Quantum Fisher information in the XXZ model with Dzyaloshinskii – Moriya interaction

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Quantum Fisher information in the XXZ model with Dzyaloshinskii – Moriya interaction

X.M. Liu

^a,b,n

, Z.Z. Du

^a

, W.W. Cheng

^c

, J.-M. Liu

^a

aLaboratory of Solid State Microstructure and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

bInstitute of Mathematical and Physical Sciences, Jiangsu University of Science and Technology, Zhenjiang 212003, China

cInstitute of Signal Processing&Transmission, Nanjing University of Posts and Telecommunication, Nanjing 210003, China

a r t i c l e i n f o

Article history:

Received 15 March 2015 Accepted 15 April 2015 by A.H. MacDonald

Available online 25 April 2015 Keywords:

A. Quantum spin model D. Quantum Fisher information D. Quantum phase transition E. Quantum renormalization-group

a b s t r a c t

We have studied the quantum Fisher information of theXXZspin chain model with Dzyaloshinskii–

Moriya interaction, using the quantum renormalization-group method. The results show that the evolution behavior of quantum Fisher information with increasing lattice size can clearly illustrate the quantum transitions of this spin model. Moreover, it is demonstrated that the scaling property of quantum Fisher information at the critical point can be used to explain the spin correlation length of this model. The present work evidences the capability of the quantum Fisher information in characterizing quantum phase transitions in condensed matters.

1. Introduction

In recent years, quantum Fisher information (QFI) [1,2] as a fundamental notion of quantum metrology [3,4] has attracted much attention due to its importance in quantum estimation and quantum information theory [5,6]. The QFI is an extension of Fisher information (FI) that was originally introduced by Fisher[7].

For Mach–Zehnder (or Ramsey) interferometer, the FI charac- terizes the sensitivity of a state with respect to perturbation of the parameters. However, the precision of parameter estimation is bounded by the quantum Cramer–Rao inequality[5,6,8]in which the quantum Fisher information (QFI) is given. From the respective metrology, it can be derived by maximizing FI over all possible positive operator valued measurement (POVM). QFI has been extensively studied in refs. [9–22]. People mainly focus on how the properties of input states, such as coherence, entanglement and spin squeezing [9–18], affect the QFI, with intent to obtain high precision. In some work[19–22], people have also investigated QFI of various quantum systems in decoherence channels, noisy quantum systems and dissipative quantum systems, explor- ing how to obtain an optimal quantum measurement. Recently, the correlation of QFI and quantum phase transition becomes an interesting theme. In the ref.[23], it is proposed that quantum phase transition can be detected by coupling the quantum-critical system to the external spin system and observing QFI of the whole

system. In our earlier work[24], we have studied QFI of several one-dimension systems, conﬁrming the evolution of QFI as a good signature of localization transition. It is believed that the applica- tion of QFI in the spin model will be valuable.

One-dimensional spin models are typical representatives of many-body systems in condensed-matter physics and also a good playground for implementation of many quantum information protocols[25,26]. For example, theXYspin model and theXXZspin model have been extensively investigated and applied in studying quantum entanglement [27,28], quantum discord [29–31], and so on. By a comprehensive investigation of these models, we can understand and analyze more complicated systems. Recently, some spin models can be supplemented with a magnetic term which is called the Dzyaloshinskii–Moriya (DM) interaction [32,33]arising from the spin–orbit coupling. The DM interaction can drive the quantumﬂuctuations, resulting in more rich phase transition phenomena in models which are interested.

On the other hand, it is known that quantum renormalization group (QRG) method[34,35]is highly powerful for studying the properties of many-body systems. It is particularly valuable in detecting the non-analytic behavior of related quantities in the vicinity of critical points[36–39]. Therefore, it would be interested to apply this method to address the behavior of the QFI in various quantum spin models. In this work, we take theXXZ spin model with DM interaction as an example and use the QRG method to investigate the QFI behavior. It will be shown that the DM interaction enriches the phase diagram and drives the original single critical point into a critical line[40]. The calculation of the QFI shows that the phase transition point will move as the DM interaction increases. We will also see that the derivative of the Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/ssc

Solid State Communications

nCorresponding author at: Laboratory of Solid State Microstructure and Innova- tion Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China.

E-mail address:[email protected](X.M. Liu).

Solid State Communications 213-214 (2015) 24–27

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QFI becomes singular around the transition point with increasing system size and the nonanalytic behavior at the critical point connects with the correlation length.

The paper is organized as follows. InSection 2, we review the basic concepts related to the QFI. Then inSection 3, the QRG is introduced to the XXZ model with DM interaction and the renormalization relation of coupling constants is given. According to the reduced density matrix of the ground states, it is easy to obtain the QFI of two sites in the spin chain. In Section 4, we analyze the behavior of the QFI, demonstrating its capability in detecting the quantum transition and critical behavior. Finally, we conclude with a summary and discussion.

2. Quantum Fisher information

In this section, we introduce the QFI[20,24]and give correlation measure deﬁned by QFI. Suppose, in the process of parameter estimation, a quantum state

ρ

^{evolves as}

ρ

_θ^¼^Uθ

ρ

^U^†_θ ^where

U_θ¼expi

θ

^J!_n

. Here J

!n is the angular momentum in the normalized!n

direction and

θ

is the parameter to be estimated.

Consequently, for the quantum state

ρ

_θ, quantum Fisher information is given by

Fð

ρ

;J

!Þ ¼n X

iaj

2ðp_ip_jÞ² p_iþp_j Ji

!n j

²¼!n

Γ

^!ⁿ^T ^ð1Þ

in which p_i and i are the eigenvalue and eigenvector of

ρ

^,

respectively and!n is the normalized direction vector[20,24]. In addition, the symmetric matrix

Γ

is a critical component and its matrix elements is

Γ

kl¼X

iaj

ðp_ip_jÞ² piþpj

½ Ji _kj Jj _l þi Ji _lj Jj _ki : ð2Þ If

ρ

is a pure state,

Γ

^klcan be simpliﬁed and the QFI is proportional to the variance ofJ

!n, having a form asFð

ρ

;J^

!Þ ¼n 4ð

Δ

^{^}^J!Þ_n ². Generally, high precision results from the large QFI. It is known that the QFI is dependent on the input states and the rotated direction. To improve the estimation precision, it is advisable to ﬁnd proper input states for a given J

!n or to choose a suitable direction!n

for a given input state

ρ

. We take the mixed state as an example and are concerned with the latter. Once given the input state

ρ

^{, we can}ﬁnd the direction!n that corresponds to the maximal QFI by diagonalizing the matrix

Γ

^[20,24]:

Γ

^d^¼^C^T

Γ

^C^¼^diagðE1;E2;E3Þ: ð3Þ in which C is a unitary matrix. If we set E1ZE2ZE3, it can be found thatE1is the maximal Fisher information.

On the other hand, similarly with the skew information[41], we employ the difference between quantum Fisher information of the two-qubit quantum state

ρ

ij and that of the

ρ

i

ρ

j with respect to the local observable of subsystemHi as a correlation measure for the two-qubit state

ρ

ij:

F⁰¼X

u

Fð

ρ

ij;Au1jÞX

u

Fð

ρ

i

ρ

j;Au1jÞ; Fð

ρ

;OÞ ¼X

m;n

ðp_mp_nÞ²

2ðpmþpnÞjh jO nm j ij² ð4Þ wherep_mandp_nare the eigenvalues of

ρ

^and^Ois an observable on a system Hilbert space. Here, for any bipartitekkdimensional state

ρ

ij, and any local orthonormal observable bases Au and Bu, QFI can be evaluated as

F¼X

u

Fð

ρ

ij;Au1_jþ1i BuÞ ð5Þ

In the work, for a two-qubit state, we can take the local orthonormal observable as follows:

Au

f g ¼f g ¼Bu 1 ffiffiffi2 p ;

σ

¹_ffiffiffi

p2;

σ

²_ffiffiffi

p2;

σ

³_ffiffiffi

p2

ð6Þ where

σ

jðj¼1;2;3Þare the Pauli matrices. Thus, we can derive the correlation measureF⁰.

3. Renormalization of the Hamiltonian and calculation of QFI The QRG scheme in real space is started by decomposing lattice sites into isolated blocks. The Hamiltonian of each block is diagonalized exactly and some of the low-lying states are kept to construct the basis for the renormalized Hilbert space. Finally, the full Hamiltonian is projected onto the renormalized space to obtain an effective Hamiltonian. In this work, we introduce the notion of“renormalization of QFI”, and we want to check whether this notion can truly capture the nonanalytic behavior close to the critical point by the derivative of QFI.

Firstly, we apply the concept of QRG to the one-dimensional anisotropicXXZmodel with DM interaction[38]. The Hamiltonian of spin 1/2XXZmodel with DM interaction in thezdirection on a periodic chain ofLsites is

HðJ;

Δ

^{Þ ¼}₄^J^X^L

i

½ð

σ

^xi

σ

^xiþ1þ

σ

^yi

σ

^yiþ1þ

Δσ

^zi

σ

^ziþ1Þ

þDð

σ

^xi

σ

^yiþ1

σ

^yi

σ

^xiþ1Þ; ð7Þ whereJis the exchange interaction strength,

Δ

is the anisotropy parameter, D is the DM interaction factor, and J;D;

Δ

40.

σ

^αi ð

α

^¼^x;y;zÞare the Pauli matrices at sitei.

To obtain a renormalized form for Hamiltonian(8), we divide the spin chain to many blocks and each block is composed of three sites. It is mentioned that this is needful because it can guarantee self-similarity after each iterative step. In reference [38], the degenerate ground states of the block Hamiltonian are given as follows:

φ

0 ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qðqþ

Δ

^Þð1þD²^Þ

q ½2ðD²þ1Þj↓↓↑i ð1iDÞð

Δ

^þqÞj↓↑↓i

2ð2iDþD²1Þj↑↓↓i

φ

⁰0 ¼ 1 ffiffiffi2

p qðqþ

Δ

^Þð1^þD²^Þ^½2ðD

2þ1Þj↓↑↑i ð1iDÞð

Δ

^þqÞj↑↓↑i

2½2iDþD²1j↑↑↓i; ð8Þ

where q¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Δ

²^þ8ð1^þD²^Þ

q

and j i↑;j i↓ are the eigenstates of

σ

^z^.

Following the standard procedure of the QRG method, the effective Hamiltonian can be written as

H^eff¼J⁰ 4

X^L⁼³

i

½

σ

^xi

σ

^xiþ1þ

σ

^yi

σ

^yiþ1þ

Δ

⁰

σ

^zi

σ

^ziþ1

þD⁰ð

σ

^xi

σ

^yiþ1

σ

^yi

σ

^xiþ1Þ ð9Þ where the iterative relationship has the form

J⁰¼J 2 q

2

ð1þD²Þ;

Δ

⁰^¼

Δ

1þD²

Δ

^þq

4

2

; D⁰¼D: ð10Þ

In calculating the QFI and other correlation quantities, we can choose one of the degenerate ground states as example. Here, we may as well take

φ

0 as reference. Accordingly, the block density matrix is given as

X.M. Liu et al. / Solid State Communications 213-214 (2015) 24–27 25

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ρ

123¼

φ

0

φ

0: ð11Þ Since we only consider the QFI of two sites, without loss of generality, we trace out site 2 to achieve the reduced density matrix between sites 1 and 3:

ρ

13¼

Δ

^þq

2q 0 0 0

0 2ðD²þ1Þ qðqþ

Δ

^Þ

2ð2iDþD²1Þ qðqþ

Δ

^Þ ⁰

0 2ð2iDþD²1Þ qðqþ

Δ

^Þ ^2ðD

2þ1Þ

qðqþ

Δ

^Þ ⁰

0 0 0 0

0 BB BB BB BB BB

@

1 CC CC CC CC CC A

: ð12Þ

According to Section 2, the QFI of the quantum state

ρ

13 is obtained as follows:

F¼2nðABÞÞ²=ðAþBÞ=ðD²þ1Þ

þ2nD²_nA=ðD²þ1Þþ2_nB=ðD²þ1Þ: ð13Þ where A¼ ð

Δ

^þqÞ=2q;B¼ ðD²þ1Þ=ð

Δ

^þqÞ=q. From the analytic results, it is shown that the QFI is determined by the anisotropy parameter and the DM interaction. In the following section, we will give detailed analysis about the evolution of QFI, to justify that the QFI is a good quantity in predicting quantum transition and critical phenomenon.

4. Computational results on QFI

The calculations show that the QFI is inﬂuenced by the anisotropic parameter

Δ

and DM interaction factorDwith increasing system size.

Obviously, the zeroth renormalization-group (RG) iteration represents a three-site model. However, theﬁrst RG iteration stands for a nine- site model which effectively describes a three-site model in the cost of renormalization coupling constants. In this case, the QFI measures the information of effective degree of freedom, i.e. two parts of the system.

In each RG iteration, we can see the variation in the QFI as a function of anisotropy parameter

Δ

^{with a}ﬁxed DM interaction factorD. The corresponding results are shown inFig. 1. It is found that the QFI is relatively small for

Δ

less than the critical value, beyond which a rapid increase of the QFI is seen. Here the critical anisotropy parameter is decided by equation

Δ

^c^¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þD²

p , which suggests that this

Δ

^c ^is

remarkablyD-dependent and the role of the DM interaction is clearly justiﬁed.

InFig. 2, we give the evolution of QFI with varyingDwhile

Δ

^is

ﬁxed at

Δ

^¼^p^ffiffiffi². In this case, the QFI has different behaviors in different parameter regions. The QFI is relatively big forDless than the critical value, while QFI is relatively small for DM interaction more than the critical value. From Fig. 1, we can find that the renormalizing behavior of QFI with the anisotropy parameter is different from that with DM interaction in Fig. 2. However,

combining the results of Fig. 1 with Fig. 2, it proves that the anisotropy effects and DM interaction are complementary.

However, from Figs. 1 and 2, it is found that the evolution behavior of QFI with the system size is not in accordance with that of other correlation quantities, for example entanglement entropy and discord[38,39]. This confuses us. Therefore, we also calculate and derive the correlation quantityF' which is defined according to QFI. The corresponding results are given in Fig. 4. From the graph, we canfind that the behavior of F' with respect to the lattice size is similar to that of entanglement entropy. This further proves the rationality of QFI as a method characterizing the quantum transition. In order to obtain the critical behavior of the spin system, thefirst derivative of QFI with respect to the DM interaction is analyzed. It is known that different RG steps manifest the size of the system. FromFig. 3, it is found that the first derivative of QFI with respect toDshows a singular behavior at the critical point. Moreover, the singular behavior of QFI becomes more and more prominent as the number of QRG iterations increases. InFig. 5, a more detailed analysis shows that the positions of the minimum ofdF=dDtends towards the critical point asDmin¼DcN⁰^:⁴⁷. Meanwhile, we have also obtained the scaling behavior ofdF=dD_D

minwith respect toNand the exponent for the scaling behavior is dF=dD

DminN^0:47 which has been plotted inFig. 6. It is more important that this exponent is related to the correlation length exponent

ν

close to the critical point.

On the other hand, based on the divergence of derivative of QFI and the scaling law, we have plotted ðdF=dDdF=dDD_min =N¹⁼^ν versusN¹⁼^νðDDminÞinFig. 7. Through this scaling method, we can ﬁnd that all graphs at iterations approximately collapse into a single curve. This is a manifestation of the existence ofﬁnite-size scaling for the quantum Fisher information. To this stage, one is in a good position to claim that the QFI is indeed a good description of the critical behavior of theXXZmodel with the DM interaction.

Fig. 1.(color online) The evolution of QFI in terms of RG iterations at aﬁxed value of DM interactionD¼1.

Fig. 2.(color online) The evolution of QFI in terms of RG iterations at afixed value of anisotropy parameterΔ¼ ffiffiffi

p2 .

Fig. 3.(color online) First derivative of QFI anddF=dDbecomes divergent as the number of the system increases.

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5. Conclusion

To summarize, we have investigated the XXZ model with DM interaction combining the quantum Fisher information and the quantum renormalization-group method. The evolution of QFI in RG

iterations gives how the properties of the system develop from a finite-size system to its thermodynamic counterpart. In other words, there exist somefinite-size scaling. Moreover, all the results shows that the QFI is relatively small for DM interaction more than the critical value while it is relatively big for DM interaction less than the critical value. Meanwhile, QFI is relatively small for the anisotropy parameter less than the critical value and it is relatively big for that more than the critical value. Through the first-order derivative of QFI, the scaling behavior is presented in detail. The results show that in the thermodynamic limit, the nonanalytic behavior of QFI is correlated with the divergence of the correlation length at the critical point. Thus, we can obtain the critical exponent of the system. In addition, we are convinced that the whole analysis in this paper can be extended to many other spin models. Finally, it has shown that the QFI is a universal method to phase transition and critical properties in one-dimension spin models.

Acknowledgments

This work was supported by the National 973 Projects of China (Grant no. 2011CB922101), the Natural Science Foundation of China (Grant nos. 11234005, 11374147), and the Priority Academic Pro- gram Development of Jiangsu Higher Education Institutions, China.

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Fig. 4.(color online) The evolution ofF⁰in terms of RG iterations at aﬁxed value of DM interactionD¼1.

Fig. 5.(color online) The scaling behavior ofDminwith respect to the system size (Dminis the position of minimum in Fig.3).

Fig. 6.(color online) The logarithm of the absolute value of minimum lnðdF=dD_D

minÞversus the logarithm of chain size lnðNÞ. The RG procedure shows the minimum diverges asdF=dD_D

minN^0:47.

Fig. 7.(color online) Theﬁnite-size scaling through RG treatment. Each curve corresponds to a different size of the model. HereN¼3^ðnþ1Þin whichnrepresents thenth step RG.