Magnetization and the Concurrence of the Spin-1/2 Ising-Heisenberg Pyrochlore Ladder A. Sadrolashrafi, N. S. Ananikian, and L. N. Ananikyan1 A. I. Alikhanyan National Science Laboratory, 0036 Yerevan, Armenia 7 1 0 2 n a Abstract J 7 We have established a quantum antiferromagnetic Heisenberg-Ising model on a spin-1/2 pyrochlore edge-shared ladder with Heisenberg intra-rung and ] h Ising inter-rung interactions as a perspicuous candidate to exhibit magneti- p zation mid and zero plateaus, characteristic peaks of magnetic susceptibility, - t n and thermal entanglement mid plateau. The model is exactly solvable and a thus, all the essential properties such as the thermal entanglement and the u q magnetic properties of the system can be exactly calculated. The calcula- [ tions are done both through the transfer matrix technique and through the 1 reduced density matrix. The magnetization plateaus are observed at zero v and half the saturation value and the magnetic susceptibility exhibits a clear 0 8 demonstrationoftheassociatedcharacteristicpeaks. Themodelalsodisplays 8 the mid plateau of the thermal entanglement as a function of the external 1 0 magnetic field at low temperatures. . 1 Keywords: quantum spin model, pyrochlore ladder, magnetization 0 7 plateaus, thermal entanglement 1 : v i 1. INTRODUCTION X r a The appearance of the magnetization plateau was predicted in the pi- oneering theoretical work of Hida [1] for a ferromagnetic - ferromagnetic - antiferromagnetic Heisenberg chain 3CuCl 2 2 dioxane compound. Mag- 2 netic ordering at low temperatures is frustrated by the geometry of the crystalline lattice, a situation known as geometrical magnetic frustration. 1A. Sadrolashrafi: [email protected] 2N. S. Ananikian: [email protected] Preprint submitted to Solid State Communications January 13, 2017 The magnetization plateau, specific heat, and magnetic susceptibility of low- dimensional quantum spin systems have attracted much attention over the last few decades both experimentally [2, 3, 4] and theoretically [5, 6, 7, 8, 9, 10, 11, 12, 13]. The frustrating properties of the corner-sharing or edge- shared tetrahedron lattice have been particularly studied for the magnetic pyrochlore oxides [14, 15, 16, 17]. Incondensedmatter,magneticmaterialsareofparticularinterest. Among these we will study the antiferromagnetism through description by Heisen- berg models. The pyrochlore edge-shared tetrahedron ladder with spin-1/2 is an excellent candidate to realize the antiferromagnetic properties [18] and test the reasonableness and validity of the theories on the quantum entan- glement in frustrated systems. The ideas have been taken from the recent article [19, 20] by exactly solving the problem through the classical transfer matrix method and figuring out the spin frustration and thermal entangle- ment of the spin-1/2 Ising-Heisenberg three-leg tube, which accounts for the Heisenberg intra-triangle and Ising inter-triangle interaction. We have applied the separation of Heisenberg intra-rung and Ising inter- runginteractionsonapyrochloreladderwithantiferromagneticspin-1/2cou- plingsusingtheclassicaltransfermatrixmethodandconstructedthethermal entanglement, magnetization plateau and magnetic susceptibility as on a di- amond chain [21, 22, 23, 24] . The entanglement properties, the correlation functions, and magnetic properties are also studied in [25, 26, 27] for a spin zigzag ladder with the generalized Majumdar-Ghosh model, a spin ladder, and a Heisenberg spin chain correspondingly. The functional dependence of the entanglement on the correlation functions and magnetic susceptibility were discussed in Ref.’s [28, 29, 30, 52, 53, 54, 57, 58, 59, 60]. Thispaperisorganizedasthefollowing: inthenextsectionwepresentthe antiferromagnetic spin-1/2 Ising-Heisenberg with frustrated magnetization plateau on a pyrochlore edge-shared ladder. Further, in the third section, we have discussed the thermal concurrence, correlation functions and magnetic susceptibility. 2. Pyrochlore edge-shared ladder: The Model and Its Exact Solu- tion Let us consider the spin-1/2 Ising-Heisenberg model on a ladder, whereas the spins belonging to the same rung are mutually coupled through the Heisenberg intra-rung interaction and the spins from the neighboring rungs 2 arecoupledthroughtheIsinginter-runginteraction(seeFIG.1). TheHamil- Figure1: Thespin-1/2Ising-Heisenbergmodelonapyrochloreedge-sharedladder;Heisen- berg intra-rung and Ising inter-rung couplings tonian of the spin-1/2 Ising-Heisenberg ladder of N rungs is then given by N N N (cid:88) (cid:88) (cid:88) (cid:88) H = J SαSα+J (Sz +Sz)(Sz Sz )−h (Sz +Sz), (1) H i i(cid:48) I i i(cid:48) i+1 i+1(cid:48) i i(cid:48) i=1 α=x,y,z i=1 i=1 where the index i labels the rungs of the ladder with the periodic boundary condition such that the site N+1 would become equal to the first site. The prime-less indices label the spins on the upper leg whereas the primed indices label them on the lower leg; Thus, the spin at site ith is the spin on the ith rung and on the upper leg whereas the i(cid:48) show the spin on the same rung but on the lower leg. J and J are the Heisenberg and Ising coupling constants H I correspondingly and h is the magnetic field strength. S is the spin operator i at site i and Sα is its α component with α = x,y,z. Figure (2) shows the i tetrahedron structure of the edge-shared pyrochlore ladder. Figure 2: tetrahedral structure of the spin-1/2 Ising-Heisenberg model on a pyrochlore edge-shared ladder The tetrahedron structure of the edge-shared pyrochlore ladder with an- tiferromagnetic behavior is also noticed in [31, 32]. 3 2.1. Transfer-Matrix Solution The total Hamiltonian (1) of the spin-1/2 Ising-Heisenberg ladder can be alternativelyrewrittenintermsofcompositespinoperators, whichdetermine the total spin of the Heisenberg rungs and its z-component 1 T = S +S(cid:48) , Tα2 = +2SαSα. (2) i i i i 2 i i(cid:48) It can be shown that the composite spin operators T2 and Tz commute i i with the total Hamiltonian (1), i.e. [H,T2] = [H,Tz] = 0, which means that i i the total spin of the rung and its z-component represent conserved quanti- ties with well defined quantum numbers. Consequently, the eigenvalues of the total Hamiltonian (1) can be related to the eigenvalues of the total spin operator of the rung T2 and its z-component Tz. i i Using the identity in (2), the total Hamiltonian (1) can be rewritten in the following form: N −3NJ (cid:88) H H = + H , (3) i 4 i=1 in which H is the Hamiltonian of the two subsequent rungs at i and i + 1 i and is written as follows in terms of the total spin operators of the rungs and their corresponding z-components: N N N J (cid:88) (cid:88) h (cid:88) H = H (T2 +T2 )+J TzTz − (Tz +Tz ). (4) i 4 i i+1 I i i+1 2 i i+1 i=1 i=1 i=1 It can be seen that the commutation relation [H ,H ] = 0 is true for any i j i and j and therefore, H ’s are actually separable from each other. Thus, i the spin-1/2 Ising-Heisenberg ladder defined by the Hamiltonian (1) can be rigorously mapped onto some classical composite spin chain model, which can be further treated by the transfer-matrix method [33] and the relative partition function can be factorized into the following form: Z = tre−βH = e3Nβ4JH tre−β(cid:80)Ni=1Hi N = e3Nβ4JH tr(cid:89)e−βHi, (5) i=1 4 where β = 1/(K T) shows the inverse of the absolute temperature with K B B being the Boltzman constant. Furthermore, we can consider the following matrixrepresentationfore−βHi inthebasisoftheeigenstatesofthecomposite spin operators T2, Tz, T2 , Tz of the two consecutive rungs, by which we i i i+1 i+1 can figure out the transfer matrix W as follows: (cid:68) (cid:69) W[i,i+1] = T2,Tz|e−βHi|T2 ,Tz i i i+1 i+1  e−β(JH+JI+h) e−β(JH+h/2) e−β(JH−JI) e−β(JH2+h)   e−β(JH+h/2) e−βJH e−β(JH−h/2) e−βJH/2  =  (.6)  e−β(JH−JI) e−β(JH−h/2) e−β(JH+JI−h) e−β(JH2−h)  e−β(JH2+h) e−βJH/2 e−β(JH2−h) 1 In the above representation, we chose (cid:104)T2,Tz|’s and (cid:104)T2 ,Tz |’s from i i i+1 i+1 the set {(cid:104)1,−1|,(cid:104)1,0|,(cid:104)1,1|,(cid:104)0,0|}. As H ’s dependence on T ’s and T ’s is i i i+1 independent of the site i and the ladder is translational invariant, one can rewrite the last line in (5) as Z = e3NβJH trWN. (7) 4 Thus, the partition function in the thermodynamic limit N → ∞ can be solely determined by the largest eigenvalue, λ , of the transfer matrix W Max given by Eq. (6) such that Z = e3NβJH λN . (8) 4 Max Then the magnetization per site and the magnetic susceptibility can be cal- culated through the following formulae: 1 ∂F 1 ∂logZ 1 ∂logλ Max m = − = = N ∂h Nβ ∂h β ∂h ∂m ∂2logλ Max χ = = (9) ∂h ∂h2 In section 2.3, we will present the corresponding graphs that show the behavior of the magnetization and also the susceptibility of the pyrochlore ladder (1) with the antiferromagnetic couplings of J = 3/2 and J = 1. But H I 5 before we proceed to the section 2.3, in 2.2 we present a second approach to calculate the magnetization m and the susceptibility χ, based on the in- formation derived from the reduced density matrix of the rung. This latter approach would be beneficial when one aims to investigate the relation be- tween the magnetic properties of the system with the quantum correlations and the entanglement. It appears that for the pyrochlore ladder (1), the results from the two approaches i.e. the transfer matrix approach and the reduced density matrix approach would fully correspond with one another. 2.2. The Reduced Density Matrix of One Rung For a rung of two spin-1/2 particles in the pyrochlore ladder (1) the thermal reduced density matrix of which can be derived from the density matrix of a block of two adjacent rungs will have the following form in the standard basis:   z 0 0 0  0 x y 0  ρ =   (10)  0 y x 0  0 0 0 w with x, y, z, and w being the following functions of the coupling constants J H and J , the external magnetic field h, and the inverse absolute temperature I β = 1 with K being the Boltzman constant: KBT B x = e−β2(2JH+h) (cid:16)eβJ2H +1(cid:17)(cid:16)eβ2(JH+h) +eβh +eβh/2 +1(cid:17) 2 y = −e−β2(2JH+h) (cid:16)eβJ2H −1(cid:17)(cid:16)eβ2(JH+h) +eβh +eβh/2 +1(cid:17) 2 (cid:16) (cid:17) z = e−β(JH+JI) eβ2(JH+2JI+h) +eβ(JI+h2) +e2βJI +eβh (cid:16) (cid:17) w = e−β(JH+JI+h) eβ2(JH+2JI+h) +eβ(2JI+h) +eβ(JI+h2) +1 As soon as the above reduced density matrix is ready, the rest of the calculation is straightforward: To achieve the magnetization per site, m, 6 one needs to calculate the expectation value of the operator S , 1 tr(ρS ), z tr(ρ) z where S is the z-component of the spin operator of one particle at a site. Z Then the next step would be the derivative with respect to the external mag- netic field, h, which yields the magnetic susceptibility of the ladder. Thus, one can follow the following formulae to figure out the magnetic properties of the system: 1 m = tr(ρS ) z tr(ρ) ∂m ∂ (cid:16) 1 (cid:17) χ = = tr(ρS ) (11) z ∂h ∂h tr(ρ) 2.3. the Magnetization Plateaus and the Magnetic Susceptibility in the Anti- ferromagnetic Couplings The phenomenon of magnetization plateau has been studied during the past decade both experimentally and theoretically. The plateau may exist in the magnetization curves of quantum spin systems in the case of a strong magnetic external field at low temperatures. The phenomenon of magnetization plateau is considered as a macroscopic manifestation of the essentially quantum effect in which the magnetization m is quantized at fractional values of the saturation magnetization m in low s dimensional magnetism [1, 34, 35, 36, 37, 38, 39, 40, 41]. Thequantumplateaustatewasactuallyfirstdiscovered, overtwodecades agointheferromagnetic-ferromagnetic-antiferromagetic(F-F-AF)chainmodel [1, 34, 36, 37]. Magnetization plateaus appear in a wide range of models on chains, lad- ders, hierarchical lattices and theoretically analyzed by dynamical, trans- fer matrix approaches as well as by the exact diagonalization in clusters [42, 43, 44, 45, 46]. To explain the experimental measurements of magnetization plateau and the double peak behavior in the natural mineral azurite, there have been pro- posed different types of theoretical Heisenberg models (the density-matrix and transfer-matrix renormalization- group techniques, density functional theory,high-temperatureexpansion,variationmean-field-liketreatment,based on the Gibbs-Bogoliubov inequality) [5, 47, 48, 8]. 7 Magnetization plateaus and the multiple peak structure of the specific heat have also been observed on an Ising-Hubbard diamond chain [9]. In the previous parts we introduced two separate approaches to achieve the magnetic properties of the system; One is based on the transfer matrix method, which is explained in 2.1 and the other one is based on the calcula- tions upon the quantum density matrix, which is talked about in 2.2. As it was mentioned earlier, the two approaches yield exactly the same results for our spin-1/2 pyrochlore ladder. Figures (3) and (4) show these results for the magnetization per site and the magnetic susceptibility of the ladder with the Heisenberg interaction and Ising interaction coupling constants being J = 3/2 and J = 1 correspond- H I ingly. The existence of the magnetization plateaus at zero and half of the saturation magnetization at low temperatures is clear in Fig.(3) and the cor- responding characteristic peaks of the magnetic susceptibility are shown in the Fig.(4). 3. Entanglement of the Antiferromagnetic Pyrochlore Ladder Entanglement is a type of correlation that is quantum mechanical in na- ture. It reflects nonlocal correlations between particles, even if they are removed and do not directly interact with each other. Studying entanglement in condensed matter systems is of great interest due to the fact that some behaviors of such systems can most probably only beexplainedwiththeaidofentanglement. Themagneticsusceptibilityatlow temperatures, quantum phase transitions, chemical reactions are examples where the entanglement is the key ingredient for a complete understanding of the system. Furthermore, in order to produce a quantum processor, the entanglement in condensed matter systems becomes an essential concept. In order to measure the entanglement between two spin-half particles sitting on the same rung in the Ising-Heisenberg pyrochlore ladder, we study the concurrence of the two Heisenberg qubits, using the definition proposed by Wootters et al. [49, 50]. The concurrence can be expressed in terms of a matrix R in the following manner: R = ρ·(σy ⊗σy)·ρ∗ ·(σy ⊗σy), (12) which is constructed as a function of the density operator ρ, given by (10), with ρ∗ being the complex conjugate of ρ and σy being the Pauli Y matrix 8 Figure 3: the magnetization behavior of the spin-1/2 Ising-Heisenberg pyrochlore edge- shared ladder with Heisenberg coupling constant J = 3/2 and Ising coupling constant H J =1,asafunctionoftheabsolutetemperatureT andthemagneticfieldh;TheBoltzman I constant K has been set to 1. The existence of the magnetization plateaus are clear in B the low temperatures. 9 Figure 4: Magnetic susceptibility of the spin-1/2 Ising-Heisenberg pyrochlore edge-shared ladderwithHeisenbergcouplingconstantJ =3/2andIsingcouplingconstantJ =1,as H I afunctionoftheabsolutetemperatureT andthemagneticfieldh;TheBoltzmanconstant K has been set to 1. The characteristic peaks of the magnetic susceptibility can be seen B in the low temperatures. 10