Global quantum discord and quantum phase transition in XY model
aa r X i v : . [ qu a n t - ph ] M a y Global quantum discord and quantum phase transition in XY model
Si-Yuan Liu,
1, 2
Yu-Ran Zhang, Wen-Li Yang, ∗ and Heng Fan
2, 3, † Institute of Modern Physics, Northwest University, Xian 710069, P. R. China Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China Collaborative Innovation Center of Quantum Matter, Beijing, P. R. China (Dated: September 12, 2018)We study the relationship between the behavior of global quantum correlations and quantum phase transitionsin XY model. We find that the two kinds of phase transitions in the studied model can be characterized by thefeatures of global quantum discord (GQD) and the corresponding quantum correlations. We demonstrate that themaximum of the sum of all the nearest neighbor bipartite GQDs is e ff ective and accurate for signaling the Isingquantum phase transition, in contrast, the sudden change of GQD is very suitable for characterizing anotherphase transition in the XY model. This may shed lights on the study of properties of quantum correlations indi ff erent quantum phases. PACS numbers: 03.67.Mn, 03.65.Ud
I. INTRODUCTION
The recent development in quantum information theory [1]has provided much insight into quantum phase transitions [2].In particular, using the quantum correlations to investigatequantum phase transitions has drawn much attention and hasbeen successful in characterizing a number of critical phe-nomena of great interest. For example, entanglements mea-sured by concurrence, negativity, geometric entanglement,von Neumann entropy, mutual information and quantum dis-cord are studied in several critical systems [3–9]. From theprevious literature, we know that the concurrence shows amaximum at the critical points of the transverse field Isingmodel and XY model [4], and the von Neumann entropy di-verges logarithmically at the critical point [3]. The quantumcritical phenomena in the XY model can also be characterizedby the divergence of the concurrence derivative or the neg-ativity derivative with respect to the external field parameter[5, 6]. Furthermore, recent studies show that entanglementspectra can be used to describing quantum phase transitions[10, 11]. The structure of the correlations is shown to be re-lated with the quantum critical phenomena [15, 16]. On theother hand, fidelity and the fidelity susceptibility of the groundstate can also be used as a good tool for detecting numerousphase transition points in some critical systems [12–14]. No-tably, the methods from quantum information may also play akey role for the topology of many-body system and the phasetransition of only one spin [17–19].Additionally, the quantum discord can be used to describ-ing the quantum phase transitions in some critical systems[20–23]. It is worth noting that some of the investiga-tions performed so far have indicated that quantum discordis more sensible than entanglement in revealing quantum crit-ical points [24], even for systems that are not at zero tem-perature [25]. As we all know, there are several multipartite ∗ Electronic address: [email protected] † Electronic address: [email protected] promotions of quantum discord [26–31]. The global quantumdiscord (GQD) proposed by Rulli and Sarandy is an widelyaccepted one [31], which can be seen as a generalization ofsymmetric bipartite quantum discord. There is an interestingquestion that if the GQD can be used as a good tool to char-acterizing the quantum phase transitions in the typical criticalsystems. On the contrary, there is another interesting questionthat if our understanding of the quantum phase transitions cantell us some useful information of the behavior of GQD in thecritical systems. Fortunately, the answers of these questionsare yes.In this paper, we investigate the relationship between thebehavior of global quantum discord and quantum phase tran-sitions in the XY model [32, 33]. Considering the local con-vertibility, the phase diagram of XY model can be dividedinto three phases, which we label phase 1A, phase 1B andphase 2 [32]. The are two kinds of quantum phase transi-tions in XY model, the phase transition between phase 1Aand phase 1B and the second order phase transition betweenphase 1 and phase 2. In order to provide a good descriptionof these quantum phase transitions, we analysis the behaviorof total global quantum discord (GQD), the sum of all the bi-partite GQDs and the residual GQD [34]. We will show thatthe sum of all the nearest neighbor bipartite GQDs is e ff ectiveand accurate for signaling the Ising quantum phase transitionsbetween phase 1 and phase 2. Moreover, it is worth notingthat the sudden change of GQD is very suitable for character-izing the phase transitions between phase 1A and phase 1B.On the other hand, since GQD can be seen as a kind of phys-ical resource for quantum information processing, the natureof these quantum phase transitions can tell us some useful fea-tures of it, such as the maximum points and sudden changes.The paper is organized as follows: In Sec. II, we intro-duce the XY model and the definitions of GQD, the sum of allthe nearest neighbor bipartite GQDs and the residual GQD. InSec. III, we study the behavior of these quantum correlationsand provide some figures of the three kinds of correlations andthe corresponding derivative, it shows that these correlationscan be used to characterizing the quantum phase transitionse ff ectively and accurately. At the same time, using the nature FIG. 1: Phase diagram of the XY model. Three phases are labelledas phase 1A, phase 1B and phase 2. For di ff erent values of h in ourmethod, the blue circles represent the critical points for phase transi-tion between phase 1B and phase 1A, and green circles represent thecritical points for phase transition between phase 1A and phase 2. of quantum phase transitions, we can get some useful featuresof the behavior of global quantum discord. In Sec. IV, we givesome conclusions and discussions. II. THE BRIEF INTRODUCTION OF THE XY MODELAND GLOBAL QUANTUM DISCORD
We study quantum phase transitions in one dimensionalXY-model [35] by using the method from quantum informa-tion theory. The Hamiltonian for our model is as follows: H = − N − X i = (cid:26) J + γ ) ˆ σ xi ˆ σ xi + + (1 − γ ) ˆ σ yi ˆ σ yi + ] + h ˆ σ zi (cid:27) , (1)with N being the number of spins in the chain, ˆ σ mi the i th spinPauli operator in the direction m = x , y , z and periodic bound-ary conditions assumed. The XX model and transverse fieldIsing model thus correspond to the the special cases for thisgeneral class of models. For the case that γ →
0, our modelreduces to XX model. When γ =
1, the model reduces totransverse field Ising model. For simplify, here we take J = h is associated with the external transversemagnetic field. For h =
1, a second-order quantum phasetransition takes place for any 0 ≤ γ ≤
1. In fact, there ex-ists additional structure of interest in phase space beyond thebreaking of phase flip symmetry at h =
1. It’s worth notingthat there exists a circle, h + γ =
1, on which the groundstate is fully separable. According to the previous literature,this circle can be seen as a boundary between two di ff eringphases which are characterized by the presence and absenceof parallel entanglement [36–39]. In fact, for each fixed γ , thesystem is only locally non-convertible when h + γ >
1. Nowwe can divide the system into three separate phases, phase 1A,phase 1B and phase 2, where the ferromagnetic region is nowdivided into two phases defined by their di ff erential local con-vertibility. These results are summarized in “phase-diagram”Fig.1. Consideration of di ff erential local convertibility sepa-rates the XY model in three phases, which we label phase 1A, phase 1B and phase 2. The phase transition from phase 1 tophase 2 is the second-order phase transition. The green pointsaround the critical line are the critical points h c obtained byour method as N → ∞ . Phase 1 has two distinct regions Aand B, the boundary is a quarter of a circle, h + γ =
1. Thephase transition from phase 1A to phase 1B can be seen asa first-order phase transition when we consider the GQD asa kind of order parameter. The blue points on the boundaryare the critical points obtained by our method. It is obviousthat our method is very accuracy for characterizing the phasetransitions.For understanding the relationship between the globalquantum correlations and quantum phase transitions in XY-model, we study the behavior of global quantum discord care-fully. The global quantum discord is a measure of multipartitequantum correlation, which can be seen as a symmetric gen-eralization of bipartite quantum discord to multipartite cases.As a well-defined multipartite quantum correlation, the GQDis always non-negative and symmetric with respect to subsys-tem exchange. Considering its applications, GQD has beenshown to be useful in many areas, such as quantum commu-nication and quantum phase transitions [31, 40–43]. In de-tail, GQD can play a role in quantum communication, in thesense that its absence means that the quantum state simplydescribes a classical probability multi distribution. That is tosay, it allows for local broadcasting of correlations [41]. Onthe other hand, the global discord has been proved to be use-ful in the characterization of quantum phase transitions. Inprevious literature, the behavior of GQD in some typical crit-ical systems, such as the Ashkin-Teller spin chain, transversefield Ising model, open-boundary XX model and cluster-Isingmodel have been studied [42, 43]. It is worth noting that bothof the transverse field Ising model and XX model can be seenas a special case of XY model, the nature of quantum phasetransitions is more complicated in this case. So there are twointeresting questions: (1) How the behavior of GQD is like inthis model; (2) If the global discord can be used to describ-ing the complicated and richness critical phenomenon in thismodel. Fortunately, the answers are yes.In order to study the behavior of global discord in ourmodel, we first review the definition of it. The definition ofglobal quantum discord is a generalization of bipartite sym-metric quantum discord. Consider a N -partite system A , A ,... , A N (each of them is of finite dimension), the GQD of state ρ A A ··· A N is defined as follows: D ( A : · · · : A N ) ≡ min Φ (cid:2) I (cid:0) ρ A ··· A N (cid:1) − I (cid:0) Φ (cid:0) ρ A ··· A N (cid:1)(cid:1)(cid:3) , (2)where Φ (cid:0) ρ A ··· A N (cid:1) = P k Π k ρ A ··· A N Π k , with { Π k = Π j A ⊗ · · · ⊗ Π j N A N } representing a set of local measurements and k denotingthe index string ( j · · · j N ). In Eq. (2), the multipartite mutualinformation I (cid:0) ρ A ··· A N (cid:1) and I (cid:0) Φ (cid:0) ρ A ··· A N (cid:1)(cid:1) are given by I (cid:0) ρ A ··· A N (cid:1) = N X k = S (cid:0) ρ A k (cid:1) − S (cid:0) ρ A ··· A N (cid:1) , (3) I (cid:0) Φ (cid:0) ρ A ··· A N (cid:1)(cid:1) = N X k = S (cid:0) Φ (cid:0) ρ A k (cid:1)(cid:1) − S (cid:0) Φ (cid:0) ρ A ··· A N (cid:1)(cid:1) , (4) h h h h h h h h h h h h h h h FIG. 2: Total GQD, the sum of all nearest neighbor bipartite GQDs and the residual GQD against h , when γ = sin θ , ( θ = ◦ , ◦ , ◦ , ◦ , ◦ ).The figures in each column correspond to a specific angle: from first column to last column, the θ increases from 15 ◦ to 75 ◦ . The first line isabout GQD, the second line shows the sum of all nearest neighbor bipartite GQDs, the last line is corresponding to the residual GQD. In eachfigure we study rings with L = , · · · ,
10, from bottom to top. where Φ (cid:0) ρ A k (cid:1) = P k ′ Π k ′ A k ρ A k Π k ′ A k . Based on the definition ofGQD, we can define two corresponding multipartite correla-tions [34]. The sum of GQDs between two nearest neighborparticles is defined as D ( A : A ) + D ( A : A ) + · · · + D ( A L − : A L ) , (5) L is the size of our system. This correlation represents allnearest neighbor bipartite GQDs contained in the critical sys-tem. Then, similar to the definition of tangle as a measure ofresidual multipartite entanglement, we can define the residualGQD corresponding to the second monogamy relation, D LR ≡ D ( A : · · · : A L ) − L − X i = D ( A i : A i + ) . (6)It is a measure for residual multipartite quantum correlation,namely, contributions to quantum correlation beyond pairwiseGQD. This measure of residual multipartite quantum corre-lation describes the total quantum correlation except for allnearest neighbor interaction of quantum correlations. In mostcases, since the bipartite GQDs do not increase under thediscard of subsystems, the second monogamy relation holds.That is to say, the residual GQD is greater than or equal tozero. In other words, the system contains non-zero long-rangecorrelation.In order to calculate the global quantum correlations men- tioned above, we first reformulate GQD as [43]: D ( A : · · · : A L ) = min { ˆ Π k } L X j = X l = ˜ ρ llj log ˜ ρ llj − L − X k = ˜ ρ kkT log ˜ ρ kkT + L X j = S ( ρ j ) − S ( ρ T ) (7)with ˜ ρ kkT = h k | ˆ R † ρ T ˆ R | k i and ˜ ρ llj = h l | ˆ R † j ρ j ˆ R j | l i , where ˆ Π k = ˆ R | k ih k | ˆ R † are the multi-qubit projective operators. Here {| k i} are separable eigenstates of N Lj = ˆ σ zj , and ˆ R is a local L -qubitrotation: ˆ R = N Lj = ˆ R j ( θ j , φ j ) with ˆ R j ( θ j , φ j ) = cos θ j ˆ I + i sin θ j cos φ j ˆ σ y + i sin θ j sin φ j ˆ σ x acting on the j -th qubit.This formula greatly reduces the computational e ff ortsneeded to evaluate GQD. III. THE BEHAVIOR OF GLOBAL QUANTUM DISCORDAND QUANTUM PHASE TRANSITIONS IN THE XYMODEL
In this section, we analysis the behavior of the three kindsof multipartite quantum correlations mentioned above in XYmodel. In order to see the features of these correlations moreclearly, we provide some figures of them and their derivatives.It shows that the property of these quantum correlations can beapplied to characterizing the two kinds of quantum phase tran-sitions e ff ectively and accurately. From another perspective,since GQD can be considered as a kind of physical resource in (a) (b) FIG. 3: Finite-size scaling analysis of the critical points for θ = ◦ and θ = ◦ in the phase transition between phase 1A and phase 2.(a) is for θ = ◦ with accuracy 0.00922 and (b) is for θ = ◦ withaccuracy 0.00263. quantum information processing, our knowledge about quan-tum phase transitions can also help us to understand and pre-dict the behavior of these correlations better. That is to say,the study of quantum critical systems can help us make a bet-ter understanding of the quantum correlations and quantuminformation processing.Now we consider the nature of these quantum correlationsas h changes at the zero temperature similar as [34]. Fig.2shows the total GQD, the sum of all nearest neighbor bipartiteGQDs D ( A : A ) + D ( A : A ) + · · · + D ( A L − : A L ) and theresidual GQD D ( A : · · · : A L ) − P L − i = D ( A i : A i + ) as a func-tion of h , when γ = sin θ , ( θ = ◦ , ◦ , ◦ , ◦ , ◦ ). Thefigures in each column correspond to a specific angle; fromfirst column to last column, the θ increases. The figures infirst line are about GQD, the figures in second line show thesum of all nearest neighbor bipartite GQDs, the figures in lastline are corresponding to the residual GQD. When θ = ◦ ,our model reduces to the transverse field Ising model. Westudy rings with L = , · · · ,
10, from bottom to top.First of all, we consider the case that γ = sin 75 ◦ = . h =
1, which is more suited to be usedto describing the second-order quantum phase transition thanthe total GQD. If we consider GQD as a resource for quantuminformation processing, these figures tell us that we can getthe most resource for quantum information and computationtasks when the second-order quantum phase transition occurs.That is to say, in order to obtain more physical resource forquantum information tasks from this system, we just need toadjust the external transverse magnetic field parameter h atthe critical point. On the other hand, the phase 1 ( h ≤ h + γ =
1, on which the ground stateis fully separable. From these figures, we find an interestingfact that the GQD sudden changes in some points. In fact,the sudden change which occurs at h = cos 75 ◦ = .
259 cancharacterize the phase transition between phase 1A and phase1B very accurately, since this point is just on the boundary h + γ =
1. From another point of view, when we observeda phase transition like this, there must be a sudden changeoccurs. In other words, we can get a sudden change of theseglobal quantum correlations just by adjusting corresponding parameter to a appropriate value. If we consider GQD as akind of order parameter, the phase transition can be seen asa first-order phase transition. Now we know that the GQDcan be used to detecting both the first-order and second-orderphase transitions in our model.The fourth column of Fig.2 shows the case that γ = sin 60 ◦ = . θ = ◦ , the sud-den change which occurs at h = cos 60 ◦ = . P L − i = D ( A i : A i + ) still reaches a maximum at nearly criticalpoint h =
1, which detecting the second-order phase transi-tion accurately.From these figures, we can find that there are many suddenchanges of GQD, only the rightmost sudden change charac-terizes the phase transition between phase 1A and phase 1B.Other sudden changes reflect the level-crossings that redefinethe ground state of the system (which are evident from thespectrum of the model). It is obvious that the sudden changewill be more apparent when we consider the smaller angle θ orsmaller system size L . That is to say, we can use this methodto detect the quantum phase transitions easier and more e ff ec-tive in these cases. As θ decreases, the two kinds of criticalpoints become closer.In particular, when θ = ◦ , our model reduces to XX model.In this case, the two kinds of phase transitions both occur at h =
1. For low magnetic field the GQD displays a step-wisebehavior, jumps occurring in correspondence of the level-crossings that redefine the ground state of the system (whichare evident from the spectrum of the model). That is, GQDtracks the structural changes in the ground state of the spinsystem as h varies. Comparing all these figures, we find thatas θ decreases, the GQD curves become more complicated.When we consider the case that θ is large enough, the max-imum of the sum of all nearest neighbor bipartite GQDs cancharacterize the second-order phase transition accurately. Onthe other hand, the phase transition between phase 1A andphase 1B can be described by the rightmost sudden change ofGQD. When θ is small enough, both phase transitions can becharacterized by the rightmost sudden change of GQD. Thesudden change will be more apparent when we consider thesmaller angle θ or smaller system size L .To get rid of the finite-size e ff ect in the phase transitions be-tween phase 1A and phase 2, we exemplify the scaling anal-ysis with the cases that θ = ◦ and θ = ◦ in Fig.3. FromFig.3 ( a ), we find that when θ = ◦ , the critical point la-beled h c obtained by our method tends to 0.995 in the ther-modynamic limit. The data can be fit to h c = − . × exp ( − L / . + .
955 with accuracy 0.00922. From Fig.3( b ), we find that when θ = ◦ , the critical point labeled h c tends to 1.020 as N → ∞ . The data can be fitted exponen-tially as h c = − . × exp( − L / . + .
020 with accuracy0.00263. For the large system, the critical point obtained byour method is very accurate.In order to see the sudden change more clearly, we showthe figures of the derivative of the sum of all nearest neighborbipartite GQDs in Fig.4. We give 6 figures about the cases that θ equals to 15 degree and 30 degree. The first column is thecase that θ = ◦ and the second column is for θ = ◦ . For h h h h h h FIG. 4: Derivative of the sum of all nearest neighbor bipartite GQDs.The first column is the case that θ = ◦ and the second column isfor θ = ◦ . In each column from top to bottom, we investigate thesystem size form 3 to 5. each case, we give three figures of L = , ,
5. The first line isabout L =
3, the second line is about L =
4, the last line is cor-responding to L =
5. It shows that when the sudden changesof the sum of all nearest neighbor bipartite GQDs occur, thecorresponding derivative exhibited bizarre behavior. Based onour discussion above, only the rightmost peak of the deriva-tive characterizes the phase transition between phase 1A andphase 1B, other peaks reflect the level-crossings that redefinethe ground state of the system. On the contrary, when we ob-served a phase transition like this, there must be a peak of thederivative occurs. This result can be used to predicting thesudden change of the corresponding correlation. It is obviousthat for the same angle, the peak is more obvious for small L .In general, when we consider the di ff erent angles, the peak ismore obvious for small θ . IV. CONCLUSIONS AND DISCUSSION
In this paper, we analyzed the global quantum discord andquantum phase transitions in quantum critical systems. Wedeveloped a proposal to describe the quantum phase transi-tions in quantum critical systems by examining the behavior of global quantum discord. We applied this proposal to thestudy of XY model. For the Ising phase transition betweenphase 1 and phase 2, we find that the sum of all nearest neigh-bor bipartite GQDs is e ff ective and accurate for signaling thephase transition point since it always reaches a maximum justaround the critical point. From another perspective, sinceGQD can be considered as a resource for quantum informa-tion processing, our result tells us that we can get most re-source for quantum information and computation tasks whenthe second-order quantum phase transition occurs. In otherwords, in order to obtain more physical resource for quan-tum information tasks from this system, we just need to adjustthe corresponding parameters to appropriate values. On theother hand, for the phase transition between phase 1A andphase 1B, it shows that the sudden change of GQD is verysuitable for detecting the critical point of this first-order phasetransition. On the contrary, when we observed a phase tran-sition like this, there must be a sudden change occurs. Thatis to say, we can get a sudden change of these global quan-tum correlations just by creating a first-order phase transitionin this model. In detail, only the rightmost sudden changecharacterizes the phase transition between phase 1A and phase1B, other sudden changes reflect the level-crossings that rede-fine the ground state of the system. It shows that the suddenchange will be more apparent when we consider the smallerangle θ or smaller system size L . That is to say, this methodcan be used to detecting the quantum phase transition moreeasier and e ff ective in these cases. When θ is small enough,both phase transitions can be characterized by the rightmostsudden change of GQD. In order to see the sudden changemore clearly, we provide some figures of the derivative of thesum of all nearest neighbor bipartite GQDs.Our proposal may help further understanding both of thecomplicated phenomena in quantum critical systems and thebehavior of global quantum correlations. This paper wouldinitiate extensive studies of quantum phase transitions fromthe perspective of quantum information processing. On theother hand, it also shows that the nature of quantum phasetransitions can be applied to predicting the behavior of corre-sponding quantum correlations. Since GQD is still meaning-ful for mixed states, this simple but e ff ective method is able tostudy finite-temperature phase transitions, which is superiorto other measure of quantum correlations. Our method is alsoworth applying to other quantum critical systems. Acknowledgments
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