# Diagnosing Quantum Phases Using Long-Range Two-Site Quantum Resource
Behaviors

Lin-Lin Su1,2 Jun Ren1,3<EMAIL_ADDRESS>Wen-Long Ma2,4
<EMAIL_ADDRESS>Z. D. Wang3<EMAIL_ADDRESS>Yan-Kui Bai1,3<EMAIL_ADDRESS>1 College of Physics and Hebei Key Laboratory of Photophysics Research and
Application, Hebei Normal University, Shijiazhuang, Hebei 050024, China
2 State Key Laboratory of Superlattices and Microstructures, Institute of
Semiconductors, Chinese Academy of Sciences, Beijing 100083, China
3 Department of Physics, HK Institute of Quantum Science & Technology, and
Guangdong-Hong Kong Joint Laboratory of Quantum Matter, The University of Hong
Kong, Pokfulam Road, Hong Kong, China
4 Center of Materials Science and Opto-Electronic Technology, University of
Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

We propose and demonstrate that the behaviors of long-range, two-site quantum
resources can effectively diagnose quantum phases. In an XX spin chain with
symmetry-breaking quantum phase transitions, we reveal that the asymptotic and
oscillating decay modes of quantum coherence or quantum discord, along with
two-site distance, can identify two spin-liquid phases. Furthermore, based on
our analytical results of spin correlation functions, we confirm the existence
of long-range entanglement in the system and establish a connection between
two-site entanglement and quantum phases. Additionally, for the extended Ising
model with topological phase transitions, we find that coherence and quantum
discord behaviors can also signify topological quantum phases. In particular,
we discover the quantum resource freezing phenomenon, where topologically
protected long-range quantum resources may have potential applications in
quantum information processing.

_Introduction._ —Quantum resources chi19rmp , including quantum coherence,
entanglement, and quantum discord, are fundamental to numerous quantum
information processing (QIP) tasks rh09rmp ; km12rmp ; str17rmp ; hu18physr .
Simultaneously, they have been developed into a powerful tool for studying
physical properties in quantum many-body systems, with the relationship
between quantum resources and quantum phases being a particularly fascinating
problem ved08rmp ; eis10rmp ; chi18rpp ; aban19rmp .

Quantum phase transitions (QPT) in many-body systems are accompanied by
dramatic changes in their physical properties sach11cam , often exhibiting
distinguishable signatures on quantum resources. In a piece of pioneering
work, Osterloh _et al_ revealed the scaling behavior of two-site entanglement
woo98prl  near the QPT point in the transverse Ising model ao02nat . Beyond
short-range, two-site entanglement, other long-range quantum resources such as
quantum coherence ple14prl ; abe06arxiv ; yao15pra ; nap16prl ; fan16pra ;
kbu17prl , quantum discord oll01prl ; hen01jpa , and multipartite entanglement
cof00pra ; mey02jmp ; wei03pra ; ou07pra ; bai09pra ; bai14prl ; elt14jpa ;
bai21njp  can also characterize various spin models exhibiting symmetry-
breaking QPTs tcw05pra ; tro06pra ; tro06prl ; ycl11pra ; sc13pra ; mh14prb ;
gia14njp ; spl18pra ; mlh20pra ; rrs21pra ; lls22pra , as well as topological
quantum phase transitions (TQPTs) described by symmetry-protected topological
order zeng19book ; nr00prb ; kit06ap ; gev03 ; xgw04 ; xyf07prl ; tre07prl ;
dhl07prl ; hl08prl ; gz15prl ; wen17rmp ; ver18prl ; gong18prx .

Although characterizing QPTs based on quantum resources has achieved
significant success, identifying different quantum phases in many-body systems
remains challenging. Recently, it was demonstrated that multipartite
entanglement, evidenced by the scaling behavior of quantum Fisher information
bra94prl ; gio06prl ; hyl12pra ; tot12pra , can characterize symmetry-
protected topological phases in an extended Kitaev chain lp17prl ; yrz18prl .
Furthermore, through varying robustness of von Neumann entanglement against
parameter perturbations, topological phases can be experimentally determined
in split-step quantum walks qqw20op . Compared to multipartite entanglement,
two-site quantum entanglement, quantum coherence, and quantum correlation are
fundamental quantum resources in QIP that are more easily detected
experimentally. However, it remains unexplored whether the behaviors of such
long-range two-site quantum resources can identify different quantum phases in
many-body systems.

In this Letter, we examine the behaviors of long-range two-site quantum
resources and demonstrate that the decay modes of these resources, along with
the two-site distance, can serve as diagnostics for quantum phases. In an XX
chain with three-spin interactions ycl11pra  exhibiting symmetry-breaking
QPTs, we show that the asymptotic and oscillating decay modes of two-site
quantum coherence and quantum correlations oll01prl ; hen01jpa  can identify
two types of spin-liquid phases. The behavior of long-range two-site
entanglement woo98prl  is also confirmed and exhibits the same functionality.
Additionally, in the extended Ising model with TQPTs gz15prl , we explore the
relationship between two-site quantum resources and topological quantum phases
by uncovering a quantum resource freezing phenomenon, which can characterize
the topological quantum phases with winding numbers $\mathcal{N}=\pm 1$.

_The extended Ising model and its two-site quantum states._ —We study the
extended Ising model with three-spin interactions gz15prl ,

$\displaystyle H=$ $\displaystyle-$
$\displaystyle\underset{j=1}{\overset{L}{\sum}}[\frac{1+\gamma}{2}\sigma^{x}_{j}\sigma^{x}_{j+1}+\frac{1-\gamma}{2}\sigma^{y}_{j}\sigma^{y}_{j+1}+\lambda\sigma^{z}_{j}$
(1) $\displaystyle+$
$\displaystyle\alpha\sigma^{z}_{j}(\frac{1+\delta}{2}\sigma^{x}_{j-1}\sigma^{x}_{j+1}+\frac{1-\delta}{2}\sigma^{y}_{j-1}\sigma^{y}_{j+1})],$

where $L$ is total spin number in the spin chain, $\gamma$ is the anisotropic
parameter of the nearest-neighbor couplings, $\lambda$ is the strength of
external magnetic field, $\alpha$ and $\delta$ represent the strength and
anisotropy of three-spin interactions, and $\\{\sigma^{x}_{j}$,
$\sigma^{y}_{j}$ and $\sigma^{z}_{j}\\}$ are the Pauli operators on the $j$th
spin. When $\gamma=\delta=0$, the system is the XXT model (an XX chain with
isotropic three-spin interactions) exhibiting the symmetry-breaking QPTs. In a
generic case for $\gamma\neq 0$ and $\delta\neq 0$, the extended Ising model
exhibits the topological QPTs.

The Hamiltonian in Eq. (1) can be exactly diagonalized and a review on the
details is presented in the Supplemental Material sup23bai . For the $i$th and
$j$th spins, the two-site reduced density matrix of the ground state has the
form,

$\begin{split}&\rho_{r}=\rho_{ij}=\begin{pmatrix}u^{+}&0&0&y^{-}\\\
0&z&y^{+}&0\\\ 0&y^{+}&z&0\\\ y^{-}&0&0&u^{-}\end{pmatrix},\end{split}$ (2)

where $r=|j-i|$ denotes the distance between the two spins, and the nonzero
matrix elements are $u^{\pm}=(1\pm
2\langle\sigma^{z}\rangle+\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/4$,
$z=(1-\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/4$ and
$y^{\pm}=(\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle\pm\langle\sigma^{y}_{0}\sigma^{y}_{r}\rangle)/4$.
The magnetization $\langle\sigma^{z}\rangle$ and two-site correlations
$\langle\sigma^{s}_{0}\sigma^{s}_{r}\rangle$ with $s=x,y,z$ are the functions
of $G_{r}$ sup23bai . For a finite chain length $L$ at zero temperature,
$G_{r}$ function can be written as be70pra ; be71pra

$G_{r}=-\frac{1}{L}{\underset{k}{\sum}}\frac{1}{\Lambda_{k}}\\{\cos(\phi_{k}r)z_{k}+\sin(\phi_{k}r)y_{k}\\},$
(3)

where $\phi_{k}=2\pi k/L$ with $k=-M,\dotsc,M$ and $M=(L-1)/2$ for odd $L$,
the energy spectra $\Lambda_{k}=\sqrt{z_{k}^{2}+y_{k}^{2}}$ with
$z_{k}=\lambda-\cos\phi_{k}-\alpha\cos(2\phi_{k})$ and
$y_{k}=\gamma\sin\phi_{k}+\alpha\delta\sin(2\phi_{k})$, respectively. In the
thermodynamic limit, the function has the following integral form el61ap

$\begin{split}G_{r}=-&\frac{1}{\pi}\int^{\pi}_{0}F(\gamma,\lambda,\alpha,\delta,r)d\phi,\end{split}$
(4)

where the kernel is $F(\gamma,\lambda,\alpha,\delta,r)=[\cos(\phi
r)z+\sin(\phi r)y]/\Lambda_{\phi}$ with the parameters being
$\Lambda_{\phi}=\sqrt{z^{2}+y^{2}}$, $z=\lambda-\cos\phi-\alpha\cos(2\phi)$,
and $y=\gamma\sin\phi+\alpha\delta\sin(2\phi)$. The $G_{r}$ function here is
vital in calculating quantum state $\rho_{r}$ and analyzing the behaviors of
two-site quantum resources.

_Characterization of quantum phases in the XXT model._ —We first consider the
XXT model for which the Hamiltonian in Eq. (1) has the anisotropic parameters
$\gamma=0$ and $\delta=0$ ycl11pra . In this case, the ground state phase
diagram is composed of ferromagnetic phase, spin-liquid I (SL-I) phase and
spin-liquid II (SL-II) phase it03epjb . The symmetry-breaking QPTs are
connected with the number of Fermi points it03epjb ; mf96prb ; ra98prb ;
sd00prb ; ca01prl ; mo15prb . By solving the energy equation
$\Lambda_{\phi}=0$ with $\phi\in\\{0,\pi\\}$ in the thermodynamic limit, we
obtain the two real Fermi points

$\phi_{\pm}=\arccos\left[\frac{-1\pm\sqrt{1+8\alpha^{2}+8\alpha\lambda}}{4\alpha}\right].$
(5)

As shown in Fig. 1, the Fermi points are plotted as the functions of magnetic
field $\lambda$ and interaction strength $\alpha$, where the cyan surface is
$\phi_{-}$, the orange surface is $\phi_{+}$, the green plane is the
background. We find the number of Fermi points can not only detect the QPTs
but also identify the three quantum phases, where there are two Fermi points
in the SL-II phase, one in the SL-I phase and zero in ferromagnetic phase.

Figure 1: (Color online) The ground state phase diagram of the XXT model
described by the number of Fermi points, where the cyan surface is
$\phi_{-}(\lambda,\alpha)$, the orange surface is $\phi_{+}(\lambda,\alpha)$,
the green plane is the background. The critical lines are
$\lambda_{c_{1}}=\alpha+1$, $\lambda_{c_{2}}=\alpha-1$ and
$\lambda_{c_{3}}=-(1+8\alpha^{2})/8\alpha$ ($\alpha\geq 0.25$ for
$\lambda_{c_{3}}$), and the regions of Ferr-I and Ferr-II are the same phase.

We now study the property of long-range two-site quantum coherence of reduced
state $\rho_{r}$ along with the distance of two spins in the XXT spin chain.
Quantum coherence is an important resource in the QIP mh16pra ; hl17pra ;
cn16prl  and quantum thermodynamics mh13nc ; fg13prl ; gg15pr ; yang22prl .
Here we adopt the $l_{1}$-norm quantum coherence ple14prl , which is expressed
as the sum of modulus of off-diagonal element:
$C_{l_{1}}(\rho_{r})=\sum_{i,j,i\neq j}|\rho_{ij}|$. In the XXT model, since
the Hamiltonian is isotropic in $x$ and $y$ direction ($\gamma=\delta=0$), we
have the matrix element
$y^{-}=(\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle-\langle\sigma^{y}_{0}\sigma^{y}_{r}\rangle)/4=0$
in Eq. (2). Therefore, the $l_{1}$-norm coherence of two-site quantum state is

$C_{l_{1}}(\rho_{r})=2|y^{+}|=|\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle|,$
(6)

where the spin correlation $\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle$ is a
determinant composed of $G_{r}$ function sup23bai . In order to obtain the
quantum coherence expediently, we derive the analytical formula of $G_{r}$
function in the thermodynamic limit for different regions of quantum phases
(see the Supplemental Material sup23bai )

$\begin{split}G_{r}=\begin{cases}-\delta_{0r},&\rm{Ferr-I}\\\
\delta_{0r},&\rm{Ferr-II}\\\ \frac{2\sin(r\phi_{+})}{\pi
r}-\delta_{0r},&\rm{SL-I}\\\ 2(\frac{\sin(r\phi_{+})}{\pi
r}-\frac{\sin(r\phi_{-})}{\pi r})+\delta_{0r},&\rm{SL-
II}\end{cases}\end{split}$ (7)

where $r$ is the distance of two spins and $\phi_{\pm}$ are the Fermi points
given in Eq. (5). According to this analytical formula, the spin correlation
$\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle$ is zero for the regions of Ferr-I
and Ferr-II in Fig. 1, so the quantum coherence $C_{l_{1}}(\rho_{r})$ is zero
in the ferromagnetic phase. However, the quantum coherence in the SL-I and SL-
II phases is nonzero in general. As shown in Fig. 2(a)-(c), we plot the
coherence $C_{l_{1}}(\rho_{r})$ as a function of parameters $\lambda$ and
$\alpha$ for spin-spin distances $r=1,6$ and $10$. We can distinguish the SL-I
and SL-II phases from the ferromagnetic phase, and the change patterns of
$C_{l_{1}}$ along with the parameters $\alpha$ and $\lambda$ are different as
shown in panels (a)-(c), where $C_{l_{1}}$ is asymptotical in SL-I phase and
oscillating in SL-II phase. The intrinsic reason is that the coherence decays
along with the distance $r$ in the asymptotical mode for SL-I phase but in the
oscillating mode for SL-II phase as shown in Fig. 2(d).

Figure 2: (Color online) The $l_{1}$-norm coherence
$C_{l_{1}}(\alpha,\lambda)$ for different two-site distances: (a) $r=1$, (b)
$r=6$, and (c) $r=10$, respectively. In panel (d), the parameters
$(\alpha=0.7,\lambda=1.0)$ in SL-I phase and $(\alpha=3.0,\lambda=0.5)$ in SL-
II phase are chosen, where the asymptotical and oscillating decay modes of
$C_{l_{1}}$ can serve as the diagnostic of quantum phases.

Then we study the long-range two-site quantum entanglement in the same model.
As quantified by concurrence woo98prl , the two-site entanglement can be
written as

$C(\rho_{r})={\rm{max}}\\{0,2(|y^{+}|-\sqrt{u^{+}u^{-}})\\},$ (8)

where $y^{+}$ and $u^{\pm}$ are the off-diagonal and diagonal elements of
reduced quantum state $\rho_{r}$ in Eq. (2), respectively. In comparison with
the $l_{1}$-norm coherence in Eq. (6), the two-site entanglement is less than
the corresponding quantum coherence. Moreover, although quantum coherence has
the long-range property, two-site entanglement is often short-range and
experiences entanglement sudden death (ESD) kz01pra ; ss03jmo ; ty04prl ;
mpa07sci ; jl04prl ; ty09sci  in multipartite spin systems without long-range
interactions ao02nat ; lls22pra . We analyze the property of two-site
concurrence in the XXT model, and confirm the existence of long-range two-site
entanglement, for which the parameters are close to the critical lines
$\lambda_{c_{1}}$, $\lambda_{c_{2}}(\alpha\leq 0.25)$, and $\lambda_{c_{3}}$
sup23bai . In SL-I and SL-II phases, we choose three sets of parameters near
the three critical lines, and calculate the concurrence as a function of two-
site distance. As shown in Fig. 3, the decay modes of long-range two-site
entanglement have the same functionality for identifying quantum phases, where
$C_{l_{1}}$ decays in the asymptotical mode for SL-I phase but in the
oscillating mode for SL-II phase.

Figure 3: (Color online) Long-range two-site entanglement in (a) SL-I phase
and (b) SL-II phase, where the distinctly different decay modes of concurrence
serve as the diagnostic of quantum phases. The hollow dots indicate the
distances with zero entanglement.

_Characterization of topological quantum phases in the extended Ising model._
— In the extended Ising system with $\gamma\neq 0$ and $\delta\neq 0$, there
are topological quantum phases in the ground state, for which the winding
number is an effective geometric order parameters gz15prl . The Hamiltonian in
Eq. (1) can be further expressed as the Bogoliubov-de Gennes form in the
momentum space gz15prl ; yrz18prl

$H=\underset{k=-M}{\overset{M}{\sum}}\begin{pmatrix}c^{\dagger}_{k}&c_{-k}\end{pmatrix}\mathcal{H}_{k}\begin{pmatrix}c_{k}\\\
c^{\dagger}_{-k}\end{pmatrix},$ (9)

where $M=(L-1)/2$ for odd $L$, $\mathcal{H}_{k}=\vec{r}(k)\cdot\vec{\sigma}$
with $\vec{r}(k)=[0,y_{k},z_{k}]$ and
$\vec{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$, and the vector $\vec{r}$
represents a two-dimensional magnetic field with the components
$y_{k}=\gamma\sin\phi_{k}+\alpha\delta\sin(2\phi_{k})$ and
$z_{k}=\lambda-\cos\phi_{k}-\alpha\cos(2\phi_{k})$. Zhang and Song established
the connection between topological quantum phase and corresponding winding
number gz15prl

$\displaystyle\mathcal{N}=\frac{1}{2\pi}\oint(ydz-zdy)/|\vec{r}|^{2},$ (10)

which is an integer representing the direction and total times that the closed
loop travels around the origin point in the $y$-$z$ plane rbs02GT . The
winding numbers can identify topological quantum phases in the extended Ising
model, and the critical points for the TQPTs are available by solving the
characteristic equations spl18pra .

Figure 4: (Color online) Energy spectra, winding vector trajectories, and
long-range quantum coherence $C_{l_{1}}(\alpha,r)$ in the extended Ising model
with the parameters $\gamma=1$, $\delta=-1$, and $\lambda=1$ for the chain
length $L=1001$. (a) The energy spectra as a function of three-spin
interaction $\alpha$ where the four critical points and five topological
phases are labeled. (b) The trajectories of the winding vectors in the $y$-$z$
plane which correspond to the winding numbers $2$, $0$, $1$, $-1$ and $-2$,
respectively. (c)-(e) The behaviors of long-range two-site quantum coherence
in different topological quantum phases.

Next, we study the connection between the long-range two-site quantum
coherence and topological quantum phases in the extended Ising model. When the
TQPTs are driven by the three-spin interaction $\alpha$, we choose the other
parameters in the Hamiltonian to be $\gamma=1$, $\delta=-1$, and $\lambda=1$,
respectively. As shown in Fig. 4(a), the energy spectra
$\Lambda_{k}=\pm\sqrt{z_{k}^{2}+y_{k}^{2}}$ are plotted for a finite chain
length $L=1001$, where the critical points $\alpha_{c_{1}}=(-\sqrt{5}-1)/2$,
$\alpha_{c_{2}}=0$, $\alpha_{c_{3}}=(\sqrt{5}-1)/2$, and $\alpha_{c_{4}}=2$
are labeled. The trajectories of the typical winding vectors in the auxiliary
$y$-$z$ plane are plotted in Fig. 4(b). As quantified by the $l_{1}$-norm
measure, the two-site quantum coherence in the extended Ising system is

$C_{l_{1}}(\rho_{r})=2(|y^{+}|+|y^{-}|),$ (11)

where
$y^{\pm}=(\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle\pm\langle\sigma^{y}_{0}\sigma^{y}_{r}\rangle)/4$
are the off-diagonal elements of two-site quantum state $\rho_{r}$ in Eq. (2).
It is noted that the first-order derivative of adjacent two-site coherence
$C_{l_{1}}$ can detect all the TQPTs in the extended Ising system sup23bai .
Here we focus on the properties of the long-range two-site coherence in
different topological phases. In Fig. 4(c)-(e), we plot $C_{l_{1}}(\rho_{r})$
as a function of three-spin interaction strength $\alpha$ and two-site
distance $r$, where the different topological phases are labeled by the
corresponding winding numbers. In the case of $\mathcal{N}=2$, the coherence
decays in the oscillating mode along with the increase of two-site distance
for a given value of $\alpha$. But, in the case of $\mathcal{N}=-2$, the
coherence decays in the asymptotical mode along with two-site distance,
although there exist initial oscillating for short two-site distance due to
the influence of adjacent three-spin interactions. Moreover, the strength of
$|\alpha|$ can change the decay rate in both topological phases of
$\mathcal{N}=\pm 2$. In the case of $\mathcal{N}=0$, the coherence can decay
in both oscillating and asymptotical modes for different values of $\alpha$ as
shown in Fig. 4(c). We note that the ground state of the XXT model we analyzed
previously is also a topological trivial ($\mathcal{N}=0$) where the quantum
coherence exhibits two kinds of decay modes too.

In the topological phases with winding numbers $\mathcal{N}=\pm 1$, the
behavior of two-site quantum coherence exhibits a remarkable property, where
the nonzero coherence will become stable for a given value of $\alpha$ after a
slight decay of short two-site distances as shown in Fig. 4(d). In addition,
the higher steady coherence in the phase $\mathcal{N}=1$ comes from spin
correlation $\langle\sigma^{y}_{0}\sigma^{y}_{r}\rangle$, while the lower
steady coherence in the phase $\mathcal{N}=-1$ is due to
$\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle$ (see the Supplemental Material
sup23bai ). The steady quantum coherence is a kind of topologically protected
long-range quantum resource, and we refer to it as quantum coherence freezing
(QCF) phenomenon along with the two-site distance, which is different from the
case of frozen quantum coherence under dynamical conditions bro15prl ;
tong16pra ; fan16sr ; sil16prl . Therefore, as shown in Fig. 4, the behaviors
of long-range two-site coherence, such as oscillating and asymptotical decay
modes or the QCF mode, can serve as an effective diagnostic of topological
quantum phases. In addition, we also investigate the topological quantum
phases driven by the external magnetic field $\lambda$, and establish the
connection between long-range quantum coherence and the topological phases,
where the coherence still exhibits the QCF phenomenon in the case
$\mathcal{N}=1$ sup23bai .

The topologically protected steady coherence is a kind of useful quantum
resource in the QIP. In order to explain further the QCF phenomenon in the
extended Ising model, we consider a special case of the Hamiltonian with
$\gamma=1$ and $\lambda=\alpha=0$, for which the winding number is
$\mathcal{N}=1$ and its ground state has an analytical form gz15prl

$|G_{1}\rangle=\frac{1}{\sqrt{2}}(\underset{j\in
e}{\Pi}|\nearrow\rangle_{j}\underset{j\in
o}{\Pi}|\swarrow\rangle_{j}+\underset{j\in
e}{\Pi}|\swarrow\rangle_{j}\underset{j\in o}{\Pi}|\nearrow\rangle_{j}),$ (12)

where
$\sigma^{x}_{j}|\nearrow\rangle_{j}(|\swarrow\rangle_{j})=|\nearrow\rangle_{j}(-|\swarrow\rangle_{j})$,
and the summing targets $e$ and $o$ denote the even and odd number of sites,
respectively. Its reduced quantum state of the $i$th and $j$th spins with
distance $r$ is

$\begin{split}&\rho_{r}^{\mathcal{N}=1}=\frac{1}{4}\begin{pmatrix}1&0&0&-1\\\
0&1&-1&0\\\ 0&-1&1&0\\\ -1&0&0&1\end{pmatrix},\end{split}$ (13)

which has the steady coherence $C_{l_{1}}(\rho_{r}^{\mathcal{N}=1})=1$ and is
a constant matrix independent of the two-site distance. Similarly, for the
topological phase $\mathcal{N}=-1$ with the parameters $\gamma=-1$ and
$\lambda=\alpha=0$, we derive the distance-independent two-site quantum state
$\rho_{r}^{\mathcal{N}=-1}$ via the analytical ground state $|G_{-1}\rangle$
gz15prl , which exhibits the QCF phenomenon with topologically protected
quantum coherence $C_{l_{1}}(\rho_{r}^{\mathcal{N}=-1})=1$.

_Discussions._ — The above results for quantum coherence and quantum
entanglement can be extended to other two-site quantum resource, such as
quantum discord that can describe quantum correlations beyond quantum
entanglement mod10prl ; str12prl ; bai13pra . The two-qubit quantum discord
can be quantified by the formula oll01prl ; hen01jpa

$D_{A}(\rho_{AB})=S(\rho_{A})-S(\rho_{AB})+\underset{\\{E^{A}_{k}\\}}{\rm{min}}\underset{k}{\sum}p_{k}S(\rho_{B|k}),$
(14)

where the measurement $\\{E_{k}^{A}\\}$ is performed on subsystem $A$ with the
minimum running over all the projection measurements, and $S(\cdot)$ is the
von Neumann entropy. Quantum discord can characterize quantum advantage
without entanglement and is a kind of key resource in the QIP kni98prl ;
lan08prl ; dat08prl ; roa11prl ; dak12natp . In the XXT model, the behaviors
of long-range two-site quantum discord $D_{A}(\rho_{r})$ is further studied,
where we find that the discord $D_{A}(\rho_{r})$ decays along with two-site
distance in the asymptotical mode for the SL-I phase and in the oscillating
mode for the SL-II phase sup23bai , which indicates that quantum discord has
the same functionality for identifying quantum phases as that of quantum
coherence. Furthermore, in the extended Ising model, we find two-site
entanglement is short-range but quantum discord is long-range which exhibits
behaviors similar to those of quantum coherence sup23bai . In particular,
unlike the frozen quantum discord in dynamical procedure maz10prl ; you12pra ,
we find a novel phenomenon of quantum discord freezing in topological phases
with the winding numbers $\mathcal{N}=\pm 1$.

_Conclusions and outlook._ —In summary, we have investigated the behaviors of
long-range two-site quantum resources and uncovered the connection between
their decay modes and various quantum phases. In the XXT model ycl11pra  with
symmetry-breaking QPTs, we demonstrate that the asymptotic or oscillating
decay modes of two-site quantum coherence and quantum discord can identify two
spin-liquid phases. We also confirm the existence of long-range two-site
entanglement in this model, which can likewise signify different quantum
phases. In the extended Ising model gz15prl , we show that the decay modes of
long-range two-site quantum resources can serve as an effective diagnostic
tool for topological phases. Compared to Fisher information or multipartite
entanglement of the global ground state in many-body systems, the long-range
resource behaviors of a set of two-site quantum states are more easily
detected experimentally. Lastly, the topologically protected two-site quantum
resources exhibit freezing phenomena, which hold promising applications in
both quantum communication and quantum metrology pez18rmp ; por22rmp .

This work was supported by NSF-China (Grants Nos. 11575051 and 11904078),
Hebei NSF (Grants Nos. A2021205020, A2019205266), and project of China
Postdoctoral Science Foundation (Grant No. 2020M670683). W.L.M. acknowledges
support from the Chinese Academy of Sciences (No. E0SEBB11, No. E27RBB11) and
the National Natural Science Foundation of China (No. 121743379, No.
E31Q02BG).

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##  I Supplemental material

###  I.1 I. The diagonalization procedure for the Hamiltonian of the extended
Ising model

The Hamiltonian of the extended Ising model given in the main text has the
form gz2015prl

$\displaystyle H=$ $\displaystyle-$
$\displaystyle\underset{j=1}{\overset{L}{\sum}}[\frac{1+\gamma}{2}\sigma^{x}_{j}\sigma^{x}_{j+1}+\frac{1-\gamma}{2}\sigma^{y}_{j}\sigma^{y}_{j+1}+\lambda\sigma^{z}_{j}$
(1) $\displaystyle+$
$\displaystyle\alpha\sigma^{z}_{j}(\frac{1+\delta}{2}\sigma^{x}_{j-1}\sigma^{x}_{j+1}+\frac{1-\delta}{2}\sigma^{y}_{j-1}\sigma^{y}_{j+1})],$

where $L$ is the total spin number in the spin chain, $\gamma$ is the
anisotropic parameter of the nearest-neighbor couplings, $\lambda$ is the
strength of external magnetic field, $\alpha$ and $\delta$ represent the
strength and anisotropy of three-spin interactions, and $\sigma^{x}_{j}$,
$\sigma^{y}_{j}$ and $\sigma^{z}_{j}$ are the Pauli operators on the $j$th
spin. Next, we will give a brief review on the diagonalization procedure for
this Hamiltonian el1961ap ; pp70ap ; eb1971pra .

The spin operators in the Hamiltonian can be mapped into the spinless fermion
operators by introducing the Jordan-Wigner transformation sac2011cam

$\displaystyle\sigma^{z}_{j}=1-2c^{\dagger}_{j}c_{j},$
$\displaystyle\sigma^{x}_{j}={\underset{l<j}{\prod}}(1-2c^{\dagger}_{l}c_{l})(c^{\dagger}_{j}+c_{j}),$
$\displaystyle\sigma^{y}_{j}=-i{\underset{l<j}{\prod}}(1-2c^{\dagger}_{l}c_{l})(c^{\dagger}_{j}-c_{j}),$
(2)

where $c^{\dagger}_{j}$ and $c_{j}$ are the spinless creation and annihilation
operators. After substituting the expressions in Eq. (2) into the Hamiltonian
in Eq. (1), we can obtain the following form

$\displaystyle H=$ $\displaystyle-$
$\displaystyle\underset{j=1}{\overset{L}{\sum}}\lambda(1-2c^{\dagger}_{j}c_{j})$
(3) $\displaystyle-$
$\displaystyle\underset{j=1}{\overset{L}{\sum}}[\gamma(c^{\dagger}_{j}c^{\dagger}_{j+1}-c_{j}c_{j+1})+c^{\dagger}_{j}c_{j+1}-c_{j}c^{\dagger}_{j+1}]$
$\displaystyle-$
$\displaystyle\underset{j=1}{\overset{L}{\sum}}\alpha(c^{\dagger}_{j-1}c_{j+1}-c_{j-1}c^{\dagger}_{j+1})$
$\displaystyle-$
$\displaystyle\underset{j=1}{\overset{L}{\sum}}\alpha\delta(c^{\dagger}_{j-1}c^{\dagger}_{j+1}+c_{j-1}c_{j+1}).$

Furthermore, the above Hamiltonian can be expressed in the momentum space by
the Fourier transformation, and the Fourier transformation of fermion
annihilation operator can be written as

$c_{j}=\frac{1}{\sqrt{L}}\underset{k=-M}{\overset{M}{\sum}}c_{k}\exp(-ij\phi_{k}),$
(4)

where $\phi_{k}=2\pi k/L$ with $k=-M,\dotsc,M$ and $M=(L-1)/2$ for odd $L$.
After some derivation, we can rewrite the Hamiltonian in the following form

$\begin{split}&H=-\lambda L+2(\lambda-1-\alpha)c^{\dagger}_{0}c_{0}\\\
&+\underset{k=1}{\overset{M}{\sum}}2[\lambda-\cos\phi_{k}-\alpha\cos(2\phi_{k})](c^{\dagger}_{k}c_{k}+c^{\dagger}_{-k}c_{-k})\\\
&+\underset{k=1}{\overset{M}{\sum}}2i[\gamma\sin\phi_{k}+\alpha\delta\sin(2\phi_{k})](c^{\dagger}_{k}c^{\dagger}_{-k}+c_{k}c_{-k}).\end{split}$
(5)

After further applying the Bogoliubov transformation, the Hamiltonian in Eq.
(5) can be transformed into the diagonal form

$\displaystyle
H=\underset{k=-M}{\overset{M}{\sum}}2\Lambda_{k}(\eta^{\dagger}_{k}\eta_{k}-\frac{1}{2}),$
(6)

for which the energy spectra are $\pm\Lambda_{k}$ and

$\displaystyle\Lambda_{k}=\sqrt{z(k)^{2}+y(k)^{2}}$ (7)

with $z(k)=\lambda-\cos\phi_{k}-\alpha\cos(2\phi_{k})$ and
$y(k)=\gamma\sin\phi_{k}+\alpha\delta\sin(2\phi_{k})$, and the mapping
operator of Bogoliubov transformation is

$\displaystyle\eta_{k}=\cos\frac{\theta_{k}}{2}c_{-k}-i\sin\frac{\theta_{k}}{2}c^{\dagger}_{k},$
(8)

in which the coefficients are

$\displaystyle\sin\frac{\theta_{k}}{2}$ $\displaystyle=$
$\displaystyle\frac{\Lambda_{k}-z(k)}{\sqrt{2}\Lambda_{k}[\Lambda_{k}-z(k)]},$
$\displaystyle\cos\frac{\theta_{k}}{2}$ $\displaystyle=$
$\displaystyle\frac{y(k)}{\sqrt{2}\Lambda_{k}[\Lambda_{k}-z(k)]},$ (9)

with the parameter being

$\displaystyle\theta_{k}$ $\displaystyle=$
$\displaystyle\arcsin\\{-[\gamma\sin\phi_{k}+\alpha\delta\sin(2\phi_{k})]/\Lambda_{k}\\}.$
(10)

###  I.2 II. Two-site reduced density matrix of ground state in the extended
Ising model

In the extended Ising model, the two-site reduced density matrix of ground
state for the $i$th and $j$th spins given in the main text has the form

$\begin{split}&\rho_{r}=\rho_{ij}=\begin{pmatrix}u^{+}&0&0&y^{-}\\\
0&z&y^{+}&0\\\ 0&y^{+}&z&0\\\ y^{-}&0&0&u^{-}\end{pmatrix},\end{split}$ (11)

where $r=|j-i|$ denotes the distance between the two spins, and the nonzero
matrix elements are $u^{\pm}=(1\pm
2\langle\sigma^{z}\rangle+\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/4$,
$z=(1-\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/4$ and
$y^{\pm}=(\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle\pm\langle\sigma^{y}_{0}\sigma^{y}_{r}\rangle)/4$,
which can be calculated from the magnetization and spin correlation functions.

For the nonzero elements in $\rho_{r}$, the magnetization
$\langle\sigma^{z}\rangle$ and two-site correlation functions
$\langle\sigma^{s}_{0}\sigma^{s}_{r}\rangle$ with $s=x,y,z$ can be written as
eb1971pra ; eb1970pra

$\displaystyle\langle\sigma^{z}\rangle=G_{0},$
$\displaystyle\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle=\begin{vmatrix}G_{-1}&G_{-2}&\cdots&G_{-r}\\\
G_{0}&G_{-1}&\cdots&G_{-r+1}\\\ \vdots&\vdots&\ddots&\vdots\\\
G_{r-2}&G_{r-3}&\cdots&G_{-1}\end{vmatrix},$
$\displaystyle\langle\sigma^{y}_{0}\sigma^{y}_{r}\rangle=\begin{vmatrix}G_{1}&G_{0}&\cdots&G_{-r+2}\\\
G_{2}&G_{1}&\cdots&G_{-r+3}\\\ \vdots&\vdots&\ddots&\vdots\\\
G_{r}&G_{r-1}&\cdots&G_{1}\end{vmatrix},$
$\displaystyle\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle=\langle\sigma^{z}\rangle^{2}-G_{r}G_{-r},$
(12)

where $G_{r}$ function for the case of a finite chain length has the
expression given in Eq. (3) of main text and its formula for the case of the
thermodynamic limit is presented in Eq. (4) of main text. Therefore, according
to Eqs. (11) and (12), the two-site quantum state $\rho_{r}$ can be obtained
by calculating the $G_{r}$ functions and related determinants. It should be
noted that, in order to analyze the properties of quantum state and its
quantum resource expediently, an analytical formula for the $G_{r}$ function
is very desirable.

###  I.3 III. The analytical formula for the $G_{r}$ function in the XXT model

We consider the XXT model, for which the Hamiltonian shown in Eq. (1) of the
main text has the parameters $\gamma=0$, $\delta=0$, and $\lambda\neq 0$. In
the thermodynamic limit, the $G_{r}$ function has the form el1961ap

$\begin{split}G_{r}=-&\frac{1}{\pi}\int^{\pi}_{0}F(\lambda,\alpha,r)d\phi,\end{split}$
(13)

where the integral kernel is

$F(\lambda,\alpha,r)=\cos(\phi r)z/\Lambda_{\phi}$ (14)

in which the parameter is $\Lambda_{\phi}=\sqrt{z^{2}}$ with
$z=\lambda-\cos\phi-\alpha\cos(2\phi)$.

The Fermi points it2003epjb ; mf1996prb  characterize the positions at which
the energy $\Lambda_{\phi}$ is equal to zero, which plays a key role in the
analysis on different quantum phases in the XXT model as shown in Fig. 1 of
the main text. There are two Fermi points in the SL-II phase, one in the SL-I
phase and zero in the Ferromagnetic phase (the regions of Ferr-I and Ferr-II
are the same phase). By solving the energy equation $\Lambda_{\phi}=0$ with
$\phi\in\\{0,\pi\\}$ in the thermodynamic limit, we can obtain the two Fermi
points $\phi_{+}$ and $\phi_{-}$ which can be written as

$\begin{split}\phi_{\pm}=\arccos[\frac{-1\pm\sqrt{1+8\alpha^{2}+8\alpha\lambda}}{4\alpha}].\end{split}$
(15)

It should be noted that the Fermi points are required to be real numbers.

Next, according to the numbers and values of Fermi points, we will analyze the
kernel functions $F(\lambda,\alpha,r)$ in different quantum phases. In the
ferromagnetic phase, there is no Fermi point and we can demonstrate that
$z(\phi)=\lambda-\cos\phi-\alpha\cos(2\phi)$ is positive in the region of
Ferr-I and negative in the region of Ferr-II. Therefore, we have

$\displaystyle F(\lambda,\alpha,r)$ $\displaystyle=$
$\displaystyle\frac{\cos(r\phi)[\lambda-\cos(\phi)-\alpha\cos(2\phi)]}{\sqrt{[\lambda-\cos(\phi)-\alpha\cos(2\phi)]^{2}}}$
(16) $\displaystyle=$ $\displaystyle\begin{cases}\cos(r\phi),&\rm{Ferr-I}\\\
-\cos(r\phi).&\rm{Ferr-II}\end{cases}$

Substituting the result into the $G_{r}$ function in Eq. (13), we can derive
that $G_{r}=0$ for the case of $r\neq 0$ and the nonzero $G_{r}$ corresponds
to the case of $r=0$. After some calculation, we obtain the analytical formula

$\displaystyle G_{r}$ $\displaystyle=$
$\displaystyle\begin{cases}-\delta_{0r},&\rm{Ferr-I}\\\ \delta_{0r}.&\rm{Ferr-
II}\end{cases}$ (17)

In the SL-I phase, there is only one Fermi point $\phi_{+}$ which divides the
function $z(\phi)$ into two parts. It can be demonstrated that $z(\phi)$ is
negative in the region $\phi\in[0,\phi_{+})$ and positive in the region
$\phi\in[\phi_{+},\pi]$. Therefore, according to Eq. (14), we can have the
kernel function

$\displaystyle F(\lambda,\alpha,r)$ $\displaystyle=$
$\displaystyle\frac{\cos(r\phi)[\lambda-\cos(\phi)-\alpha\cos(2\phi)]}{\sqrt{[\lambda-\cos(\phi)-\alpha\cos(2\phi)]^{2}}}$
(18) $\displaystyle=$
$\displaystyle\begin{cases}-\cos(r\phi),&\phi\in[0,\phi_{+})\\\
\cos(r\phi).&\phi\in[\phi_{+},\pi]\end{cases}$

By substituting the above expression into Eq. (13), we can derive the
analytical formula of $G_{r}$. In the case of $r\neq 0$, we have

$\displaystyle G_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\phi_{+}}_{0}\cos(r\phi)d\phi-\frac{1}{\pi}\int^{\pi}_{\phi_{+}}\cos(r\phi)d\phi$
(19) $\displaystyle=$ $\displaystyle\frac{2\sin(r\phi_{+})}{\pi r}.$

When $r=0$, the function is

$\displaystyle
G_{r}=\frac{1}{\pi}\int^{\phi_{+}}_{0}d\phi-\frac{1}{\pi}\int^{\pi}_{\phi_{+}}d\phi=\frac{2\phi_{+}}{\pi}-1.$
(20)

Combining Eqs. (19) and (20), we can obtain the formula of $G_{r}$ in the SL-I
phase

$\displaystyle G_{r}=\frac{2\sin(r\phi_{+})}{\pi r}-\delta_{0r}.$ (21)

In the SL-II phase, there are two Fermi points $\phi_{+}$ and $\phi_{-}$,
which divide the function $z(\phi)$ into three parts where $z(\phi)>0$ in the
region $\phi\in[\phi_{+},\phi_{-}]$ and $z(\phi)<0$ in the regions
$\phi\in[0,\phi_{+})\cup(\phi_{-},\pi]$. Then the kernel function in the SL-II
phase has the form

$\displaystyle F(\lambda,\alpha,r)$ $\displaystyle=$
$\displaystyle\frac{\cos(r\phi)[\lambda-\cos(\phi)-\alpha\cos(2\phi)]}{\sqrt{[\lambda-\cos(\phi)-\alpha\cos(2\phi)]^{2}}}$
(22) $\displaystyle=$
$\displaystyle\begin{cases}-\cos(r\phi),&\phi\in[0,\phi_{+})\\\
\cos(r\phi),&\phi\in[\phi_{+},\phi_{-}]\\\
-\cos(r\phi).&\phi\in(\phi_{-},\pi]\end{cases}$

According to Eq. (13), we can obtain the $G_{r}$ function in the case of
$r\neq 0$

$\displaystyle G_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\phi_{+}}_{0}\cos(r\phi)d\phi-\frac{1}{\pi}\int^{\phi_{-}}_{\phi_{+}}\cos(r\phi)d\phi$
(23) $\displaystyle+$
$\displaystyle\frac{1}{\pi}\int^{\pi}_{\phi_{-}}\cos(r\phi)d\phi$
$\displaystyle=$ $\displaystyle\frac{2[\sin(r\phi_{+})-\sin(r\phi_{-})]}{\pi
r},$

and the formula in the case of $r=0$ is

$\displaystyle G_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int^{\phi_{+}}_{0}d\phi-\frac{1}{\pi}\int^{\phi_{-}}_{\phi_{+}}d\phi+\frac{1}{\pi}\int^{\pi}_{\phi_{-}}d\phi$
(24) $\displaystyle=$ $\displaystyle\frac{2(\phi_{+}-\phi_{-})}{\pi}+1.$

Combining Eqs. (23) and (24), we can obtain the analytical formula of $G_{r}$
function in the SL-II phase

$\displaystyle G_{r}=\frac{2[\sin(r\phi_{+})-\sin(r\phi_{-})]}{\pi
r}+\delta_{0r}.$ (25)

The expressions in Eqs. (17), (21) and (25) constitute the analytical form of
the $G_{r}$ function in the XXT model, which is given in Eq. (7) of the main
text. Based on the analytical formula, we can expediently obtain the two-site
reduced state $\rho_{r}$ and study further the properties of long-range two-
site quantum resources in the reduced states, such as two-site quantum
coherence, quantum entanglement and quantum discord.

###  I.4 IV. The confirmation of long-range two-site entanglement in the XXT
model

In multipartite spin systems, two-site entanglement is often short-range along
with two-site distance and exhibits the phenomenon of entanglement sudden
death ty2009sci . Here, we study the entanglement property of the reduced
state $\rho_{r}$ in the XXT model and analyze its long-range behavior of two-
site entanglement. As quantified by the concurrence woo1998prl , we have

$C(\rho_{r})={\rm{max}}\\{0,2(|y^{+}|-\sqrt{u^{+}u^{-}})\\},$ (26)

where $y^{\pm}$ and $u^{\pm}$ are the off-diagonal and diagonal elements of
reduced state $\rho_{r}$ which can be calculated by the analytical formula of
$G_{r}$. We mainly consider the spin-liquid phases in the XXT model, since the
ground state in the ferromagnetic phase is product states for which the
concurrence of reduced state is zero.

After some calculation, we can obtain the two-site concurrence in the SL-I
phase

$\displaystyle C(\rho_{r})$ $\displaystyle=$
$\displaystyle{\rm{max}}\\{0,2|y^{+}|-\frac{2}{\pi^{2}}\sqrt{(C_{1}-E_{1})(D_{1}-E_{1})}\\}$
(27) $\displaystyle\approx$
$\displaystyle{\rm{max}}\\{0,C_{l_{1}}(\rho_{r})-\frac{2}{\pi^{2}}\sqrt{C_{1}\cdot
D_{1}}\\},$

where the parameters are $C_{1}=\phi^{2}_{+}$, $D_{1}=(\pi-\phi_{+})^{2}$ and
$E_{1}=\sin^{2}(r\phi_{+})/r^{2}$ with $\phi^{+}$ being the Fermi point, and,
in the second equation, we use the two-site coherence
$C_{l_{1}}(\rho_{r})=2|y^{+}|$ in Eq. (6) of the main text and the
approximation that the parameter $E_{1}$ tends to zero in the long-range case
of two-site distance. Therefore, we know that the two-site concurrence is less
than the corresponding two-site coherence. Furthermore, since the coherence is
long-range and decays in the asymptotical mode, we can infer the result that
the two-site entanglement in Eq. (27) can be long-range in the case of
$C_{1}\cdot D_{1}$ tending to zero.

Next, we analyze the two-site entanglement in the SL-II phase and, according
to the formula in Eq. (26), we can derive

$\displaystyle C(\rho_{r})$ $\displaystyle=$
$\displaystyle{\rm{max}}\\{0,2|y^{+}|-\frac{2}{\pi^{2}}\sqrt{(E_{2}-G_{2})(F_{2}-G_{2})}\\}$
(28) $\displaystyle\approx$
$\displaystyle{\rm{max}}\\{0,C_{l_{1}}(\rho_{r})-\frac{2}{\pi^{2}}\sqrt{E_{2}\cdot
F_{2}}\\},$

where the parameters are $E_{2}=(\pi+\phi_{+}-\phi_{-})^{2}$,
$F_{2}=(\phi_{+}-\phi_{-})^{2}$ and
$G_{2}=[\sin(r\phi_{+})-\sin(r\phi_{-})]^{2}/r^{2}$ with $\phi^{\pm}$ being
the two Fermi points, and we use the approximation that the parameter $G_{2}$
tends to zero in the long distance. After a similar analysis as that in the
SL-I phase, we can obtain the result that the two-site entanglement is long-
range in the case of $E_{2}\cdot F_{2}$ tending to zero.

Figure 1: (Color online) $C_{1}\cdot D_{1}$ (a) and $E_{2}\cdot F_{2}$ (b) as
the functions of the parameters $\alpha$ and $\lambda$, where the red lines on
the bottom indicate the zero values for the two functions and correspond to
the critical lines $\lambda_{1}$, $\lambda_{2}(\alpha\leq 0.25)$ and
$\lambda_{3}$.

In Eqs. (27) and (28), the functions $C_{1}\cdot D_{1}$ and $E_{2}\cdot F_{2}$
are independent of the two-site distance. As shown in Fig. 1, we plot the two
functions along with the change of the parameters $\alpha$ and $\lambda$,
where the red lines in the bottom indicate the cases of $C_{1}\cdot D_{1}=0$
and $E_{2}\cdot F_{2}=0$, and the three red lines just correspond to the
critical lines $\lambda_{c_{1}}=\alpha+1$, $\lambda_{c_{2}}=\alpha-1$ with the
parameter $(\alpha\leq 0.25)$ and
$\lambda_{c_{3}}={-(1+8\alpha^{2})}/(8\alpha)$, respectively. It should be
noted that the two-site coherence $C_{l_{1}}$ is zero when the functions
$C_{1}\cdot D_{1}$ or $E_{2}\cdot F_{2}$ are zero. Therefore the long-range
two-site entanglement occurs at the regions of the two functions tending to
zero. In the main text, we choose three sets of parameters and confirm the
existence of long-range two-site entanglement as shown in Fig. 3.

###  I.5 V. Characterization of quantum phases via the long-range two-site
quantum discord in the XXT model

Quantum discord is a kind of typical quantum correlation in quantum
information processing and can be written as ho2001prl ; lh2001jpa

$\displaystyle
D_{A}(\rho_{AB})=S(\rho_{A})-S(\rho_{AB})+\underset{\\{E^{A}_{k}\\}}{\rm{min}}\underset{k}{\sum}p_{k}S(\rho_{B|k}),$
(29)

where the measurement $\\{E_{k}^{A}\\}$ is performed on subsystem $A$ with the
minimum running over all the projection measurements,
$S(\sigma)=-\mbox{Tr}\sigma\mbox{log}\sigma$ is the von Neumann entropy and
$\rho_{B|k}=\mbox{Tr}_{A}[E_{k}^{A}\rho_{AB}E_{k}^{A}/\mbox{Tr}(E_{k}^{A}\rho_{AB}E_{k}^{A})]$
is the output state of subsystem $B$ after the measurement $E_{k}^{A}$ with
the probability being $p_{k}=\mbox{Tr}(E_{k}^{A}\rho_{AB}E_{k}^{A})$.

In the XXT model, the two-qubit reduced state $\rho_{AB}=\rho_{r}$ has the
form given in Eq. (2) of the main text, for which the single qubit reduced
state for subsystem $A$ is

$\rho_{A}=\left(\begin{array}[]{cc}u^{+}+z&0\\\ 0&u^{-}+z\\\
\end{array}\right).$ (30)

After some derivation, we can obtain the von Neumann entropies for the reduced
states $\rho_{A}$ and $\rho_{AB}$ which can be written as

$\displaystyle S(\rho_{A})$ $\displaystyle=$ $\displaystyle-
Tr\rho_{A}\log_{2}\rho_{A}$ (31) $\displaystyle=$
$\displaystyle-\frac{1}{2}(1+\langle\sigma^{z}\rangle)\log_{2}[\frac{1}{2}(1+\langle\sigma^{z}\rangle)]$
$\displaystyle-\frac{1}{2}(1-\langle\sigma^{z}\rangle)\log_{2}[\frac{1}{2}(1-\langle\sigma^{z}\rangle)],$

and

$\displaystyle S(\rho_{AB})$ $\displaystyle=$ $\displaystyle-
Tr\rho_{r}\log_{2}\rho_{r}$ (32) $\displaystyle=$ $\displaystyle-
z_{1}\log_{2}z_{1}-z_{2}\log_{2}z_{2}$
$\displaystyle-u^{+}\log_{2}u^{+}-u^{-}\log_{2}u^{-},$

where
$z_{1}=z+y^{+}=(1+2\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle-\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/4$,
$z_{2}=z-y^{+}=(1-2\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle-\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/4$
and $u^{\pm}=(1\pm
2\langle\sigma^{z}\rangle+\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/4$,
respectively.

Figure 2: (Color online) The change of quantum discord $D_{A}(\alpha,\lambda)$
for different two-site distances: (a) $r=1$, (b) $r=6$, and (c) $r=10$,
respectively. In panel (d), two sets of parameters are chosen in SL-I and SL-
II phases, where the asymptotical and oscillating decay modes of $D_{A}$ can
serve as the diagnostic of quantum phases.

The third term in Eq. (29) is the measurement-induced conditional entropy,
where the optimal measurement on subsystem $A$ can be determined via the
method presented by Chen _et al_ qc11pra . The two-qubit reduced state
$\rho_{AB}=\rho_{r}$ in the XXT model has the $X$-shape for which the optimal
measurement for $D_{A}(\rho_{AB})$ is $\sigma_{z}$ if

$\begin{split}(y^{+})^{2}\leqslant(u^{+}-z)(u^{-}-z),\end{split}$ (33)

and the optical measurement is $\sigma_{x}$ if

$\begin{split}|\sqrt{u^{+}u^{-}}-\sqrt{z^{2}}|\leqslant|y^{+}|,\end{split}$
(34)

where the parameters $y^{\pm}$, $u^{\pm}$ and $z$ are the matrix elements of
$\rho_{r}$ given in Eq. (2) of the main text. Therefore, the measurement-
induced conditional entropy in Eq. (29) is

$\widetilde{S}(B|A)=\underset{\\{\sigma_{x}^{k}~{}\texttt{or}~{}\sigma_{z}^{k}\\}}{\rm{min}}\underset{k}{\sum}p_{k}S(\rho_{B|k}),$
(35)

where $\sigma_{x}^{k}$ and $\sigma_{z}^{k}$ with $k=0$ and $1$ are
corresponding measurement operators for the Pauli measurements, and the
related conditional entropies are

$\begin{split}\underset{\\{\sigma^{k}_{x}\\}}{\sum}p_{k}S(\rho_{B|k})=-\lambda_{+}\log_{2}(\lambda_{+})-\lambda_{-}\log_{2}(\lambda_{-}),\end{split}$
(36)

in which the parameters are
$\lambda_{\pm}=(1\pm\sqrt{\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle^{2}+\langle\sigma^{z}\rangle^{2}})/2$,
and

$\begin{split}\underset{\\{\sigma^{k}_{z}\\}}{\sum}p_{k}S(\rho_{B|k})=-\sum_{i=+,-}p_{i}(\xi_{i}\mbox{log}_{2}\xi_{i}+\eta_{i}\mbox{log}_{2}\eta_{i}),\end{split}$
(37)

in which the probabilities and the corresponding eigenvalues of subsystem $B$
are

$\displaystyle p_{\pm}$ $\displaystyle=$
$\displaystyle(1\pm\langle\sigma^{z}\rangle)/2,$ $\displaystyle\xi_{\pm}$
$\displaystyle=$
$\displaystyle(1-\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/[2(1\pm\langle\sigma^{z}\rangle)],$
$\displaystyle\eta_{\pm}$ $\displaystyle=$ $\displaystyle(1\pm
2\langle\sigma^{z}\rangle+\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/[2(1\pm\langle\sigma^{z}\rangle)].$
(38)

Combining Eq. (35) and Eqs. (31)-(32), we can calculate the two-site quantum
discord for the reduced state $\rho_{r}$ in the XXT model.

Next, we study the long-range property of quantum discord in the XXT model. As
shown in Fig. 2, the change of $D_{A}(\alpha,\lambda)$ along the parameters
$\alpha$ and $\lambda$ for the spin-spin distance $r=1,6$ and $10$ are plotted
in panels (a)-(c), where the quantum discord in SL-II phase is magnified by a
factor of 3 in panels (b) and (c). The quantum correlation $D_{A}$ is zero in
the ferromagnetic phase, and the nonzero $D_{A}$ can distinguish SL-I and SL-
II phases from the ferromagnetic phase. Moreover, similar to the case of two-
site coherence in the main text, the change pattern of quantum discord is
different in the two spin-liquid phases as shown in panels (a)-(c), where
$D_{A}$ is asymptotical in SL-I phase and oscillating in SL-II phase. The
intrinsic reason is that $D_{A}$ decays along with the distance $r$ in the
asymptotical mode for the SL-I phase and in the oscillating mode for SL-II
phase, which can serve as an effective diagnostic tool for the two phases. In
panel (d), we choose the parameters $(\alpha=0.7,\lambda=1.0)$ in SL-I phase
and $(\alpha=3.0,\lambda=0.5)$ in SL-II phase, where the quantum discord
exhibits distinctly different decay modes.

###  I.6 VI. Detection of the TQPTs by the $l_{1}$-norm measure of quantum
coherence

In the main text, we pointed out that the first-order derivative of adjacent
two-site coherence $C_{l_{1}}(\rho_{r})$ can detect the topological quantum
phase transition in the extended Ising model. Here, we first consider the
TQPTs driven by the three-spin interaction $\alpha$, and the other parameters
in the model are chosen to be $\gamma=1$, $\delta=-1$ and $\lambda=1$,
respectively. In Fig. 3(a) and 3(b), we plot the $l_{1}$-norm coherence and
its first-order derivative as a function of the parameter $\alpha$ for the
adjacent spin pair. As shown in the figures, the quantum coherence
$C_{l_{1}}(\rho_{r})$ is a continuous function, but its first-derivative
exhibits the divergent behaviors where the divergent points just indicate the
critical points $\alpha_{c_{1}}=(-\sqrt{5}-1)/2$, $\alpha_{c_{2}}=0$,
$\alpha_{c_{3}}=(\sqrt{5}-1)/2$ and $\alpha_{c_{4}}=2$ for the TQPTs.

Figure 3: (Color online) The $l_{1}$-norm of quantum coherence $C_{l_{1}}$ and
its derivative $\partial_{\alpha}C_{l_{1}}$ versus the three-spin interaction
$\alpha$ [panels (a) and (b)] and external magnetic field $\lambda$ [panels
(c) and (d)] for adjacent two-spin pair. Figure 4: (Color online) Spin
correlations (the first row) and quantum coherence (the second row) as a
function of two-site distance in topological phases with winding numbers
$\mathcal{N}=2,0,1,-1$ and $-2$, where the TQPTs are driven by three-spin
interaction $\alpha$ (other parameters are chosen to be $\gamma=1$,
$\delta=-1$ and $\lambda=1$) and the long-range $l_{1}$-norm coherence is
dominated by spin correlation $\langle\sigma_{0}^{x}\sigma_{r}^{x}\rangle$
(red-dot line) or $\langle\sigma_{0}^{y}\sigma_{r}^{y}\rangle$ (blue-triangle
line) in different phases.

We also analyze the TQPTs driven by external magnetic field $\lambda$, and the
other parameters in the model are chosen to be $\alpha=1.5$, $\gamma=1$ and
$\delta=1$, respectively. In Fig. 3(c) and 3(d), we plot the $l_{1}$-norm
coherence and its first-order derivative as a function of the parameter
$\lambda$ for the adjacent spin pair. As shown in the figures, the divergent
points of the first-order derivative can indicate the critical points
$\lambda_{c_{1}}=-1.5$, $\lambda_{c_{2}}=0.5$ and $\lambda_{c_{3}}=2.5$ for
the TQPTs in the extended Ising model. Moreover, similar to the above cases,
the TQPTs driven by the parameters $\gamma$ and $\delta$ can also be detected
by the first-order derivative of $l_{1}$-norm quantum coherence for the
adjacent two-spin pair.

###  I.7 VII. The relation between spin correlations and quantum coherence in
topological quantum phases

According to Eq. (11) of the main text, the long-range two-site quantum
coherence $C_{l_{1}}(\rho_{r})$ in the extended Ising model can be expressed
as the function of spin correlations
$\langle\sigma_{0}^{x}\sigma_{r}^{x}\rangle$ and
$\langle\sigma_{0}^{y}\sigma_{r}^{y}\rangle$. It is pointed out in the main
text that the behaviors of long-range quantum coherence (such as the
oscillating, asymptotical and freezing modes) can serve as the diagnostic
tools for topological phases with different winding numbers. Here, we further
study the relation between spin correlations and quantum coherence in the
extended Ising model.

For the TQPTs driven by the three-spin interaction, we choose six typical
values for the parameter $\alpha$, which correspond to the topological phases
with the winding numbers $\mathcal{N}=0,\pm 1,\pm 2$, respectively. As shown
in Fig. 4, two-site spin correlations
$\langle\sigma_{0}^{x}\sigma_{r}^{x}\rangle$ (red-dot lines) and
$\langle\sigma_{0}^{y}\sigma_{r}^{y}\rangle$ (blue triangle lines) in five
topological phases are plotted along with the increasing spin-spin distance in
the first row of the figure, and the corresponding $l_{1}$-norm coherence
$C_{l_{1}}(\rho_{r})$s (black-square lines) in different phases are plotted in
the second row of the figure. In the topological phase $\mathcal{N}=2$, the
quantum coherence decays in the oscillating mode (see the first panel in the
second row) and mainly comes from the spin correlation
$|\langle\sigma_{0}^{x}\sigma_{r}^{x}\rangle|$, where the spin correlation
$|\langle\sigma_{0}^{y}\sigma_{r}^{y}\rangle|$ will disappear after a short
two-site distance (see the first panel in the first row). The coherence in the
topological phase $\mathcal{N}=0$ can decay in both oscillating and
asymptotical modes, where the oscillating-damped coherence comes from spin
correlation $|\langle\sigma_{0}^{x}\sigma_{r}^{x}\rangle|$, but the
asymptotical-damped coherence is due to the spin correlation
$|\langle\sigma_{0}^{y}\sigma_{r}^{y}\rangle|$ (see the panels and its insets
in the second column of Fig. 4). In the topological phase $\mathcal{N}=-2$,
the quantum coherence decays in the asymptotical mode and is mainly
contributed by the spin correlation
$\langle\sigma_{0}^{x}\sigma_{r}^{x}\rangle$, as shown in the two panels in
the fifth column of Fig. 4.

Moreover, it was pointed out in the main text that the quantum coherence in
topological phases $\mathcal{N}=\pm 1$ will exhibit freezing phenomenon after
a short-range distance. In the panels of the third and forth columns of Fig.
4, we plot the quantum coherence and its corresponding spin correlations,
where we find that the freezing coherence in the case of $\mathcal{N}=1$ is
contributed by the spin correlation
$|\langle\sigma_{0}^{y}\sigma_{r}^{y}\rangle|$ (red-dot line) but the freezing
resource in the case of $\mathcal{N}=-1$ comes from the spin correlation
$|\langle\sigma_{0}^{x}\sigma_{r}^{x}\rangle|$ (blue-triangle line).

###  I.8 VIII. Behaviors of long-range quantum discord in the topological
phases driven by three-spin interactions

In the main text, we have analyzed the properties of different topological
phases with the behaviors of long-range two-site quantum coherence. Here, we
will further demonstrate that the behaviors of long-range quantum discord have
the same functionality for characterization on the topological phases driven
by three-spin interactions. The expression of two-site reduced state
$(\rho_{r}=\rho_{AB})$ is given in Eq. (2) of main text, and the related
quantum discord $D_{A}$ can be calculated according to the formula in Eq. (14)
of main text where the von Neumann entropies $S(\rho_{A})$ and $S(\rho_{AB})$
can be obtained via some direct calculation but the measurement-induced
conditional entropy

$\widetilde{S}(B|A)=\underset{\\{E^{A}_{k}\\}}{\rm{min}}\underset{k}{\sum}p_{k}S(\rho_{B|k})$
(39)

needs to be optimized by all the projective measurements $\\{E_{k}^{A}\\}$ on
the subsystem $A$.

Although the two-site reduced state $\rho_{r}$ is still $X$-shape in the
extended Ising model ($\gamma\neq 0$ and $\delta\neq 0$), the method presented
in Ref. qc11pra  for selecting the optimal measurement of $D_{A}(\rho_{AB})$
is non-applicable due to the condition on matrix elements
$\left|y^{+}+y^{-}\right|\geq\left|y^{+}-y^{-}\right|$ not being satisfied.
Therefore, in order to calculate the measurement-induced conditional entropy
in quantum discord, we will consider the situation for all the projective
measurements in which the measurement operator can be written as mss09pra ;
ycl2011pra

$\begin{split}E^{A}_{i}=V|i\rangle\langle i|V^{{\dagger}},\end{split}$ (40)

where $\\{|i\rangle\\}$ is the standard computational basis
$\\{|0\rangle,|1\rangle\\}$ and the unitary transformation matrix $V$ has the
form

$\begin{split}&V=\begin{pmatrix}\cos\frac{\theta}{2}&e^{-i\phi}\sin\frac{\theta}{2}\\\
e^{i\phi}\sin\frac{\theta}{2}&-\cos\frac{\theta}{2}\end{pmatrix},\end{split}$
(41)

in which the parameters satisfy $0\leq\theta\leq\pi$ and $0\leq\phi<2\pi$,
respectively. In this case, the quantum discord $D_{A}$ can be obtained via
the optimization of parameters $\theta$ and $\phi$ in measurement-induced
conditional entropy. After carefully numerical verifications similar to that
of Ref. ycl2011pra , we find that the optimal projective measurements in Eq.
(39) correspond to the following three cases

$\displaystyle E_{k(\texttt{I})}^{A}:~{}~{}\theta=\pi/2,~{}\phi=\pi/2,$
$\displaystyle E_{k(\texttt{II})}^{A}:~{}~{}\theta=\pi/2,~{}\phi=0,$
$\displaystyle E_{k(\texttt{III})}^{A}:~{}~{}\theta=0,~{}\phi=0,$ (42)

with $k=1,2$, and then the measurement-induced conditional entropy is

$\begin{split}\widetilde{S}(B|A)=\underset{\\{E_{k(\texttt{I})}^{A},E_{k(\texttt{II})}^{A},E_{k(\texttt{III})}^{A}\\}}{\rm{min}}\underset{k}{\sum}p_{k}S(\rho_{B|k}).\end{split}$
(43)

In Fig. 5, we choose three typical quantum states of $\rho_{r}$ for which the
optimal measurements for $\widetilde{S}(B|A)$ correspond to
$E_{k(\texttt{I})}^{A},E_{k(\texttt{II})}^{A}$ and $E_{k(\texttt{III})}^{A}$,
respectively.

Figure 5: (Color online) The measurement-induced conditional entropies
$\widetilde{S}(B|A)$ for three typical quantum states $\rho_{r}$ ($r=2$) which
correspond to optimal projective measurements: (a)
$\\{E_{k(\texttt{I})}^{A}\\}$ for $\rho_{r}$ with parameters $\gamma=1$,
$\delta=-1$, $\lambda=1$ and $\alpha=0.1$, (b) $\\{E_{k(\texttt{II})}^{A}\\}$
for $\rho_{r}$ with parameters $\gamma=1$, $\delta=-1$, $\lambda=1$ and
$\alpha=-2.5$, and (c) $\\{E_{k(\texttt{III})}^{A}\\}$ for $\rho_{r}$ with
parameters $\gamma=1$, $\delta=-2$, $\lambda=-0.3$ and $\alpha=1$,
respectively.

Based on the previous analysis, we can derive the two-site quantum discord

$D_{A}(\rho_{r})=\widetilde{S}(B|A)-S(B|A),$ (44)

where the quantum condition entropy $S(B|A)=S(AB)-S(A)$ and the measurement-
induced conditional entropy $\widetilde{S}(B|A)$ is given by Eq. (43) in which

$\displaystyle\widetilde{S}^{(\texttt{I})}(B|A)$ $\displaystyle=$
$\displaystyle\underset{E_{k(\texttt{I})}^{A}}{\sum}p_{k}S(\rho_{B|k})$ (45)
$\displaystyle=$
$\displaystyle-\beta_{+}\log_{2}(\beta_{+})-\beta_{-}\log_{2}(\beta_{-})$

with
$\beta_{\pm}=[1\pm(\langle\sigma^{y}_{0}\sigma^{y}_{r}\rangle^{2}+\langle\sigma^{z}\rangle^{2})^{1/2}]/2$,

$\displaystyle\widetilde{S}^{(\texttt{II})}(B|A)$ $\displaystyle=$
$\displaystyle\underset{E_{k(\texttt{II})}^{A}}{\sum}p_{k}S(\rho_{B|k})$ (46)
$\displaystyle=$
$\displaystyle-\lambda_{+}\log_{2}(\lambda_{+})-\lambda_{-}\log_{2}(\lambda_{-})$

with
$\lambda_{\pm}=[1\pm(\langle\sigma^{x}_{0}\sigma^{x}_{r}\rangle^{2}+\langle\sigma^{z}\rangle^{2})^{1/2}]/2$,
and

$\displaystyle\widetilde{S}^{(\texttt{III})}(B|A)$ $\displaystyle=$
$\displaystyle\underset{E_{k(\texttt{III})}^{A}}{\sum}p_{k}S(\rho_{B|k})$ (47)
$\displaystyle=$
$\displaystyle-\sum_{i=+,-}p_{i}(\xi_{i}\mbox{log}_{2}\xi_{i}+\eta_{i}\mbox{log}_{2}\eta_{i})$

with $p_{\pm}=(1\pm\langle\sigma^{z}\rangle)/2$,
$\xi_{\pm}=(1-\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/[2(1\pm\langle\sigma^{z}\rangle)]$
and $\eta_{\pm}=(1\pm
2\langle\sigma^{z}\rangle+\langle\sigma^{z}_{0}\sigma^{z}_{r}\rangle)/[2(1\pm\langle\sigma^{z}\rangle)]$,
respectively.

In Fig. 6(a)-(c), we plot quantum discord $D_{A}(\rho_{r})$ as a function of
three-spin interaction $\alpha$ and two-site distance $r$, where the different
topological phases are labeled by the corresponding winding numbers. In the
case of $\mathcal{N}=2$, the quantum discord decays in the oscillating mode
along with two-site distance for a given value of $\alpha$. However, in the
case of $\mathcal{N}=-2$, quantum discord decays in the asymptotical mode
after a short two-site distance as shown in Fig. 6(c). Moreover, the quantum
discord in topological phase $\mathcal{N}=0$ can decay in both oscillating and
asymptotical modes as shown in Fig. 6(a). In particular, given a value of
$\alpha$, the nonzero quantum discord in topological phases $\mathcal{N}=\pm
1$ exhibits quantum correlation freezing phenomenon after a short two-site
distance as shown in Fig. 6(b). In generic case, the freezing quantum discord
in topological phase $\mathcal{N}=1$ has a larger value than that of quantum
discord in topological phase $\mathcal{N}=-1$. In Fig. 6(d)-(f), we select
some typical values in different topological phases and plot the long-range
behaviors of quantum discord, where the oscillating-damped, asymptotical-
damped and freezing modes of $D_{A}(\rho_{r})$ can serve as the effective
diagnostic tools for quantum phases. It should be noted that the steady
quantum discord is a kind of topologically protected long-range quantum
correlation and has the potential applications in quantum computation and
quantum communication.

Figure 6: (Color online) (a)-(c) The quantum discord $D_{A}(\rho_{r})$ as a
function of three-spin interaction $\alpha$ and two-site distance $r$ in
different topological phases with winding numbers $\mathcal{N}=0,\pm 1$ and
$\pm 2$ where the other parameters are chosen to be $\gamma=1$, $\delta=-1$,
and $\lambda=1$, respectively. (d)-(f) The behaviors of long-range two-site
discord for some typical values of $\alpha$ in the five quantum phases. Figure
7: (Color online) Long-range properties of quantum coherence (the first row)
and quantum discord (the second row) as a function of two-site distance and
external magnetic field accompanied with the corresponding energy spectra
$\Lambda_{k}(\lambda)$ and trajectories of winding vectors, and, in the third
row, the oscillating-damped, asymptotical-damped and freezing two-site quantum
resources can serve as a set of effective diagnostic tools for topological
phases in the extended Ising model.

###  I.9 IX. Behaviors of long-range two-site quantum resources in the
topological phases driven by external magnetic field

In this section, we will study the long-range behaviors of quantum coherence
and quantum discord in the topological phases driven by external magnetic
field $\lambda$, where the other parameters in the extended Ising model are
chosen to be $\alpha=1.5,\gamma=1,\delta=1$, respectively. The critical points
of the topological quantum phase transitions can be obtained via solving the
characteristic equation spl2018pra

$3\xi^{-2}/2+\xi^{-1}-\lambda=0$ (48)

for which the solutions are $\lambda_{c_{1}}=-1.5$ corresponding to
$\xi_{1}=\exp[\pm i\arccos(-1/3)]$, $\lambda_{c_{2}}=0.5$ corresponding to
$\xi_{2}=-1$, and $\lambda_{c_{3}}=2.5$ corresponding to $\xi_{3}=1$,
respectively ykw19qip . The two-site quantum coherence can be calculated
according to the formula in Eq. (11) of the main text, and the corresponding
quantum discord can be obtained by the method given in Eq. (44) of the
supplemental materials.

In Fig. 7, we plot the energy spectra of the system with a chain length
$L=1001$ (the three critical points and four regions of topological phases are
labeled), the trajectories of the winding number vectors in the $x$-$y$ planes
(corresponding to winding numbers $\mathcal{N}=0,1$, and $2$), and the long-
range properties of two-site quantum coherence and quantum discord. As shown
in the first row of the figure, the $l_{1}$-norm coherence
$C_{l_{1}}(\rho_{r})$ in different regions of topological phases exhibit the
oscillating-damped, asymptotical-damped and freezing modes along with two-site
distance. In particular, the topologically protected quantum coherence in the
phase $\mathcal{N}=1$ will attain to a steady value after a short two-site
distance. The case for two-site quantum discord $D_{A}(\rho_{r})$ is similar,
which exhibits the oscillating-damped, asymptotical-damped and freezing
behaviors along with two-site distance in different regions of topological
phases as shown in the second row of the figure. In the third row of the
figure, we choose some typical values of external magnetic field $\lambda$ and
plot the long-range behaviors of quantum coherence and quantum discord, where
the two kinds of quantum resources exhibit the similar decay modes which
further demonstrate that these long-range behaviors of quantum resources can
serve as a set of effective diagnostic tools for topological quantum phases.
Moreover, similar to the case in the topological quantum phases
$\mathcal{N}=\pm 1$ driven by three-spin interaction $\alpha$, the
topologically protected quantum resources in topological phase $\mathcal{N}=1$
driven by magnetic field $\lambda$ still exhibit the freezing phenomenon along
with two-site distance.

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