Nonclassicality and criticality in symmetry-protected magnetic phases
Matthew J. M. Power, Steve Campbell, Maria Moreno-Cardoner, Gabriele De Chiara
NNonclassicality and criticality in symmetry-protected magnetic phases
Matthew J. M. Power, Steve Campbell, Maria Moreno-Cardoner, and Gabriele De Chiara
Centre for Theoretical Atomic, Molecular and Optical Physics,Queen’s University Belfast, Belfast BT7 1NN, United Kingdom (Dated: September 27, 2018)Quantum and global discord in a spin-1 Heisenberg chain subject to single-ion anisotropy (uniaxial field) arestudied using exact diagonalisation and the density matrix renormalisation group (DMRG). We find that thesemeasures of quantum nonclassicality are able to detect the quantum phase transitions confining the symmetryprotected Haldane phase and show critical scaling with universal exponents. Moreover, in the case of thermalstates, we find that quantum discord can increase with increasing temperature.
I. INTRODUCTION
The study of strongly-correlated magnetic systems has ex-perienced a tremendous boost thanks to inputs from quantuminformation processing [1]. In particular, the analysis of var-ious forms of entanglement has revealed deep connections toquantum phase transitions [2–4] and order parameters [5, 6],conformal field theory [7] and numerical simulations of quan-tum many-body systems [8, 9]. The fidelity approach has beensuccessful for the analysis of quantum phase transitions (QPT)with no order parameters or with infinite order [10, 11].An alternative approach based on quantum correlations hasallowed the quantum information / condensed matter commu-nity to analyse the “quantumness” of not only the ground stateof magnetic Hamiltonians, but also thermal states. For the lat-ter, entanglement tends to disappear quite rapidly with tem-perature and is subject to mathematical pathologies such asthe “sudden death” [12]. Quantum discord (QD) is a mea-sure of nonclassicality that has attracted a lot of attention re-cently [13–15]. While the resource nature of QD is still anopen question [16], it has shown to be a remarkably e ff ectivetool in studying a range of quantum phenomena and protocolsincluding QPTs [17, 18], entanglement distribution [19], andwork extraction [20]. While most of the literature is devotedto the discord of two-level systems and continuous variableGaussian states, very little has been done for higher dimen-sional quantum states [21]. This can be understood in light ofthe di ffi culty in evaluating QD [22]. Thus, this work aims atproviding a concrete analysis of the QD in a realizable physi-cal system beyond those typically examined, namely a spin-1chain.In this paper, we study the quantum discord of spin-1 chainsgoverned by a Heisenberg Hamiltonian and in the presence ofsingle-ion anisotropy, i.e. a uniaxial (quadratic Zeeman) field.The model gives rise to three gapped magnetic phases (N´eel,Haldane, Large-D) separated by a second order transition anda Gaussian one respectively. The Haldane phase, a symmetryprotected phase, is quite peculiar as it is characterised by theabsence of local order but by the establishment of a hiddenstring order parameter.While this model has been studied extensively in the con-densed matter and quantum information communities usingthe entanglement and fidelity [23–37], a systematic investiga-tion of its discord content is missing. The aim of this paperis to fill this gap by analysing the two-spins discord and the global version introduced in Ref. [38]. We employ numericalsimulations based on exact diagonalisation and density matrixrenormalisation group (DMRG) [39, 40]. We find that discordis able to locate very accurately the two transitions by showingsingular behaviour as a function of the single-ion anisotropy.Moreover, for the Gaussian transition we are able to extractan estimate for the critical exponent for the correlation length.Finally, we analyse the thermal behaviour of discord and findthat while it normally decays with increasing temperature, inthe large-D phase and for two non nearest-neighbour spins itactually increases.The paper is organised as follows: in Sec. II we revise themeasures of nonclassicality used in the rest of the paper; inSec. III we revise the magnetic properties of the spin-1 modelwe consider; finally in Sec. IV we show our numerical resultsand in Sec. V we conclude. II. MEASURES OF NONCLASSICALITY
The quantum discord (QD) between two systems A and B described by a density matrix ρ AB can be defined as the di ff er-ence of two distinct ways to measure correlations in a quan-tum system that would otherwise give the same result clas-sically [13, 14]. The first way is to use the quantum mutualinformation I ( ρ AB ) = S ( ρ A ) − S ( ρ A | ρ B ) , (1)where S ( ρ A ) = − Tr[ ρ A log ρ A ] is the von Neumann entropyof the reduced state ρ A = Tr B ρ AB of system A and analogouslyfor system B . The quantity S ( ρ A | ρ B ) = S ( ρ AB ) − S ( ρ B ) , (2)is the conditional entropy. An alternative definition of cor-relations can be given in terms of information acquired on A after performing a measurement of B with a set of projectors { Π jB } . Let us call ρ A | j = (1 / p j )Tr B [ Π jB ρ AB ] the state of system A after outcome j is obtained measuring system B with prob-ability p j = Tr[ Π jB ρ AB ]. We thus define the one-way classicalinformation as J ( ρ AB ) = S ( ρ A ) − (cid:88) j p j S ( ρ A | j ) . (3)QD is the di ff erence of the quantum mutual information andthe classical one-way information, minimized over the set of a r X i v : . [ qu a n t - ph ] J un orthogonal projective measurements on B D B → A ( ρ AB ) = inf { Π jB } [ I ( ρ AB ) − J ( ρ AB )] . (4)This definition is not symmetric under the exchange of A and B as the measurements are performed on system B only.A symmetrized version of the QD can be obtained with abi-local measurement Π i j = Π iA ⊗ Π jB such that Π ( ρ AB ) = (cid:80) i j Π i j ρ AB Π i j . We define the symmetric QD D ( ρ AB ) = min { Π } S ( ρ AB || Π ( ρ AB )) − (cid:88) α = A , B S ( ρ α || Π α ( ρ α )) , (5)where we have introduced the relative entropy: S ( ρ || σ ) = Tr[ ρ log ρ ] − Tr[ ρ log σ ] , (6)which vanishes as σ approaches ρ . Eq. (5) can be interpretedas the di ff erence between the first term, that is global on A and B , and the second term, which is the sum of two localcontributions. Eq. (5) was shown to be generalizable to mul-tipartite states [38]. For a quantum system comprising of N subsystems we define the global quantum discord (GQD) D N ( ρ N ) = min { Π } (cid:26) S ( ρ N || Π ( ρ N )) − N (cid:88) α = S ( ρ α || Π α ( ρ α )) (cid:27) . (7)In order to evaluate Eqs. (5) and (7) for the spin-1 system pre-sented in the following section, we require suitable projectivemeasurements. In Ref. [21] the parametrization of local or-thogonal measurements for spin-1 particles was given. We usethe spin-1 operators S x , y , z fulfilling the normal angular mo-mentum commutation relations. For simplicity, we define theeigenstates of the z -component of the angular momentum as: S z | m (cid:105) = m | m (cid:105) with m = − , , +
1. A projective measure-ment for three-level systems is specified by three orthogonalprojectors summing to the identity matrix (cid:88) m = , ± | m A (cid:105) (cid:104) m A | = , (8)where A is a unitary matrix and we have defined the trans-formed basis states as | m A (cid:105) = A | m (cid:105) . (9)Contrary to spin-1 / P = (cid:104) S (cid:105) is not always one for spin-1 systems,i.e. they are not always coherent states. This is related to thefact that, while for spin-1 / | m r (cid:105) = exp (cid:104) i (cid:16) γ ( S z ) − γ − φ S z (cid:17)(cid:105) ×× exp (cid:2) − i α ( S x S y + S y S x ) (cid:3) ×× exp (cid:34) i β √ S y + S y S z + S z S y ) (cid:35) | m (cid:105) . (10)Then the most general basis is obtained by rotating the states | m r (cid:105) in any possible direction using the following combinationof rotations: | m A (cid:105) = e − i ψ S x e − i θ S y e − i φ S z | m r (cid:105) . (11)It is therefore su ffi cient to parametrize the most general or-thonormal basis of spin-1 systems with six coe ffi cients (since φ is constrained by the other parameters [21]). III. THE MODEL
We conduct our analysis on the ground and thermal statesof a spin-1 chain described by the Heisenberg Hamiltonianwith uniaxial anisotropy of strength U H = (cid:88) i S xi S xi + + S yi S yi + + S zi S zi + + U (cid:88) i (cid:16) S zi (cid:17) , (12)where the subscript i runs over the sites in the chain and wewill consider both open and periodic boundary conditions.When U < U >
0) the anisotropy is usually referred to as“easy-axis” (“easy-plane”) anisotropy. The ground state phasediagram consists of three phases: for U (cid:38) .
968 the systemis in the “large D” phase [41] in which the ground state has astrong component onto the | . . . (cid:105) state. For U < − . U ∈ [ − . , . H can be chosen real. It follows that the optimalbasis is always real as in Ref. [18]. This further reduces thenumber of parameters to optimize over since we impose γ = ψ = φ = φ = , so that the basis | m A (cid:105) is a real superposition of the states | m (cid:105) .We remark that this constraint is only applied to the optimiza-tion of the GQD, which due to computational complexity, ne-cessitates such simplifications. However, when dealing with (cid:45) U (cid:45) (cid:45) (cid:45) E N FIG. 1: (Color online) Lowest three energy levels of HamiltonianEq. (12) with periodic boundary conditions as a function of U for L =
5. For odd length chains the ground state changes suddenly dueto energy level crossings. Here we see the first crossing at U (cid:39) − . U (cid:39) . the symmetric QD, Eq. (5), a full minimization with the fullset of angles is tractable. IV. NONCLASSICALITY IN THE SPIN-1 HEISENBERGMODEL
When considering finite-length chains, we find an even / oddparity e ff ect with the total length, the origin of which liesin a geometrical frustration for odd lengths. In fact, in therepulsively interacting Heisenberg chain we are considering,nearest-neighbor spins tend to form strongly correlated pairs,an e ff ect observed in the alternating behavior of the block en-tropy and other correlations. Thus, while for an even chain allspins are paired, for odd chains there is always an unpairedspin. This frustration gives rise to energy crossings as ob-served in Fig. 1. This means that, at these energy crossings,the ground state of the system changes discontinuously with U . For this reason, we examine even and odd lengths sepa-rately, as for the latter we will observe discontinuities in thediscord measures. However, we remark that this is a finite sizee ff ect that will vanish in the thermodynamic limit, as the threephases of the model are all gapped. A. Nearest-neighbor spins
We begin by analyzing the reduced state of the two centralspins for the thermal ground state when open ended-boundaryconditions are imposed on Eq. (12). Further, to capture thepertinent features of the model, we consider only even- L and,thus avoid pathological features due to energy level crossingsin the ground state when L is odd. Through DMRG calcula-tions, we are able to determine the reduced state of the twocentral spins and calculate the symmetric discord Eq. (5). InFig. 2 we plot D for L =
8, 16, 32, 64, 128 and 256 spins.We see several interesting features emerging for increasinglylarge chains. For U < − . U > .
6, we see the curves for (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) 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(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) U (cid:68) FIG. 2: (Color online) Nearest-neighbor symmetric QD, D , for thereduced state of the two central spins of an open-ended chain oflength L = L =
32, 64, 128, and 256 are almost indistinguishable except near U ∼− . all lengths have collapsed on top of each other and the reducedstates are virtually identical. This indicates that, in these re-gions (which are su ffi ciently far from the QPTs of the model),already with 8 spins, we are close to the properties of the ther-modynamic limit. In the intermediate region, U ∈ [ − . , . L the value of the QD increases. A striking feature is the cuspat U =
0. This corresponds to the point when the optimiz-ing angles required to minimize D change. When U < D is optimized when both spins are measured using theangles θ = α = β =
0, corresponding to a projection ontothe eigenbasis of S z , while for U > θ = π/ α = β =
0, corresponding to a projection onto the eigenbasisof S x . For U = D . Such a sudden change is unsurprising considering that atthis point we are switching from easy-axis for U < U > U = ff ering only bya spin rotation give the same discord. Indeed, such behavioris not uncommon when dealing with nonclassicality indicatorsthat involve complex parameter optimizations. A similar be-havior was recently reported in the spin-1 / XY model whenexamining measures of local quantum coherence [42].In Fig. 3 (a) we examine the derivative of the QD with re-spect to U for chains of increasing length. We see a peakedbehavior appearing near U = − .
3, which becomes more pro-nounced for larger L . Such behavior is consistent with thesignature of a second order QPT [17]: a discontinuity of thesecond order derivative of the ground state energy or of thefirst derivative of the state (and therefore of discord). Wecan accurately predict the critical value for L → ∞ throughfinite size extrapolation. In Fig. 3 (b) we disregard the twosmallest sized chains ( L = U = − . (a) (cid:45) U (cid:45) (cid:182) U (cid:68) (b) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) x (cid:45) L (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) U Max (cid:64) (cid:182) U (cid:68) (cid:68) FIG. 3: (Color online) (a) Derivative of the nearest neighbor sym-metric QD against U for L = U where the derivative is maximum against inverse chain length. precisely inline with the value determined in [33].Turning our attention to the Haldane-large D QPT, this tran-sition is Gaussian and expected to be a third order transition.Furthermore, the critical region is known to be very tight, andnot extending more than ± . ffi cult.In fact, in Ref. [36] the authors employed a refined DMRGtechnique in order to access lengths of up to 20,000 spins todetermine the critical point to a high degree of accuracy, find-ing U = . U = .
96 by studying the finite size scaling (FSS) ofthe entanglement spectrum for up to L = U = .
971 [30]. As the Haldane-large D transition is a thirdorder continuous phase transition, we expect a point of inflec-tion in the second derivative of the energy, and consequently,in the first derivative of the ground state and therefore of D .Thus, we anticipate, by examining the second derivative of D , to find a minimum. Using finite size extrapolation, as be-fore, even with the best quadratic rather than linear fit, we findthe QPT predicted at U = .
994 (results not shown) which is afew percent o ff the value predicted in Ref. [36], indicating thatthe nature of this QPT will require larger sizes to accuratelylocate its critical point using discord. However, a curious re-sult appears when studying the second derivative. In Fig. 4 FIG. 4: (Color online) Second derivative of the symmetric QDwith respect to U in the critical region of the Gaussian QPT. Thepoint markers are the numerically calculated values L =
16 (down-ward green triangles), 32 (blue diamonds), 64 (gray circles), 128(black squares), and 256 (upward orange triangles). The solid linesare quadratic functions of best fit for each data set. The vertical reddashed line at U = . D using the estimate for the critical point U = . ν = . ± . we show the behavior of ∂ D /∂ U within the critical region.The various symbols correspond to the numerically calculatedvalues, while the solid curves are quadratic lines of best fit foreach data set. For all chains with L >
32 we see the secondderivatives cross each other near the same point located ap-proximately at U = . ff erent, although this is in keeping with the behavior of theN´eel-Haldane transition where L = ∂ D ∂ U = f [( U − . L /ν ] , (13)where f is an analytic function close to f [0] and ν is the criti-cal exponent associated with the divergence of the correlationlength. That is, the critical exponent associated to the sec-ond derivative of discord equals to zero, and thus, the thirdderivative vanishes in the thermodynamic limit. This way allthe dependence on the parameter U is reabsorbed in the cor-relation length. By fitting our results we find that the value ν = . ± . ff erent lengths as shownin the inset in Fig. 4. The value we find is in agreement withthe more accurate result ν = .
47 found in Ref. [36]. There-fore, if our conjecture is correct, discord is not only able tolocate the position of the QPT but also the universal scalingexponents associated with it.There is an additional parity e ff ect for these even lengthchains. In the above cases we consider chains such that( L − / L , the smaller this di ff erence be-comes and we see the behavior reported in Fig. 2 (a), i.e. in-creasing QD for increasing L. In contrast, when ( L − / L =
6, 10, 14, . . . ) the two central spins corre-spond exactly to a dimer formed in the chain, and this resultsin larger values for the QD that decreases as we increase L .The qualitative behavior remains una ff ected and, in fact, for L >
30 any di ff erences between these situations are negligi-ble. B. Global measures
While the previous section highlighted that using measuresof nonclassicality applied to reduced states can capture thethermodynamic properties of systems beyond the paradig-matic spin-1 / / L = θ = α = β = U < θ = π/ α = β = U >
0. We conjecture that for a chain withperiodic boundaries these will be the optimizing angles for alleven and odd length chains. For L = , U andconfirmed the conjecture. For larger systems, we calculatethe GQD using these known fixed values for the optimizingangles. Despite not being accessible to analytic proof, thisapproach of using numerical confirmation has proven fruitfulwhen calculating such involved quantities [43].In Fig. 5 (a) we show the GQD, Eq. (7), for finite sizedchains of length L = , , ,
8. Consistent with the behaviorof the reduced state of large chains, we see a cusp at U =
0, which is once again a consequence of the change in thenature of the uniaxial anisotropy. We see a significant increasein the rate of change of the GQD by increasing the length.In panel (b), we examine the behavior of the GQD for odd (a) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● - � � � � � ����� � (b) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:45) U (cid:68) N FIG. 5: (Color online) (a) GQD of the thermal ground state for L = L =
3, 5, and 7 from bottom to top. See text fordiscussion. length chains L = , ,
7. Here we see the sudden changes inthe GQD occur when there is an energy level crossing in theground state. For L = , U =
0. This feature is absent for the L = (cid:37) ( T ) = e −H / T Tr (cid:2) e −H / T (cid:3) , (14)where T is the temperature and we have assumed units suchthat Boltzmann’s constant is one. In Fig. 6 we fully explorethe thermal e ff ects for L = U = U = T where GQD isresilient to the thermal e ff ects until T ∼ .
5, when a quickdecay occurs. In contrast, when U = − (a) (b) (c) (d) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) U (cid:61) U (cid:61) U (cid:61) (cid:45) T (cid:68) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) T (cid:68) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) T (cid:68) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236)(cid:236) T (cid:68) FIG. 6: (Color online) (a) GQD against T for a closed ring of 6 spins with U = small gap between the ground and first excited energy levels inthis region. For comparison, we examine the various di ff erenttwo-spin reduced states, and in panel (b) we see the nearest-neighbors behave qualitatively the same as the GQD. Interest-ingly, for next-nearest [panel (c)] and next-next-nearest neigh-bors [panel (d)], the large D region shows an initial increasein D of the reduced state with increasing T . This unusualbehavior, rarely observed for entanglement, was also notedin [44] where states of increasing mixedness can have increas-ing QD. In the present situation it can be explained by thepresence of highly correlated excited states (similar to doblon-holon states in a Bose-Mott insulator) above a near factorisedground state in the large-D phase. A further comment is inorder, as reported in [45, 46], QD can be created by the ac-tion of local non-unital channels. While the full thermal statecan be considered as the action of local channels, each suchchannel acts at the same rate, and therefore would appear tobe incompatible with the conditions outlined in [45]. V. CONCLUSIONS
We have examined the nonclassical properties of thesymmetry protected spin-1 Heisenberg chain with uniaxialanisotropy. Through DMRG, we were able to explore thesymmetric quantum discord (QD) of the reduced state of two central spins in an open ended chain. By examining the be-havior of the QD and its derivative, through finite size extrap-olation, we were able to pinpoint the N´eel-Haldane QPT inexcellent agreement with the recent literature. Interestingly,we found the second derivative of the symmetric QD appearedto detect the Haldane-large D transition thanks to the third or-der character of this transition, and allowed us to connect itsscaling with universal critical exponents. While the use ofQD has been extensively applied to spin-1 / in-creases with temperature. Note:
The data used in generating all figures in this articleis available from the link in Ref. [47].
Acknowledgments
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