Phase diagrams, quantum correlations and critical phenomena of antiferromagnetic Heisenberg model on diamond-type hierarchical lattices
Pan-Pan Zhang, Zhong-Yang Gao, Yu-Liang Xu, Chun-Yang Wang, Xiang-Mu Kong
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Critical phenomena and quantum discord of quantumantiferromagnetic Heisenberg model on diamond-typehierarchical lattices
Pan-Pan Zhang , Zhong-Yang Gao , Yu-LiangXu , Chun-Yang Wang and Xiang-Mu Kong , ∗ School of Physics and Optoelectronic Engineering,Ludong University, Yantai 264025, China College of physics and Engineering,Qufu Normal University, Qufu 273165, China (Dated: February 25, 2021)
Abstract
The anisotropic spin-1 / d f = 1 . , .
58, respectively. For system A, using the real-space renormalization group approach, wecalculate the phase diagram, the critical exponent and quantum discord, and find that there existsa reentrant behavior in the phase diagram. We also find that the quantum discord reaches itsmaximum at T = 0 and the thermal quantum discord decreases with the increase of L , and it isalmost zero at L ≥
30. No matter how large the size of system is, quantum discord will change to0 when anisotropic parameter ∆ = 1. For systems B and C, using the equivalent transformationand the real-space renormalization group method, we obtain phase diagrams and find that: theN´eel temperature tends to zero in the isotropic Heisenberg limit on d f = 2 system; there exists aphase transition in the isotropic Heisenberg model on system C. By studying quantum discord, wefind that there is a certain degree of twist of the relation curve between quantum discord and T when ∆ = 0. Moreover, as an example, we discuss the quantum effect in system A, which can beresponsible for the existence of the reentrant behavior in the phase diagram. PACS numbers: ∗ Corresponding author. E-mail address: [email protected] (X.-M. Kong). . INTRODUCTION The research on fractal lattice system is always a hot topic because of its unique structure,there are already massive researches about classical systems[1–3], but the research workabout quantum system are relatively less[4, 5]. In 2003, de Sousa et al. studied the phasediagram of the two-dimensional quantum spin anisotropic Heisenberg model[5]. Recently,Sierpinski fractal materials have been synthesized in the laboratory[6]. It is necessary tofurther study the fractal lattice.As an important parameter to describe phase transition, the critical temperature hasan important effect on critical behavior. The research shows that the critical temperature T c = 0 and T c > d f = 2 and d f = 3, respectively[5, 7, 8],but antiferromagnetic (AF) Heisenberg systems need further study. Various renormalizationgroup approaches[9–12], Green’s function technique[13, 14], spin-wave theory [15] and MonteCarlo method [16, 17] have been applied to study the quantum AF Heisenberg systems.de Sousa et al. calculated the phase diagram of the AF Heisenberg model by the mean-field RG approach and found that the AF N´eel temperature T N < T c for 2D lattices and T N > T c for 3D lattices [12, 18]. However, results from spin-wave theory and Green’s functiontechnique on the isotropic Heisenberg model obtained T N = T c for 2D and 3D lattices in AFHeisenberg system[14, 15, 19, 20]. Quantum Monto Carlo results indicate that an orderedlow temperature phase is obtained even at much smaller anisotropies ∆ ∼ − for 2Dlattices in AF Heisenberg system[16, 21]. From these results, we can see that the situationfor fractal lattice system is still unclear and it is worth our further study.Quantum correlation, as an important quantum resource, has been applied to quantumcommunication, quantum teleportation, and quantum computation, etc[22, 23]. Quantumcorrelation contains more quantum information than quantum entanglement which is oneparticular quantum correlation[24, 25]. The quantum discord (QD) which captures nonclas-sical correlation even without entanglement is proved as an effective measure for all aspectsof quantum correlation[26, 27]. As a wider class of measures than entanglement, QD has be-come an active research topic over the past few years[28–30], which has important applicationvalue in quantum information processing, quantum dynamics and even biophysics[31–34].But, it isn’t difficult to find that the present studies have focused on the one-dimensional2pin chain, the studies of fractal lattices are less[29, 35].In this work, we investigate the phase diagrams and QD of the anisotropic AF Heisen-berg model on three kinds of diamond-type hierarchical lattices (systems A, B and C), withfractal dimensions d f = 1 . , .
58, respectively, which is shown in Fig. 1. Using thereal-space RG method straightforwardly, we study the phase diagram, some critical behav-iors and quantum correlations under different temperature and the anisotropic parameterson system A. For systems B and C, we apply an effective combination method of the equiv-alent transformation (ET) and the real-space RG method, which has been used for classicalsystems in Ref. [3] but for quantum systems this is done for the first time as far as we know.The structure of this paper is organized as follows. In Sec. II we outline the method andthe critical behaviors for system A. In Sec. III the method and the critical behaviors arepresented for systems B and C. Sec. IV discusses the quantum correlations for the threelattices. In Sec. V the analyses of the quantum effect is discussed. The summary is given inthe the last section.
II. THE METHOD AND CRITICAL BEHAVIORS FOR SYSTEM A
The anisotropic spin-1 / H = K X h ij i (cid:2) (1 − ∆)( σ xi σ xj + σ yi σ yj ) + σ zi σ zj (cid:3) , (1)where K = J/k B T , J is the exchange coupling constant (with J <
J > k B is the Boltzmann constant, T is the temperature, h ij i denotes nearest-neighbor spin pairs, ∆ ∈ [0 ,
1] is the anisotropicparameter, and σ αi ( α = x, y, z ) are components of a spin-1 / i . Note that, asparticular cases, the Hamiltonian (1) describes the Ising model (∆=1), isotropic Heisenbergmodel (∆=0) and XY model (∆=- ∞ ), respectively.In this section, using the real-space RG method[4, 36], we study the phase diagramand some critical behaviors of the anisotropic Heisenberg model on the DH lattice A. Suchkind of lattices are constructed in an iterative manner, starting from a two-point lattice[37,38] (Fig. 1). We take out the generator from system A to perform the RG transformation,which is shown in Fig. 2( a ). As we can see, after summation of the internal spins, thegenerator (Fig. 2( a σ and σ , joined3y a single bond (Fig. 2( a H ′ ) = Tr exp( H ) , (2)where H ′ = K s [(1 − ∆ s )( σ x σ x + σ y σ y ) + σ z σ z ] + K and H = K (1 − ∆)[( σ x σ x + σ y σ y ) + ( σ x σ x + σ y σ y ) + ( σ x σ x + σ y σ y )+( σ x σ x + σ y σ y ) + ( σ x σ x + σ y σ y ) + ( σ x σ x + σ y σ y )]+ K ( σ z σ z + σ z σ z + σ z σ z + σ z σ z + σ z σ z + σ z σ z )are the Hamiltonians of the renormalized two-site cluster and of the six-site cluster (seeFig. 2( a )), respectively. Tr denotes a partial trace over the internal σ i ( i = 3 , , , K has been included in H ′ . The RG recursionrelations between the renormalized parameters ( K s , ∆ s ) and the original parameters ( K, ∆)are determined by Eq. (2). Notice that the noncommutativity between the Hamiltonians ofthe neighboring clusters is neglected, and therefore our results are approximations, and adiscussion for this approximation will be presented in Sec. V.In order to calculate the partial trace in Eq. (2), we expand exp( H ′ ) asexp( H ′ ) = a ′ + b ′ ( σ x σ x + σ y σ y ) + c ′ σ z σ z , (3)where a ′ , b ′ and c ′ are functions of K ′ , ∆ ′ and K . In the representation of the directproduct of σ z and σ z , both the left-hand side and the right-hand side of Eq. (3) can beexpressed as 4 × K s ) = ( a ′ + c ′ ) ( a ′ − c ′ ) − b ′ , (4)exp(4 K s ∆ s ) = ( a ′ + c ′ ) ( a ′ + 4 b ′ − c ′ ) (5)and exp( K ) = a ′ + c ′ exp( K s ) . (6)Similarly, we expand exp ( H ) asexp( H ) = a + X h i 4n which A = e ij,kl ( σ xi σ xj + σ yi σ yj )( σ xk σ xl + σ yk σ yl ) + d ij,kl ( σ xi σ xj + σ yi σ yj ) σ zk σ zl + f ij,kl σ zi σ zj σ zk σ zl and B = g ij,kl,mn ( σ xi σ xj + σ yi σ yj ) σ zk σ zl σ zm σ zn + p ij,kl,mn ( σ xi σ xj + σ yi σ yj )( σ xk σ xl + σ yk σ yl ) σ zm σ zn + q ij,kl,mn ( σ xi σ xj + σ yi σ yj )( σ xk σ xl + σ yk σ yl )( σ xm σ xn + σ ym σ yn ) , where a , { b ij } , { c ij } , . . . { q ij,kl,mn } and r depend on K and ∆. From Eqs. (2), (3) and (7),it can be obtained that a ′ = 16 a , b ′ = 16 b , and c ′ = 16 c , and the recursion relationsbecomes exp(4 K s ) = ( a + c ) ( a − c ) − b , (8)exp(4 K s ∆ s ) = ( a + c ) ( a + 2 b − c ) . (9)By numerically iterating Eqs. (8) and (9), we obtain the AF phase diagram which isshown in Fig. 3. The inset shows the critical curve of the F system in Fig. 3 for comparison.We can see that there exists two phases, namely, an ordered phase and a disordered onecharacterized by fully stable fixed points ( ∞ , 1) and (0 , . , 1) and (0 , ν = ln b ln λ = 1 . , (10)where b = 3 is the scaling factor and λ ≡ ( ∂K s /∂K ) | ∆=1 ,K =0 . . At the isotropic Heisenbergfixed point of the F model, we obtain ν = ∞ which is consistent with the exact result [39].For the AF model, the phase diagram is different from the F one: the transition line goesto zero at a value of ∆ c = 0 . T → 0. It is found that the criticaltemperature behaves as T c ∼ ∆ (11)5or the F case (Fig. 4( a )), and the N´eel temperature T N behaves as T N ∼ ln (∆ c − ∆) (12)near ∆ c = 0 . 703 (Fig. 4( b )). III. THE METHOD AND CRITICAL BEHAVIORS FOR SYSTEMS B AND C In this section, we use an effective combination method of the equivalent transforma-tion (ET) and the real-space RG method to the AF Heisenberg model on systems B and C,which has been used for classical systems in Ref. [3] but for quantum systems this is donefor the first time as far as we know. As we can see, Fig. 2( b ) has antiferromagnetic interac-tions ( K < 0) and after the ET, it has ferromagnetic interactions ( K ′ > b )): (1) The ETtechnique is used to transform the AF system into an equivalent F one and the ET equationsare obtained. (2) By numerically iterating the RG recursion equations of the equivalent Fsystem, the critical points can be obtained. (3) The results obtained from the second stepare substituted into the ET equations in the first step and the AF phase diagram can beobtained.In the following, we give the calculation procedure of the AF Heisenberg model on systemB. To construct the ET equations, we transform the cluster of Fig. 2( b 1) (which containsfour σ i ( i = 1 , , , b 2) (the cluster contains two spins-1 and 2). Following thesame steps as in Sec. II, we can obtain the ET equations in the same form as Eqs. (4), (5)and (6), as well as the 16 × 16 matrix exp( H )exp ( H ) = a + X i 0. For the equivalent ferromagnetic system, we obtain the RG recursionrelations in the same form as Eqs. (14) and (15)exp(4 K ′′ ) = ( a + c ) ( a − c ) − b , (16)exp(4 K ′′ ∆ ′′ ) = ( a + c ) ( a + 2 b − c ) , (17)where a , b , c are functions of K ′ and ∆ ′ . Then, we substitute K ′ and ∆ ′ which are ob-tained by numerically iterating the ferromagnetic RG equations (16) and (17) into Eqs. (14)and (15), and finally get the phase diagram of the AF system.Using the same procedure, we calculate the phase diagram of system C. However, theanalytical expressions of the right-hand side of the ET equations of this system are difficultto calculate. In order to get the expressions, we express exp ( H ) approximately asexp ( H ) ≈ H + H 2! + · · · + H , (18)in which we adopt the first twelve terms for convenient calculation.Fig. 5 shows the phase diagram of systems B and C in ( k B T / | J | , ∆) space. The AFcritical line of system A is also depicted for comparison. There are two unstable fixed pointsfor system B at (1 . , 1) and (0 , 0) (for system C at (2 . , 1) and (2 . , . , 1) and (2 . , 1) are the attractors of all ∆ = 0 points in the critical lines.Physically, this indicate that the critical behavior along the lines are the same as that of theIsing model. Moreover, from Fig. 5, we find that:(1) For low-dimensional system A, there exists a reentrant behavior in the phase diagram.(2) For 2D system B, the N´eel temperature tends to zero when decreasing ∆.(3) For higher-dimensional system C, there exists phase transition in the isotropic Heisen-berg limit ∆ = 0. 7hich is in accordance with the previous quantum Monte Carlo[16, 21, 40, 41] and meanfield renormalization group[11, 12, 18, 42] results. From our results, we can see that as thevalue of d f increases the phase transition will present on the isotropic Heisenberg limit andas the value of d f decreases the reentrant behavior will appear, and 2D maybe the criticaldimension. We consider this phenomena to be the case for different fractal dimensions andthe quantum effects which will be discussed in Sec. V. Moreover, we also calculate the criticalbehavior when T → T N ∼ , (19)the critical temperature behaves is shown in Fig. 6. IV. THE QUANTUM CORRELATIONS FOR THE THREE LATTICES In this section, we study the QD between two non-nearest-neighbor end spins of AFHeisenberg model in three diamond-type hierarchical lattices at finite temperatures anddiscuss the relations of QD with temperature and anisotropic parameter. It is widely ac-cepted that quantum mutual information is a measure of the total correlation contained ina composite quantum system which includes classical and quantum correlations[43, 44]. Wewant to get quantum correlation by subtracting classical correlation from quantum mutualinformation. As a common representation of quantum correlation, QD captures nonclassicalcorrelation even without entanglement. For a quantum state ρ of the composite systemcontaining subsystems 1 and 2, the QD[29, 45] is defined as D ( ρ ) = I ( ρ ) − C ( ρ ) , where I ( ρ ) is the quantum mutual information and C ( ρ ) is the classical correlation.Because the density matrix ρ of two spins in AF Heisenberg model exhibits an X structure,it is called X state, as ρ = a a a a a a a a , a = a = e k e k + e k (2∆ − + e k (1 − ,a = a = 0 ,a = a = e k (1 − e k (2∆ − e k + e k (2∆ − + e k (1 − ) ,a = a = e k (1 − − e k (2∆ − e k + e k (2∆ − + e k (1 − ) . For two-qubit X state, there are some effective numerical and analytical expressions[45–47]. In this paper, we adopt the method in the Ref.[45] to obtain the QD of the two-pointlattice, we will not given the analytical expression here, because the form is complicated.Then the QD is obtained between two end sites on the fractal lattices (given in Refs.[48–50])by implementing the decimation RG method.Fig. 7 shows the QDs versus anisotropy parameter ∆ and the temperature T (with theunit | J | /k B ) in system A containing L sites. The variation trend of the QD between two endspins with temperature T is shown in Figs. 7 ( a ) and ( b ). It can be seen that the QD existsmaximum at T = 0 and decreases with the increase of k B T / | J | ; the thermal QD decreaseswith the increase of L , and it is almost zero at L ≥ 30. From Figs. 7 ( c ) and ( d ) we can findthat the QD decreases with the increase of ∆ , no matter how large the size of system is, QDwill change to 0 when ∆ = 1. And with the increase of L , the QD is smaller and smaller.Fig. 8 shows a sequential change of QD with anisotropy parameter ∆ and the temperature T . We can find that for L = 2 case, there are a certain mutation in the contour at ∆ = 0;For L > T in system B and C. Their basic rule is similar to system A, but we find that there is aturning point at ∆ = 0 by the calculation results. When ∆ > T is zero or lower,the QD is reduced in turn with the increase of L until L ≥ ≤ T is zero or lower, the QD of L > a ) and ( b ). Unlike system A, QD still exists when the the sizeof system is large ( L = 2732) in system B. From Figs. 9 ( c ) and ( d ), we can find that therelation curve between QD and T no longer tends to smoothly decline when ∆ = 0 whichis an unstable fixed point, but there are a certain degree of twist. When ∆ > 0, the system9ends to be the Ising model, it tend to XY model when ∆ < 0. As shown in Fig. 10, insystem C not only there are a certain degree of twist but also there is a cross point of therelation curve between QD and T at ∆ = 0, the degree of twist gradually increases withthe increase of L . In order to further study the relation of QD with T and ∆, we preparethe contour plots of QD on system B (Fig. 11) and C (Fig. 12), which show a sequentialchange of QD with ∆ and T . We can find that for L = 2 case, there are a certain mutationin the contour at ∆ = 0; For L > V. THE QUANTUM EFFECT In this section, we discuss the effect of quantum fluctuation and analyze the approx-imation which is induced by the noncommutativity. This effect has been examined byTakano and Suzuki [51], who have analyzed the discrepancy existing between the exactresult and the approximate one. We note that, if we deal with a classical system on alinear chain, the exact solution corresponding to the entire chain can be straightforwardlyachieved by solving finite pieces of the same chain. However, for quantum systems, it can-not be solved by such simplifying process because of the noncommutativity between theHamiltonian of the finite pieces. In the following, we take system A as an example to dis-cuss this effect. Firstly, we assume the generator of system A (Fig. 2( a K ′ ( K, ∆) , ∆ ′ ( K, ∆)). For the same system, weapply the Migdal-Kadanoff method and obtain the approximative result which is defined as (cid:0) K ′′ ( K, ∆; K, ∆; K, ∆) , ∆ ′′ ( K, ∆; K, ∆; K, ∆) (cid:1) . Finally, using the convenient ratios pro-posed in Ref. [36], we define the errors as E K ≡ K ′′ ( K, ∆; K, ∆; K, ∆) K ′ ( K, ∆) − , (20) E ∆ ≡ ∆ ′′ ( K, ∆; K, ∆; K, ∆)∆ ′ ( K, ∆) − . (21)The T -dependence of E KF (for the F case) and E KAF (for the AF case) for typical values of∆ is presented in Fig. 13. As we can see, both E KF and E KAF tend to zero in the T → ∞ limit,10nd in the Ising limit (∆=1), both E KF and E KAF equals zero at all temperatures. In the rangeof low temperatures, the fluctuation of the AF system is stronger than that of the F systemwhich is in accordance with the previous conclusion [52]. With the decrease of ∆, the effect ofthe fluctuation will be strengthened for both F and AF systems. The ordering is destroyedby the quantum fluctuation and this fact is responsible for the reentrant behavior in thephase diagram of system A. Furthermore, the competition between the quantum fluctuationand the thermal one is very important. At finite temperature, quantum fluctuation is usuallysuppressed in comparison with thermal fluctuation. However, when the temperature is closeto zero, the quantum fluctuation dominate and strongly influence the critical behavior ofthe system. For systems B and C, we analyze the phase diagrams and consider that whenthe N´eel temperature is relatively high, i.e., the thermal fluctuation dominate, the orderingwill not be destroyed and there exists phase transition when ∆ is close to zero. VI. SUMMARY We investigate the criticality of the spin-1 / d f = 1 . T c ∼ ∆ when ∆ is close to zero. Moreover,we apply an effective combination method to study the phase diagrams of systems B and C(with d f = 2 and 2 . 58, respectively). The results show that the N´eel temperature of systemB tend to zero in the ∆ → J for bothF and AF systems, the N´eel temperature of system B is bigger than that of system C forthe same values of ∆ and there exists phase transition in the isotropic Heisenberg modelon system C. We also find that the thermal QC decreases with the increase of L , and thethermal correlation is almost zero in system A when L is large. For system B and systemC, this is similar to system A when ∆ > 0, but QD exists when the the size of system islarge and ∆ < 0. And there is a certain degree of twist of the relation curve between QDand T in system B and system C when ∆ = 0, we suspect that this may be related to theirunstable fixed point. In the end, as an example, we discussed the quantum fluctuation ofsystem A and our results indicate that, for low dimensional lattice, the fluctuation stronglyinfluence the critical behavior of the AF system when T → cknowledgments This work is supported by the National Natural Science Foundation of China underGrants No. 11675090, No. 11847086, and No. 11905095; the Shandong Natural ScienceFoundation under Grant No. ZR2019PA015. P.-P.Z. would like to thank Rong-Tao Zhang,Zhe Wang, Pan-Pan Fang, Qi-Ming Wang, Li-Yuan Wang, Yue Li, Hai-Jun Ma, and Li-DongXiao for fruitful discussions and useful comments. [1] Y. Gefen, B. B. Mandelbrot, A. Aharony, Phys. Rev. Lett. , 855 (1980).[2] Y. Gefen, A. Aharony, B. B. Mandelbrot, J. Phys. A: Math. Gene. , 1267 (1983).[3] Y. Qin, and Z. R. Yang, Phys. Rev. B , 8576 (1991).[4] A. O. Caride, C. Tsallis, and S. I. Zanette, Phys. Rev. Lett. , 145 (1983).[5] J. Ricardo de Sousa, N. S. Brancob, B. Boechatc, C. Cordeiroc, Physica A 167 (2003).[6] S. N. Kempkes, M. R. Slot, S. E. Freeney, S. J. M. Zevenhuizen, D. Vanmaekelbergh,I. Swart and C. Morais Smith, Nat. Phys. , 127 (2018).[7] N. D. Mermin, H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).[8] J. Ricardo de Sousa, Physica A 138 (1998).[9] R. Chitra, S. Pati, H. R. Krishnamurthy, D. Sen, and S. Ramasesha, Phys. Rev. B , 6581(1995).[10] N. S. Branco and J. Ricardo de Sousa, Phys. Rev. B , 5742 (2000).[11] J. Ricardo de Sousa, and I. G. Araujo, Phys. Lett. A , 333 (2000).[12] I. G. Araujo, J. Ricardo de Sousa, and N. S. Branco, Physica A , 585 (2002)[13] P. J. Jensen, K. H. Bennemann, D. K. Morr and H. Dreyss´e, Phys. Rev. B , 144405 (2006).[14] R. P. Singh, Z. C. Tao, and M. Singh, Phys. Rev. B , 1244 (1992).[15] C. M. Soukoulis, S. Datta, and Y. H. Lee, Phys. Rev. B , 446 (1991).[16] A. Cuccoli, T. Roscilde, V. Tognetti, R. Vaia and P. Verrucchi, Phys. Rev. B , 104414(2003).[17] T. Krokhmalskii, V. Baliha, O. Derzhko, J. Schulenburg, and J. Richter, Phys. Rev. B95,094419 (2017).[18] J. Ricardo de Sousa and J. A. Plascak, Phys. Lett. A , 66 (1997). 19] B. G. Liu, Phys. Rev. B , 9563 (1990).[20] J. Ricardo. de Sousa and I. G. Ara´ujo, Solid State Commun. , 265 (2000).[21] T. Roscilde, A. Cuccoli, and P. Verrucchi, physica status solidi (b) , 433 (2004).[22] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev.Lett. , 1895 (1993).[23] A. Niezgoda, M. Panfil, and J. Chwede´nczuk, Phys. Rev. A , 042206 (2020).[24] A. Szasz, Phys. Rev. A , 062313 (2019).[25] S. Siwiak-Jaszek, T. P. Le, and A. Olaya-Castro, Phys. Rev. A , 032414 (2020).[26] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. , 017901 (2001).[27] L. Henderson and V. Vedral, J. Phys. A: Math. Gen. , 6899 (2001).[28] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, Rev. Mod. Phys. , 1655 (2012).[29] Y. L. Xu, X. Zhang, Z. Q. Liu, X. M. Kong, and T. Qi, Ren Eur. Phys. J. B 132 (2014).[30] G. Adesso, T. R. Bromley, and M. Cianciaruso, J. Phys. A: Math. Theor. , 473001 (2016).[31] F. G. S. L. Brand˜ao, M. Piani, and P. Horodecki, Nat. Commun. , 7908 (2015).[32] V. Madhok and A. Datta, Int. J. Mod. Phys. B , 1345041 (2013).[33] D. Girolami, A. M. Souza, V. Giovannetti, T. Tufarelli, J. G. Filgueiras, R. S. Sarthour, D.O. Soares-Pinto, I. S. Oliveira, and G. Adesso, Phys. Rev. Lett. , 210401 (2014).[34] K. Br´adler, M. M. Wilde, S. Vinjanampathy, D. B. Uskov, Phys. Rev. A , 062310 (2010).[35] M. Usman, K. Khan, The European Physical Journal D 181 (2020).[36] A. M. Mariz, C. Tsallis, and A. O. Caride, J. Phys. C , 4189 (1985).[37] Z. R. Yang, Phys. Rev. B , 728 (1988).[38] X. M. Kong and Z. R. Yang, Phys. Rev. E , 016101 (2004).[39] N. D. Mermin and H. Wagner, Phys. Rev. Lett. , 1133 (1966).[40] S. S. Aplesnin, J. Phys: Condens. Mat. , 10061 (1998).[41] W. G. Clark and L. C. Tippie, Phys. Rev. B , 2914 (1979).[42] J. Ricardo de Sousa, Phys. Lett. A , 321 (1996).[43] B. Groisman, S. Popescu, and A. Winter, Phys. Rev. A , 032317 (2005).[44] B. Schumacher and M. D. Westmoreland, Phys. Rev. A , 042305 (2006).[45] M. Ali, A. R. P. Rau, and G. Alber, Phys. Rev. A , 042105 (2010).[46] J. Maziero, T. Werlang, F. F. Fanchini, L. C. C´eleri, R.M. Serra, Phys. Rev. A , 022116(2010). 47] Q. Chen, C. Zhang, S. Yu, X. X. Yi, C. H. Oh, Phys. Rev. A , 042313 (2011).[48] A. N. Berker, S. Ostlund, J. Phys.: Condens. Matter 12, 4961 (1979).[49] R. B. Griffiths, M. Kaufman, Phys. Rev. B , 5022 (1982).[50] M. Kaufman, R.B. Griffiths, Phys. Rev. B , 244 (1984).[51] H. Takano and M. Suzuki, J. Stat. Phys , 635 (1981).[52] J. Fr¨ohlich and E. H. Lieb, Phys. Rev. Lett. , 440 (1977).[53] J. Hove, and A. Sudbø, Phys. Rev. B , 104405 (2007). Figure Captions Fig. 1 The structure of three DH lattices. P denotes the number of branches and L denotes the number of bonds per branch. They all start from a two-point lattice andgenerate the lattices from the middle generators in an iterative manner.Fig. 2 Self-dual two-terminal clusters used in the RG transformation. (a) System A. (a1)Cluster that generates, through six interactions. (a2) The renormalized two-site cluster. (b)System B. (b1) The third step in the construction of the 2D antiferromagnetic DH lattice.(b2) Cluster that generates, through four interactions. (b3) Two-site cluster. The first stepfrom (b1) to (b2) is the equivalent transformation and the second step is the renormalizationgroup transformation.Fig. 3 Phase diagrams of the Heisenberg AF and F models on system A. The red linecorrespond to the AF anisotropic Heisenberg model and the inset shows the phase diagramof F anisotropic Heisenberg models. O(D) stands for ordered (disordered) phase. The opencircle and the full square denote the semistable and fully stable fixed points, respectively.Fig. 4 The critical temperature behaves of system A. (a) and (b) respect the criticaltemperature T c of F case and the N´eel temperature T N of AF varies with ∆, respectively.Fig. 5 Phase diagrams of the AF Heisenberg model on systems B and C, and the phasediagram on system A is also depicted for comparison. There exists a reentrant behavior insystem A, the N´eel temperature tends to zero when decreasing ∆ in system B and thereexists phase transition in the isotropic Heisenberg limit ∆ = 0 in system C.14ig. 6 The critical temperature behaves of system B for low temperatures in the ∆ → T N is directly proportional to ln ∆.Fig. 7 The QD between two end spins on system A with L site varies with T (with theunit | J | /k B , the same below) and anisotropy parameter ∆. (a) and (b) the QD varies with T , (c) and (d) the QD varies with ∆. The QD curves for different L cases have a cross pointand a certain degree of twist at ∆ = 0.Fig. 8 The contour plots of QD on system A with L site varies with T and anisotropyparameter ∆. The contour of QD for different L cases have a cross point and a certaindegree of twist at ∆ = 0. When L = 2, there are a mutation in the contour of QD at ∆ = 0;When L > 2, there exists the cusp in the contour of QD at ∆ = 0.Fig. 9 The quantum discord between two end spins on system B with L site varies withwith T and ∆. (a) and (b) the QD varies with T , (c) and (d) the QD varies with ∆. TheQD curves for different L cases have a certain degree of twist at ∆ = 0.Fig. 10 The quantum discord between two end spins on system B with L site varies with T and ∆. (a) and (b) the QD varies with T , (c) and (d) the QD varies with ∆. The QDcurves for different L cases have a cross point and a certain degree of twist at ∆ = 0.Fig. 11 The contour plots of QD on system B with L site varies with T and anisotropyparameter ∆. The contour of QD for different L cases have a cross point and a certaindegree of twist at ∆ = 0. When L = 2, there are a mutation in the contour of QD at ∆ = 0;When L > 2, there exists the cusp in the contour of QD at ∆ = 0.Fig. 12 The contour plots of QD on system C with L site varies with T and anisotropyparameter ∆. The contour of QD for different L cases have a cross point and a certaindegree of twist at ∆ = 0. When L = 2, there are a mutation in the contour of QD at ∆ = 0;When L > 2, there exists the cusp in the contour of QD at ∆ = 0.Fig. 13 Thermal dependence of the errors E KF and E KAF defined by Eq. (21), for typicalvalues of ∆. It can be seen obviously that the error for the F system are bigger than theAF system at low temperatures. Our calculation also shows E KF = E KAF = 0 at the Isinglimits (∆ = 1). 15 ig.1 ig.2 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 k B T/|J| D O k B T/J D O Fig.3 .0000 0.0002 0.0004 0.0006 0.0008 0.00100.0000.0020.0040.0060.008 k B T C /|J| (a) -7.94 -7.92 -7.90 -7.88 -7.86 -7.84 -7.820.0290.0300.0310.0320.033 k B T N /|J| ln[ C - ] (b) Fig.4 .0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.53.0 k B T/|J| d f =1.63 d f =2 d f =2.58 Fig.5 k B T N /|J| D ) Fig.6 D T L=2 L=6 L=30 L=174 L=1038 L=6222 (a) D =0.5 D T L=2 L=6 L=30 L=174 L=1038 L=6222 (b) D =-1 -0.5 0.0 0.5 1.00.00.20.40.60.81.0 (c)T=0.8 D L=2 L=6 L=30 L=174 L=1038 L=6222 -1.5 -1.0 -0.5 0.0 0.5 1.00.00.20.40.60.81.0 (d)T=1 D L=2 L=6 L=30 L=174 L=1038 L=6222 Fig.7 T D L=2 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.51.01.52.0 T -5.000E-040.014690.029870.045060.060250.075440.090620.10580.1210 D L=6 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.51.01.52.0 T -2.000E-051.000E-030.0020200.0030400.0040600.0050800.0061000.0071200.008140 D L=30 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.00.51.01.52.0 T -2.000E-084.850E-079.900E-071.495E-062.000E-062.505E-063.010E-063.515E-064.020E-06 D L=174 Fig.8 .5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.0 D T L=2 L=4 L=12 L=44 L=172 L=684 L=2732 (a) D =0.5 (b) D =-1 D T L=2 L=4 L=12 L=44 L=172 L=684 L=2732 -0.5 0.0 0.5 1.00.00.20.40.60.81.0 (c) T=0.8 D L=2 L=4 L=12 L=44 L=172 L=684 L=2732 -0.5 0.0 0.5 1.00.00.20.40.60.81.0 (d) T=1 D L=2 L=4 L=12 L=44 L=172 L=684 L=2732 Fig.9 .0 1.5 2.0 2.5 3.0 3.5 4.00.00.10.20.30.40.50.60.7 D T L=2 L=5 L=23 L=131 L=779 (a) D =0.5 (b) D =-1 D T L=2 L=5 L=23 L=131 L=779 -0.2 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 D L=2 L=5 L=23 L=131 L=779 (c) T=0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 D L=2 L=5 L=23 L=131 L=779 (d) T=1 Fig.10 T D L=2 -1.0 -0.5 0.0 0.5 1.00.40.60.81.01.21.41.61.82.0 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=4 -1.0 -0.5 0.0 0.5 1.00.60.81.01.21.41.61.82.02.2 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=12 -1.0 -0.5 0.0 0.5 1.00.60.81.01.21.41.61.82.02.2 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=44 -1.0 -0.5 0.0 0.50.60.81.01.21.41.61.82.02.2 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=172 -1.0 -0.5 0.0 0.50.60.81.01.21.41.61.82.02.2 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=684 Fig.11 T D L=2 -1.5 -1.0 -0.5 0.0 0.5 1.00.60.81.01.21.41.61.82.02.22.4 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.01.01.21.41.61.82.02.22.42.6 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=23 -1.0 -0.5 0.0 0.5 1.01.01.21.41.61.82.02.22.42.62.8 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=131 -1.0 -0.5 0.0 0.5 1.01.41.61.82.02.22.42.62.83.0 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=779 -1.0 -0.5 0.0 0.51.61.82.02.22.42.62.83.03.2 T -0.0010000.040880.082750.12460.16650.20840.25030.29210.3340 D L=4667 Fig.12 E KF k B T/J D =0 D =0.3 D =0.5 D =0.8 E KAF k B T/J D =0 D =0.3 D =0.5 D =0.8=0.8