Tuning interaction strength leads to ergodic-nonergodic transition of quantum correlations in anisotropic Heisenberg spin model
aa r X i v : . [ qu a n t - ph ] M a r Tuning interaction strength leads to ergodic-nonergodic transition of quantumcorrelations in anisotropic Heisenberg spin model
Utkarsh Mishra, R. Prabhu, Aditi Sen(De), and Ujjwal Sen
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
We investigate the time dynamics of quantum correlations of the anisotropic Heisenberg model ina time-dependent magnetic field, in one-dimensional, ladder, and two-dimensional lattices. We findthat quantum correlation measures in the entanglement-separability paradigm are ergodic in thesesystems irrespective of system parameters. However, information-theoretic quantum correlationmeasures can also be nonergodic, and exhibit a transition from nonergodic to ergodic behavior withthe change of interaction strength in the direction of the magnetic field. We also observe thatthe transition point changes drastically as we go from one-dimensional and ladder lattices to thetwo-dimensional one.
I. INTRODUCTION
Statistical mechanical models provide a quantitativeway to understand physical phenomena involving a largenumber of particles which are interacting among them-selves. These models have been established as promis-ing substrates in different physical systems for imple-menting many quantum information protocols which in-clude, for example, one-way quantum computation [1]and quantum communication tasks [2]. The character-ization, quantification and realization of quantum cor-relations in many-body systems are some of the mainchallenges in quantum information [3–6].Quantum correlation concepts in multiparty sys-tems can broadly be classified into two categories– entanglement-separability paradigm measures andinformation-theoretic ones. Quantum correlations ofthe first kind are established to be useful resources formany quantum information tasks which include quan-tum dense coding [7], quantum teleportation [8], and se-cure quantum cryptography [9]. Recently however, sev-eral non-classical phenomena have been discovered inwhich entanglement is absent [10–14]. To understandand quantify the resource necessary for exhibiting suchnon-classicality, information-theoretic quantum correla-tion measures like quantum discord [15, 16] and quantumwork-deficit [17] have been proposed.Measures of both the paradigms have proven to be ad-vantageous in investigations of cooperative physical phe-nomena observed in many-body systems [3–6, 18]. Due tothe paucity of analytical as well as numerical methods tosolve quantum spin models, most of these considerationsare restricted to the ground state or the thermal stateof the system. While it is important to understand thequantum correlation properties of these “static states” ofthe system, the time-evolution of the system is an inte-gral part of several quantum information processing pro-tocols, a prominent example being the one-way quantumcomputer [1].Properties like magnetization, susceptibility, classicaland quantum correlations in the static states of theisotropic Heisenberg model have been studied exten-sively, both theoretically and experimentally [5, 6, 19, 20]. The model can be exactly solved by the Bethe ansatz [21].Variation of different physical parameters in this modelleads to the appearance of rich phases [22, 23], like spin-liquid, resonating valence bond states, etc. Moreover,such models can now be created in the laboratories in acontrolled way by using e.g., photons [24], trapped ions[25], and cavity QED [26]. However, numerical simula-tions or approximate methods are the only techniquesthat can be used to investigate properties of the time-evolved states of this model. Here we investigate thebehavior of quantum correlations of the evolved state aswell as the equilibrium state in the anisotropic Heisen-berg model in low-dimensional systems, under the influ-ence of time-dependent magnetic fields and temperature.In particular, we observe collapse and revival of quantumcorrelation measures of the evolved state in this system.The usual statistical mechanical description of a phys-ical quantity is valid only when the time-average of thequantity matches with its ensemble average, and in thatcase, the physical quantity is termed as ergodic. Ergod-icity of physical quantities in spin models has been ofinterest to researchers for a long time [27–31]. In par-ticular, the question of ergodicity of physical quantitieslike magnetization, classical correlations, and entangle-ment in quantum XY spin chains have been investigated[27–30, 32].Here we consider the validity of the statistical me-chanical description of quantum correlation measures ofanisotropic Heisenberg models in one-dimension (1 D ),ladder and two-dimension (2 D ). Specifically, we findthat the entanglement measures remain ergodic, irrespec-tive of the initial strength of the applied magnetic fieldin the z -direction and the interaction strengths, whereasfor intermediate values of the initial magnetic field, theinformation-theoretic measures like quantum discord andquantum work-deficit show a transition from nonergodicto ergodic behavior, with the tuning of the strength ofthe two-body interaction in the z -direction. The resultshold irrespective of the relative strength (“anisotopy”)of the xx - and yy -interactions. However, the transitionpoint depends on the xy -anisotropy (i.e., the parameterthat controls the relative strength of the xx - and yy -interactions) and the strength of the magnetic field.The paper is organized as follows. In Sec. II, we give abrief description of the model under investigation. In Sec.III, we discuss about the canonical equilibrium state, thetime-evolved state, and ergodicity. Calculation of bipar-tite quantum correlations requires the two-site densitymatrix of the system. We discuss properties of single-site and two-site density matrices for the equilibrium aswell as the time-evolved state in Sec. IV. Measures ofquantum correlations for both the paradigms are definedin Sec. V. We present our results in Sec. VI (for 1 D ),in Sec. VII (for ladder), and in Sec. VIII (for 2 D ). Weconclude in Sec. IX. II. THE MODEL
We consider a system of N quantum spin- particlesarranged in a lattice with unequal nearest-neighbor in-teractions along x, y, and z directions. It is therefore theantiferromagnetic anisotropic Heisenberg model or theXYZ model, and is given by H int = 14 X [ J x σ x~i σ x~j + J y σ y~i σ y~j + J z σ z~i σ z~j ] , (1)where σ a~i ( a = x, y, z ) are the Pauli spin matrices atthe site ~i of the spin lattice, and J x , J y , and J z repre-sent the coupling constants in the x, y and z directionsrespectively. The summation in Eq. (1) runs over allnearest-neighbor pairs on the lattice. Periodic bound-ary conditions are assumed in all cases considered in thispaper. We will consider systems of quantum spins ar-ranged in lattices in different low dimensions. Assign-ing different relations among J x , J y , and J z , in the aboveHamiltonian lead to various other well-known spin mod-els, including the isotropic Heisenberg model for which J x = J y = J z , and the anisotropic XY model for which J x = J y , J z = 0. To check for ergodic properties of differ-ent physical quantities of these Heisenberg spin models,we will consider the initial state of the evolution to bethe canonical equilibrium state at the initial instant (seediscussion in the succeeding section). A non-trivial evo-lution of the system can be obtained in this case by intro-ducing a magnetic field represented by H mag , in such away that [ H int , H mag ] = 0. Hence the total Hamiltoniancan now be written as H ( t ) = H int − h ( t ) H mag . (2)For the present paper, we choose J x = J (1 + γ ) J, J y = J (1 − γ ) , J z = Jδ and H mag = J P σ z~i , with the sum-mation running over all sites of the lattice. Here J > γ and δ are dimensionless system parameters. Here γ represent xy -anisotropy. For brevity, we will sometimes call it sim-ple as “anisotropy”. The time-dependence of the appliedmagnetic field is of the form h ( t ) = (cid:26) a, t ≤ , t > , (3) where a = 0 is a dimensionless parameter. Here t repre-sents the time. Therefore, the total Hamiltonian is givenby H ( J, γ, δ, h ( t )) = J P [(1 + γ ) σ x~i σ x~j + (1 − γ ) σ y~i σ y~j + δσ z~i σ z~j ] − J h ( t ) P σ z~i . (4)When h ( t ) = 0 and J x = J y = J z , the above Hamiltonianis exactly solvable by using Bethe ansatz [21] by whichthe ground state energy can be obtained [33]. However,there exists no such exact solution for the anisotropicHeisenberg model. Moreover, we wish to study the evo-lution of the system and hence require the single site- andtwo-site properties of the entire energy spectrum of thesystem at a given time. Hence, to study the statisticalmechanical properties of such systems at finite temper-ature, we opt for exact diagonalization using numericalsimulations. III. STATISTICAL MECHANICALPROPERTIES
In this paper, we aim to study the statistical mechan-ical properties of the anisotropic Heisenberg model intime-dependent external magnetic fields. The statisti-cal mechanical notions like canonical equilibrium state,time-evolved state and ergodicity will be briefly definedin this section, mainly to set the terminology and the no-tations. In particular, we introduce a quantity called the“ergodicity score” which helps us to quantify the degreeto which a physical quantity is possibly nonergodic.
A. Time-evolution
For the quantum spin system, described by the Hamil-tonian in Eq. (4), we denote the canonical equilibriumstate of the system, at time t , as ρ βeq , and is given by ρ βeq ( t ) = exp( − βH ( t )) Z , (5)where Z is the partition function, Z = tr[exp( − βH ( t ))] , and β = k B T , with k B being the Boltzmann constant. T represents the absolute temperature.The canonical equilibrium state can evolve due to theapplication of external “disturbances”, like switching onof the magnetic field across the system. In our case,the evolution of the system is governed by the Hamilto-nian given in Eq. (4). We assume that the system is incontact with a heat bath at temperature T ′ for a longtime until t = 0. We assume that the contact is in thecanonical sense, so that the system and the heat bathexchange energy (under the normal average energy con-straint), but do not exchange particles. We assume thatthis contact leads the system to the canonical equilibriumstate at t = 0, i.e., the state of the system at t = 0 is ρ αeq (0), where α = k B T ′ . For t >
0, the magnetic fieldis switched off, and we consider the situation where thecontact with the heat bath is also cut off for all times t >
0. The system therefore starts evolving according tothe Schr¨odinger equation governed by the Hamiltonianin Eq. (4), with the initial state of this evolution being ρ αeq (0), and we denote the corresponding evolved state as ρ α ( t ). Note here that ρ αeq ( t = 0) = ρ α (0). B. Ergodicity and Ergodicity Score
To check whether a given physical quantity Q is er-godic, we consider the value of Q in the evolved state ata “large time”. The time of evolution, t l , is termed aslarge, for the physical quantity Q , if (i) there are no fluc-tuations in the physical quantity Q with respect to timefor t > t l , or if (ii) the fluctuation amplitude of Q with re-spect to time is smaller than the required precision level,for t > t l , or if (iii) the fluctuations of Q with respect totime is of a constant amplitude. We are interested in thetime-average of the physical quantity Q at large-times.For the cases (i) and (ii), an explicit time-averaging isnot required, as the system dynamics brings the quan-tity Q to its time-averaged value. For the case (iii), anexplicit time-averaging for times t > t l is required. Wenow ask whether there exists a temperature T ′ , at whichthe large-time time-averaged value of a physical quantity Q in the evolved state is equal to the value of same phys-ical quantity in the equilibrium state at temperature T at large-time. The physically relevant range of T can beconsidered as up to an order of magnitude of the initialtemperature T ′ . This difference between T and T ′ is, forexample, to allow for possible errors in an experimentalrealization of the physical system or some typical theo-retical effective standard deviation in that system.If the time-average of a physical quantity is the same asthe ensemble average, the quantity is said to be ergodic.Such a study is therefore based on the comparison ofthe large-time time-averaged value, Q ∞ ( T ′ , a ), with thecanonical equilibrium value, Q can ( T, h ( t = ∞ )). Notethat these quantities also depend on the system param-eters J, γ, and δ . The physical quantity Q is thereforesaid to be ergodic if Q ∞ ( T ′ , a ) = Q can ( T, h ( t = ∞ )) . (6)Otherwise, it is termed as nonergodic.Let us now introduce a quantity which can quantifythe degree to which a given physical quantity, Q fails tobe ergodic. We call it the “ergodicity score”, and defineit as Q (˜ δ, α ) = max[0 , Q ∞ ( T ′ , a ) − max T Q can ( T, h ( t = ∞ ))](7)where ˜ δ denotes the aggregate of all physical parametersrequired to define the Hamiltonian of the system under consideration. For the system considered in this paper, ˜ δ consists of J, h, δ and γ . We remember that α = k B T ′ .The maximization over T is for all T that falls in thephysically relevant range around T ′ , as discussed earlier.Note therefore that a non-zero value of η Q implies that Q is nonergodic and that a vanishing Q indicates ergodicity. IV. SINGLE- AND TWO-SITE DENSITYMATRICES OF TIME-DEPENDENTHEISENBERG MODEL
To analyze the ergodic properties of quantum corre-lations, let us now find the general form of the single-and two-site density matrices of equilibrium and evolvedstates of the Hamiltonian given in Eq. (4). The generalsingle-site density matrix is given by ρ = 12 [ I + ~m.~σ ] , (8)where I is the 2 × ~m = tr[ ρ ~σ ] is themagnetization vector. If the entire system is of N qubits,then the single-site density matrix can be obtained bytracing out N − N − ρ β ∗ eq ( t ) = ρ βeq ( t ), where the complex conjugation has been taken inthe computational basis, m x = 0. Moreover, in this case, m y = 0, since [ H, Q i σ zi ] = 0. Therefore, the single-sitedensity matrix for the equilibrium state reduces to ρ eq ( t ) = 12 [ I + m eqz ( t ) σ z ] . (9)where we have hidden the dependence on temperaturein the notation. The single-site density matrix for theevolved state also turns out to be ρ ( t ) = ( I + m z ( t ) σ z ),using the Wick’s theorem [28, 34].The nearest-neighbor two-site density matrix can bewritten, in general, as ρ = 14 [ I ⊗ I + ~m.~σ ⊗ I + I ⊗ ~m.~σ + X i,j = x,y,z T ij ( σ i ⊗ σ j )]where T ij = tr[( σ i ⊗ σ j ) ρ ] represent the two-site cor-relation functions. Since periodic boundary conditionsare assumed, the nearest-neighbor state ρ is indepen-dent of which two neighboring sites are chosen for con-structing the nearest-neighbor state. Due to the form ofthe single-site density matrices that has already been de-rived, the two-site density matrices for both equilibriumand evolved states reduces to ρ = 14 [ I ⊗ I + m z ( σ z ⊗ I + I ⊗ σ z ) + X i,j = x,y,z T ij ( σ i ⊗ σ j )] . Using Wick’s theorem, we can show that all off-diagonalcorrelations vanish for the equilibrium state. However,for the evolved state, only xz - and yz -ones vanish. V. MEASURES OF QUANTUM CORRELATION
We will now quickly define the measures of quan-tum correlations used in this paper. We will intro-duce two measures within the entanglement-separabilityparadigm, namely logarithmic negativity and concur-rence. We will subsequently define two information-theoretic quantum correlation measures, viz. quantumdiscord and quantum work-deficit.
A. Logarithmic Negativity
Given a bipartite quantum state, ρ AB , shared betweentwo parties A and B , the logarithmic negativity [35]quantifies the amount of entanglement present in the bi-partite state. The definition of logarithmic negativityis based on negativity, N ( ρ AB ), which is defined as thesum of the absolute values of the negative eigenvalues ofthe partial transposed density matrix [36] of the bipartitestate ρ AB . The logarithmic negativity (LN) is defined as E N ( ρ AB ) = log [2 N ( ρ AB ) + 1] . (10)For two qubit states, LN is positive if and only if thestate is entangled [36]. B. Concurrence
For two-qubit states, concurrence is another useful en-tanglement measure [37]. For a two-qubit mixed bipartitestate ρ AB , it is defined as C ( ρ AB ) = max[0 , λ − λ − λ − λ ] , (11)where λ , λ , λ , λ are the square roots of the eigenval-ues of ρ AB ˜ ρ AB in decreasing order and ˜ ρ AB = [ σ y ⊗ σ y ) ρ ∗ AB ( σ y ⊗ σ y ], with the complex conjugation beingtaken in the computational basis. The maximum is takento ensure that concurrence is zero for separable states.The measure is non-zero for all entangled states. C. Quantum Discord
Quantum discord [15, 16] is an information-theoreticmeasure of quantum correlation. It is defined as D ( ρ AB ) = I ( ρ AB ) − J ( ρ AB ) . (12)Here I ( ρ AB ) and J ( ρ AB ) are equivalent in classical in-formation theory, where they both represent the mutualinformation between two random variables. In the quan-tum world, the first term represents the total correlationof the bipartite state ρ AB , and is given by I ( ρ AB ) = S ( ρ A ) + S ( ρ B ) − S ( ρ AB ) , where S ( ρ ) = − tr[ ρ log ρ ] is the von Neumann entropyof a quantum state ρ , and ρ A and ρ B are the reduceddensity matrices of ρ AB . The second term, J ( ρ AB ) inthe definition of quantum discord can be argued as theamount of classical correlation present in ρ AB , and isdefined by J ( ρ AB ) = S ( ρ A ) − S ( ρ A | B ) . Here S ( ρ A | B ) = min { B i } P i p i S ( ρ A | i ) is the conditionalentropy of ρ AB , when the rank-1 projection-valued mea-surement, { B i } , is performed on the B -part of the sys-tem, with ρ A | i = tr B [( I A ⊗ B i ) ρ AB ( I A ⊗ B i )], p i =tr AB [( I A ⊗ B i ) ρ ( I A ⊗ B i )], and with I A being the identityoperator on the Hilbert space of A . D. Quantum Work-Deficit
Another information-theoretic measure of quantumcorrelation is the quantum work-deficit, which is definedas the difference between the amount of work extractablefrom a shared state by global and local quantum heat en-gines [17]. It is possible to quantify the amount of workthat can be extracted from a bipartite state ρ AB by globaloperations as I G ( ρ AB ) = N − S ( ρ AB ) , (13)where N is the logarithm (base 2) of the dimension ofthe Hilbert space on which ρ AB is defined. It can beinterpreted as the number of pure qubits that can be ex-tracted from ρ AB by global operations on the state, andthat consists of an arbitrary sequence of unitary and de-phasing operations. Such operations are called “closedglobal operations”. Let us now define “closed local oper-ations and classical communication (CLOCC)”. It con-sists of local unitaries, local dephasing, and sending thedephased state from one party to other. The number ofpure qubits that can be extracted by CLOCC is given by, I L ( ρ AB ) = N − inf Λ ǫ CLOCC [ S ( ρ ′ A ) − S ( ρ ′ B )] , (14)where S ( ρ ′ A ) = S (tr B [Λ( ρ AB )]) and S ( ρ ′ B ) = S (tr A [Λ( ρ AB )]). The quantum work-deficit is defined as W D ( ρ AB ) = I G ( ρ AB ) − I L ( ρ AB ) . (15)In the next sections, our aim is to study the ergod-icity of these quantum correlations in the anisotropicHeisenberg models of different lattice geometries. Thelattices considered are the chain, the ladder, and the two-dimensional square lattice. Periodic boundary conditionsis used in all cases. VI. QUANTUM HEISENBERG XYZ SPINCHAIN WITH MAGNETIC FIELD
In this section, we investigate the statistical me-chanical properties of quantum correlation measures inthe one-dimensional quantum spin- lattice describedby the Hamiltonian in Eq. (4). The isotropic anti-ferromagnetic Heisenberg model in one-dimension pro-vides an understanding of the spin-spin correlationfunctions and suppression of long range magnetic or-der in spin-liquids. Moreover, some materials likeSr CuO and SrCuO mimic the Heisenberg spin chain[19]. Recently developed techniques make it possible torealize this model in physical systems like photons [24],trapped ions [25], and cavity QED [26]. Entanglementin the ground and the thermal states of the Heisenbergmodel have been studied [38]. A. Quantum correlations in equilibrium andevolved states
For any system, that is in its canonical equilib-rium state, all quantum correlations vanish when thetemperature goes to infinity. Measures that are de-fined within the entanglement-separability paradigm typ-ically vanish even for moderately high temperatureswhile information-theoretic measures like quantum dis-cord goes to zero asymptotically with the increase oftemperature. This feature is retained by the systemdescribed by the Hamiltonian in Eq. (4), on an one-dimensional lattice with periodic boundary conditions.This shows that information-theoretic quantum correla-tion measures are more robust to temperature when com-pared to entanglement-separability measures. Moreover,we observe that the entanglement of the nearest-neighborreduced state of the canonical equilibrium state behavesdifferently with temperature in different ranges of γ and δ . See Figs. 1(a) and 1(b). In particular, we find thatfor fixed low values of the anisotropy, γ , the entangle-ment saturates to a value with increasing β , and thissaturated value is more or less independent of δ , the rel-ative strength of the zz -interaction. However, when γ isrelatively high, entanglement saturates to a low value forsmall δ , while for high δ , it saturates to a higher value.On the other hand, quantum discord saturates to a lowvalue with decreasing temperature for small δ as well asfor high δ , while it saturates to a high value for interme-diate values of δ (see Figs. 1(c) and 1(d)). This behaviorof quantum discord is true for all values of γ . However,with the increasing of the value of γ , the point where themaximum value of quantum discord is obtained, shiftsto higher values of δ . We have performed calculationsalso for concurrence and quantum work-deficit, and theyhave qualitatively similar features as logarithmic nega-tivity and quantum discord respectively.Let us now discuss the time-dynamics of entanglementand other quantum correlations in the nearest-neighborstate. For the discussion, we choose γ = 0 .
8. However,the behavior remains the same for other moderate val-ues of γ . The entanglement measures collapse and revivenon-periodically with time, when δ is small. See Fig.2(a), where we can view this feature for logarithmic neg-
0 5 10 15 20J β δ
0 0.1 0.2 0.3 0.4 0.5 0 5 10 15 20J β δ
0 0.1 0.2 0.3 0.4 0 5 10 15 20J β δ
0 0.1 0.2 0.3 0 5 10 15 20J β δ FIG. 1. (Color online) Behavior of quantum correlations inthe equilibrium state. We plot quantum correlation measuresof nearest-neighbor reduced states of the canonical equilib-rium states, for a system of 12 quantum spin- particles ar-ranged as a ring and described by the Hamiltonian H withrespect to Jβ , and the relative strength of the zz -interaction, δ , for different values of γ . The top plots are for logarithmicnegativity and the bottom ones are for quantum discord. Theleft plots are for γ = 0 . γ = 0 . ativity. For intermediate values of δ , revival of entangle-ment occurs less frequently (Fig. 2(b)). For very high δ ,the model is “Ising-like”, and the entanglement as well asother quantum correlation measures collapse and reviveperiodically with time. The non-periodic collapse andrevival behavior persists up to moderate values of δ forthe information-theoretic quantum correlation measureslike quantum discord. See Figs. 2(c) and 2(d). B. Statistical mechanical properties of quantumcorrelation measures
We now examine the ergodicity properties of the quan-tum correlation measures. From Figs. 1 and 2, by analyz-ing the behavior of the entanglements of the equilibriumand evolved states, we find that entanglement measuresare ergodic for all values of δ , γ ( = 0), and a . We haveanalyzed this for logarithmic negativity as well as for con-currence. Hence, the ergodicity score is vanishing for allsystem parameters for all such measures.Quantum discord and quantum work-deficit, bothinformation-theoretic measures, also remain ergodic,when δ ≥ γ . However, for δ < γ these measures ex-hibit nonergodicity for a large range of the magnetic field.In Fig. 3, we plot η D with respect to the δ and thefield strength, a , for γ = 0 .
8, where we assume that thetime-evolution starts off from the canonical equilibriumstate for the Hamiltonian in Eq. (4) at t = 0 and fortemperature given by Jα = 20. To plot η D , we choose Jβ = 20 for the equilibrium state, in the calculation of FIG. 2. (Color online) Quantum correlations of the time-evolved states. The system under consideration is the same as inFig. 1, but for 8 spins. The evolution is assumed to begin in the equilibrium state at t = 0 and at an exemplary value of thetemperature given by Jα = 20. Logarithmic negativity (top plots) and quantum discord (bottom plots) of the nearest-neighborreduced states of the time-evolved states, are plotted against the initial magnetic field, a , and Jt ~ , for different values of δ . Herewe choose γ = 0 .
8. The left plots are for δ = 0 . δ = 0 .
8. All axes correspond to dimensionlessquantities except those for quantum discord, which is measured in bits. Q can ( T, h ( t = ∞ )), since we find that the quantum dis-cord of the equilibrium state is a monotonically increas-ing function with respect to Jβ and saturates for a Jβ much below Jβ = 20.The trends, with respect to δ , of ergodicity scores ofquantum discord and quantum work-deficit for different γ , are depicted in Fig. 4. For a fixed anisotropy γ , therealways exists a certain value of δ , for which quantum dis-cord changes from being nonergodic to being ergodic. Wedenote that critical value of δ as δ γc , remembering thatit pertains to quantum discord, and that there is a sim-ilar critical δ , at a possible different value, for quantumwork-deficit. We observe that the δ γc increases with theincrease in γ , and in Fig. 4, δ γ =0 . c < δ γ =0 . c < δ γ =0 . c forboth quantum discord and quantum work-deficit.The general behavior, of the quantum correlation mea-sures in this system, that is emerging, is as follows.Entanglement measures exhibit ergodic behavior in allrelevant parameter domains. The picture is richer forinformation-theoretic quantum correlation measures, andin particular, for a given anisotropy γ and a given mea-sure, there is a critical δ = δ γc at which the system transitsfrom nonergodic to ergodic behavior for that measure. VII. QUANTUM HEISENBERG XYZ SPINLADDER WITH MAGNETIC FIELD
It is interesting to study whether the two quantumcorrelation paradigms showing opposing statistical me-chanical behavior persists in higher-dimensional systems.To find this, we first consider the spins in a ladder ar-rangement, which is made up of two Heisenberg XYZspin- chains, coupled by the same interactions alongthe rungs [39]. There is the time-dependent z -field at allsites. Periodic boundary condition is assumed along therails. Such systems can be found in solid state materi-als like Sr CuO and Sr Cu O [19]. Recently it wasfound that the entanglement spectrum [40] of the groundstate of this model is related to the energy spectrum ofits two single Heisenberg chain [41].In this model, the quantum correlation measures ofthe evolved and equilibrium states behave in a similarfashion as for the XYZ chain. In particular, entangle-ment of the nearest-neighbor states remain ergodic in thiscase. And there exists a critical δ , above which the time-averaged value of the information-theoretic correlationmeasures, quantum discord and quantum work-deficit, ofthe nearest-neighbor reduced states of the evolved statesmatch with the same measure of the equilibrium state,for some β , in a given magnetic field and a given γ (seeFig. 6 for the states along the rails). Quantum discord ofthe long-time equilibrium state does not remain a mono- FIG. 3. (Color online) Ergodicity score for quantum discord.The ergodicity score for quantum discord of the anisotropicHeisenberg XYZ chain (with a magnetic field) of 8 spins, ar-ranged in a ring, is plotted against δ and the applied initialmagnetic field a , for a fixed γ = 0 .
8. The initial state of thetime-evolution is the t = 0 canonical equilibrium state at atemperature given by Jα = 20. The ergodicity score is mea-sured in bits. All other physical parameters used in the figureare dimensionless. η D δ η W D δ FIG. 4. (Color online). Comparing ergodicity scores for quan-tum discord and quantum work-deficit. The ergodicity scorefor quantum discord (left) and quantum work-deficit (right) ofthe nearest-neighbor reduced state of the time-evolved statesof the anisotropic Heisenberg XYZ chain (with a magneticfield) of 12 spins, arranged in a ring, is plotted against δ ,for different values of γ and a for fixed initial magnetic field a = 0 .
6. Here we choose Jα = 20 for the t = 0 canonicalequilibrium state from which the evolution starts off. Thedepicted curves are for γ = 0 . γ = 0 . γ = 0 . tonically increasing function with β like in the 1 D model.See Fig. 5. To calculate η D , we choose Jβ = 60, at whichthe maximum value of Q can ( T, h ( t = ∞ )) is attained, forall values of δ . δ γc increases with the increase in γ , whileit is independent of the choice of the initial applied mag-netic field for a fixed γ . These qualitative features ofquantum discord remain the same, when a rung of theladder is considered. A similar feature is observed forquantum work-deficit of the rung and rail states. SeeFig. 6(b) in this respect. We therefore again find that the strength of the zz -interaction, as quantified by δ , canbe adjusted in such a way that the nonergodic nature ofthe information-theoretic measures, that persists in thissystem for low δ , gets washed off, and we obtain ergodicbehavior for high δ . VIII. 2D QUANTUM HEISENBERG XYZMODEL WITH MAGNETIC FIELD
The two-dimensional Heisenberg model describes im-portant systems, including materials like SrCu (BO ) and CaV O [42]. Experimental studies of the Heisen-berg model in 2 D lattices have been proposed e.g., intrapped ions [43] and optical lattices [44].We consider a quantum Heisenberg XYZ spin modelon a square lattice with antiferromagnetic interactionsbetween the nearest-neighbor spins. Periodic boundarycondition is assumed and hence, geometrically, the sys-tem forms a spin-arrangement on a torus. The time-dependent magnetic field is assumed to be active at allsites. Like in the ladder and 1 D models, we again findthat the entanglement measures are ergodic for all val-ues of γ , δ , and the initial magnetic field a . Interest-ingly, unlike in the 1 D and ladder systems, the transi-tion from nonergodicity to ergodicity of the information-theoretic measures, occurs for relatively low values of the zz -interaction strength, i.e., for low values of δ (Fig. 7).For example, when γ and h are 0.6, in the ladder and1 D systems, both quantum discord and quantum work-deficit remain nonergodic till δ ≈ .
8, while they bothbecome ergodic in 2 D at δ ≈ .
16. These observationslead us to infer that information-theoretic measures aremore sensitive to the dimension of the lattice, than theentanglement measures, with respect to their statisticalmechanical properties.
IX. DISCUSSION
Quantum Heisenberg models have created lot of in-terest due to their rich physical properties and the pos-sibility of realizing such systems in artificial materialsas well as in inorganic compounds. However, investi-gations into the dynamics of such models, for example,under the influence of time-dependent magnetic fields,are limited by the fact that the system cannot be diago-nalized analytically. Here, we have studied the behaviorof quantum correlations, both from the entanglement-separability paradigm and the information-theoretic one,of the equilibrium state as well as the evolved state of thequantum Heisenberg anisotropic XYZ model, by numeri-cal simulations. In particular, we found that althoughentanglement measures are ergodic irrespective of thesystem parameters, information-theoretic measures ex-hibit a rich picture, with respect to their statistical me-chanical properties. Specifically, we find that the zz -interaction strength has a cross-over value, for a given FIG. 5. (Color online) Behavior of quantum correlations in the equilibrium state. We plot quantum correlation measures ofnearest-neighbor reduced states of the canonical equilibrium states, for a system of 8 quantum spin- particles arranged as aladder and described by the Hamiltonian H with respect to Jβ , and the relative strength of the zz -interaction, δ , for differentvalues of γ . The top plots are for logarithmic negativity and the bottom ones are for quantum discord. The left plots are for γ = 0 . γ = 0 .
8. Quantum discord is measured in bits. All other axes in the figures correspond todimensionless parameters. η D δ
0 0.2 0.4 0.6 0.8 1 η W D δ FIG. 6. (Color online). Ergodicity curves in the HeisenbergXYZ ladder. The ergodicity scores of quantum discord (left)and quantum work-deficit (right) of a nearest-neighbor re-duced state, along a rail, of the time-evolved state, in theladder, of 8 spins is plotted with respect to the relativestrength of the zz -interactions. The transition points, wherethe system moves from nonergodic to ergodic behavior ofthe information-theoretic measures are qualitatively similar tothose in one-dimension, for a fixed γ . The depicted plots arefor γ = 0 . γ = 0 . γ = 0 . a = 0 . , Jα = 20. Theunits are the same as in Fig. 4. xy -anisotropy and a given information-theoretic quan-tum correlation measure, that indicates a transition fromnonergodic to ergodic behavior for that measure. Thequalitative features of the measures in the entanglement-separability paradigm and the information-theoretic oneare the same in the one-dimensional, ladder, and two- η D δ η W D δ FIG. 7. (Color online). Ergodicity scores in the 2 D Heisen-berg XYZ model. The ergodicity scores of quantum dis-cord (left) and quantum work-deficit (right) in the nearest-neighbor reduced state of the time-evolved state, with re-spect to the strength of the zz -interaction for the anisotropicHeisenberg XYZ model on a 2 D square lattice, consisting of12 spins in a torus. The plots are for γ = 0 . γ = 0 . a = 0 . , Jα = 20. The units are the same as in Fig. 4. dimensional square lattices. However, in the square lat-tice, the information-theoretic measures are more sensi-tive to the change of the zz -interaction strength than inother dimensions. Such dimension-dependent change ofergodic behavior is absent for entanglement measures. ACKNOWLEDGMENTS
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