\pi)+\sin(\frac{mk\pi}{N+1})\right]\left[\frac{e^{-i\lambda_{+}^{(k)}t}}{\sqrt{A_{+}^{(k)}}}|V_{+}^{(k)}\rangle+\frac{e^{-i\lambda_{-}^{(k)}t}}{\sqrt{A_{-}^{(k)}}}|V_{-}^{(k)}\rangle\right].$
(S17)

where

$\displaystyle\langle
V^{(0)}|m,A\rangle=\frac{1}{\sqrt{A_{0}}}(-\nu)^{m-1},~{}~{}~{}\langle
V_{\pm}^{(k)}|m,A\rangle=\frac{1}{\sqrt{A_{\pm}^{(k)}}}\left[\frac{1}{\nu}\sin\left(\frac{m-1}{N+1}k\pi\right)+\sin\left(\frac{mk\pi}{N+1}\right)\right],$
(S18)

and

$\displaystyle\langle
V^{(0)}|\psi_{0}\rangle=\frac{1}{\sqrt{MA_{0}}}(\nu^{0}+\nu^{1}+\nu^{2}+\dots+\nu^{M-1})=\frac{1}{\sqrt{MA_{0}}}\sum_{m=1}^{M}\nu^{m-1},$
(S19) $\displaystyle\langle
V_{\pm}^{(k)}|\psi_{0}\rangle=\frac{1}{\sqrt{MA_{\pm}^{(k)}}}\sum_{m=1}^{M}(-1)^{m-1}\left[\frac{1}{\nu}\sin\left(\frac{m-1}{N+1}k\pi\right)+\sin\left(\frac{mk\pi}{N+1}\right)\right].$
(S20)

Figure S1:  The values of (a) $\mathcal{O}_{2}(t)$, (b) $\mathcal{O}_{3}(t)$,
(c) $\mathcal{O}_{4}(t)$, and (d) $R_{12}(t)$ versus $\epsilon t$ and $\nu$
when $W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{3}a_{n}$ and
$|\psi_{0}\rangle=|1,A\rangle$. Other system parameters are $N=200$ and
$d_{1}=d_{2}=0$.

When the OTOC operator
$W=\sum_{l=1}^{L}a_{l}^{\dagger}(\sigma_{3}+\sigma_{0})a_{l}/2=\sum_{l=1}^{L}a_{l,A}^{\dagger}a_{l,A}$
($L=1$ corresponds to the single-site operation), we obtain

$\displaystyle\mathcal{O}(t)={\rm tr}[\rho_{0}e^{iH_{\rm
s}t}W^{\dagger}e^{-iH_{\rm s}t}\rho_{0}e^{iH_{\rm s}t}We^{-iH_{\rm s}t}]$
$\displaystyle\\!\\!=\left|\sum_{l=1}^{L}\\!\left\\{\\!\frac{(-\nu)^{l-1}}{A_{0}\sqrt{M}}\\!\\!\sum_{m=1}^{M}\\!\nu^{m-1}\\!\\!+\\!\\!\\!\sum_{k=1}^{N}\\!\sum_{m=1}^{M}\\!\\!\frac{2(-1)^{m-1}\\!\cos(\lambda_{+}^{(k)}\\!t)}{\sqrt{M}A_{+}^{(k)}}\\!\left[\\!\frac{1}{\nu}\sin(\frac{m\\!-\\!1}{N\\!+\\!1}k\pi)\\!+\\!\sin(\frac{mk\pi}{N\\!+\\!1})\\!\right]\\!\\!\left[\\!\frac{1}{\nu}\sin(\frac{l\\!-\\!1}{N\\!+\\!1}k\pi)\\!+\\!\sin(\frac{lk\pi}{N\\!+\\!1})\\!\right]\\!\right\\}^{2}\right|^{2}.$
(S21)

This equation is reduced to Eq. (3) of the main text under the conditions of
$L=1$ and $M=1$. When the OTOC operator
$W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{3}a_{n}=\sum_{n=1}^{N}a_{n}^{\dagger}\sigma_{3}a_{n}-a_{N}^{\dagger}\sigma_{3}a_{N}$,
the OTOC function is reduced to

$\displaystyle\mathcal{O}(t)={\rm tr}[\rho_{0}e^{iH_{\rm
s}t}W^{\dagger}e^{-iH_{\rm s}t}\rho_{0}e^{iH_{\rm s}t}We^{-iH_{\rm
s}t}]=|\mathcal{O}_{1}+\mathcal{O}_{2}(t)-\mathcal{O}_{3}(t)+\mathcal{O}_{4}(t)|^{2}$
(S22)

where

$\displaystyle\mathcal{O}_{1}=\frac{1}{MA_{0}}\left(\sum_{m=1}^{M}\nu^{m-1}\right)^{2},$
$\displaystyle\mathcal{O}_{2}(t)=\sum_{k=1}^{N}\frac{2\cos(2\lambda_{+}^{(k)}t)}{MA_{\pm}^{(k)}}\left[\sum_{m=1}^{M}(-1)^{m-1}\left(\frac{1}{\nu}\sin(\frac{m-1}{N+1}k\pi)+\sin(\frac{mk\pi}{N+1})\right)\right]^{2},$
$\displaystyle\mathcal{O}_{3}(t)=\left|\\!\frac{(-\nu)^{N-1}}{A_{0}\sqrt{M}}\\!\\!\sum_{m=1}^{M}\\!\nu^{m-1}\\!+\\!\\!\sum_{k=1}^{N}\\!\sum_{m=1}^{M}\\!\frac{2(-1)^{m-1}\cos(2\lambda_{+}^{(k)}t)}{\sqrt{M}A_{\pm}^{(k)}}\\![\frac{\sin(\frac{m-1}{N+1}k\pi)}{\nu}\\!+\\!\sin(\frac{mk\pi}{N+1})]\\![\frac{\sin(\frac{N-1}{N+1}k\pi)}{\nu}\\!+\\!\sin(\frac{Nk\pi}{N+1})]\right|^{2},$
$\displaystyle\mathcal{O}_{4}(t)=\left|\sum_{k=1}^{N}\\!\sum_{m=1}^{M}\\!\frac{-2i(-1)^{m-1}\lambda^{(k)}_{+}\sin(2\lambda_{+}^{(k)}t)}{\sqrt{M}A_{\pm}^{(k)}\epsilon\nu}\\![\frac{1}{\nu}\\!\sin(\frac{m-1}{N+1}k\pi)+\\!\sin(\frac{mk\pi}{N+1})]\\!\sin(\frac{Nk\pi}{N+1})\right|^{2}.$

The terms $\mathcal{O}_{2}(t)$, $\mathcal{O}_{3}(t)$ and $\mathcal{O}_{4}(t)$
trend to zero over time, and
$\mathcal{O}_{3}(t),\mathcal{O}_{4}(t)\ll\mathcal{O}_{2}(t)$ [see Figs.
S1(a-c)]. Then the above OTOC function can be approximately reduced to
$\mathcal{O}(t)\approx|\mathcal{O}_{1}+\mathcal{O}_{2}(t)|^{2}$, and we can
obtain the Eq. (4) of the main text under the conditions of $L=1$ and $M=1$.
We define the ratio $R_{12}(t)=\mathcal{O}_{1}/\mathcal{O}_{2}(t)$. As shown
in Fig. S1(d), in the trivial phase $\nu>1$, $R_{12}(t)\ll 1$ means that the
analytical solution can be approximatively reduced to
$\mathcal{O}(t)\approx\mathcal{O}_{2}^{2}(t)$, which evolves to almost zero in
the long-time limit. In the non-trivial phase $\nu<1$, we have
$R_{12}(t\to\infty)\gg 1$ and the OTOC is reduced to
$\mathcal{O}(t\to\infty)\approx\mathcal{O}^{2}_{1}$, which is a finite value.
At the critical point $\nu=1$, $R_{12}(t\to\infty)\approx 1$ corresponds to
$\mathcal{O}(t\to\infty)\approx 0$. This can also be seen in the phase diagram
Fig.1(e) of the main text. Note that $\nu\neq 0$ in the above equations, and
$\nu=0$ means that the hopping cannot occurs in the intercells. Here we have
extended the size of the SSH model to $2N+1$ sites, but these solutions Eqs.
(I-I) are still valid for the system including $2N$ sites when the value of
$N$ is large enough. In the main text, we have shown the numerical results
obtained by numerically calculating Eq. (S2) for the system including $2N$
sites and the analytical results obtained by Eqs. (I-I) for the system
including $2N+1$ sites, respectively. The excellent agreement between the
analytical solutions and fully numerical simulations demonstrates the validity
of our approximation.

Figure S2: (a) Energy spectrum of the 1D nearest-neighbor SSH model. (b) The
evolution of the OTOC for different values of $\nu$, when the system is
initially in the eigenstate $|\psi_{E}\rangle_{\rm 1d}$ of the system whose
eigenvalue has the lowest absolute value, corresponding to the red curve of
(a). (c) The average of the OTOC evolution $\bar{\mathcal{O}}(t)$ versus
$\nu$. System parameters are $N=200$ and
$W=\sum_{n=1}^{N}a_{n,A}^{\dagger}a_{n,A}$. The gray and white areas
correspond to the topological non-trivial phase (TNP) and topological trivial
phase (TTP), respectively.
Figure S3: Topological phase diagrams: (a) the averaged OTOC
$\bar{\mathcal{O}}(t)$ versus disorder strength $d$ and $\nu$ when
$W=\sum_{n=1}^{N}a_{n,A}^{\dagger}a_{n,A}$ and
$|\tilde{\psi}_{0}\rangle=Q_{\rm 1d}|\psi_{E}\rangle_{\rm 1d}$; (b) the order
parameter defined in Refs. S_MondragonShem2014HS ; S_Meier2018AD  versus
disorder strength $d$ and $\nu$. Here all data are averaged over 30
independent disorder configurations, and $d_{2}=2d_{1}=d$ and $N=1000$.

##  II The OTOC witness when the initial state is the eigenstate of system

In the section, we discuss the connection between OTOC dynamics and TPTs when
the initial state is the eigenstate $|\psi_{E}\rangle_{\rm 1d}$ of the system
whose eigenvalue has the lowest absolute value [see the red curve in Fig.
S2(a)]. Let’s take the system described by the the nearest-neighbor SSH model
as an example. Applying an operation on the initial state, we obtain
$|\tilde{\psi}_{0}\rangle=Q_{\rm s}|\psi_{E}\rangle_{\rm 1d}$, where $Q_{\rm
s}=\sum_{n}a_{n}^{\dagger}(\sigma_{0}+\sigma_{3})a_{n}/2$. In the topological
non-trivial phase $\nu<1$, the state $|\psi_{E}\rangle_{\rm 1d}$ is naturally
an edge state of the system, e.g., the left edge state
$\sum_{n}\alpha_{n}|n,A\rangle$ with $\sum_{n}|\alpha_{n}|^{2}=1$, and then
$|\tilde{\psi}_{0}\rangle=Q_{\rm 1d}|\psi_{E}\rangle_{\rm
1d}=\sum_{n}\alpha_{n}|n,A\rangle=|\psi_{E}\rangle_{\rm 1d}$. The OTOC is
reduced to $\mathcal{O}(t)={\rm tr}[\tilde{\rho}_{0}^{2}]=1$, which is a
conserved quantity. The averaged OTOC function $\bar{\mathcal{O}}(t)=1$. In
the topological trivial phase $\nu>1$, $|\psi_{E}\rangle_{\rm 1d}$ is not the
edge state of the system, but a bulk state $|\psi_{E}\rangle_{\rm
1d}=\sum_{n}(\alpha_{n}|n,A\rangle+\beta_{n}|n,B\rangle)$, where
$\sum_{n}(|\alpha_{n}|^{2}+|\beta_{n}|^{2})=1$ and
$\sum_{n}|\alpha_{n}|^{2}=\sum_{n}|\beta_{n}|^{2}=1/2$. In Figs. S2(b,c), we
show the OTOC dynamics and averaged OTOC in different topological phases when
the OTOC operator
$W=\sum_{n=1}^{N}a_{n,A}^{\dagger}a_{n,A}=\sum_{n=1}^{N}a_{n}^{\dagger}(\sigma_{0}+\sigma_{3})a_{n}/2$
and $|\tilde{\psi}_{0}\rangle=Q_{\rm s}|\psi_{E}\rangle_{\rm 1d}$. The results
are obtained by numerically calculating the OTOC function Eq. (S2), which
demonstrates that the OTOC periodically oscillates in the topological trivial
phase and becomes a conserved quantity in the topological non-trivial phase.
The averaged OTOC $\bar{\mathcal{O}}(t)$ is thus discrete at the critical
point, i.e., $\bar{\mathcal{O}}(t)=0.375$ and $\bar{\mathcal{O}}(t)=1$ in the
trivial and non-trivial phases, respectively.

##  III Additional discussion of the application of OTOC witness in the
disordered systems

We remind the Hamiltonian of the disordered nearest-neighbor SSH model

$\displaystyle H_{\rm
s}=\sum_{n}\left\\{\nu_{n}a_{n}^{\dagger}\sigma_{1}a_{n}+\frac{\omega_{n}}{2}\left[a_{n+1}^{\dagger}(\sigma_{1}+i\sigma_{2})a_{n}+{\rm
h.c.}\right]\right\\},$ (S24)

where $\omega_{n}=\epsilon(1+d_{1}r_{n})$ [or
$\nu_{n}=\epsilon(\nu+d_{2}r_{n}^{\prime})$] is the intercell (or intracell)
hopping strength. Disorder with the dimensionless strengths $d_{1}$, $d_{2}$
has been included here, and $r_{n}$, $r_{n}^{\prime}$ are independent random
real numbers chosen from the uniform distribution $[-0.5,0.5]$. In the clean
system (i.e., $d_{1}=d_{2}=0$), the above equation is reduced to a standard
SSH Hamiltonian Eq. (S1). Indeed, the symmetry-protected boundary state has
strong robustness to weak disorder, but the topological features disappear as
the disorder is too large. Moreover, the disorder can also induce the
appearance of the non-trivial topology when it is added in a topological
trivial structure. This disorder-driven topological phase is called as
topological Anderson insulator phase. To fully show the effects of disorder on
the TPTs, we numerically calculate the OTOC function Eq. (S2) with different
system parameters by choosing the OTOC operator
$W=\sum_{n=1}^{N}a_{n,A}^{\dagger}a_{n,A}$ and
$|\tilde{\psi}_{0}\rangle=Q_{\rm 1d}|\psi_{E}\rangle_{\rm 1d}$. Figure S3(a)
displays the dependence of the averaged OTOC $\bar{\mathcal{O}}(t)$ on the
disorder strength $d$ and $\nu$, which can be considered as a topological
phase diagram for the systems including disorders. There still exists an
obvious step transition from $0.375$ to $1$ at the critical point in the
presence of weak disorder. When the disorder is increased, the
distinguishability of the OTOC dynamics disappear. Besides confirming the
robustness of the OTOC witness to weak disorder, this phase diagram also shows
topological Anderson insulator phase. For example, the system enters into the
topological non-trivial phase from the trivial phase along with increasing the
disorder strength, when the value of $\nu$ is slightly larger than 1. The
physical mechanism for this result can be explained as follows. The relative
strong disorder can induce an addition locality on the system, which leads a
shift of the critical point of TPTs, as the red curve in Fig. S3(a). Then, in
the phase diagram, there appears a range of $\nu$ corresponding to the
occurrence of the TPT from the trivial to non-trivial phase with increasing
disorder. Figure S3(b) shows the similar phase diagram from Refs.
S_MondragonShem2014HS ; S_Meier2018AD . Here, the topological phase diagram
obtained by the OTOC witness is consistent with previous works, which verify
the validity of our results.

##  IV The application of OTOC witness in two-dimensional lattice described by
the Qi-Wu-Zhang model

In this section, we discuss the connection between OTOC dynamics and TPTs of
2D lattice system described by the Qi-Wu-Zhang model. As shown in Fig. S4(a),
the Qi-Wu-Zhang model on the 2D square lattice has Hamiltonian S_Qi2006WZ ;
S_Asboth2016OP

$\displaystyle H_{\rm
qwz}=\sum_{x,y}\left\\{\eta_{0}\left[a_{x+1,y}^{\dagger}\left(\frac{\sigma_{3}+i\sigma_{1}}{2}\right)a_{x,y}+a_{x,y+1}^{\dagger}\left(\frac{\sigma_{3}+i\sigma_{2}}{2}\right)a_{x,y}+{\rm
h.c.}\right]+\mu^{\prime}a_{x,y}^{\dagger}\sigma_{3}\sigma_{x,y}\right\\},$
(S25)

where $a_{x,y}^{\dagger}=(a_{x,y,A}^{\dagger},a_{x,y,B}^{\dagger})$ is the
creation operator of the unit cell with sublattices $A$ and $B$, $\eta_{0}$ is
the hopping strength, and $\mu^{\prime}$ is the sublattice potential. The
system does not have chiral symmetry. The topological trivial and non-trivial
phases in the system can be identified by Chern number in momentum space. The
parameter ranges $-2<\mu^{\prime}/\eta_{0}<2$ and
$\mu^{\prime}/\eta_{0}=other$ correspond to topological trivial and non-
trivial phases, respectively. In Fig. S4(b), we show that the OTOC of the
system in the long-time limit $\mathcal{O}(t\to\infty)$ as a function of
$\mu^{\prime}/\eta_{0}$ when
$W=\sum_{x,y}a_{x,y}^{\dagger}(\sigma_{0}-\sigma_{3})a_{x,y}/2$ and
$|\psi_{0}\rangle=|1,A\rangle$, where $[W,H_{\rm qwz}]_{\pm}\neq 0$. The
distinct behavior of the OTOC dynamics in the trivial and non-trivial phases
is still observed in the system. The value of $\mathcal{O}(t\to\infty)$
changes from almost zero to the finite values, when the system enters into the
non-trivial phase from the trivial phase.

Figure S4:  (a) Scheme of the Qi-Wu-Zhang model. The unit cell consists of
sublattices $A$ and $B$. (b) The OTOC in the long-time limit
$\mathcal{O}(t\to\infty)$ versus $\mu^{\prime}/\eta_{0}$. The TNP and TTP are
indicated by the gray shading and write area, respectively. The cell numbers
of $x$ and $y$ directions $N_{x}=N_{y}=20$, and other system parameters are
$W=\sum_{x,y}a_{x,y}^{\dagger}(\sigma_{0}-\sigma_{3})a_{x,y}^{\dagger}/2$ and
$|\psi_{0}\rangle=|1,A\rangle$.

##  V The application of OTOC witness in non-Hermitian systems

In this section, we discuss the connection between TPTs and OTOC dynamics in
non-Hermitian systems S_Leykam2017BH ; S_Gneiting2020KV ; S_Minganti2021AM ;
S_Leefmans2022DW ; S_Jin2021Song . The Hamiltonian of the 1D non-Hermitian SSH
model reads S_Lieu2018 ; S_Chen2019DS ; S_Kunst2018EB ; S_Yao2018Wang

$\displaystyle H_{\rm
nh}=\sum_{n}\left\\{\frac{\epsilon(\nu+\delta)}{2}a_{n}^{\dagger}(\sigma_{1}+i\sigma_{2})a_{n}+\frac{\epsilon(\nu-\delta)}{2}a_{n}^{\dagger}(\sigma_{1}-i\sigma_{2})a_{n}+\frac{\epsilon}{2}\left[a_{n+1}^{\dagger}(\sigma_{1}+i\sigma_{2})a_{n}+{\rm
h.c.}\right]\right\\},$ (S26)

where $\sigma_{j}$ ($j=0,1,2,3$) are the Pauli matrices, corresponding to the
identity matrix $I$, $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$,
respectively. Here $a_{n}^{\dagger}=(a_{n,A}^{\dagger},a_{n,B}^{\dagger})$ is
the annihilation operator of the unit cell $n$ with sublattices $A$, $B$, and
$\epsilon$ is the intercell hopping strength, and $\epsilon(\nu+\delta)$ and
$\epsilon(\nu-\delta)$ are the intracell hopping strength. When $\delta=0$,
Eq. (S26) is reduced to the standard SSH Hamiltonian (S1), with the phase
transition point $\nu_{c}=1$. When $\delta\neq 0$, Eq. (S26) is a non-
Hermitian Hamiltonian, and the phase transition point is
$\nu_{c}=\sqrt{1+\delta^{2}}$, where $\nu<\nu_{c}$ and $\nu>\nu_{c}$
correspond to the topological non-trivial and trivial phases, respectively.

Figure S5: (a–b) Dominated by $H_{\rm nh}$, the evolution of the OTOC for
different values of $\nu$. (c–d) The OTOC in the long-time limit
$\mathcal{O}(t\rightarrow\infty)$ versus $\nu$. The gray and white areas
correspond to the TNP and TTP, respectively. Other system parameters are
$\delta=0.4$, $|\psi_{0}\rangle=|1,A\rangle$, (a,c)
$W=a_{1,A}^{\dagger}a_{1,A}$, and (b,d)
$W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{3}a_{n}$.

Similar to the discussion in Hermitian systems, here we choose
$|\psi_{0}\rangle=|1,A\rangle$, and take the OTOC operators
$W=a_{1,A}^{\dagger}a_{1,A}$ and
$W=\sum_{n=1}^{N-1}a_{n}^{\dagger}\sigma_{3}a_{n}$ as examples, where
$|n,A/B\rangle$ represents the system occupying in the sublattice $A/B$ of the
unit cell $n$. As shown in Figs. S5(a,b), the OTOC becomes almost zero along
with the time evolution in the trivial phase, while it trends to a non-zero
finite value in the topological non-trivial phase. This means that the
distinguished OTOC dynamics in the trivial and non-trivial phases can be
obtained for two choices of the operator $W$. In Figs S5(c,d), we present the
OTOC in the long-time limit $\mathcal{O}(t\rightarrow\infty)$ during a wide
range of parameters, which shows a sudden change of the values of
$\mathcal{O}(t\rightarrow\infty)$ at the phase transition point. The above
results further demonstrate that the constructed OTOC can still be a witness
for detecting TPTs even in non-Hermitian systems.

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