Holographic entanglement entropy and complexity in St$\ddot{u}$ckelberg superconductor

TL;DR

This paper investigates how holographic entanglement entropy and complexity behave across different phase transitions in St"uckelberg superconductors, suggesting they can serve as probes for the nature of these transitions.

Contribution

It analyzes the behavior of holographic entanglement entropy and complexity during first and second order phase transitions in St"uckelberg superconductors, highlighting their potential as diagnostic tools.

Findings

Entanglement entropy and complexity behave differently in first and second order transitions.

They can serve as probes to distinguish the type of superconducting phase transition.

Abstract

The holographic superconductors, as one of the most important application of gauge/gravity duality, promote the study of strongly coupled superconductors via classical general relativity living in one higher dimension. One of the interesting properties in holographic superconductor is the appearance of first and second order phase transitions. Recently, another active studies in holographic framework is the holographic entanglement entropy and complexity evaluated from gravity side. In this note, we study the properties of the holographic entanglement entropy and complexity crossing both first and second order phase transitions in St$\ddot{u}$ckelberg superconductor. We find that they behave differently in two types of phase transitions. We argue that holographic entanglement entropy and complexity conjectured with the volume can also be a possible probe to the type of superconducting phase transition.

Tables

Table 1. Table 1: The critical temperature for the first order phase transition which grows with the coupling parameter.

$c_4$	$0.52$	$0.55$	$0.60$	$0.70$	$0.80$
$T_c$	$0.0365796$	$0.0372203$	$0.0381431$	$0.0403314$	$0.0423508$

Equations

$$S$$

$$L_M$$

$$d{s}^2$$

$$T=-\frac{f^{\prime }(z_H)e^{-\chi (z_H)/2}}{4\pi },$$

$${\psi }^{\prime \prime }-\left(\frac{2}{z}+\frac{{\chi }^{\prime }}{2}-\frac{f^{\prime }}{f}\right){\psi }^{\prime }+\frac{q^2{\varphi }^2{e}^{\chi }}{f^2}\left(\psi +2q^2{c}_4{\psi }^3\right)-\frac{m^2}{z^{2}f}\psi =0,$$

$${\varphi }^{\prime \prime }+\frac{{\chi }^{\prime }}{2}{\varphi }^{\prime }-\frac{q^2\varphi }{z^{2}f}\left({\psi }^2+q^2{c}_4{\psi }^4\right)=0,$$

$${\chi }^{\prime }-\frac{z}{2}\left({\psi }^{\prime 2}+\frac{q^2{\varphi }^2{e}^{\chi }}{f^2}\left({\psi }^2+q^2{c}_4{\psi }^4\right)\right)=0,$$

$$f^{\prime }-\frac{zf}{4}\left({\psi }^{\prime 2}+\frac{12}{z^2}+\frac{z^2{e}^{\chi }}{f}{\varphi }^{\prime 2}\right)-\frac{{\psi }^2}{4z}\left(m^2+\frac{z^2{q}^2{\varphi }^2{e}^{\chi }}{f}\left({\psi }^2+q^2{c}_4{\psi }^4\right)\right)+\frac{3}{z}=0.$$

$$\psi ={\psi }_{1}z+{\psi }_2{z}^2\cdot \cdot \cdot ,\varphi =\mu -\rho z+\cdot \cdot \cdot ,f=1+f_3{z}^3\cdot \cdot \cdot ,\chi ={\chi }_0+{\chi }_{1}z\cdot \cdot \cdot .$$

$$S_E$$

$$\frac{F}{V_2}=f_3$$

$$S_A=\frac{\text{mbox}Area({\gamma }_A)}{4G_N^{(d+2)}}.$$

$$C=\frac{\text{mbox}Volume({\gamma }_A)}{8G_N^{(d+2)}},$$

$$ds_{induced}^{2}=\frac{1}{z^{2}}\Big{[}(\frac{1}{f}+x^{\prime}(z))dz^{2}+dy^{2}\Big{]}~{}.$$

$$4G_{4}S=Area({\gamma }_A)=\int dy{\int }_{-\ell /2}^{\ell /2}\frac{dx}{z^2}\sqrt{1+\frac{1}{f(z)}(\frac{dz}{dx}{)}^2}.$$

$$x(z)={\int }_z^{z_{\ast }}\frac{z^{2}dz}{\sqrt{(z_{\ast }^4-z^4)f(z)}}.$$

$$S=\frac{L}{2G_4}{\int }_{ϵ}^{z_{\ast }}\frac{z_{\ast }^2}{z^2}\frac{dz}{\sqrt{(z_{\ast }^4-z^4)f(z)}}=\frac{L}{2G_4}(\frac{1}{ϵ}+s),$$

$$C=\frac{1}{4\pi G_4}{\int }_{ϵ}^{z_{\ast }}dz\frac{x(z)}{z^3\sqrt{f(z)}},$$

Full text

I Introduction

In the last decades, the AdS/CFT correspondence Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj , has made significant progress and development. In many applications of gauge/gravity duality, AdS and condensed matter physics duality (AdS/CMT) has attracted plenty of interest, which provides a powerful approach to study the strongly coupled field theory in terms of weakly coupled gravitational systems. In particular, physicist has investigated kinds of holographic superconductors which was first proposed in Gubser:2008px ; Hartnoll:2008vx including s-wave model Hartnoll:2008kx , p-wave model Gubser:2008wv and d-wave model Chen:2010mk ; Benini:2010pr . Considerable efforts in holographic superconductors have been made in the related research and discussion, and for reviews, readers can see for example, Horowitz:2010gk ; Cai:2015cya and references therein.

In the study of holographic superconductors and other condensed matter physical models like holographic superfluidity, many study on the types of phase transition analysis have been done. The first proposal on holographic s-wave superconductor in probe limit was found to undergo a second order phase transition from normal stateHartnoll:2008vx . Later, It was addressed in Franco:2009if that the holographic superconductor via the St$\ddot{u}$ckelberg mechanism allows the first order phase transition to occur when the model parameter surpasses a critical value, which was extended in Franco:2009yz ; Ma:2011zze ; Hafshejani:2018svs . First order phase transition in holographic s-wave superconductor has also been observed in superfluidity modelHerzog:2008he , $N=8$ gauged supergravity modelAlbash:2012pd and so on. Besides, in p-wave holographic superconductor models, some works for instance Cai:2013pda ; Cai:2013aca ; Li:2013rhw ; Giordano:2015vsa showed the zero-order, first-order and second-order phase transition in their models, and discovered the retrograde condensation in certain parameter space.

On the other hand, holographic entanglement entropyRyu:2006bv as a measure of the degrees of freedom in a strongly coupled system, has been firstly evaluated in holographic metal/superconductor phase transition in Albash:2012pd . It was addressed that the entanglement entropy in superconducting case is always less than that in the normal phase. Beside the entanglement entropy is continuous but its slope in terms of temperature is discontinuous at the transition temperature $T_{c}$ for the second order phase transition, while for the first order phase transition, the entanglement entropy presents a discontinuous drop at the critical temperature. Thus, the authors of Albash:2012pd argued that the holographic entanglement entropy can be used to determine the orders of superconducting phase transition. Later, the authors of Kuang:2014kha found that the entanglement entropy has a different behavior near the contact interface of the superconducting to normal phase due to the proximity effect. Further effort in using holographic entanglement entropy as a probe of phase transition has been made in Cai:2012nm ; Das:2017gjy ; Peng:2014ira ; Yao:2014fwa ; Yao:2016ils ; Dey:2014voa ; Romero-Bermudez:2015bma ; Peng:2015yaa ; Momeni:2015iea ; Ling:2016wyr ; Ali:2018aon and therein. The authors studied the behaviors of holographic entanglement entropy in different orders phase transition and also in quantum phase transition. They argued that the holographic entanglement entropy may not be universal in holographic superconduction models, which explains the existence of the difference between in first order phase transition and second order phase transition.

In this paper, we will investigate the properties of holographic complexity measured by the volumeAlishahiha:2015rta in St$\ddot{u}$ckelberg holographic superconductor model. The complexity essentially measures the difficulty of turning a quantum state into another and so it could reflect a phase transition on the boundary field theory. Holographic complexity has been studied in one dimensional s-wave superconductor in Momeni:2016ekm ; Zangeneh:2017tub ; Chakraborty:2019vld , p-wave superconductor in Fujita:2018xkl and QCD phase transitionGhodrati:2018hss . It was found that holographic complexity behave in the different way with holographic entanglement entropy and both of them can reflect the one dimensional phase transitions.

Here we are aiming to study the properties of holographic complexity for a strip subregion when the system crosses the first order and second order St$\ddot{u}$ckelberg superconducting phase transition. We will consider only the $\psi^{4}$ coupling term in St$\ddot{u}$ckelberg model, which brings in first order phase transition when the coupling parameter is bigger than a critical value in probe limit case Franco:2009if and in backreaction casePeng:2014ira . We will see that the holographic complexity increases monotonously as the temperature becomes lower for the second order phase transition, while it present a multivalue area in high temperature region in first order case which appears in the same temperature region in free energy diagram. Besides, similar to the holographic entanglement entropy, at the second order phase transition temperature, the complexity is continuous and its slope in terms of the temperature have a jump, while in the first order phase transition case, the complexity presents a jump at the critical temperature. Thus, we are expecting that similar to the holographic entanglement entropy, the holographic complexity can also be a good probe to the order of superconducting phase transition.

Our paper is organized as follows. We briefly show the St$\ddot{u}$ckelberg superconducting phase transition, the condensation and computing the free energy of the dual system in section II. In section III, we first review the holographic setup of entanglement entropy and complexity for a strip subregion in AdS/CFT framework, and then numerically evaluate them in the superconducting model. The last section contributes to our conclusion and discussion.

II St$\ddot{U}$ckelberg Superconducting phase with backreaction

The generalized four dimensional action containing a $U(1)$ gauge field and the scalar field coupled via a generalized St$\ddot{u}$ckelberg Lagrangian is Franco:2009yz

[TABLE]

where $l$ is the AdS radius which will be set to be unity in the following discussion. $L_{M}$ is the generalized St$\ddot{u}$ckelberg Lagrangian

[TABLE]

where $\mathscr{F}(\psi)$ is a function of $\psi$. In this paper, we will consider $\mathscr{F}(\psi)=\psi^{2}+c_{4}\psi^{4}$ where $c_{4}$ is a model parameter111We note that the holographic superconductor with $\mathscr{F}(\psi)=\psi^{2}+c_{6}\psi^{4}$ coupling with backreaction has been studied in Cai:2012es .. Considering the gauge symmetry $\tilde{A}_{\mu}\rightarrow~{}\tilde{A}_{\mu}+\partial\Lambda$ and $p\rightarrow~{}p+\Lambda$, we fix the gauge $p=0$ by using the gauge freedom.

To include the backreaction, we consider the metric ansatz

[TABLE]

The Hawking temperature is expressed as

[TABLE]

and the event horizon $z_{H}$ satisfies $f(z_{H})=0$. Then, considering the ansatz of the matter fields as $\psi=\psi(z)~{},~{}A=\phi(z)dt$, we obtain the equations of motion from the action (1) under the metric (3)

[TABLE]

In our following study, we will take $q=1$ and $m^{2}=-2$ without loss of generality, though our analysis can be extended into other proper parameters. Thus, the behavior of various field in the asymptotical AdS boundary is

[TABLE]

According to AdS/CFT dictionary, $\mu$ and $\rho$ are considered as the chemical potential and charge of the boundary field theory, respectively. $\psi_{1}$ and $\psi_{2}$ can be dual to the source while the other is the vacuum expectation value due to the choice of standard quantization or alternative one. As is pointed out in Hartnoll:2008kx that $\psi_{1}$ as expectation value is always divergent at zero temperature limit, i.e., it may be not physical. So we will choose $\psi_{1}$ as the source and $\psi_{2}$ is related to the expectation value $\langle O_{2}\rangle~{}$. Moreover, near the horizon, the regular condition requires that the Maxwell field satisfies $\phi(r_{H})=0$ and the scalar field is not vanished.

We numerically solve the equations of motion (5)-(8) by setting the source $\psi_{1}=0$. Our numerical results of the condensation are explicitly shown in figure 1 which indicates that the type of phase transition is affected by the values of coupling parameter $c_{4}$. There exists a critical value of $c_{4c}\sim 0.5$ below which the system undergoes a second superconducting phase transition while above which a first order phase transition occurs in the system. Similar phenomena was also found in Franco:2009if .

For the second order phase transition, the critical temperature $T_{c}$ is straightforward to be read off, at which the condensation appears. We find that the critical temperature is around $0.0358912$ which is not explicitly dependent on coupling parameter. This is an obvious result indicating that the physical process is stable when the system has the second-order phase transition. Similar phenomena have also been found in the studiesFranco:2009if ; Peng:2014ira . However, for the first order phase transition, it is not that direct to fix $T_{c}$ because of the multivalue of the condensation. Then, a usual way to fix it is to calculate the free energy of the solutions.

In order to further determine $T_{c}$ for the phase transitions and its order, we will compute the free energy of the system. We will work in the canonical ensemble, in which the charge is fixed. According to Balasubramanian:1999re ; Skenderis:2002wp , the free energy of the boundary field theory is connected with on-shell Euclidean bulk action as $F=-TS_{E}$ and in our model

[TABLE]

where $g_{\infty}$ is the induced metric on the boundary, and $K$ is the trace of extrinsic curvature. Then considering the equations of motion (5)-(8) and the boundary behaviors (9), the free energy density in our model is evaluated by

[TABLE]

where $V_{2}=\int dxdy$ which will be set to be unit and $f_{3}$ is a coefficient in (9). We notice that for the normal state, the free energy is $f_{3}=\mu/4-1$.

The results of $\Delta F=F-F_{RN}$, which is the difference between the free energy of the condensed state and normal state, are shown in figure 2. The blue solid lines are for the superconducting state while the black lines represent the normal state. For $c_{4}=0$ in the left plot, it is explicit that when $T>T_{c}$, the RN solution is physically favorable; when $T=T_{c}$ which is marked by the red dashed vertical line, second order phase transition occurs and the superconducting phase become physically favorable as the temperature further decreases. For $c_{4}=0.6$ in the right plot, the multivalue region of the $\Delta F$ appears which implies that first order phase transition occurs. Though the temperature $T=0.03559$ represents the appearance of condensation function, but physical phase transition happens at the interaction point $T\simeq 0.038143$, below which the superconducting state is thermodynamically favorable.

Thus, from the free energy, we can read off $T_{c}$ for the first order transition for different $c_{4}$ which are listed in table 1. $T_{c}$ for the first order phase transition grows as the coupling parament, which behaves differently with that in the second order phase transition. Comparing the two types of the superconducting phase transition, the second-order phase transitions has lower but stable critical temperatures, so more drastic condensation and phase transition process is hinted in the first order case.

Next, we shall study holographic entanglement entropy and holographic complexity in the two types of phase transitions. We expect that the holographic entropy and complexity would perform different properties in the two types of phase transition, and suggest more deeply physics about the superconducting phase transitions in the dual systems.

III Holographic entanglement entropy and holographic complexity

III.1 The setup

We consider the subsystem $A$ with a straight strip geometry which is described by $-\frac{l}{2}\leq x\leq\frac{l}{2}\,,0\leq y\leq L$(see Fig.3). Here $l$ is the size of region $A$ and $L$ is a regulator which can be set to be infinity. Recalling the proposal by Ryu and TakayanagiRyu:2006bv , the radial minimal extended surface $\gamma_{A}$ bounded by $A$ with a turning point $z_{\star}$ determine the entanglement entropy $S_{A}$ as via

[TABLE]

In holographic framework, there are two different proposals on how to evaluate the complexity. One is the CV conjecture (Complexity=Volume) Stanford:2014jda ; Susskind:2014jwa . Another is the CA conjecture (Complexity=Action) Brown:2015bva ; Brown:2015lvg . CV conjecture is that the holographic complexity is proportional to the volume of a codimension-one hypersurface with the AdS boundary and the Holographic Ryu- Takayanagi surface. While in CV conjecture, one identifies the HC with the gravitational action evaluated on the Wheeler-DeWitt patch in the bulk. Here we will employ CV conjecture and following Alishahiha’s proposalAlishahiha:2015rta , the holographic complexity for this system $A$ is measured by the volume enclosed by $\gamma_{A}$ as

[TABLE]

where $G^{(d+2)}_{N}$ is the Newton constant of the general gravity in AdSd+2 theory.

Following Albash:2012pd , the induced metric of the hypersurface $\gamma_{A}$ who has the same boundary as the stripe does is

[TABLE]

Then, the holographic entanglement entropy connecting with the area of the surface can be evaluated as

[TABLE]

The condition $\left(\frac{dz}{dx}\right)^{2}=f(z)\left(\frac{z_{\ast}^{4}}{z^{4}}-1\right)$ gives the minimal area in (15) where $z_{\ast}$ is the location in $z$ with the smooth extremal surfaceBen-Ami:2016qex . Integrating the condition gives us

[TABLE]

Subsequently, the entanglement entropy (12) is geometrized in terms of AdS/CFT dictionary as

[TABLE]

where $L=\int dy$ and $\epsilon\rightarrow 0$ is the UV cutoff. And according to Alishahiha:2015rta ; Momeni:2016ekm , the holographic complexity (13) is holographically related to the volume in the bulk enclosed by $\gamma_{A}$ as

[TABLE]

where $x(z)$ is expressed in (16).

It is noticed that from the definitions (12) and (13), one may expect that the analytic features of entanglement entropy and complexity should be similar because they are both determined in terms of the analytic features of the extremal geometry. Specially, according to the formulas (17) and (18), it is obvious that the profile of $f(z)$ mainly determines the analytic features, which we will study in detail later. However, the entanglement entropy is usually not enough to describe the rich geometric structure because it grows in a very short time during the thermalization process of a strongly coupled systemSusskind:2014moa . So it is still significant to study the complexity which may help us to enhance and supplement the description of the phase transition in the holographic superconductor model. Next, we will present the behavior of the two physical quantities numerically.

III.2 The results

We shall analyze the properties of the holographic entanglement entropy and complexity of the dual system in different types of superconducting phase transitions.

The numerical results of holographic entanglement entropy are shown in figure 4 where the red dashed lines are for the normal state and the blue lines are for the condensed state. For the second order phase transition with $c_{4}=0$ (left plot), $s$ decreases monotonously as the temperature becomes lower. When the temperature reaches the critical temperature $T_{c}=0.0358912$, the holographic entropy separates into two branches which implies a phase transition. And the values for the superconducting state is always lower than that for the normal state. This is reasonable because the cooper pairs form in the superconducting state which suppress of the degree of freedom of the system. Contrasting to the monotonous behavior of second order phase transition, the entanglement entropy are always non-monotonous for the first order phase transition with $c_{4}=0.6$(right plot), and has a discontinuous drop to the minimal entropy at the critical temperature $T_{c}=0.038143$. Similar phenomena has also been found in the first order of insulator/superconducting phase transitionCai:2012es ; Peng:2014ira . We note that we always expect new degrees of freedom to emerge in new phases, so that the discontinuity or drop of the entropy at transition point may imply non-trivial reorganization of the degrees of freedom in the system.

We turn to study the properties of holographic complexity shown in figure 5. Different from the entanglement entropy, the complexity grows up as the temperature decreases which is similar as that found in Zangeneh:2017tub . For $c_{4}=0$, the holographic complexity in the second order phase transition case increases monotonously as $T$ becomes lower, until $T$ reaches the critical value, then $c$ in superconducting state is always larger than that in normal state. Similarly, in the case of first order phase transition with $c_{4}=0.6$, When the temperature decreases, the numerical holographic complexity smoothly increases firstly and presents multi-values until the transformation point $T=0.03559$. But the real process of complexity has a jump to the maximal complexity at $T_{c}=0.038143$ in the superconducting state.

Thus, we argue that the drop of the entanglement entropy and the jump of the complexity at the critical point may be quite general features for the first order superconducting phase transition, while for second order phase transition, both of them are continue. Thus, both of holographic entanglement entropy and complexity can be a possible probe of phase transition in our superconductor model. This numerical similarity is expected as we have mentioned in last subsection that the features of entanglement entropy and complexity are both determined by the features of extremal geometry.

IV conclusion and discussion

In this paper, we firstly reviewed the holographic superconductor phase transition in the St$\ddot{u}$ckelberg model with $\mathscr{F}(\psi)=\psi^{2}+c_{4}\psi^{4}$. For the value of coupling parameter is less than the critical value, there is the second order phase transition; or else, when the coupling parameter is larger than the critical one, the first order phase transition occurs. According to the analysis of free energy for the system, it is obvious that the system is experiencing more drastic condensation during the first order phase transition. Comparing to the critical temperature of phase transition, for the second order phase transition, the system reflects lower but has more stable critical temperature which is not affected by the coupling strength. For the first order case, the relation between critical temperature and coupling strength is almost linearly increasing. Our results are similar to that found in the previous literaturesFranco:2009if ; Peng:2014ira .

Then we studied the holographic entanglement entropy and holographic complexity for the two types of phase transition. For the second order phase transition, the holographic entanglement entropy monotonously decreases while the complexity increases as the temperature becomes lower. While they have a multivalue area in high temperature region in first order case, which appears in the same temperature region in free energy diagram. One of the multivalue lines is not physical and does not exist in the real system. According to (17) and (18), the behavior of the metric function $f(z)$ takes charge of the multivalue region of holographic entanglement entropy and complexity. We show the profiles of $f(z)$ for samples of temperatures in both cases in figure 6. It is obvious that in the first order case, the metric function $f(z)$ has unusual behaviors in low temperature like the case in Albash:2012pd .

Both holographic entanglement entropy and complexity behaves differently crossing the second order and first order superconducting phase transition point. At the second order phase transition point, they are continuous and the slopes in terms of the temperature have a jump. While in the first order superconducting phase transition, there exists the drop for the entanglement entropy and the jump for the complexity at the critical temperature. Thus, we argued that both of them can be used as a probe to the order of phase transition in our holographic superconducting model. The discontinued behaviour of the slope around the transition temperature means a reorganization of the degrees of freedom in the system. This is a characteristic of second order phase transition since the entanglement entropy encodes the freedom degrees of the system. However, as we see that the situation is more complicated in first order phase transition. The existence of discontinuity in the behavior possibly indicates that not only the entangled pairs contributes to the entanglement entropy, but extra degrees of freedom turning from other entanglement are also reorganized. This extra pseudo-degree of freedom may come from the quantum effect, for instance, the thermodynamic system is excited to higher energy levels and certain symmetry is broken, released in the strongly coupled systemAlbash:2010mv . Thus, the new scales may be detectable by holographic entanglement entropy and complexity, but the physical origins and deep insight from the field theory side deserve further study. We note that the complexity is deeply connected with fidelity susceptibility, which is known to be able to probe phase transitionFlory:2017ftd ; Gan:2017qkz ; MIyaji:2015mia ; Moosa:2018mik . It would be very interesting to pursuit the deep physics of difference in the probes by studying the fidelity susceptibility in holographic superconducting models.

In this paper, we mainly focused on the difference of the qualitative behavior of holographic complexity for different orders of superconducting phase transitions. Very recently, in the paper Yang:201902 , the authors studied the scaling of complexity on the temperature effected by the superconductor model parameters, as well as the time dependent complexity when the system undergoes second order phase transition. Study on these phenomena in superconducting models with different types of phase transition so as to further understand the deep physics in holographic superconductor should be another interesting direction. We note that the time evolution of complexity under a quench in normal state have also been studied in Moosa:2017yvt ; Chen:2018mcc ; Chapman:2018dem ; Ling:2018xpc ; Fan:2018xwf ; Ageev:2019fxn .

It is noticed that here we computed the holographic complexity via complexity$=$volume conjecture and resolved its UV divergence tactfully. As is known that another independent proposal to evaluate the complexity is complexity$=$action conjecture Brown:2015bva ; Brown:2015lvg ; HosseiniMansoori:2018gdu . Then an important question involved is whether the physical features we found depends on the chosen conjecture. Especially, it was pointed out in Carmi:2016wjl ; Moosa:2017yiz that new UV divergences may arise in complexity$=$action conjecture. We hope to study the computation via complexity$=$action conjecture and do the comparison between the results from the two conjectures in the superconducting model in the near future.

The holographic entanglement and complexity are determined by the near horizon geometry. Thus, another possible direction is to analytical study their near horizon behavior to judge the order of phase. For example, one can study the holographic entanglement and complexity over the St$\ddot{u}$ckelberg superconductor at zero temperature limit (zero temperature limit of minimal coupling holographic superconductor can be seen in Horowitz:2009ij ) which may provide more insights to judge the order of superconduction phase transition.

Acknowledgements.

We appreciate Mahdis Ghodrati and Yan Peng for helpful correspondence. This work is supported by the Natural Science Foundation of China under Grants No. 11705161 and No.11835009. X. M. Kuang is also supported by Natural Science Foundation of Jiangsu Province under Grant No.BK20170481.