Multi-partite entanglement and quantum phase transition in the one-, two-, and three-dimensional transverse field Ising model
aa r X i v : . [ qu a n t - ph ] S e p Multi-partite entanglement and quantum phase transition in the one-, two-, andthree-dimensional transverse field Ising model
Afshin Montakhab ∗ and Ali Asadian Department of Physics, College of Sciences, Shiraz University, Shiraz 71454, Iran (Dated: October 22, 2018)In this paper we consider the quantum phase transition in the Ising model in the presence ofa transverse field in one, two and three dimensions from a multi-partite entanglement point ofview. Using exact numerical solutions, we are able to study such systems up to 25 qubits. TheMeyer-Wallach measure of global entanglement is used to study the critical behavior of this model.The transition we consider is between a symmetric GHZ-like state to a paramagnetic product-state. We find that global entanglement serves as a good indicator of quantum phase transitionwith interesting scaling behavior. We use finite-size scaling to extract the critical point as well assome critical exponents for the one and two dimensional models. Our results indicate that suchmulti-partite measure of global entanglement shows universal features regardless of dimension d .Our results also provides evidence that multi-partite entanglement is better suited for the study ofquantum phase transitions than the much studied bi-partite measures. PACS numbers: 03.67.Mn, 03.65.Ud, 64.70.Tg, 74.40Kb
I. INTRODUCTION
There has been much work on entanglement in the pasttwenty years [1–8]. Entanglement is a purely quantumphenomenon with no classical counterpart. It is thoughtto hold the key to a deeper understanding of the theoret-ical aspects of quantum mechanics. From a more prac-tical aspect, entanglement is the key ingredient in manyinformation processing applications including quantumcomputation and quantum cryptography [9]. On theother hand, there has been much cross-fertilization inthe fields of condensed-matter physics and quantum in-formation theory in recent years [10–18]. Here, many tra-ditional condensed-matter systems including fermionicand bosonic gases, and in particular lattice spin mod-els have been investigated in the light of new develop-ments in quantum information theory and entanglementin particular [10]. It has been found that entanglementplays a crucial role in the low-temperature physics ofmany of these systems, particularly in their ground (zero-temperature) state [16–19]. A very fruitful avenue alongthese lines has been the relation of entanglement andphase transitions in general, and quantum (ground state)phase transitions in particular. This is a bit surprisingsince entanglement was originally thought to be some-what fragile and thus easily destroyed by fluctuations.In a quantum phase transition (QPT) [20], a ther-modynamic system described by a Hamiltonian H ( λ )changes its macroscopic phase at the critical value of thecontrol parameter λ c . In recent studies of many ther-modynamic systems exhibiting QPT, in particular quan-tum spin models[16–19], it has become clear that theonset of transition is accompanied by a marked changein the entanglement. Depending on the model, entan- ∗ Electronic address: [email protected] glement could peak, or show discontinuous behavior, orshow diverging derivatives with scaling behavior at thecritical point[10, 21]. What is less clear is the exact role(or the general mechanism) through which entanglementand QPT are related. In such studies, various quantita-tively different measures of entanglement have been used.Therefore, for example, one would like to know if thereare universal features in the entanglement of various spinmodel exhibiting QPT?Another important feature which is emerging out ofrecent studies of condensed-matter systems from quan-tum information perspectives is the need for multi-partitemeasures of entanglement [22–26] . This, by the way, isan example of cross-fertilization referred to earlier. Sincethe root of quantum theory[27] is originally in bi-partitesystems like Bell states, it has been natural to studymacroscopic systems using bi-partite measures such asthe von Neumann entropy or concurrence. In fact, withvery few exceptions, the general body of the current lit-erature has used such bi-partite measures to study manyparticle systems. Although this has been so because of amatter of tradition and/or convenience, there is increas-ing evidence that such measures are generally inadequateto study QPT in condensed-matter systems[28, 29]. Af-ter all, it is natural to use multi-partite entanglement ifone is to study the role of entanglement in multi-partite(many-particle) systems, as important types of entangle-ment in such systems (e.g. various n -tangles) may not becaptured by a bi-partite measure, but would be includedin an (ideal) multi-partite measure. Additionally, somemulti-partite measures (as will be discussed in Sec. III)have thermodynamic properties (e.g. extensivity) whichmake them more suitable for studies of such thermody-namic phenomenon as QPT. Another equally importantshortcoming is that most such studies have been carriedout for one dimensional (1d) models. Although this isperhaps because of computational difficulties, it is cer-tainly not well-justified. As is well-known, spatial dimen-sion (d) plays an important role in the physics of ther-modynamic systems, phase transitions in particular[30].Here, we propose to study QPT in prototypical trans-verse field quantum Ising model using the Meyer andWallach [31] measure of global entanglement in one, two,as well as three dimensions. Such global entanglementmeasure seems to be well-suited for studies of many par-ticle systems [22, 32]. Since analytic results are usuallydifficult to come up with, numerical results with finite-size systems are typically the way to proceed. However,solving quantum lattice spin systems numerically is alsocomputationally expensive as only a few qubits (spins)can be solved exactly and approximation techniques havelimited success in one dimension and are more limited inhigher dimensions [33]. Very recently, however, such sys-tems have been studied using efficient numerics[34].In this article, we solve the transverse field quantumIsing model numerically ( exact ) for up to 25 qubits inone, two and three dimensions. Our main result is thatglobal entanglement is a well-suited measure to studyQPT with some universal features in any dimension. Weshow that global entanglement has interesting scalingproperties near the critical point. Using finite-size scalingarguments, we extract critical points as well as some crit-ical exponents for the 1d and 2d models consistent withprevious studies. Due to system-size limits, we are onlyable to study the smallest 3d system and thus cannotperform finite-size studies. However, the general shapeof global entanglement in the 3d model (Fig. 7) indicatesthat our 1d and 2d results easily generalize to 3d systems.More importantly, our results provide a general frame-work for computation of an accessible measure of en-tanglement and its relevance to QPT’s in many-particlethermodynamic systems.This paper is structured as follows: in Section II, wediscuss the multi-dimensional quantum Ising model inthe presence of a transverse field and its ground stateproperties relevant to our study here. In Section III, wediscuss some key concepts regarding the Meyer-Wallachmeasure of global entanglement, while our main resultsare presented in Section IV. Our concluding remarks, in-cluding suggestions for further work is presented in Sec-tion V. II. TRANSVERSE FIELD ISING MODEL
The system under consideration here is the ferromag-netic Ising model in a transverse field given by the Hamil-tonian: H = − λ X
Global entanglement, defined by the Meyer-Wallachentanglement measure of pure-state [31], and henceforthdenoted by E gl , is a monotone[40], and a very usefulmeasure of multi-partite entanglement. As we will showbriefly, E gl is a measure of total non-local informationper particle in a general multi-partite system. There-fore, E gl gives an intuitive meaning to multi-partite en-tanglement as well as being an experimentally accessiblemeasure [40–42].Finite amount of information can be attributed to N-qubit pure state which is N bit of information accordingto Brukner-Zeilinger operationally invariant informationmeasure [43]. This information can be distributed in lo-cal as well as non-local form, which is associated withentanglement [44]. This information has a complimen-tary relation: I total = I local + I non − local . (3)The total information is conserved unless transferredto environment through decoherence. The amount ofinformation in local form is I local = P Ni =1 I i where, I i = 2 T rρ i − ρ i is single particle reduceddensity matrix obtained by tracing over the other par-ticles’ degrees of freedom. Therefore, according to Eq.(3) I non − local = P Ni =1 − T rρ i ) which is distributedin different kinds of quantum correlations, the tangles,among the system, I non − local = 2 X i
Using Eq. (6), we can therefore easily calculate E gl exactly for any dimension d, up to the limitations set bycomputational limits of our numerics.We start by showing our results for the 1d model. Fig-ure 1 shows E gl vs λ and the inset shows its derivativefor various system sizes up to N = 24. The general be-havior shown here is that of E gl increasing slowly fromits zero value at λ = 0 with a sharp transition to its large λ value of 1 around λ c = 1. The critical point is betterseen in the derivative (inset) which peaks at the maximalvalue λ m ( N ). As the system size increases, the peak ofthe derivative sharpens and moves closer to the criticalpoint λ c = 1. λ E g l d E g l / d λ λ N=4N=8N=12N=16N=20N=24
FIG. 1: (Color online) Global entanglement as a function of λ for the 1d transverse Ising model. The inset shows thederivative and the system sizes used. Increasing N sharpensthe peak and moves it closer to the critical point. The extrapolation to the infinite system size along withthe convergence to the critical point (inset) is shown inFigure 2. As one can see, λ m ( ∞ ) = 1 .
01, very closeto the well-known result of λ c = 1, showing that E gl is a good indicator of the critical behavior of this model.The inset shows that the convergence to the critical pointis in accordance with | λ c − λ m | ∼ N − α with exponent α = 1 . E gl near thecritical point. According to scaling ansatz [45], we have dE gl /dλ ∼ Q ( N ν ( λ − λ m )) where ν is the correlationlength critical exponent, and Q ( x ) ∼ ln( x ) is generallyassumed. As is seen in Figure 3, an acceptable collapseoccurs for various N’s using the scaling ansatz with thecritical exponent ν = 1 .
06 in line with previous studies[17], and close to the exact result of ν = 1. The insetshows the logarithmic divergence of the maximum (peak) λ m log(N) l og ( λ m − λ c ) FIG. 2: (Color online) Convergence of λ m to the critical pointas N → ∞ , for the 1d transverse Ising model. The y-interceptis 1 .
01. The inset shows the relation | λ c − λ m | ∼ N − . . −1.5 −1 −0.5 0 0.5 1 1.500.050.10.150.20.25 N ( λ − λ m ) − e x p ( d E g l / d λ − d E g l / d λ | λ = λ m ) N=16N=18N=20N=22N=24 log(N) d E g l / d λ | λ = λ m FIG. 3: (Color online) Finite-size scaling of global entangle-ment for the 1d transverse Ising model. The inset shows thelogarithmic divergence of the value of the derivative at themaximal point λ m . of the derivative of E gl . Hence, the general shape of E gl ,the logarithmic divergence of its derivative at the criticalpoint, along with its consistency with finite-size scalingansatz provides strong evidence for the well-suitednessof such measure for the 1d Ising model. The questionnow is, if such features also hold for higher dimensionalmodels?We next turn to the 2d model. Using periodic bound-ary conditions we have been able to study such model forup to L = 5 = 25 = N qubits. Figures 4, 5 and 6 showsimilar results as that of Figures 1, 2 and 3. We notethe following: The general shape of the E gl still remains(Figure 4), with a (logarithmic) divergence of the deriva-tive at the critical point (insets of Figure 4 and 6). Thecritical point is now identified as λ m ( ∞ ) = 0 . λ c = 0 . λ c = 0 . λ c = 0 . | λ c − λ m | ∼ L − α with exponent α = 1 .
00 being exactlythe same as the 1d case. The difference here is that thisconvergence occurs from above the critical point as op-posed to the 1d case. The finite-size scaling ansatz is alsovalid (Figure 6), giving the correlation length exponent ν = 0 .
51. The inset of Figure 6 shows the logarithmicdivergence of the derivative at the critical point.In Figure 7, we show our result for the 3d version ofthis system for the only system size we are able to study.The general behavior of E gl seen in the 1d and 2d modelis clearly seen here for the 3d case as well. While we arenot able to perform scaling analysis similar to the 1d and2d models, it seems reasonable to assume that the samegeneral behavior carries over to the 3d model. We notethat λ m ( L = 2) = 0 .
26 here, which would understand-ably be different from the infinite-size limit, but in theright ball-park of λ c ≈ . λ E g l d E g l / d λ λ FIG. 4: (Color online) Global entanglement as a function of λ for the 2d transverse Ising model. The inset shows thederivative and the system sizes used. Increasing system sizessharpens the peak and moves it towards the critical point. λ m log(L) l og ( λ m − λ c ) FIG. 5: (Color online) Convergence of λ m to the critical pointas L → ∞ , for the 2d transverse Ising model. The y-interceptis 0 . | λ c − λ m | ∼ L − . . −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.400.050.10.150.20.250.3 L ( λ − λ m ) − e x p ( d E g l / d λ − d E g l / d λ | λ = λ m ) L=3L=4L=5 log(L) d E g l / d λ | λ = λ m FIG. 6: (Color online) Finite-size scaling of global entangle-ment for the 2d transverse Ising model. The inset shows thelogarithmic divergence of the value of the derivative at themaximal point λ m . λ E g l λ d E g l / d λ FIG. 7: (Color online) Global entanglement and its derivative(inset) for the 2 × × Finally, it is worth considering another important formof multi-partite entanglement, namely, genuine entangle-ment. Genuine entanglement in a many-particle systemrepresents the amount of entanglement shared by all par-ticles. Therefore, genuine entanglement is equal to the N -tangle, the last term in Eq. (5). One might expectthat such a term would gradually lose its significance as N increases. However, due to the GHZ nature of ourground state, this term ( N -tangle) is the dominant termin E gl and its dominance increases with increasing λ .This is shown in Figure 8 for both the 1d and 2d models,where one can easily see that increasing λ , increases theshare of N -tangle in E gl . It is also worth noting that thestructure of genuine entanglement is very similar to anorder parameter. It is zero on one side of the transitionand becomes non-zero around the critical point rising toits maximum at 1 /λ = 0. This behavior becomes morepronounced as system-sizes get larger, however, we notethat the transition is not sharp and is in fact “rounded”.However, since net magnetization cannot be used as anorder parameter here, such behavior deserves further at- λ E λ E FIG. 8: (Color online) Global entanglement (continuouscurve) and genuine entanglement (dashed curve) for the 1d(main figure) and 2d (inset) transverse Ising model. The 1dresult is for N = 16 and the 2d result is for a 4 × tention. V. CONCLUDING REMARKS
In this paper we have studied the quantum phase tran-sition in the transverse field Ising model from a multi-partite entanglement point of view on a one, two andthree dimensional square lattice. Our work is interestingfrom various points of view. First we use a multipartiteglobal entanglement as a measure. Secondly, we studythe symmetric GHZ-like ground state and its transitionto the paramagnetic product-state. Thirdly, by study-ing QPT in various dimensions we are able to establishcommon features of such a transition in different univer-sality classes. We find that global entanglement is a goodindicator of such transitions with universal aspects, in-cluding scaling, in any dimension. The well-suitedness ofsuch a measure is displayed in the nice fits obtained infigures such as Figure 2 or Figure 5, for example. As aby-product, we find critical points and various exponentsfor the 1d and 2d models consistent with previous stud-ies. We note that our estimation of the critical pointsfor the 1d and 2d models are to within one percent ofthe generally accepted values, an impressive result giventhe limited size of the systems studied here, providingfurther evidence for well-suitedness of our measure whencompared with similar studies using bi-partite measures.Our estimation of ν , although acceptable, are under-standably less impressive as finite-size scaling collapsesrequire larger system sizes to obtain better estimates for ν [47]. We note that our main goal is to investigate the(universal) features of global entanglement in quantumphase transitions, and not to produce reliable exponentsfor such models. Since the parameter d determines theuniversality class of the systems considered here, the factthat we see similar behavior of global entanglement atthe QPT regardless of d , shows what we have thus farreferred to as universal features of global entanglement.We close by mentioning that similar studies could becarried out for more general spin models exhibiting morecomplicated quantum phase transitions. It would be in-teresting to see if such universal features of global entan-glement carry over to other models. Acknowledgments
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