A Realization of the Haldane-Kane-Mele Model in a System of Localized Spins

# A Realization of the Haldane-Kane-Mele Model in a System of Localized Spins

## Abstract

We study a spin Hamiltonian for spin-orbit-coupled ferromagnets on the honeycomb lattice. At sufficiently low temperatures supporting the ordered phase, the effective Hamiltonian for magnons, the quanta of spin-wave excitations, is shown to be equivalent to the Haldane model for electrons, which indicates the nontrivial topology of the band and the existence of the associated edge state. At high temperatures comparable to the ferromagnetic-exchange strength, we take the Schwinger-boson representation of spins, in which the mean-field spinon band forms a bosonic counterpart of the Kane-Mele model. The nontrivial geometry of the spinon band can be inferred by detecting the spin Nernst effect. A feasible experimental realization of the spin Hamiltonian is proposed.

###### pacs:
85.75.-d, 75.47.-m, 73.43.-f, 72.20.-i

Introduction.— Electronic systems with spin-orbit coupling (SOC) can exhibit spin Hall effects, in which a longitudinal electric field generates a transverse spin current and vice versa ?; ?. In particular, Kane and Mele (2005) showed that a single layer of graphene has a topologically nontrivial band structure with an SOC-induced energy gap, which gives rise to a quantum spin Hall effect characterized by helical edge states. This identification of graphene as a quantum spin Hall insulator has served as a starting point for the search for other topological insulators ?; ?; ?; ?.

SOC magnets with no charge degrees of freedom can also exhibit various Hall effects Haldane and Arovas (1995); ?; ?; ?; ?; ?; ?; ?; ?; Katsura et al. (2010); Lee et al. (2015); Matsumoto and Murakami (2011a); ?. By exploiting the ubiquitous spin-heat interactions ?; ?, thermal Hall effect, in which a longitudinal temperature gradient induces a transverse heat current, has been used to probe SOC in such insulating magnets. For ordered mangets, thermal Hall effects are often accounted for by geometrically nontrivial band structures of magnons, quanta of spin-wave excitations. For example, Matsumoto and Murakami (2011a) showed that certain engineered thin-film ferromagnets can have chiral edge states of magnetostatic spin waves (that are associcated with topologically nontrivial bulk band structures) and thus exhibit a thermal Hall effect. For magnets that are disordered due to either thermal or quantum fluctuations, thermal Hall effects have been predicted by using the Schwinger boson or fermion representation of spins Arovas and Auerbach (1988); ? that do not need a putative symmetry-breaking underlying state Katsura et al. (2010); Lee et al. (2015); Okamoto (2016).

In this Letter, we propose a simple spin Hamiltonian for SOC ferromagnets on the honeycomb lattice, in which we can find the bosonic counterparts of both the Haldane Haldane (1988) and the Kane-Mele model Kane and Mele (2005). In the ordered phase supported at sufficiently low temperatures, we show that the effective Hamiltonian for magnons is equivalent to the Haldane model Haldane (1988). For elevated temperatures, where the system is disordered, we take an alternative Schwinger-boson (or bosonic-spinon) representation of spins, in which the mean-field spinon band is identified as a bosonic counterpart of the Kane-Mele model Kane and Mele (2005). The nontrivial geometry of the spinon band gives rise to the spin Nernst effect, in which a longitudinal temperature gradient generates a transverse spin current Bauer et al. (2012). Lastly we propose a feasible way to realize the underlying Hamiltonian.

Model.—We consider a ferromagnetic material with SOC whose localized spins are arranged on a honeycomb lattice. The corresponding model Hamiltonian reads

 (1)

where the first and second terms represent the isotropic Heisenberg interaction () and the Ising interaction between nearest neighbors, respectively 1. The third term is the Dzyaloshinskii-Moriya (DM) interaction Dzyaloshinsky (1958); ? between next-nearest neighbors, where the constants characterize the dependence of the interaction on the relative position of two next-nearest spins [Fig. 1(a)]. Notice that Eq. (1) represents the minimal Hamiltonian describing the above interactions that are invariant under point-group symmetry of the lattice with concurrent spatial and spin rotations. For the particular experimental realization that we propose in this Letter (see below), the Ising contribution in Eq. (1) can be safely neglected compared to the other terms. As we are interested in the topological properties of the model for both ordered and disordered phases, we introduce an external magnetic field applied along the direction [and therefore a Zeeman coupling term in Eq. (1)] to stabilize the ferromagnetic ground state. We shall denote by and by the distances between nearest neighbors and next-nearest neighbors, respectively.

Magnon picture.—The uniform state represents the classical ground state of the model Hamiltonian (1) for and . Application of the Holstein-Primakoff transformation , , and with yields the following effective magnon Hamiltonian

 Hm= (3JS+B)∑id†idi−JS∑⟨i,j⟩[d†idj+h.c.] (2) −DS∑⟨⟨i,j⟩⟩[iνijd†idj+h.c.],

up to second order in the magnon operators and . In this approximation, the Hamiltonian reduces to the Haldane model Haldane (1988).

The topological features of the magnon bands can be readily captured in the momentum representation. Let be the spinor operators in the Fourier space, where and represent magnon annihilation operators on the sublattices and , respectively. Fourier transform of the Hamiltonian (2) then reads

 Hm=∑k∈B.Z.Ψ†k[(3JS+B)I+h(k)⋅τ]Ψk, (3)

where is a pseudovector of the Pauli matrices and

 h(k)=∑j⎛⎜⎝−JScos[k⋅αj]JSsin[k⋅αj]2DSsin[k⋅βj]⎞⎟⎠={3√3DS^zk=K−3√3DS^zk=K′, (4)

where and are defined in Fig. 1(a), , and . The dispersions of the upper and the lower energy band are given by

 E±m(k)=3JS+B±|h(k)|. (5)

In the absence of SOC (), the upper and the lower band meet at two points, and , forming linearly-dispersed bands Fransson et al. (); Owerre (). SOC opens an energy gap at these points, making the band structure topologically nontrivial Kane and Mele (2005). Figures 1(b) and (c) show the one-dimensional projection of the magnon bands and the direction of , respectively, for the values and 2.

The Berry curvatures of the upper and the lower magnon bands can be calculated according to the formula , where is the unit vector along Xiao et al. (2010). Figure 1(d) plots the Berry curvature of the upper band. Notice that the Berry curvature is large around the corners of the Brillouin zone, and , where the vector exhibits nontrivial topological textures that wrap a half of the unit sphere. The Chern numbers Shindou et al. (2013b) of the bands are evaluated as .

Spinon picture.—While the magnon picture is valid at sufficiently low temperatures where the system is ordered, it fails when the system is disordered due to thermal fluctuations. For high temperatures comparable to the exchange strength , the Schwinger-boson representation of spins Arovas and Auerbach (1988) provides an alternative approach to study the topological features of the spin system. The corresponding transformation reads , and . Here () represents the annihilation (creation) operator of spin- up- or down bosons at the site , which are referred to as Schwinger bosons or bosonic spinons. The local number constraint, , needs to be imposed to fulfill the spin- algebra. The Hamiltonian (1) in the spinon picture reads

 Hs= −2J∑⟨i,j⟩χ†ijχij−B2∑i(c†i,↑ci,↑−c†i,↓ci,↓) (6) +∑iλi(c†i,↑ci,↑+c†i,↓ci,↓−2S),

up to a constant, where are operators defined for pairs of sites for each spin , , and is the Lagrange multiplier related to the above holonomic constraint.

As the first and third terms are quartic in the spinon operators, we take the mean-field approach 3. We choose the Hartree-Fock decoupling Bruus and Flesberg (2004) that retains the symmetries of the original Hamiltonian by conserving the total number of spinons and the -component of the total spin by following Ref. Lee et al. (2015). First, we use the mean field for nearest neighbors and . While is a complex number generally, we can assume it real by absorbing its factor into the operator . We then make the substitution, in the Hamiltonian. Secondly, for next-nearest neighbors, we define two mean fields for : for the symmetric part (with respect to ) and for the antisymmetric part. The two mean fields are real owing to . We then perform the necessary substitution. Thirdly, we replace the local Lagrange multipliers by the global one . The resultant mean-field Hamiltonian is given by

 Hs= −ηJ∑⟨i,j⟩,s[c†i,scj,s+h.c.] (7) +D2∑⟨⟨i,j⟩⟩,s[iνijsζ−sc†i,scj,s+h.c.] +D2∑⟨⟨i,j⟩⟩,s[sξ−sc†i,scj,s+h.c.] +∑i,s(λ−sB2)c†i,sci,s.

The mean fields and represent short-ranged spin correlations Sarker et al. (1989). The first two terms in the above Hamiltonian then correspond to the Kane-Mele model Kane and Mele (2005), from which we can infer the nontrivial topology of the spinon-band structure and the existence of edge states for both spin-up and spin-down spinons. As we shall discuss below, the third term of Eq. (7) does not affect the topological features of the spinon bands.

The spinon Hamiltonian in the momentum representation reads

 Hs=∑k∈B.Z.,sΨ†k,s[gs(k)I+hs(k)⋅τ]Ψk,s, (8)

where is the spinor of annihilation operators, , and

 hs(k)=∑j⎛⎜⎝−Jηcos[k⋅αj]Jηsin[k⋅αj]−Dsζ−ssin[k⋅βj]⎞⎟⎠, (9)

where the vectors and are depicted in Fig. 1(a). The corresponding upper and lower energy bands for each spin are then given by

 E±s(k)=gs(k)±|hs(k)|. (10)

Notice that the spin-down spinon bands mimic the magnon bands owing to the similarity between the momentum dependence of [Eq. (9)] and [Eq. (4)].

Self-consistency of the mean-field approach is guaranteed through the equations in momentum space 4:

 2S =12N∑k,s[ρ−s(k)+ρ+s(k)], (11) 12N =J∑k,sρ−s(k)−ρ+s(k)|hs(k)|∣∣∑ieik⋅αi∣∣2, ζs =16N∑k[ρ−s(k)+ρ+s(k)][∑icos(k⋅αi)], ξs =sDζ−s6N∑kρ−s(k)−ρ+s(k)|hs(k)|∣∣∑isin(k⋅αi)∣∣2,

where is the Bose-Einstein distribution of spin- spinons in the band and is the number of unit cells. Note that the total number of spinons is fixed by the first condition. This enables the Bose condensation of spinons in the limit of zero temperature, which corresponds to magnetic ordering Sarker et al. (1989). The mean-field spinon bands are obtained by solving self-consistently Eqs. (10) and (11), which are shown in Fig. 2(a) at the temperature for the values , and . The SOC induces an energy gap between the spin- spinon bands, whose Chern numbers read . Therefore, the topological nontriviality of the bulk bands for each spin supports the edge states. The thermal dependence of the mean fields and for the parameters is shown in Figs. 2(b) and (c). The vanishing of these fields at a finite temperature marks a transition between phases with finite (low-) and zero (high-) correlation lengths of spins. Our mean fields represent short-ranged (between 1st and 2nd nearest neighbors) spinon correlation functions, and thus this phase transition is not associated with an onset of long-ranged ordering. The presence of the seriously disordered high- phase with no correlation between spins is expected to be an artifact of mean-field treatments, i.e., the limit of in the generalization of the symmetry group from SU(2) to SU(N), as for the case of the pure Heisenberg model () where it has been shown to disappear after accounting for fluctuations from the mean fields Tchernyshyov and Sondhi (2002).

A connection of the spinon picture to the magnon picture can be established by taking the zero-temperature limit , where the spin-down spinon bands are equivalent to the magnon bands and the spin-up spinons form the Bose-Einstein condensation that is the ordered ground state in the magnon picture. To see this, let us apply an external magnetic field , which makes the system completely polarized along the axis, , as . This polarization corresponds in the spinon picture to the Bose-Einstein condensation of spinons into the lowest-energy mode localized at the state for the spin-up spinon band Sarker et al. (1989). The mean fields associated with this polarized state are and , for which the spin-down spinon bands are equivalent to the magnon bands, 5. .

Spin Nernst effect.—Spin-up and spin-down spinons experience opposite Berry curvatures, , in the absence of an external magnetic field. This can induce the spin Nernst effect Bauer et al. (2012), in which a transverse spin current is generated by applying a longitudinal temperature gradient, . The spinon picture is well suited to compute the spin Nernst conductivity due to its applicability over a broad range of temperatures. We use the expression for derived in Ref. Kovalev and Zyuzin (2016) for the free magnon bands, , where is the volume of the system and .

Figure 3 shows the thermal dependence of the spin Nernst conductivity for the physical parameters . At zero temperature due to the absence of thermal excitations. As the temperature increases, spinons are thermally populated and becomes finite. As the temperature approaches the ferromagnetic-exchange strength , the magnitudes of the mean fields start decreasing. The bands thereby flatten more and have smaller Berry curvatures, which in turn results in the suppression of . Application of a finite magnetic field increases the energies of the spin-down spinons, which in turn decreases the magnitude of the spin Nernst effect. The magnon picture should give similar numerical results for the spin Nernst effect for low temperatures and small magnetic field , owing to the equivalence between the magnon and the spin-down spinon bands in the limit and also the relation between two spinon bands in the limit .

Discussion—Although, to the best of our knowledge, the proposed Hamiltonian does not correspond to any existing material, the model may be engineered by depositing magnetic impurities on metals with strong spin-orbit coupling. The minimal Hamiltonian consists of two terms, . The first term describes the dynamics of surface electrons, whereas the second one describes the coupling between the localized spins and the spin density of the metal evaluated at the position of the impurities, . Magnetic interactions are mediated by itinerant electrons through the Ruderman-Kittel-Kasuya-Yosida Ruderman and Kittel (1954); ?; ? interaction. When the system respects both mirror () and C point-group symmetries, the effective Hamiltonian reduces to Eq. (1). We provide two exemplary realizations of such system in the Supplementary Material 6. It is worth remarking that the Ising-like coupling appears as a second order effect in the SOC, which justifies neglecting it over the other first order terms in the SOC.

Breaking the mirror symmetry, e.g., by an external electric field , can generate a DM interaction between the nearest neighbors Fransson et al. (2014),

 H′=D′∑⟨i,j⟩(^z×αij)⋅(Si×Sj). (12)

The term translates into a Rashba-like hopping term in the spinon mean-field Hamiltonian:

 (13)

This term competes with the intrinsic DM interaction in Eq. (1) and can close the topological gaps at the Dirac points if sufficiently strong as in the Kane-Mele model Kane and Mele (2005). In the presence of a strong magnetic field , however, the effect of the term is of order in the perturbative treatment due to the energy separation between the and bands, which allows us to neglect its effect on the gaps .

Another possibility would be using chromium tri-halides like CrBr, which consist of weakly-coupled ferromagnetic honeycomb layers ?; ?, with the DM interaction induced by the proximity effect with strong spin-orbit coupled materials, e.g., Pt.

In the spinon picture, we have neglected fluctuations of the Lagrangian multiplier and the bond operators from their mean-field values, which can be taken into account by, e.g., performing corrections (the mean-field treatment corresponds to generalizing the spin symmetry group from SU(2) to SU(N) and taking limit) Trumper et al. (1997). In particular, the phase fluctuations of the bond operators couple to the spinons as the U(1) gauge fields, which has been shown to result in confining the spinons in the ordered phases of some frustrated magnets, e.g., the Heisenberg antiferromagnet on the square lattice Read and Sachdev (1991); ?. Investigating effects of the mean-field fluctuations in our spinon picture is a topic for future research.

After the completion of the manuscript, we became aware of recent related works Owerre (), in which the author studied the topological property of the magnon band on the honeycomb lattice and the associated thermal Hall effects. The spinon bands and spin Nernst effects, however, are not discussed in the reference.

###### Acknowledgements.
We are grateful to Elihu Abrahams, Yong P. Chen, Fenner Harper, and Jing Shi for insightful discussions. This work was supported by the Army Research Office under Contract No. 911NF-14-1-0016, by the U.S. Department of Energy, Office of Basic Energy Sciences under Award No. DE-SC0012190, and by the NSF-funded MRSEC under Grant No. DMR-1420451. RZ thanks Fundación Ramón Areces for the postdoctoral fellowship within the XXVII Convocatoria de Becas para Ampliación de Estudios en el Extranjero en Ciencias de la Vida y de la Materia.

### Footnotes

1. Each nearest-neighbor pair contributes only once to the summation in the first and the second term; so does each next-nearest-neighbor pair to the third term.
2. To be precise, Fig. 1(b) is obtained by combining the plots of as a function of for all .
3. The mean-field configuration corresponds to the saddle-point solution of the action in the path-integral formulation Arovas and Auerbach (1988).
4. Formally, the self-consistency conditions can be obtained by demanding the functional derivative with respect to the mean fields of the free energy to vanish.
5. The agreement between the magnon and the spin-down spinon bands in the zero-temperature limit can be also understood by considering the operator in the ground state . Specifically, in the magnon picture, where we used ; in the spinon picture, where we used in the condensed state Sarker et al. (1989). In this approximation, the magnon and spin-down spinon operators are equal.
6. See Supplemental Material for details.

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