Spin superfluidity in the anisotropic XY model in the triangular lattice

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Spin superfluidity in the anisotropic XY model in the triangular lattice L.S. Lima n Departamento de Física e Matemática, Centro Federal de Educação Tecnológica de Minas Gerais, 30510-000 Belo Horizonte, MG, Brazil

art ic l e i nf o

a b s t r a c t

Article history: Received 31 March 2016 Accepted 5 April 2016 by Prof. A.H. MacDonald

We use the SU(3) Schwinger's boson theory to study the spin transport properties in the twodimensional anisotropic frustrated Heisenberg model in the triangular lattice at T ¼0. We have investigated the behavior of the spin conductivity for this model which presents an single-ion anisotropy. We study the spin transport in the Bose-Einstein condensation regime where we have that the tz bosons are condensed and the following condition is valid: 〈t z 〉 ¼ 〈t †z 〉 ¼ t. Our results show a metallic spin transport for ω 40 and a superfluid spin transport in the limit of DC conductivity, ω-0, where σ ðωÞ tends to infinity in this limit of ω. & 2016 Published by Elsevier Ltd.

1. Introduction The antiferromagnet in the triangular lattice plays a fundamental role in the understanding of frustrated quantum systems. It is one of the most fundamental systems of geometrically frustrated magnets and has been studied using several techniques such as quantum Monte Carlo [1]. The main effects of frustrating interactions, in the neighborhood of a Neel state, are the increase of the coupling and the decrease of the spin-wave velocity. It is believed that for S ¼ 1=2, the system displays classical Néelordered ground state with a 120º spiral order [2]. It was argued in Ref. [3] that the triangular lattice has a magnetization of the lattice more reduced from its classical value than the square lattice and its Néel order may be destabilized more easily. The progress in the investigations of spin supercurrent and magnon BEC was recently described in the review [4]. Particularly there was overviewed the spin supercurrent Josephson Effect which is the response of the current to the phase between two weakly connected regions of coherent quantum states. For quasiparticles such as magnons and excitons in Bose–Einstein condensation (BEC), it is demonstrated that the interference between two quasiparticles condensates. Spin current as a function of the phase difference across the junction, α2  α1 , where α1 and α2 are phases precession in two coherently precessing domains. It is the response of the current to the phase between two weakly connected regions of coherent quantum states [4]. This was described by Josephson in [5]. The Bose–Einstein condensation of quasiparticles whose number is not conserved is an important phenomena of condensed matter physics. In thermal equilibrium the chemical potential of n

Corresponding author.

excitations vanishes and, as a result, their condensate does not form [4]. The only way to overcome this situation is to create a non-equilibrium but dynamically steady state, in which the number of excitations is conserved, since the loss of quasiparticles owing to their decay is compensated by pumping of energy. Thus the Bose-condensation of quasiparticles belongs to the phenomena of second class, when the emerging steady state of the system is not in a full thermodynamic equilibrium [4]. The spin superfluid transport and Bose condensation (BEC) are related but a different phenomenon. The spin superfluid transport is well known from 1984 when it was discovered in superfluid 3He [6,7]. The superfluid 3He is an antiferromagnet. All magnetic properties, including magnon BEC and spin superfluid in superfluid 3He are indeed the properties of magnetically ordered system. 3He has just a small relaxation rate. This simplified the discovery of BEC and spin supercurrent. Later these phenomena were found in many other systems, like antiferromagnets with Suhl–Nakamura interaction (CsMnF3, MnCO3 and so on.) [8]. It was also found in YIG films [9]. The gradients of excited magnons wave function lead to spin superfluidity, which is the quantum transport of magnetization by magnons. The properties of spin superfluidity and magnon BEC has been recently well discussed in Ref. [4]. The spin transport properties of materials are the corner stones for many applications. Once having these properties determined, it is possible to calculate the parameters for devices which can operate on the basis of these structures. Recently, the transport phenomenon by magnetic excitations such as magnons, excitons and so on, has been much studied due to its connection with spintronics [10–13]. The injection of the spin current into a magnetic film can generate a spin-transfer torque that acts on the magnetization collinearly to the damping torque [11]. The spin transport properties in the spin systems has been studied theoretically by Sentef et al. [14] who have analyzed the

http://dx.doi.org/10.1016/j.ssc.2016.04.005 0038-1098/& 2016 Published by Elsevier Ltd.

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spin transport in the easy-axis Heisenberg antiferromagnetic model in two and three dimensions, at T¼ 0. Damle and Sachdev [15] have treated the two-dimensional case using the non-linear sigma model in the gapped phase. Pires and Lima [16–18] treated the two-dimensional easy plane Heisenberg antiferromagnetic model. Lima and Pires [19] studied the spin transport in the twodimensional anisotropic XY model using the SU(3) Schwinger boson theory in the absence of impurities, Lima [20] has studied the case of the Heisenberg antiferromagnetic model in two dimensions with Dzyaloshinskii–Moriya interaction. Zewei Chen et al. [21] analyzed the effect of spatial and spin anisotropy on spin conductivity for the S ¼ 1=2 Heisenberg model on a square lattice and more recently, Kubo et al. [22] studied the spin conductivity in two-dimensional non-collinear antiferromagnets at T ¼0 using spin wave theory and Lima et al [23] have studied the spin transport in the site diluted two-dimensional anisotropic Heisenberg model in the easy plane, using the self consistent harmonic approximation. The aim of this paper is to study the spin transport in the twodimensional anisotropic frustrated Heisenberg model on a honeycomb lattice using the SU(3) Schwinger's boson approximation. Recently, this formalism has been used to study the spin transport in [24–27]. The critical properties of this model were studied using this method in [2]. This work is divided in the following way. In Section 2, we discuss about the method used, in Section 3 we develop the Kubo formalism of the linear response to calculate the spin conductivity of this model, Section 4 is dedicated to our conclusions and final remarks. The model that we are interested is represented in Fig. 1 and is defined by the following Hamiltonian  X X  x x J ij Si Sj þ Syi Syj þ D ðSzi Þ2 : ð1Þ H¼ 〈i;j〉

i 0

where J ij ¼ J on all horizontal bonds, J ij ¼ J on all the other bonds, J ij ¼ J 2 on the next-nearest neighbors. We consider the value of spin S ¼1. The frustration here is due to the triangular lattice. This model is interesting in view of its connections to the insulating phase of some layered organic superconductors [30]. The isotropic model with D ¼0 has an ordered ground state. In the limit of large D, the model will be in a disordered ground state with total magnetization null separated by a gap from the first excited states, where there is a critical Dc denoting a quantum phase transition, described by the condensation of magnons, from the large D phase to the ordered phase [2].

2. Method The SU(3) Schwinger boson formalism has been derived to treat systems with single ion anisotropy by Papanicolau [31] being

a generalization of the SU(2) formalism. In this formalism we choose the basis: i j x〉 ¼ pffiffiffiðj 1〉  j  1〉Þ; 2

j z〉 ¼ ij 0〉

where j n〉 are eigenstates of Sz. The spin operators are written via a set of three boson operators t α ðα ¼ x; y; zÞ defined as [29] j x〉 ¼ t †x j v〉;

j y〉 ¼ t †y j v〉;

j z〉 ¼ t †z j v〉;

ð2Þ

where j v〉 is the vacuum state. We also have the constraint condition t †x t x þ t †y t y þt †z t z ¼ 1. In terms of the t operators we can write Sx ¼  iðt †y t z  t †z t y Þ; t †x j v〉

Sy ¼  iðt †z t x  t †x t z Þ;

Sz ¼ iðt †x t y  t †y t x Þ

t †y j v〉,

ð3Þ

z

the states and both consist of eigenstates S ¼ 7 1 and have the average 〈Sz 〉 ¼ 0. This property will preserve the disorder of the ground state. To study disordered phases, it is convenient to † introduce other two bosonic operators u† and d [31] 1 u† ¼  pffiffiffiðt †x þ it †y Þ; 2

1 † d ¼ pffiffiffiðt †x  it †y Þ; 2

ð4Þ

and so j 1〉 ¼ u† j v〉;

j 0〉 ¼ t †z j v〉;

†

j 1〉 ¼ d j v〉; †

ð5Þ

d þ t †z t z

¼ 1. The spin operators can also with the constraint u† u þ d be written as [29] pffiffiffi pffiffiffi † † ð6Þ S þ ¼ 2ðt †z d þ u† t z Þ; S  ¼ 2ðd t z þ t †z uÞ; Sz ¼ u† u þ d d: The Schwinger's boson formalism is a mean field approximation that becomes accurate in the N-1 limit. For the SU (3) Schwinger boson approach for spins S ¼1, the order parameter has eight components which correspond to the eight generators of the SU(3) group [29]. Substituting Eq. (6) into the Hamiltonian (1) and supposing that the tz bosons are condensed, i.e. 〈t z 〉 ¼ 〈t †z 〉 ¼ t, we obtain [29] H¼

J0 X 2 † † ½t ðdr dr þ δ þu†r þ δ ur þur dr þ δ þ dr u†r þ δ þ h:c:Þ 2 r;δ †

†

†

†

þðu†r ur  dr dr Þðu†r þ δ ur þ δ  dr þ δ dr þ δ Þ JX 2 † † þ ½t ðdr dr þ d þ u†r þ d ur þ ur dr þ d þ dr u†r þ d þ h:c:Þ 2 r;d þðu†r ur  dr dr Þðu†r þ d ur þ d dr þ d dr þ d Þ J X 2 † † þ 2 ½t ðdr dr þ 2d þ u†r þ 2d ur þ ur dr þ 2d þdr u†r þ 2d þ h:c:Þ 2 r;d †

†

þðu†r ur  dr dr Þðu†r þ 2d ur þ 2d  dr þ 2d dr þ 2d Þ X X † μr ðu†r ur þ d†r dr þ t 2  1Þ; þ D ðu†r ur þ dr dr Þ  r

ð7Þ

r

where a temperature dependent chemical potential μr is introduced to impose the local constraint S2r ¼ SðS þ 1Þ ¼ 2. One solves the Hamiltonian Eq. (7) using a mean-field approach, replacing the local parameter μr by a single parameter μ and making the mean field decoupling for the remaining operators as in Refe. [29]. We make the Fourier transform of the operators u and d and after this, we write the Hamiltonian in a matrix form as [29] X X H¼ ωk ðα†k αk þ β†k βk Þ þ ðωk  Λk Þ þ μNð1  t 2 Þ; ð8Þ k

Fig. 1. Representation of the flow of a spin current in the horizontal direction on a triangular lattice. The nearest-neighbor exchange J 0 on all horizontal bonds and J are the other bonds. The next-nearest neighbor bonds J2 are not shown in the figure.

1 j y〉 ¼ pffiffiffiðj 1〉þ j  1〉Þ; 2

where the α and and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωk ¼ Λ2k  Δ2k

k

η parameters are defined as α ¼ J 0 =J, η ¼ J 2 =J

Λk ¼  μ þ D þ Δk ;

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ð9Þ ð10Þ

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Δk ¼ zt 2 γ k ;

ð11Þ

where z is the number of nearest neighbors, and pffiffiffi  ð1 þ ηÞ 1 gðkx ; ky Þ; γ k ¼ α cos kx þ η cos 3ky þ 3 3

ð12Þ

pffiffiffi !   3ky kx cos gðkx ; ky Þ ¼ 2 cos 2 2

ð13Þ

At zero temperature Eq. (9) can be simplified as [29,28,2] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ ωk ¼ ð  μ þ DÞ 1 þ yf k ; where y ¼ 2t 2 =ð  μ þDÞ. In this case, the self-consistent equations are given as 1 μ ¼ ½I 2 ðyÞ  I 1 ðyÞ; y

1 t ¼ 2  ½I 2 ðyÞ þ I 1 ðyÞ; 2 2

ð15Þ

where I 1 ðyÞ ¼

Λð! q ; ω þ i0 þ Þ is analytic in the upper half of the complex plane and extrapolation along the imaginary axis can be reliably done. The continuity equation for the lattice allows us to write the discrete version of the current as J n þ x J n ¼ 

and

1 V

Z πZ π 0

0

2

d k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; 1 þ yf k

I 2 ðyÞ ¼

1 V

Z πZ π

2

d k 0

0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ yf k ;

ð16Þ

where f k ¼ zγ k . The integral is performed in the first Brillouin zone of the honeycomb lattice. From these equations we have 2 D ¼ ð2  I 1 ðyÞÞ: y

ð18Þ

When D oDc , the system is in the ordered phase. Following Ref. [32], we assume part of the excitations are condensed at k0. Keeping ωk0 ¼ 0, one finds t 2 ¼ 2  n0 

 1 I 1 ðyc Þ þI 2 ðyc Þ ; 2

μ¼

 1  2n0 þ I 2 ðyc Þ  I 1 ðyc Þ ; yc ð19Þ

where n0 ¼ yc ðDc  DÞ=4. The excitation gap located at k0 is given by Δ ¼ ω0 and it goes to zero when y-yc ¼ 1=j gðk0 Þj . At this point one has a quantum phase transition from the disordered large D phase to the small D ordered phase.

3

∂Szn ; ∂t

ð22Þ

where n þ x is the nearest neighbor site of site n in the positive xz direction. The Heisenberg equation of motion S_ n ¼ i½H; Szn  can be used with Eq. (22) to obtain the spin current operator pffiffiffi t 2 X½ð1 þ αÞ sin kx þ η sin 3kx  † J ¼ J ðαk þ βk Þðα†k þ β k Þ: ð23Þ 2 k ωk Green's function for non-interacting excitons is given by Gðk; iωn Þ ¼

1 ; iωn  ω

ð24Þ

where ωn is the Matsubara frequency. We obtain the temperature dependent Green's function, which is obtained from the zerotemperature Green's function by replacing ω by iωn, where ωn ¼ 2π n=β and Z 1 iX : ð25Þ 2π β n After performing the sum using X n

1 β ; ¼ ðiωn xÞ eβx  1

ð26Þ

where β ¼ 1=T. A simple analytical continuation yields the frequency and temperature dependent Green's function, from Eq. (24), making z ¼ iω. The real part of σ ðωÞ, σ 0 ðωÞ can be written in a standard form as [36] 0

σ ðωÞ ¼ σ 0 ðωÞ þ σ reg ðωÞ;

ð27Þ

where σ 0 ðωÞ gives a measure of the ballistic transport and is given by σ 0 ðωÞ ¼ DS δðωÞ, where DS is the Drude's weight ! 0 ! DS ¼ π ½〈K〉 þ Λ ð q ¼ 0; ω- 0 Þ: and σ

reg

ð28Þ

ðωÞ, the regular part of σ ðωÞ, is given by [36] 0

Λ” ð ! q ¼ 0; ωÞ ; ω

3. Spin transport

σ reg ðωÞ ¼

We use the SU(3) Schwinger's boson approximation [31] with objective to determine the regular part of the spin conductivity (AC conductivity) or continuum conductivity at T ¼0. A spin cur! rent appears if there is a gradient of magnetic field ∇ B , through the system. It plays the role of a chemical potential for spins. One connects a low dimensional magnet with two bulk ferromagnets. They act as reservoirs of spins [13,12]. One flow of spin current ! appears if there is a difference, Δ B , between the magnetic fields at the two ends of the sample. As we are interested in calculating the longitudinal spin conductivity, we will add an external space, ! one dependent on time, magnetic field B ðx; tÞ, applied along the axis z^ , direction of the Hamiltonian Eq. (1). In the Kubo formalism [35,36,14,16] the spin conductivity is given by

that represents the continuum contribution to the conductivity. In 0 Eqs. (28) and (29), Λ and Λ” stand for the real and imaginary parts of Λ. Using the Matsubara's Green's function, we obtain σ reg ðωÞ at zero temperature as

! 〈K〉 þ Λð q ; ωÞ σ ðωÞ ¼ lim ; þ ! q -0 iðω þi0 Þ

ð20Þ

! where 〈K〉 is the kinetic energy and Λð q ; ωÞ is the current–current correlation function defined by Z 1 i ! ! iωt Λð ! q ; ωÞ ¼ dte 〈½J ð q ; tÞ; J ð  q ; 0Þ〉: ð21Þ ℏN 0

σ reg ðωÞ ¼

Λðk ¼ 0; ωÞ ω 2π

¼ ðg μB Þ

ℏ

Z πZ π 2 d k 0

0

ð29Þ

h pffiffiffi i2 ð1 þ αÞ sin kx þ η sin 3kx

ð2π Þ2

ω3k

δðω  ωk Þ: ð30Þ

We can integrate analytically Eq. (30) to obtain " # pffiffiffi pffiffiffi 1 π η2 3π 2  ηð1 þ αÞ sin 3π ; σ  ðωÞ ¼ 3 ð1 þ αÞ þ 4 ω 4

ð31Þ

and

σ reg ðωÞ ¼

ðg μB Þ2 1 σ  ðωÞ: ℏ 4π 2

ð32Þ

In Fig. 2, we present the behavior of σ reg ðωÞ with ω. we must have a spin current flowing in the horizontal as depicted in Fig. 1 over the material. Since the system is gapless we have obtained a

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67 68 69 70 71 Uncited references Q5 72 73 74 [33,34]. 75 76 77 Acknowledgement 78 79 I am grateful the Brazilian agencies FAPEMIG, CNPq, CAPES and 80 CEFET-MG. 81 82 83 References 84 85 [1] C. Lacroix, P. Mendels, F. Mila, Introduction to Frustrated Magnetism, Springer, Berlin, 2011. 86 [2] A.S.T. Pires, Physica A 391 (2012) 5433. 87 [3] Th. Jolicoeur, E. Dagotto, E. Gagliano, S. Bacci, Phys. Rev. B 42 (1990) 4800. 88 [4] Yu. M. Bunkov, G. E. Volovik, Novel Superfluids, eds. K. H. Bennemann and J. B. Ketterson, Oxford University Press, arXiv:1003.4889v3 (2013). 89 [5] B.D. Josephson, Phys. Rev. Lett. 1 (1962) 251. 90 [6] A.S. Borovik-Ramanov, Yu.M. Bunkov, V.V. Dmitriev, Yu.M. Mukharskiy, Long91 lived, J. Exp. Theor. Phys. Lett. 40 (1984) 1033; 92 A.S. Borovik-Ramanov, Yu.M. Bunkov, V.V. Dmitriev, Yu.M. Mukharskiy, Sov. Phys. J. Exp. Theor. Phys. 61 (1985) 1199. 93 [7] I.A. Fomin, J. Exp. Theor. Phys. Lett. 60 (1984) 1037. 94 [8] Yu.M. Bunkov, E.M. Alakshin, R.R. Gazizulin, A.V. Klochkov, V.V. Kuzmin, V. 95 S. L'vov, M.S. Tagirov, Phys. Rev. Lett. 108 (2012) 177002. [9] S.O. Demokritov, V.E. Demidov, O. Dzyapko, G.A. Melkov, A.A. Serga, 96 B. Hillebrands, A.N. Slavin, Nature 443 (2006) 430. 97 [10] A.V. Chumak, V.I. Vasyuchka, A.A. Serga, B. Hillebrands, Nat. Phys 11 (2015) 98 453. [11] V. Lauer, D.A. Bozhko, T. Bracher, P. Pirro, V.I. Vasyuchka, A.A. Serga, M. B. 99 Jungfleisch, M. Agrawal, Yu. V. Kobljanskyj, G.A. Melkov, C. Dubs, B. Hilleb100 rands, A.V. Chumak, arXiv:1508.07517v1 (2015). 101 [12] F. Meier, D. Loss, Phys. Rev. Lett. 90 (2003) 167204. [13] Yuri V. Kobljanskyj, Gennadii A. Melkov, Alexander A. Serga, Andrei N. Slavin, 102 Burkard Hillebrands Phys. Rev. Applied 4 (2015) 014014. 103 [14] M. Sentef, M. Kollar, A.P. Kampf, Phys. Rev. B 75 (2007) 214403. 104 [15] K. Damle, S. Sachdev, Phys. Rev. B 56 (1997) 8714. [16] A.S.T. Pires, L.S. Lima, Phys. Rev. B 79 (2009) 064401. 105 [17] A.S.T. Pires, L.S. Lima, J. Phys.: Condens. Matter 21 (2009) 245502. 106 [18] A.S.T. Pires, L.S. Lima, J. Magn. Magn. Mater. 322 (2010) 668. 107 [19] L.S. Lima, A.S.T. Pires, Eur. Phys. J. B 70 (2009) 335. [20] L.S. Lima, Phys. Status Solidi B 249 (2012) 1613. 108 [21] Zewei Chen, Trinanjan Datta, Dao-Xin Yao, Eur. Phys. J. B. 86 (2013) 63. 109 [22] Yurika Kubo, Susumu Kurihara, J. Phys. Soc. Jpn. 82 (2013) 113601. 110 [23] L.S. Lima, A.S.T. Pires, B.V. Costa, J. Magn. Magn. Mater. 371 (2014) 89. 111 [24] L.S. Lima, Solid State Commun. 228 (2016) 6. [25] L.S. Lima, Solid State Commun., 2016, in Press. 112 [26] L.S. Lima, J. Magn. Magn. Mater. 411 (2016) 108. 113 [27] L.S. Lima, J. Magn. Magn. Mater. 405 (2016) 332. 114 [28] A.S.T. Pires, Solid State Commun. 217 (2015) 61. [29] A.S.T. Pires, Physica B 479 (2015) 130. 115 [30] A.E. Trumper, Phys. Rev. B 60 (1999) 2987. 116 [31] N. Papanicolaou, Nucl. Phys. B 305 (1988) 367. 117 [32] H.T. Wang, Y. Wang, Phys. Rev. B 71 (2005) 104429. [33] A.S.T. Pires, J. Mag. Magn. Matter, 323, 1977 (2011); A.S.T. Pires, Solid State 118 Commun. 152, 1838 (2012); A.S.T. Pires, Solid State Commun. 196, 24 (2014). 119 [34] P. Li, G.M. Zhang, S.Q. Shen, Phys. Rev. B 75 (2007) 104420. 120 [35] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Springer-Verlag, New York, 1985. 121 [36] G.D. Mahan, Many Particles Physics, Plenum, New York, 1990. 122 [37] S. Sachdev, Science 336 (2012) 1510. 123 [38] E. Fradkin, Field Theories of Condensed Matter Physics, second edition, Cambridge, UK, 2013. 124 125 126 127 128 129 130 131 132

often only the sign differences between related quantities like magnetic fields and generated spin and charge currents are determined.

ðgμb Þ2 σ  ðωÞ for α ¼ 0:3. In the range of Fig. 2. Behavior of σ  ðωÞ where σ reg ðωÞ ¼ ℏ Dc ¼0 and η ¼ 0:15 at the T ¼ 0, the behavior of the spin conductivity does not vary a lot in the region of the quantum phase transition. As the AC conductivity tends to infinity in the limit ω-0, we have an ideal spin conductor in that limit.

behavior of the AC conductivity tending to infinity when ω-0, what correspond to the DC limit. Consequently we obtain a superconductor behavior to spin transport for the DC spin current. This behavior is similar one recently obtained for the honeycomb lattice and for the two-dimensional ferroquadrupolar model [25,24]. This is characteristic of spin systems without gap in the excitation spectrum. As this system does not present gap in the spectrum, we have excitons to form the spin current in all the ω values of the spectrum. Besides, if there is no scattering mechanism, as in this model, it is expected that the conductivity is divergent; the reason is that spin–spin scattering is not treated properly in a mean field approach.

4. Conclusions and final remarks We studied the spin transport properties in the twodimensional anisotropic frustrated Heisenberg model in the triangular lattice using the SU(3) Schwinger's boson theory. Because of the triangular symmetry of the lattice, we must have a spin current flowing in the J 0 (horizontal direction) through the material. How we obtain a spin conductivity tending to infinity when ω-0, we must have, in this limit, ideal spin transport, independent of the value of the Drude's weight DS found. Recently, the critical properties of this model were studied using the SU(3) Schwinger boson method at [2]. From a general way, the two-dimensional Heisenberg model is a very important model due the intense tentative to understand the interplay of antiferromagnetism and superconductivity [37,38]. From an experimental point of view, recently there is an intense research about the quantum Hall effect for spins and magnon spintronics [12,13,10,11]. In the studies of these effects,

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