Global quantum discord and thermal tensor network in XXZ chains at finite temperatures

Accepted Manuscript Global quantum discord and thermal tensor network in XXZ chains at finite temperatures Long-hui Shen, Bin Guo, Zhao-yu Sun, Mei Wang, Yu-Ying Wu PII:

S0921-4526(19)30253-4

DOI:

https://doi.org/10.1016/j.physb.2019.04.021

Reference:

PHYSB 311429

To appear in:

Physica B: Physics of Condensed Matter

Received Date: 16 February 2019 Revised Date:

14 April 2019

Accepted Date: 19 April 2019

Please cite this article as: L.-h. Shen, B. Guo, Z.-y. Sun, M. Wang, Y.-Y. Wu, Global quantum discord and thermal tensor network in XXZ chains at finite temperatures, Physica B: Physics of Condensed Matter (2019), doi: https://doi.org/10.1016/j.physb.2019.04.021. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Graphical Abstract Global quantum discord and thermal tensor network in XXZ chains at finite temperatures

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Long-hui Shen, Bin Guo, Zhao-yu Sun, Mei Wang, Yu-Ying Wu

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We evaluate thermal-state discord of 16-site XXZ chains efficiently with thermal tensor net-

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works. The low-temperature dots are from thermal-tensor-network calculation, and the hightemperature solid lines are from linear scaling analysis. In the intermediate regions, they are quite compatible with each other. It indicates that the results are reliable in the full temperature

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regions.

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Highlights Global quantum discord and thermal tensor network in XXZ chains at finite temperatures Long-hui Shen, Bin Guo, Zhao-yu Sun, Mei Wang, Yu-Ying Wu

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• Thermal tensor networks are used to study thermal-state global discord of 1D chains

• Thermal-state discord shows footprint of the quantum phase transition of XXZ model

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• Thermal-state discord shows a broad plateau at finite temperatures

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• In the large N limit, thermal-state discord shows a linear scaling

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Global quantum discord and thermal tensor network in XXZ chains at finite temperatures Long-hui Shena , Bin Guoa,∗, Zhao-yu Sunb,∗∗, Mei Wangb , Yu-Ying Wub

a Department of Physics, Wuhan University of Technology, Wuhan 430070, China. of Electrical and Electronic Engineering, Wuhan Polytechnic University, Wuhan 430023, China.

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b School

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Abstract

We characterize multi-site quantum correlations in 1D finite-size XXZ chains at finite temperatures with global quantum discord. With the help of an exact diagonalization method and a thermal-tensor-network algorithm, the thermal-state discord G(ρˆ T ) is evaluated efficiently by

well-developed optimization algorithms. Firstly, we find that in a finite temperature region, G(ρˆ T ) shows some footprint of the quantum phase transition of the model. The underlying mechanism is that ρˆ T captures the level crossing in the low-lying excited states. Secondly, we study the tem-

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perature dependence of G(ρˆ T ). When the anisotropy is strong, G(ρˆ T ) will show a broad thermal plateau. We offer a quantitative explanation of the plateau by truncating the thermal-state operator ρˆ T with respect to several low-lying states. Thirdly, we investigate the scaling behavior of G(ρˆ T ). We find that when N is large enough, G(ρˆ T ) would show a linear scaling. Finally, combined with the thermal tensor networks and the linear scaling behavior, we successfully figure

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out reliable results for G(ρˆ T ) in the full-temperature regions with N up to 16. We believe that the thermal tensor networks will play a role in studying general multi-site quantum correlations in

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1D chains at finite temperatures.

Keywords: Quantum phase transitions, Global quantum discord, thermal tensor network

1. Introduction

An important task in condensed matter physics is to understand various quantum phases and phase transitions [1]. In traditional methods, we have to construct order parameters to capture ∗ Corresponding

∗∗ Corresponding

author: [email protected] author: [email protected]

Preprint submitted to Physica B: condensed-matter

April 20, 2019

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the basic properties of these phases, based upon some prior knowledge about the specific phases. Recently, it is realized that some concepts from quantum information theory (such as quantum entanglement) can offer us an interesting perspective to characterize these quantum phases and critical phenomena, and no prior knowledge about the phases is required [2, 3, 4, 5, 6, 7]. Further-

more, the quantum information perspective plays a key role in the fast developments of numerical

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simulation methods for quantum models, such as tensor network algorithms [8, 9].

Among various concepts of quantum information, quantum discord has attracted much attention [10, 11, 12, 13, 14, 15]. Quantum (pairwise) discord is a measure of quantum correlation

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between two sites in a quantum system. An important advantage of discord is that, it can survive in quantum separable states where quantum entanglement vanishes [16]. In addition, its concept can be naturally generalized into multi-site setting, namely, global quantum discord [17, 18]. In many-body quantum systems, it is quite intuitive to consider multi-site correlations which spread in the entire systems, rather than just in two-site subsystems. Consequently, global discord has been used to describe multi-site quantum correlations in various low-dimensional quantum models [19, 20, 21, 22]. Since global discord captures multi-site quantum correlations, it has shown

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some attractive properties. For instance, global discord can capture the infinite-order quantum phase transition in the Ashkin-Teller spin chains, where the pairwise discord fails [18]. It needs mention that the calculation of global discord is rather non-trivial. In very few situations, global discord can be figured out analytically [23]. For general quantum states, never-

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theless, numerical optimizations are involved [19]. Recently, in order to overcome the problem in evaluating global discord, some tensor-network-based algorithms have been proposed, both for pure states and mixed states. For instance, combined with the matrix-product-state algorithm,

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the global discord of the ground states of several 1D chains has been studied [20]. It is found that in the large-N limit, global discord is a linear function of the size of the chains. Quite recently, with the help of matrix product density operators, the above algorithm is upgraded and can be applied to mixed states [21]. Then the algorithm has been used to calculate the global discord of the reduced density matrices for subchains (which are mixed states naturally) on infinite-size N × N lattices at zero temperature. Again, a linear scaling is observed. Thereby, one can use the

“discord density” to describe the global correlations in the ground states of infinite-size lattices. These tensor networks are used to study 1D and 2D quantum models at zero temperature. However, there are good reasons for us to consider systems at finite temperatures. Firstly, ac2

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cording to the third law of thermodynamics, zero temperature cannot be reached, thus real materials are always at finite temperatures. Secondly, in some situations, as we will show in this

paper, because of the energy-level reconstruction, the excited states of finite-size models may be related to the quantum phase transition in the (infinite-size) models. It would be feasible to use

ˆ the thermal state ρˆ T = e−βH to capture these excited states, where Hˆ is the Hamiltonian of the

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models and β is the inverse temperature. We would like to mention that global discord at finite temperatures has already been studied in several models, such as the 1D transverse-field Ising model, the cluster-Ising model and the open-chain XX model [19, 24]. In these papers, a full

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diagonalization method is used to study short chains with N ≤ 10.

In this paper, we will study the global discord in 1D finite-size XXZ chains at finite temperatures. For such a purpose, for N ≤ 12, we will use a full diagonalization method to figure out ˆ

the thermal-state operator ρˆ T = e−βH exactly. For N > 12, we will use thermal tensor networks combined with the imaginary-time evolution algorithm to figure out ρˆ T . As one will see, thermal tensor networks offer us a unified and seamless framework to calculate the thermal-state operator ρˆ T and consequently the thermal-state discord G(ρˆ T ) for general 1D long chains. Specifically, we

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will use thermal-state discord to characterize the infinite-order quantum phase transition in the XXZ models. We will also pay our attention to the robustness of thermal-state discord against thermodynamic fluctuations in the models. Furthermore, we will study the scaling behavior of thermal-state discord, and verify whether or not the “discord density” is still meaningful at finite

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temperatures.

This paper is organized as follows. In Sec. 2, we will introduce the concept of global discord. ˆ

In Sec. 3, we will show how to express the thermal-state operator ρˆ T = e−βH into thermal tensor

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networks. In Sec. 4, we will investigate the thermal-state discord G(ρˆ T ) in the finite-size XXZ models. A summary and some discussions would be given in Sec. 5.

2. Global quantum discord Global quantum discord is a valuable measure of multi-site quantum correlations. For a

general N-site state ρ, ˆ the global discord G(ρ) ˆ can be given by [23] G(ρ) ˆ = min[I(ρ) ˆ − I(Φ(ρ))], ˆ Φ

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(1)

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where Φ(∙) denotes a multi-site local projective measurement performed on the system, the min function is with respect to all these measurements, and I(∙) is the mutual information of the corresponding density matrices, i.e., I(ρ) ˆ =

N X i=1

S (ρˆ [i] ) − S (ρ), ˆ

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with S (∙) the von Neumann entropy, and ρˆ [i] the reduced density matrix of site i.

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One can see that global discord is just the minimum difference between the mutual information of the original state ρˆ and the mutual information of the state Φ(ρ). ˆ By introducing the

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relative entropy, global discord can also be interpreted as the minimum difference between the relative entropy of the whole system and the sum of the relative entropies of every single site [17, 18]. Global discord has many valuable features. For instance, for classical states, it is clear that G = 0, since ρˆ = Φ(ρ) ˆ [18]. On the other hand, a non-zero discord would indicate some quantum correlations in ρ. ˆ In addition, since it captures multi-site correlations, global discord may be used in quantum communication [25].

Eq. (1) is very compact. However, in order to improve the efficiency of numerical calcula-

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tions, let us re-write the formula as [19] G(ρ) ˆ

=

P N P2 ss ss ˆ˜ [i] min{Rˆ i } { i=1 log2 ρˆ˜ [i] s=1 ρ P N ss ss , − 2s=1 ρˆ˜ log2 ρˆ˜ } PN + i=1 S (ρˆ [i] ) − S (ρ) ˆ

(3)

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where s and s denote the basis of the Hilbert space for a single site and the whole lattice, respec-

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tively, and the operators ρˆ˜ [i] and ρˆ˜ are defined as ρˆ˜ [i] = Rˆ †[i] ρˆ [i] Rˆ [i] ,

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N ˆ† N ˆ R[i] ∙ ρˆ ∙ ⊗i=1 R[i] , ρˆ˜ = ⊗i=1

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with Rˆ [i] = sin θ2i cos φi σ ˆ x +sin θ2i sin φi σ ˆ y +cos θ2i σ ˆ z . Eq. (3) is the starting point in our calculation of the global discord.

Since we are interested in the global discord in 1D chains at finite temperatures, the density

matrix ρˆ in Eq. (3) would be the thermal-state density matrix ρˆ T . For an N-site chain described ˆ the thermal-state density matrix ρˆ T at finite temperature T can be regarded by a Hamiltonian H, ˆ that is, as a mixed state for all the eigen states {|ϕi i} of the Hamiltonian H, P2N −βEi e |ϕi ihϕi | , ρˆ T = i=1P2N −βEi i=1 e 4

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Figure 1: (Color online) (a) Tensor network for the matrix-product-state purification |Ψi for the density operator ρˆ T . ˆ ˆ (b) Imaginary-time evolution of the purification. The operator e−ΔτH is expressed by the product of local operators U.

˜ of the imaginary-time evolution. The bond dimension is increased and some truncation (c) The output purification |Ψi

˜ (d) Tensor network for the density operator ρˆ T in the terms algorithms will be used to reduce the bond dimension of |Ψi. of the purification |Ψi. (e) Local tensor of the purification for the density operator ρˆ˜ T . (f) Tensor network for the density operator ρˆ˜ T . Both ρˆ T and ρˆ˜ T are heavily involved in the definition of global discord.

where β =

1 kB T

is the inverse temperature (we set the Boltzmann constant as kB = 1 in this paper),

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Ei is the eigen energy arranged in ascending order, and e−βEi is the weight of the state |ϕi i.

There are some difficulties in evaluating the thermal-state discord G(ρˆ T ). First of all, we need

some method to figure out the thermal-state operator ρˆ T . When the size of the chain is small, ˆ as shown in Eq. (6). However, ρˆ T can be figured out exactly by a full diagonalization of H,

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when N is slightly large, a full diagonalization of Hˆ would become highly inefficient. Secondly, in the numerical optimization in Eq. (3), a 2N × 2N matrix ρˆ˜ T is involved, thus the optimization would be quite expensive for large N. Thirdly, for a general mixed state ρˆ T , an exact calculation

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of the von Neumann entropy S (ρˆ T ) depends upon the full eigen-value spectrum of ρˆ T , which is

also non-trivial for large N. In previous papers, some techniques have been proposed to solve

these problems in the framework of matrix product states and matrix product density operators [20, 21]. In this paper, we will investigate the thermal-state discord in quantum chains with the help of thermal tensor networks. As one will see, thermal tensor networks provide a seamless framework for us to (1) figure out the thermal-state density matrix ρˆ T , (2) represent the operator ρˆ˜ T , and (3) then calculate the global discord for general 1D quantum chains at finite temperatures.

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3. Thermal tensor networks 3.1. Thermal tensor networks and imaginary-time evolution

In this section, we will introduce how to express the thermal-state operator ρˆ T of 1D quantum chains as tensor networks. There are several alternative ways to do the job [26, 27, 28]. In this

paper, we will adopt the matrix-product-state (MPS) purification to express the operator ρˆ T ,

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combined with the standard imaginary-time-evolution algorithm [26]. A key advantage of this algorithm is that, MPS and imaginary-time evolution have been widely used in searching for

ground states of 1D quantum systems. Thereby, by making quite minor modifications to existing

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codes, one is ready to figure out ρˆ T of these 1D chains. We would like to mention that some other advanced thermal tensor networks have been developed recently, both for 1D lattices and for 2D lattices [27, 28].

First of all, a density matrix ρˆ can always be expressed in term of the so-called MPS purification |Ψi. The network for a purification |Ψi of a 6-site model has been shown in Fig. 1(a). One

can see that every tensor Ai of the purification |Ψi has three types of indices, i.e., the inner indices

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(the black solid line) connecting other tensors, an external index (the red dashed line) indicating the physical degree of freedom of the site, and an external auxiliary index (the blue dashed line). The corresponding density matrix ρˆ can be obtained from the purification just by tracing over all the auxiliary indices as

ρˆ = tra (|ΨihΨ|).

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ˆ at some finite temperaLet’s consider an 1D quantum chain (described by a Hamiltonian H) ˆ

ture T . Its thermal state e−βH can be expressed as [26]

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ˆ

e−βH = =

ˆ ˆ (e−ΔτH ) M ∙ 1ˆ ∙ (e−ΔτH ) M ˆ

ˆ

(e−ΔτH ) M ∙ tra (|Ψ0 ihΨ0 |) ∙ (e−ΔτH ) M

where Δτ denotes a rather small (imaginary) time slice, M =

β 2Δτ

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denotes the number of steps

ˆ is just the purification required to reach the target temperature T , and |Ψ0 i (its tensors are Ai = 1)

ˆ for the identity operator 1.

According to Eq. (8), we will define a series of purifications in a recursive way as ˆ

|Ψm i = e−ΔτH |Ψm−1 i, 6

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with m = 1, 2, ..., M labeling the step of the evolution. Then it is clear that at the m-th step, the Hˆ

Tm =

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with T m =

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purification |Ψm i can be used to construct the thermal-state operator e− Tm with the temperature Hˆ

Thereby, as the imaginary-time evolution progresses, we are able to figure out e− Tm

1 1 1 1 1 2Δτ , 4Δτ , 6Δτ , 8Δτ , 10Δτ ...,

and so on.

Next, we show how to carry out a single step of evolution. Since Δτ is rather small, with ˆ

the help of the Trotter-Suzuki approximation, e−ΔτH can be decomposed as the product of local

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operators Uˆ [26], as shown in Fig. 1(b). The tensor contractions between these local operators Uˆ and local tensors Ai can be implemented easily, and we denote the resulting purification as

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˜ m i [Fig. 1(c)]. It is clear that the bond dimension of |Ψ ˜ m i is larger than the original purification |Ψ

˜ m i into a |Ψm i with some predefined |Ψm−1 i. Thereby, we need to compress the purification |Ψ bond dimension D0 . About the compression of a 1D tensor network, some mature algorithms can be found in Ref. [29]. This “compressed” purification |Ψm i will be used as the input for the next step of evolution.

3.2. Thermal-state discord and thermal tensor networks

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In Campbell’s formula for global discord in Eq. (3), there are three terms, that is, (1) the PN single-site entropy term i=1 S (ρˆ [i] ), (2) the N-site entropy term S (ρ), ˆ and (3) the min term. For the (1) term, it is clear that S (ρˆ [i] ) can be figured out easily, since only single-site operators are

involved. For the (2) term, the N-site von Neumann entropy S (ρ) ˆ has been widely studied by density matrix renormalization group [29]. We will return to this issue in Sec. 4.3.

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For the (3) term, we will show how to carry out the numerical optimization efficiently with the help of the thermal tensor networks. Our original optimization problem in Eq. (3) can be

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described as min{Rˆ [i] } f ({Rˆ [i] }; ρˆ T ), where {Rˆ [i] } are the variables of the objective function, and ρˆ T

is just a 2N × 2N constant operator which has already been expressed in term of the purification

[Fig. 1(d)]. Thus, in practice, the optimization problem is transformed as min{Rˆ [i] } f ({Rˆ [i] }; {Ai }),

with {Ai } the local tensors for the purification of the thermal-state operator ρˆ T . Furthermore, according to Eq. (5), one can see that the operator ρˆ˜ T can also be expressed in term of the ss purification, which we have shown in Fig. 1 (e) and (f). Moreover, only diagonal elements ρˆ˜ [i] ss and ρˆ˜ are involved in the optimization of Eq. (3). They will be calculated from {Ai } on the

fly. In summary, in every step of our optimizations, only small local tensors {Ai } are involved. Thereby, the numerical efficiency is greatly improved, making it possible to deal with large N. 7

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We will consider the 1D spin- 12 XXZ models described by Hˆ =

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x σ ˆ ix σ ˆ i+1 +σ ˆ yi σ ˆ yi+1 + Δσ ˆ zi σ ˆ zi+1 ,

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4. Models and main results

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where σ ˆ ix,y,z denote Pauli matrices on site i, and Δ is the anisotropic parameter. In our calcula-

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tions, we just consider open boundary conditions. The low-lying energy spectrum for several finite-size chains with N = 10 and N = 12 has been shown in Fig. 2(a) and (b), respectively. In

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the low-lying excited states, there are several level crossings at Δ = 1. These level crossings in the excited states of finite-size chains play a role in the quantum phase transition of the infinitesize models. That is, in the large-N limit, the spectrum would be re-constructed, and becomes gapless for Δ < 1 and gapped for Δ > 1 [30]. An infinite-order quantum phase transition occurs at Δc = 1.

We will use the thermal-tensor-network algorithm to calculate the thermal-state operator ρˆ T when N is large. The algorithm has two sources of error. The first one is the error of the Trotter-

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Suzuki decomposition. In our calculation, we have adopted a second-order Trotter-Suzuki decomposition with the (imaginary) time slice Δτ = 0.01. In addition, the maximum number of steps for the imaginary-time evolution is M = 500, thus the lowest temperature is T = 0.1. The second source of error is the finite bond dimension D0 of the MPS purification. In our calculation, we set the bond dimension as D0 = 80. In order to check the validity of the thermal-tensor-

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network code, we have defined an error quantity E(ρˆ T T N , ρˆ exact ) = tr[(ρˆ T T N − ρˆ exact )2 ] [26], where ρˆ T T N denotes the approximate thermal-state operator from the thermal-tensor-network method,

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and ρˆ exact denotes the exact thermal-state operator from Eq. (6). We have calculated the error quantity E(ρˆ T T N , ρˆ exact ) with various temperatures and various Δs for N = 10, and the results

have been shown in Fig. 3. One can see that the error is smaller than 10−6 for most parameters, thus the validity of our code is confirmed. In order to avoid possible misleading of the numerical results, in this section, for N ≤ 12,

we will just present the results with ρˆ T calculated exactly from the full diagonalization method. For N > 12 where the exact ρˆ T is unreachable, we will present the results with ρˆ T calculated approximately by the thermal-tensor-network algorithm.

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Figure 2: (Color online) Low-lying energy spectrum as a function of the anisotropic parameter Δ in the finite-size XXZ chains with (a) N = 10 and (b) N = 12. Level crossing occurs in the excited states, rather than in the ground state. In the large-N limit, the spectrum would be gapless for Δ < 1 and gapped for Δ > 1, and Δc = 1 is the quantum phase transition

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Figure 3: (Color online) The error between the approximate thermal-state operator (ρˆ T T N ) from the thermal-tensor-

network method and the exact thermal-state operator (ρˆ exact ) from the full diagonalization method in 10-site chains with various temperatures and anisotropic parameters Δ. In the thermal-tensor-network algorithm, we have set bond dimension of the MPS purification as D0 = 80, and the (imaginary) time slice is Δτ = 0.01.

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Figure 4: (Color online) Thermal-state discord as a function of the anisotropic parameter Δ in the XXZ chains at finite temperatures with (a) T = 0.001 and (b) T = 1. In both figures, from bottom to top, the size of the chains are according to N = 2, 4, 6, ..., 12.

4.1. strength and stability of thermal-state discord

In Fig. 4 (a), we have illustrated the thermal-state discord G(ρˆ T ) as a function of Δ for several chains with T = 0.001. From bottom to top, the curves correspond to chains with N =

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2, 4, 6, ..., 12, respectively. Since the temperature is extremely low, the thermal state ρˆ T (unnormalized) can be regarded as the mixed state for several lowest-lying eigen states, i.e., ρˆ T ≈ e−βE1 |ϕ1 ihϕ1 | + e−βE2 |ϕ2 ihϕ2 |. It is clear that the weight (e−βE2 ) of the first excited state is much

smaller than the weight (e−βE1 ) of the ground state, since E2 > E1 and β = 1/T = 1000. Thereby,

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the effect of the level crossing, which occurs in the excited states, would be rather weak in the thermal-state discord. One can see that thermal-state discord shows a slight cusp-like singular point at Δc = 1 in Fig. 4 (a). On the other hand, as the increase of the temperature, the weight

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e−βE2 of the excited state would become un-negligible, thus the effect of the level crossing would become significant in the thermal-state discord. As shown in Fig. 4 (b), when the temperature is T = 1, the thermal-state discord shows an unambiguous peak at the critical point Δc = 1. It

is quite interesting that at finite temperatures, the thermal-state discord of finite-size chain can shows some footprint of the quantum phase transition of the models. Comparing the results for T = 1 with T = 0.001, one can see that increasing the temperature

tends to weaken the thermal-state discord. In order to illustrate the effect of thermal fluctuations in more detail, we have shown thermal-state discord as a function of the temperature for various Δ in Fig. 5, with (a) N = 8, (b) N = 10 and (c) N = 12. First of all, it is clear that when the 10

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Figure 5: (Color online) Thermal-state discord as a function of the temperature T in the finite-size XXZ chains for (a) N = 8, (b) N = 10, and (c) N = 12 for various Δ.

temperature is high enough, the thermal-state discord would always vanish gradually, regardless of the size N or the anisotropic parameter Δ of the chains. Thus thermal fluctuation would finally destroy all the quantum correlations in the models. In addition, one can see that in a finite lowtemperature regions, the G(T ) curve for Δ = 1 is above other curves, which is consistent with the

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peak point of the G(Δ) curve in Fig. 4.

Next, we analyze the low-temperature behavior of the thermal-state discord. We will take Δ = 0 and Δ = 5 as two typical instances for the gapless phase and the gapped phase, respectively. For any N in Fig. 5, when Δ = 0, one can see that as the increase of the temperature, the thermal-

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state discord decreases rapidly. Similar result can also be observed for Δ = 0.5. This behavior is related to the property of the low-lying spectrum of the models. As we have mentioned, in the large-N limit, the system is gapless for Δ ≤ 1. Thus, when T is slightly larger than zero,

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various excited states can quickly play a role in the thermal state ρˆ T , thus destroy the quantum correlations. One can see that the analysis of thermal-state discord helps us to capture the gapless feature in the energy spectrum of the systems. Then we pay our attention to the gapped regions, i.e., Δ = 5. For N = 8 (see Fig. 5(a)),

we find that the thermal-state discord shows a sharp fall in the low-temperature regions, then keeps a constant approximately with the temperature up to T ≈ 2. It shows that the thermal-state discord is quite robust against thermal fluctuations. In addition, the robustness behavior survives in larger chains (see Fig. 5(b) and (c)). This robustness of thermal-state discord against thermal fluctuation may be valuable in quantum information tasks. The behavior is also determined by 11

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Figure 6: (Color online) (a) Low-lying energy spectrum of the XXZ chains with Δ = 5 and N = 10. i labels the eigen states. Thfig:Spectrumfig:energy gape energies for the ground state and the first excited state are very close to each other, and there is a large energy gap (denoted by ΔE) between the first excited state and the second excited state. (b) Global ( j)

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discord of the truncated thermal states ρˆ T as a function of j, where ρˆ T is an approximation for the thermal state ρˆ T 0 0

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with respect to the j lowest-lying energy states, with T 0 = 1. The red dash line indicates the exact value of G(ρˆ T 0 ). It is clear that a good approximation is obtained even if we just keep the j = 2 lowest-lying states.

the energy spectrum of the models. In Fig. 6(a), we have shown the eigen energies {Ei } of the

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chain with Δ = 5 and N = 10. One can see that the energy for the ground state and that for the first excited state are very close to each other 1 . Moreover, there is a rather large energy gap (denoted by ΔE) between the first excited state and the second excited state. Thereby, in the low-temperature regions, the high-lying excited states always have rather little weight in the

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thermal state ρˆ T . That’s why G(ρˆ T ) is robust in the temperature regions 0.1 . T . 2 in Fig. 5. In order to verify the theory, let us consider the thermal plateau in the vicinity of T 0 = 1 in Fig. 5 (b). Firstly, according to Eq. (6), we will “truncate” the thermal state ρˆ T0 by keeping j

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lowest-lying states as

ρˆ T0 ≈

ρˆ (Tj)0

=

Pj

e−βEi |ϕi ihϕi | , P j −βE i i=1 e

i=1

(11)

ˆ (2) with j = 1, 2, ..., 2N . For instance, ρˆ (1) T 0 is just the ground state, ρ T 0 is the mixed state for the

ground state and the first excited state, and so on. As the increase of j, ρˆ (Tj)0 approaches the exact thermal state ρˆ T0 gradually. We have calculated the corresponding discord for these truncated states ρˆ (Tj)0 and our results are shown in Fig. 6(b). The exact value of thermal-state discord for

ρˆ T0 is illustrated as the red dash line. One can see that a good approximation has already been 1 That

explains the sharp fall of the thermal-state discord in the T ≈ 0 regions for Δ = 5 chains in Fig. 5.

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Figure 7: (Color online) Scaling of thermal-state discord in the finite-size XXZ chains for various temperatures with the anisotropic parameter (a) Δ = 0, (b) Δc = 1 and (c) Δ = 2. It indicates that, for any given finite temperature T and anisotropic parameter Δ, the discord would present a linear scaling in the large-N limit.

obtained even if we just keep the j = 2 lowest-lying states. Therefore, the contribution of the high-lying states is indeed quite small. When the temperature is slightly larger (or smaller) than T 0 , it is expected that the weights for the high-lying states are still ignorable. Consequently, the

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thermal state ρˆ T is not very sensitive to the change of the temperature. That’s why the thermalstate discord presents an approximate plateau in the G(T ) curve for Δ = 5 in Fig. 5. The robustness is determined by the energy gap between the lowest-lying states and the highlying states of the chains. In the gapped regions Δ > 1, increasing Δ will induce an even larger energy gap (see Fig. 2(b)). Consequently, the thermal-state discord would become even more

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robust. On the other hand, according to the G(Δ) curves in Fig. 4, it is expected that increasing Δ will weaken the thermal-state discord. Thereby, the strength and the thermal stability of the

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thermal-state discord seem to be a contradiction in the XXZ models. 4.2. scaling of thermal-state discord Since real materials always contain thousands of atoms, one may be interested in the behavior

of the thermal-state discord in the large-N limit. However, as we have shown, when N is very large, it becomes difficult to calculate the thermal-state discord. Alternatively, we can analyze the

scaling behavior of the discord, which may offer us some valuable information about the large-N limit. In Fig. 7 we have shown the size dependence of the thermal-state discord for various temperatures, with (a) Δ = 0, (b) Δ = 1 and (c) Δ = 2. On one hand, for any Δ, when the temperature 13

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is high enough, we find that the G(N) curve shows a linear scaling. On the other hand, when the

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temperature is low, the G(N) curve shows some kind of even-odd oscillation for small N, i.e., T = 0.3 in Fig. 7(a), and T = 0.7 in Fig. 7(b) and (c). One can see that as the increase of N, the oscillation becomes weak gradually. Thereby, it is expected that when N is large enough, the corresponding G(N) curve will also tend to a linear function. In summary, Fig. 7 indicates limit, i.e., G ∼ k ∙ N + c,

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that, at finite temperatures, the thermal-state discord would present a linear scaling in the large-N

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where k and c depend on the anisotropic parameter Δ and the temperature T of the models. Obviously, the value of G would diverge in the large-N limit. Thus, it may be more convenient to use the “discord density”

k = lim

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G N

(13)

to describe the global correlations in the infinite systems. It is clear that the “discord density” is a well defined quantity, since it is a finite number in the large-N limit.

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4.3. thermal-state discord in long chains in full temperature regions In return, the linear scaling of the thermal-state discord can help us to improve the algorithm to calculate the thermal-state discord G(ρˆ T ) in long chains. As we have shown, there are two major difficulties in calculating discord. The first one is the efficient calculation and storage of

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the density matrices ρˆ T and ρˆ˜ T , which has been solved by the thermal tensor networks. The second one is the calculation of the von Neumann entropy S (ρˆ T ) in Eq. (3). According to Eq. (6), one can see that S (ρˆ T ) is determined by the eigenvalue spectrum {e−βEi } of ρˆ T . On one

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hand, when the temperature is very low, the weights {e−βEi } for high-lying states would be rather

small, which is a good property for numerical truncation. In such a situation, S (ρˆ T ) can be

calculated accurately by the density matrix renormalization group (DMRG) method. A detailed algorithm has been proposed in Ref. [21]. On the other hand, when the temperature is rather high, most of the eigenvalues in the spectrum cannot be ignored. In such a case, it is expected that DMRG would fail in calculating S (ρˆ T ), and consequently, the value of G(ρˆ T ) would be

incorrect. Fortunately, as shown in Fig. 7, in high temperature regions, G(ρˆ T ) shows a rather

good linear scaling. Thereby, we can just figure out G(ρˆ T ) accurately by analyzing short chains. 14

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Figure 8: (Color online) Thermal-state discord as a function of the temperature T in the finite-size XXZ chains for (a) N = 14 and (b) N = 16 for various Δ. The (low-temperature) dot curves are from thermal tensor network calculations, and the (high-temperature) solid lines are from linear scaling analysis. In the intermediate regions, they are quite compatible with each other. It indicates that the results are reliable in the full temperature regions.

To verify the idea, we have figured out the thermal-state discord for chains with N = 14 and N = 16 in Fig. 8. In the high-temperature regions, the results (i.e., the solid lines) are

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obtained by analizing the data from small N, combined with the linear scaling. It is expected that the results tend to be rather accurate for high temperatures. In the low-temperature regions, the results (i.e., the dot curves) are obtained by using the thermal tensor networks, combined with the DMRG algorithm. The results tend to be quite accurate for low temperatures. One can see that in the intermediate temperature regions, the dot curves meet the solid lines, and

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they are quite compatible with each other. Thereby, the obtained results are reliable in the full temperature regions. We believe that the method is valuable in studying other 1D long chains at

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finite temperatures.

Especially, comparing the results for N = 8, 10, 12 in Fig. 5 and the results for N = 14, 16 in

Fig. 8, one can see that the broad thermal plateau for Δ = 5 is preserved, and the value of G(ρˆ T )

is enhanced gradually as the increase of N. Thereby, it is expected that the thermal plateau can be observed in the large N limit.

5. Summary and discussions In previous studies, several tensor networks have been proposed to investigate global quantum discord for low-dimensional quantum lattices [20, 21]. In this paper, we expand the studies 15

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to finite temperatures, and use thermal tensor networks to investigate the global discord of the ˆ

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thermal state ρˆ T = e−βH in 1D finite-size XXZ chains. Especially, the thermal tensor networks are expressed in terms of matrix-product-state purification, thus only rather small tensors are involved in the calculations.

Our first observation is that, in a finite temperature region, the thermal-state discord G(ρˆ T ) of

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finite-size XXZ chains shows some footprint of the quantum phase transition of the (infinite-size)

models. The behavior can be explained as follows. In the large-N limit, the energy spectrum of the model is gapped for Δ > 1 and gapless for Δ < 1, and the phase transition just locates at

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Δc = 1. However, for finite-size chains, the energy spectrum is re-constructed into a quite special pattern. That is, there is always a finite energy gap between the ground state and the excited states for any finite Δ, and the level crossing occurs just in the excited states. The level crossing, which is closely related to the phase transition of the models, cannot be captured by the analysis of the ground states for finite-size chains. On the other hand, the thermal state ρˆ T naturally captures these low-lying energy states and the corresponding level crossing. Thereby, G(ρˆ T ) shows an unambiguous peak at Δc = 1 at finite temperatures.

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Then we have studied the temperature dependence of G(ρˆ T ) in the models. Especially, in the

gapped phase with Δ = 5, we find that G(ρˆ T ) is quite robust against thermodynamic fluctuations.

In fact, it shows a rather broad thermal plateau with the temperature up to T ≈ 2. The plateau

is explained quantitatively by truncating the thermal state ρˆ T0 according to the lowest j energy

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states. This robustness of multi-site quantum correlations may have some applications in quantum information tasks. It needs mention that, increasing Δ tends to enlarge the energy gap thus stabilize the thermal-state discord, meanwhile weaken the strength of the thermal-state discord.

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Strength and thermal stability of the thermal-state discord seem to be a contradiction in the XXZ models.

Furthermore, we have studied the scaling behavior of G(ρˆ T ), and find that G(ρˆ T ) would

present a linear scaling in the large-N limit. Thus, one can use the “discord density”, i.e.,

limN→∞ GN , to describe the thermal-state discord in the infinite systems. We would like to mention that in previous papers, “discord density” has been proposed to describe the global correlations in low-dimensional models at zero temperature [20, 21]. Thereby, our observation in this paper generalizes the conclusion to finite temperatures. In addition, the scaling behavior can help us to improve our algorithm. Combined with the linear scaling behavior and the thermal tensor net16

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work techniques, we have proposed a reliable method to calculate G(ρˆ T ) in the full temperature regions in long chains.

In this paper, we have just considered the XXZ chains. It would be interesting to investigate thermal-state discord in some other low-dimensional spin models. For instance, in an infinitesize spin ladder model with four-spin exchange [31, 32, 33, 34], the ground state undertakes a

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quantum phase crossover, which is related to the “avoided level crossing” between the ground state and the first excited state of the model. However, for finite-size ladders, the “avoided level crossing” is re-constructed into a level crossing between the first excited state and the second

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excited state. Thereby, it is expected that the analysis of the ground state for finite-size ladders cannot lead to the true information about the quantum phase crossover of the infinite-size model. We believe the thermal-state discord and the thermal tensor network would also be helpful to characterize this quantum phase crossover. Furthermore, for other critical phenomena, such as the second-order quantum phase transition in the transverse-field Ising chains, thermal-state discord may also be informative. The Ising models are gapped in the non-critical regions, thus the discord would be quite robust against thermal fluctuations. On the other hand, the Ising

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models are gapless in the vicinity of the critical point, and the discord would be very sensitive when the temperature increases. Thereby, thermal-state discord may be valuable in detecting the critical points of low-dimensional spin models at finite temperatures. It is an interesting question that can we measure global quantum discord in low-dimensional

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quantum materials? There is always a delay between experimental realizations and theoretical studies. Nevertheless, thanks to the experimental physicists, techniques are developing fast in laboratories, and many quantum information quantities can now be measured in experiment.

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For instance, for some typical spin chains, there is an analytical formula between the quantum information measures Ci,i+1 (such as the concurrence and the entanglement of formation) and the

spin correlation functions hSi ∙Si+1 i. Thereby, one can extract hSi ∙Si+1 i with advanced experiment

methods, i.e., neutron diffraction scattering and magnetic susceptibility measurement [35, 36,

37], and finally estimate the quantum information measures. Especially, tripartite entanglement has been observed in the compound Na2 Cu5 Si4 O14 by Souza et al. [36]. In addition, quantum nonlocality (a measure of quantum correlation, indicated by the violation of Bell’s inequality) has been observed above room temperature in the copper carboxylate by the same group [37]. There are also some achievements in quantum discord. Recently, Benedetti et. al. have experimentally 17

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estimated the quantum discord G[i, j] for two polarization qubits. In their work, they have used

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the full tomography technique to reconstruct the density states ρˆ of the system, and explicitly

performed the optimal measurement on ρˆ [38]. Quite recently, Coto et.al. have proposed several methods to estimate global quantum discord [39]. They have associated global discord with an

experimentally measurable quantity, i.e., the degree of excitations of the sub-systems. Another approach is to use the monogamy inequalities obeyed by global discord. Under some conditions

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[24], the following inequality holds for N-qubit systems, G[1,...,N] ≥ G[1,2] + G[2,3] + G[3,4] + ∙ ∙ ∙ +

G[N−1,N] . For solid states, the system is generally translationally invariant, thus one shall simply

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have G[1,...,N] ≥ (N − 1)G[i,i+1] . Thereby, for systems satisfying the conditions in Ref. [24], one

may use the above mentioned experiment [38] to figure out the two-site discord G[i,i+1] , then use

G[i,i+1] to estimate the lower bound of global discord G[1,...,N] . We hope the rapid advancements of both the theoretical understanding and the experimental techniques will make it possible to measure global discord in solid materials in the near future.

We would like to mention that some other measures of multi-site quantum correlations have also attracted much attention, for instance, multipartite quantum nonlocality and multipartite en-

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tanglement [40, 41]. These quantities have been widely used to characterize the ground-state quantum correlations in low-dimensional quantum models. We believe the thermal tensor networks can play a role in studying these multi-site quantum correlations in low-dimensional mod-

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els at finite temperatures.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (Grant

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Nos. 11675124 and 11704295).

Authors contributions

All the authors were involved in the preparation of the manuscript. All the authors have read

and approved the final manuscript.

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