Spin-1/2 anisotropic Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interaction via mean-field approximation
Walter E. F. Parente, J. T. M. Pacobahyba, Minos A. Neto, Ijanílio G. Araújo, J. A. Plascak
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Spin- / anisotropic Heisenberg antiferromagnet withDzyaloshinskii-Moriya interaction via mean-field approximation Walter E. F. Parente
Universidade Estadual de Roraima, Av. Senador Helio Campo,s/n - Centro, Caracara - RR, 69360-000
J. T. M. Pacobahyba
Departamento de F´ısica, Universidade Federal de Roraima, BR 174,Km 12. Bairro Monte Cristo. CEP: 69300-000 Boa Vista/RR
Minos A. Neto ∗ Universidade Federal do Amazonas, Departamento de F´ısica,3000, Japiim, 69077-000, Manaus-AM, Brazil
Ijan´ılio G. Ara´ujo
Departamento de F´ısica, Universidade Federal de Roraima, CEP: 69300-000, Boa Vista/RR.
J. A. Plascak
Universidade Federal da Para´ıba, Centro de Ciˆencias Exatas e da Natureza - Campus I,Departamento de F´ısica - CCEN Cidade Universit´aria58051-970 - Jo˜ao Pessoa-PB, Brazil andDepartment of Physics and Astronomy,University of Georgia, 30602 Athens GA, USA (Dated: February 21, 2018) ∗ Corresponding author.E-mail address: [email protected] (Minos A. Neto)Contact number: 55-92-3305-4019 bstract ABSTRACT
The spin-1 / PACS numbers : 64.60.Ak; 64.60.Fr; 68.35.Rh
PACS numbers: . INTRODUCTION The theoretical study of quantum magnetic systems has gained increased attention in thelast decades not only because of their quite interesting intrinsic quantum phase transitions,which are induced by quantum fluctuations, but because these magnetic systems can also bewell suited to describe some superconducting materials. For instance, the cuprate La CuO compound has been shown to exhibit metamagnetic behavior [1, 2] and, when doped withSr, forming the La − x Sr x CuO compound, it becomes a high-temperature superconductorfor x > .
5% [3].On the theoretical point of view, the spin-1 / planes of these cuprates superconductors. The interest inthe AHM comes from the early Anderson suggestion that these quantum fluctuations shouldbe responsible for the superconductivity in this class of compounds [4]. Therefore, the AHMhas been treated from several different techniques and some aspects of its phase diagramare well stablished (see, for instance, references [5, 6] and references therein). However, inreal materials, anisotropies are expected, not only regarding the exchange interactions butalso from spin-orbit couplings, which may lead to the Dzyaloshinskii-Moriya interaction.In this sense, the AHM in the presence of an external field and a Dzyaloshinskii-Moriyainteraction turns out to be a very interesting model with applications in several experimentalrealizations, going from cuprates superconductors to spin-glass behavior in magnetic systems[5–7].The AHM in the presence of an external field has been treated by the effective-fieldtheory (EFT) [5] and the phase diagram in the temperature versus external field presentsa reentrant behavior at low temperatures. This same model with an external longitudinalfield and an extra Dzyaloshinskii-Moriya interaction has also been studied by employing theEFT. The results show that the phase diagram also exhibits re-entrancies and, in addition,some anomaly behavior at low temperatures [6, 8]. As no exact solution is still availablefor this model, the results obtained by the EFT are still questionable and, in some way,controversial.Our purpose in this work is thus to study the thermodynamic behavior of the AHM withXXZ exchange anisotropy in the presence of a longitudinal external magnetic field and a3zyaloshinskii-Moriya interaction placed in the z direction. We employ the usual mean-fieldapproximation, based on Bogoliubov variational approach using a two-spin cluster, in orderto analyze the order parameter and the corresponding phase transition. A two-spin likeapproximation has been proven to be quite efficient in treating the transverse Ising model[9] and the Blume-Capel model in a transverse crystal field [10]. Although still being a mean-field approach, it would be very interesting to compare the results so obtained in treatingthe AHM with external field and Dzyaloshinskii-Moriya interactions with the previous onesfrom EFT.The plan of the paper is as follows. In the next section, the model and the variationalprocedure formalism are presented. The results are discussed in Section III and some finalremarks are commented in the last section. II. MODEL AND FORMALISM
The model studied in this work is the anisotropic antiferromagnetic Heisenberg model ina magnetic field and with a Dzyaloshinskii-Moriya interaction, which can be described bythe following Hamiltonian H = J X h i,j i (cid:2) (1 − ∆) (cid:0) S xi S xj + S yi S yj (cid:1) + S zi S zj (cid:3) − X h i,j i D · ( S i × S j ) − N X i =1 H · S i , (1)where the first term represents the nearest-neighbors anisotropic exchange interaction, with∆ being the anisotropy parameter, the second term represents the nearest-neighbors anti-symmetric Dzyaloshinskii-Moriya (DM) interaction placed along the z direction ( D = D ij = − D ji = D ˆz ), the third term corresponds to the Zeeman interaction with an external mag-netic field H = H ˆz , and finally S νi ( ν = x, y, z ) are the spin-1 / i -sites ina hypercubic lattice of N spins. On such a lattice, one has coordination number c given by c = 2 , , , · · · for, respectively, the one-dimensional lattice, square two-dimensional lattice,simple cubic three-dimensional lattice, and so on. This model has no exact solution and thetopology of the phase diagram in the H − D plane, as a function of the anisotropy ∆, stillremains a fundamental problem. For ∆ = 0 one has the isotropic Heisenberg model, whilefor ∆ = 1 we recover the spin-1 / F ( H ) ≤ F ( H ) + hH − H i ≡ Φ( η ) , (2)where F and F are the free energies associated with two systems defined by the Hamiltonians H and H ( η ), respectively, the thermal average h ... i should be taken in relation to thecanonical distribution associated with the trial Hamiltonian H ( η ), with η standing forvariational parameters. The approximated free energy F is given by the minimum of Φ( η )with respect to η , i.e. F ≡ Φ min ( η ).With D and H defined above, the hamiltonian H in eq. (1) can be rewritten as H = J X h i,j i (cid:2) (1 − ∆) (cid:0) S xi S xj + S yi S yj (cid:1) + S zi S zj (cid:3) − D X h i,j i (cid:0) S xi S yj − S yi S xj (cid:1) − H N X i =1 S zi . (3)Otherwise, for the trial Hamiltonian H , we have chosen the simplest cluster for thismodel, which corresponds to a sum of N/ A or B sub-lattice. We then have H = N/ X ℓ =1 (cid:8) J (cid:2) (1 − ∆) (cid:0) S xA ℓ S xB ℓ + S yA ℓ S yB ℓ (cid:1) + S zA ℓ S zB ℓ (cid:3) − D (cid:0) S xA ℓ S yB ℓ − S yA ℓ S xB ℓ (cid:1) − ( η A + H ) S zA ℓ + ( η B + H ) S zB ℓ (cid:9) , (4)where S αA ℓ and S αB ℓ are the spins of the ℓ -th pair on the A and B sub-lattices, respectively,and η A and η B are variational parameters, which are different in each sub-lattice.It is not difficult to diagonalize the above trial Hamiltonian and to obtain the corre-sponding free-energy F . The same holds for the mean value hH − H i , where we still have N c/ − N/ H . Thus, after minimizing the right hand side of Eq. (2)with respect to the variational parameters η A and η B we obtain the approximated mean-fieldHelmholtz free energy per spin, f = Φ /N , which can be written as f = − β ln (cid:8) e − K cosh ∆ + 2 e K cosh 2 K ∆ (cid:9) − J c − m A m B , (5)with ∆ = 2 h − ( c − m A + m B ) K, (6)∆ = p (1 − ∆) + ( c − ( m A − m B ) + d (7)5nd the corresponding sub-lattice magnetizations m A and m B given by m A = sinh ∆ + e K ∆ sinh K ∆ cosh ∆ + e K cosh K ∆ (8)and m B = sinh ∆ − e K ∆ sinh K ∆ cosh ∆ + e K cosh K ∆ , (9)where ∆ = ( c − m A − m B ) / ∆ . (10)In the above expressions, we have defined the reduced quantities K = J/k B T , h = H/J , d = D/J , and β = 1 /k B T , where k B is the Boltzmann constant.In order to analyze the criticality of this system, it will be more convenient to definean order parameter that characterizes the antiferromagnetic phase transition. The two newquantities that we will use here are the total ( m ) and staggered ( m s ) magnetizations (thelatter one being the desired order parameter of the antiferromagnetic transition), which aregiven by m = 12 ( m A + m B ) (11)and m s = 12 ( m A − m B ) . (12)Substituting Eqs. (8) and (9) in (11) and (12) we have m = sinh ˜∆ cosh ˜∆ + e K cosh 2 K ˜∆ (13)and m s = ˜∆ e K sinh 2 K ˜∆ cosh ˜∆ + e K cosh 2 K ˜∆ , (14)where now ˜∆ = 2 K [ h − ( c − m ] , (15)˜∆ = p (1 − ∆) + ( c − m s + d , (16)and ˜∆ = ( c − m s / ˜∆ . (17)Accordingly, the free energy defined in (5) can be written in terms of the new order parameteras f ( m, m s ) = − β ln n e − K cosh ˜∆ + 2 e K cosh 2 K ˜∆ o − J c − m − m s ) . (18)6hus, for a given value of the set of Hamiltonian parameters, namely the reduced exchangeanisotropy ∆ /J , the reduced Dzyaloshinskii-Moriya interaction d , and the reduced externalfield h , all in units of the exchange interaction J , we can obtain the temperature dependenceof m and ms by simultaneously solving the two nonlinear coupled equations (13) and (14).Below the antiferromagnetic phase transition the magnetizations of sub-lattices A and B areopposite and nonzero, while above the transition temperature one has m s = 0 and m = m ,which is the paramagnetic phase. In this paramagnetic phase one has m = sinh(2 h − c − Km )cosh(2 L − c − Km ) + e K cosh 2 K p (1 − ∆) + d (19)and f ( m ) = − β n e − K cosh(2 h − c − Km ) + 2 e K cosh 2 K p (1 − ∆) + d o − J c − m . (20)Close to the second-order transition line, one has m s << T N (in fact k B T N /J ) as a function of the Hamiltonianparameters. We can also obtain the criticality by noting that at the transition m s = 0 andfrom relations (13) - (17) we obtain two coupled equations m = sinh ˜∆ cosh ˜∆ + e K cosh 2 K ˜∆ , (21)and 1 = (cid:18) c − (cid:19) e K sinh 2 K ˜∆ cosh ˜∆ + e K cosh 2 K ˜∆ , (22)where ˜∆ = p (1 − ∆) + d . Thus, given the values of ∆, d , h , and c , these two equationsfurnishes m and k B T N /J .In order to seek for first-order transitions, we have to compare the antiferromagnetic freeenergy with m s = 0 and the paramagnetic one with m s = 0. The first-order phase transitionis located when they have the same value. However, for the present model, as discussed inthe next section, no first-order transitions have been detected.7 II. RESULTS
The reduced critical transition temperature k B T N /J , as a function of the Hamiltonianparameters, is shown in Fig. 1 for the simple cubic lattice and the isotropic Heisenbergmodel (∆ = 0) in the presence of external field and DM interaction. All the transitionlines are second order and we have no indication of any first-order transition lines in thesediagrams. We can see that the increase of either the external field or the DM interactiontends to decrease the corresponding transition temperature. k B T N /J k B T N /J FIG. 1: Critical transition temperature k B T N /J of the simple cubic lattice c = 6 and isotropicHeisenberg model ∆ = 0 as a function of: (left) external field h and several values of the DMinteraction d ; and (right) DM interaction d and several values of the external field h . We can see that phase diagram in the temperature versus external field presents a reen-trancy for DM interactions d . . h . In addition, at T = 0, the valueof the critical external field is given by h c = c and is independent of the DM interaction.On the other hand, from Eqs. (21) and (22) it can be shown that the critical value of theDM interaction d c is given by d c = p ( c − − (1 − ∆) , (23)and is independent of the external field. 8he critical temperature at zero external field, namely k B T N /J ( H = 0) = 5 .
78, is alsocomparable to those obtained from different methods, such as EFRG-12 k B T N /J = 4 . k B T N /J = 3 .
54 from Monte Carlo simulations [13],high-temperature expansion k B T N /J = 3 .
59 [14], variational cumulant expansion k B T N /J =4 .
59 [15], EFT-2 k B T N /J = 4 .
95 and EFT-4 k B T N /J = 4 .
81 [5]. k B T N /J k B T N /J FIG. 2: The same as Fig. 1 for c = 8. The same sort of picture can be seen in Fig. 2 for higher dimensions, which is anexample for the case c = 8. For this value of the coordination number, at zero field, ourresults coincide with those by Bublitz et al. MFA-2 k B T N /J = 7 .
83 [16]. This numericalresult is also comparable to other different methods, such as high-temperature expansion k B T N /J = 5 .
53 [17], EFT-2 k B T N /J = 6 .
94 , and EFT-4 k B T N /J = 6 .
89 [5].For the particular case ∆ = d = h = 0 we have the Ising limit. In this way m = 0from Eq. (21), as expected, and the Eq. (22) provides exactly the equation of the pairapproximation for Ising model [18] which is given by2 K ( c − e K = 1 + e K . (24)As a final comment, it should be stressed that the qualitative behavior of the phasediagrams are also the same for different values of the anisotropy ∆, even in the limit of theIsing case ∆ = 1 [8]. 9 V. FINAL REMARKS
We have studied the anisotropic antiferromagnetic Heisenberg model, in a external mag-netic field and with a Dzyaloshinskii-Moriya interaction, using a mean-field procedure basedon Bogoliubov inequality for the free energy by employing a cluster of two spins. We haveobtained the phase diagram as a function of the Hamiltonian parameters.When comparing our results to those obtained from the effective field theory, which are,up to our knowlwdge, the only results for this system, we notice that there is no anomalousbehavior of the transition lines at low temperature, although some re-entrancies are stillpresent in some region of the phase diagram.Of course, this treatment is still a mean field one. It is in some sense different fromthe EFT, although the latter one having also an intrinsic mean field character. The presentresults strongly suggest that the behavior of the thermodynamic properties of the anisotropicHeisenberg model with Dzyaloshinskii-Moriya interaction is far from having a completeunderstandable behavior.
ACKNOWLEDGEMENTS
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