Elastic Gauge Fields and Hall Viscosity of Dirac

Yago Ferreiros1 and Mar´ıa A. H. Vozmediano2 1Department of Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden∗ 2Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco; 28049 Madrid, Spain. (Dated: July 10, 2021) We analyze the coupling of elastic lattice deformations to the degrees of freedom of magnon Dirac materials. For a Honeycomb ferromagnet we find that, as it happens in the case of , elastic gauge fields appear coupled to the magnon pseudospinors. For deformations that induce constant pseudomagnetic fields, the spectrum around the Dirac nodes splits into pseudo- Landau levels. We show that when a Dzyaloshinskii-Moriya interaction is considered, a topological gap opens in the system and a Chern-Simons effective action for the elastic degrees of freedom is generated. Such a term encodes a Hall viscosity response, entirely generated by quantum fluctuations of magnons living in the vicinity of the Dirac points. The magnon Hall viscosity vanishes at zero temperature, and grows as temperature is raised and the states around the Dirac points are increasingly populated.

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I. INTRODUCTION pseudospin is associated to the sublattice degree of free- dom. Still, the will have a non trivial Berry Since the synthesis of graphene, Dirac materials are phase in momentum space and may acquire topological a common trend in condensed matter [1]. They appear properties when gapped, leading to magnon responses in a variety of compounds with the characteristic that such as the thermal Hall effect and the Nernst effect their low energy elementary excitations are described by [14, 25–36]. a Dirac Hamiltonian. Most of these materials are based Historically, the most interesting properties of the on moving on a periodic lattice and their el- Dirac Hamiltonian in two spatial dimensions have been ementary excitations are fermionic degrees of freedom. associated with the responses to vector fields minimally Interesting exceptions are the optical lattices [2], pho- coupled to the spinors, in particular the electromagnetic tonic crystals [3], or lattice constructions like the arti- field. A very appealing phenomenon in graphene and ficial graphene described in [4]. Magnons are becom- similar Dirac materials is the coupling of the lattice de- ing interesting players in the new condensed matter sys- formations to the spinorial degrees of freedom. This led tems. The proposal of topological magnons [5–10] has to the prediction of elastic gauge fields in two and three- been followed by that of magnon Dirac matter [11–20]. dimensional Dirac materials, such as graphene [37] and Dirac magnons have been very recently observed in three- Weyl [38]. In the case of graphene, the pres- dimensional Dirac antiferromagnets [21]. Although de- ence of elastic gauge fields was confirmed experimentally scribed by a Dirac Hamiltonian, the constituent elements in [39] -see also the realization in [4]). The general in- are with no electric charge. In two dimensions, fluence of elastic lattice deformations on the spinor de- Dirac magnons can arise as excitations of Honeycomb grees of freedom was analyzed in [40] and has been re- lattices of localized spins. Honeycomb ferromagnets are viewed recently in [41]. In this work we study the in- realized in chromium trihalides CrX3 (X = F, Cl, Br, fluence of lattice deformations on the magnon physics I) [15], with recent theoretical and experimental works in the magnon Dirac materials. The main aspect of demonstrating the robustness of the Honeycomb ferro- this work is related to the vector coupling of elasticity magnetic phase in CrI3 [22–24]. In addition, they can be with the magnons. As an immediate consequence, under artificially engineered by depositing magnetic impurities the particular strain giving rise to constant pseudomag- on metallic substrates [14]. netic fields, the magnon spectrum will be organized in An interesting question is what properties of ”Dirac “magnon Landau levels”, an unexpected issue that can arXiv:1711.08653v2 [cond-mat.mes-hall] 5 Feb 2018 physics” will survive in these bosonic constructions and be probed experimentally. what will be the physical interpretation of the ”spinor” Furthermore, when a Dzyaloshinskii-Moriya interac- response functions. Irrespective of the nature of the ele- tion [42–44] for spins is considered, a topological gap mentary excitations, the solutions of the opens and the Dirac magnon system resembles a Hal- will be spinors in a general sense. In the particular case dane model [45] for magnons [13, 14]. For fermionic ex- of the Honeycomb ferromagnet lattice constructions, the citations, the Haldane model has been recently shown to generate a phonon Hall viscosity response when coupled to elasticity, arising from a Chern-Simons term for the elastic gauge fields [46]. The Hall viscosity is a topologi- ∗ [email protected] cal viscoelastic response first described in the context of ⌫ =+1 ⌫ = 1 ij ij

2

B A 2

↵2 (a) (b) ⌫ij =+1 ⌫ij = 1 ↵1 3 ↵3 1

B FIG. 1. (a) Honeycomb lattice with the sites corresponding to the two sublattices A and BA represented by red and blue circles, respectively. The nearest and next-nearest neighbor vectors, αi and βi, are represented by blue and2 red arrows, respectively. (b) Relative sign νij of the Dzyaloshinskii-Moriya interaction. ↵2 ↵1 3 quantum Hall fluids[47], and subsequently investigated where Ψk = (ak, bk), a and b are the magnon annihilation ↵3 in a number of works [48–54] (please note that this list operators on the two sublattices, σ=1 (σx, σy, σz), and is just a small sample and not exhaustive). In this work  JS cos[k α ] we show that for magnon excitations, the Haldane model X − · i d(k) =  JS sin[k αi]  , (6) presents a Hall viscosity response which increases with · temperature and vanishes at T = 0. We call it magnon i 0 Hall viscosity as it is entirely generated by magnon fluc- with the vectors αi defined in Fig.1(a). The dispersion tuations. relations for the upper and lower energy bands are given by II. THE MODEL E±(k) = 3JS + B d(k) . (7) ± | | The upper and lower bands meet at six Dirac points, We consider a ferromagnetic material whose localized with only two of them being inequivalent. Let us pick spins are arranged on a Honeycomb lattice. The corre- Ks = (s4π/3√3a, 0), with s = and where a is the sponding model Hamiltonian reads distance between nearest neighbors.± Expanding around X X z these two Dirac points we get H = J Si Sj B Si , (1) − · −   hi,ji i svJ kx/2 s d (k) =  vJ ky/2 , (8) where the first term represents the isotropic Heisenberg D − interaction (J > 0) between nearest neighbors, and the 0 second term is a Zeeman coupling to a magnetic field with vJ = 3JSa/2~. The continuum Dirac Hamiltonian applied along the z direction. By applying the Holstein- is then Primakoff transformation Z 2 s d k † + x y p S = S + iS = 2S n d , (2) HD = 2 Ψk ~ vJ (skxσx kyσy)Ψk. (9) i i i − i i (2π) − S− = (S+)†,Sz = S n , (3) i i i − i † with ni = di di, we arrive at the following effective III. ELASTIC GAUGE FIELDS AND MAGNON magnon Hamiltonian LANDAU LEVELS X X H = (3JS + B) d†d JS (d†d + H.c.). (4) i i − i j i hi,ji To introduce elastic deformations of the lattice, we al- low for small displacements of the spins away of their up to second order in the magnon operators. Although initial positions. We can write J Ji, where i stands higher order corrections may account for anharmonic for the three nearest-neighbors. Now≡ the value of the magnon effects, they preserve the relevant symmetries exchange coupling changes according to variations of the and therefore do not destroy the Dirac magnon proper- positions of the spins. Applying the same methodology ties [11]. Fourier transforming the Hamiltonian we get as for graphene [37, 55], expanding the exchange coupling [11, 13, 14] around its equilibrium value J X † H = Ψk[(3JS + B) 1 + d(k) σ]Ψk, (5) βJ · Ji = J αi δui, (10) k∈B.Z. − a2 · 3

(a) (b)

FIG. 2. Total Berry curvature (a) as computed from the vector dDM , versus Dirac Berry curvature (b), obtained by adding the Berry curvatures of the six gapped Dirac cones located at the six Dirac points.

where δui is the displacement of the spins around their where q is a parameter denoting the strength of the equilibrium configurations, and β is a parameter that strain, with units of inverse length. This exact arrange- controls the expansion and has to be computed from the ment was experimentally realized for molecular graphene microscopic properties of the system, we obtain the elas- [4]. The value of the pseudomagnetic field associated to tic vector fields this distortion is 8βJq 16~βq el βJS eB = = . (16) Ax = (uxx uyy), (11) vJ − | | vJ 3a This is, magnons react to the distortion of the lattice in 2βJS a similar way as particles of charge e would react to a Ael = u . (12) y − v xy magnetic field of magnitude given by Eq. (16). Solving J the Dirac equation in the presence of this constant pseu- Here uab = (∂aub + ∂bua)/2 is the strain tensor, a, b = domagnetic field gives rise to the quantized spectrum x, y, and u is the displacement vector, which arises when q 2 taking the continuum limit δu (α ∂) u [37, 55]. The En = 2~ eB v n, (17) i i ± | | J continuum Dirac Hamiltonian now∼ reads· so that the magnons arrange into Landau levels around Z 2 2 0 s d k d k † n el 0 the Dirac points. Since the pseudomagnetic field is of H = Ψ vJ [s~kx A (k k )]σx D (2π)4 k − x − opposite sign at the two valleys, there are counterprop- el 0 o agating edge states [56] and the total Chern number is [~ky sA (k k )]σy Ψk0 . (13) − − y − zero. Finally we can estimate the size of the gap be- tween Landau levels as a function of J. Taking a lattice One could engineer Dirac magnons using STM, by de- spacing of a 10 A˚ and a maximum strain strength of positing magnetic defects on a metallic substrate form- −∼1 q 10−3 A˚ [4], setting S = 1, and doing ak 1 ing a Honeycomb lattice of spins. For large separation F in∼ Eq. (14) so that β 1, we obtain E √nJ/ 2. between the spins, the exchange coupling between differ- n For the artificial Honeycomb∼ lattice of magnetic∼ ± defects ent sites can be written as J sin(2k R )/ R 2 [14], i F i i we are considering, the value of J will depend on the ex- as described by the RKKY interaction.∼ | Here| |k | is the F change interaction between the localized spins and the Fermi wave vector of the substrate and R is the dis- i spin density of the Fermi sea, and on the distance be- tance between nearest neighboring spins.| Allowing| for tween nearest neighboring spins [14]. small displacement around the equilibrium configuration Ri = αi + δui, we can expand Ji to first order in δui and obtain Eq. (10) with β given by IV. MAGNON HALL VISCOSITY

β = 2 2akF cot(2akF ), (14) − Let us consider an extension of the model of Eq. (1) by 2 and with J sin(2akF )/a . adding an inversion symmetry breaking Dzyaloshinskii- ∼ By playing with the position of the spins on the sub- Moriya (DM) term [13, 14] strate, one can engineer specific forms of Ael . A con- x,y X X z stant pseudomagnetic field, of opposite sign in the two HDM = J Si Sj B Si − · − valleys s = , can be obtained by the following arrange- hi,ji i ± X ment of the spins [56] +D ν zˆ (S S ). (18) ij · i × j u = 2qxy, u = q(x2 y2), (15) hhi,jii x y − 4

The DM interaction (last term) depends on the relative In what follows, the coupling to elasticity will be position of two next-nearest neighboring spins through treated perturbatively, with the equilibrium configura- the constants νij = νij = 1 [Fig.1(b)]. The effec- tion given by the undeformed Honeycomb lattice. Under tive Hamiltonian after− the application± of the Holstein- minimal coupling to gauge fields, all quantum fluctua- Primakoff transformation turns out to be tions in bands with non zero Chern number contribute to the generation of Cherns-Simons terms. As an exam- X † X † HDM = (3JS + B) d di JS (d dj + H.c.) ple, electronic systems with non zero Chern number, this i − i i hi,ji is fermionic Chern insulators, give rise to the quantum X † anomalous Hall effect, which at the level of the action DS (iν d d + H.c.), (19) − ij i j is given by a Chern-Simons term for the electromagnetic hhi,jii field. In our case, the minimal coupling is to elasticity, so which basically is the Haldane model for magnons pro- quantum fluctuations in our Haldane model for magnons posed in Refs. [13, 14]. (which basically is a bosonic Chern insulator) will give The Fourier transform of Hamiltonian (19) is captured rise to a Chern-Simons term for the elastic gauge fields. by Eq. (5) with the vector d given now by The mechanism is analogous to what happens in the Hal- dane model of electrons coupled to elasticity [46], and the  JS cos[k α ] Chern-Simons term reads X − · i dDM (k) =  JS sin[k αi]  , (20) · C˜ Z i 2DS sin[k β ] 3 µρν el el i SCS = d x  Aµ ∂ρAν , (25) · 4π~ with the next-nearest vectors βi defined in Fig.1(a). el ˜ The dispersion of the upper and lower energy bands is where µ, ρ, ν = t, x, y, At = 0, and C is given by the con- tributions to the Chern number of the occupied bands. ± EDM (k) = 3JS + B dDM (k) , (21) The difference with the electronic case, which at half fill- ± | | ing is given by C˜ = C− = 1, is just that magnons are − with at the Dirac points, Ks, of ∆ = bosons, so they contribute with the Bose-Einstein distri- | | 6√3DS. In analogy with the trivial honeycomb magnon bution function system of Eq. (4), when coupled to elasticity and ex- 1 X Z panded around the Dirac points, the magnon Haldane C˜ = Ωτ (k)ρτ (k) d2k, (26) 2π model gives rise to the elastic gauge fields of Eqs. (11) τ=± B.Z. and (12). The topological nature of the Hamiltonian of Eq. (19) is captured by the Berry curvatures of the upper with and lower magnon bands [57] 1 ρτ (k) = . (27) 1 Eτ (k)/κT Ω±(k) = dˆ (∂ dˆ ∂ dˆ), (22) e DM 1 ∓2 · kx × ky − el dDM However, Eq. (26) is only valid if A couples minimally dˆ = , (23) µ dDM to magnons in the entire Brillouin zone. This is not the | | case though, as the coupling to elasticity is only mini- with Chern numbers mal near the Dirac points, where the physics of the sys- tem is dominated by the Dirac approximation, so strictly 1 Z C± = Ω± d2k = 1. (24) speaking Eq. (26) is not totally accurate. By moving 2π B.Z. ± away from the Dirac points, the “Diracness” is lost and One may wonder what is the physical picture when pseu- magnons do not contribute to Eq. (25) anymore. To domagnetic fields coexist with a DM topological gap. If take this into account we shall define a Dirac Berry cur- vature ΩD, i. e. a Berry curvature that only captures the we look at the Landau level spectrum, we get []: En = (2 eB v2 n + ∆2)1/2 for n 1, and E = sign(eB)∆ Dirac contributions. It can be constructed by linearizing ~ J 0 d for± the| lowest| Landau level. In≥ one valley the lowest Lan- DM around all six Dirac points, and adding the Berry dau level is lowered in energy, while in the other valley curvatures of the six resulting gapped Dirac cones. As a it is raised. Regarding the spectrum at the boundary, check, it is apparent by comparison of Ω and ΩD [Figs. due to the non-zero Chern number there will be a chiral 2(a) and2(b)], that such a Dirac Berry curvature is miss- edge state coexisting with the counterpropagating edge ing the information coming from the “non-Diracness” of states due to the psudomagnetic field. The spectrum the bands. ˜ in the presence of pseudomagnetic fields and non-zero The correct value of C can therefore be obtained by Chern number has been discussed in detail in two [58] substituting Ω by ΩD in Eq. (26) and three- [59, 60] dimensional systems (spin-orbit cou- 1 X Z pled graphene and time-reversal breaking Weyl semimet- C˜ = Ωτ (k)ρτ (k) d2k, (28) D 2π D als, respectively). τ=± B.Z. 5

become finite. The magnitude of ηH continues to grow as the population around the Dirac points, where the Berry curvature is maximal, increases, until the temperature rises to a point where the magnon picture is no longer valid. The magnon description is reasonable for temper- atures considerably smaller than the Curie temperature Tc, above which the system is disordered by thermal fluc- tuations. If we take a monolayer of CrI3 as a candidate material hosting Dirac magnons, its Curie temperature is Tc = 45 K [23] and its exchange coupling and spin have been estimated to be J = 2.2 meV and S = 3/2 [24]. If we assume the validity of the magnon picture up to a temperature of around T Tc/10 = 4.5 K, our ˜ ≈ FIG. 3. CD as a function of temperature for a value of the calculations are valid up to κT/JS 0.12, which corre- DM interaction of D = J/10 and for three different values of ≈ sponds to a maximum value of roughly C˜D 0.1. For the magnetic field. higher temperatures, an alternative picture based∼ on the Schwinger- representation of spin can be invoked to correctly capture the topological properties of the sys- so that the Chern-Simons term becomes tem [14]. As a final note, we see that the application of a ˜ Z magnetic field increases the energies of magnons, which CD 3 µρν el el SCS = d x  Aµ ∂ρAν . (29) leads to a decrease of the Hall viscosity. 4π~ Inserting the explicit form of the elastic gauge fields, given by Eqs. (11) and (12), in the Chern-Simons term, V. CONCLUSIONS AND DISCUSSION we get

2 ˜ Z 4~β CD We have studied the influence of lattice deformations S = d3x (u u )u ˙ . (30) CS 9πa2 xx − yy xy on the magnon physics of a Honeycomb ferromagnet. We have proven that, in the vicinity of the Dirac points, elas- The stress tensor can be obtained by differentiating the ticity couples at lowest order as vector fields to Dirac action with respect to the strain tensor magnons. For strain configurations giving rise to con- stant pseudomagnetic fields, magnons arrange in Landau δS levels. Such strain configurations can be realized using Tab = , (31) −δuab STM, by depositing magnetic defects on a metallic sub- strate. The presence of Landau levels could be tested with a, b = x, y, while the viscosity tensor, η, charac- experimentally by using, among other techniques, inelas- terizes the dependence of the stress tensor on the strain tic neutron scattering [21]. rate By including a Dzyaloshinskii-Moriya interaction, a topological gap opens and a Chern-Simons effective ac- T = η u˙ . (32) ab − abcd cd tion for the elastic gauge fields is generated. Such a term η is symmetric under the interchange of a and b, and encodes a phonon Hall viscosity response, which is gener- c and d, due to the symmetry of the strain tensor. The ated entirely by quantum fluctuations of magnons living Hall viscosity, ηH , is given by the antisymmetric part of η in the vicinity of the Dirac points. Its value vanishes under the interchange of the pairs a, b and c, d [47]. For at zero temperature, and grows as temperature is raised an isotropic 2D system, ηH has only one independent and the states around the Dirac points are increasingly H populated. Having in mind that measuring the Hall vis- component: ηxxxy = ηxyxx = ηyyxy = ηxyyy = η [47]. Then, by applying− Eq. (31−) to the Chern-Simons cosity is not an easy task, a possible natural direction is action (30) we get to measure changes in the phonon structure generated by ηH [46, 54, 61]. 2 ˜ 4~β CD We can compare the Hall viscosity of the Haldane ηH = . (33) − 9πa2 model for magnons obtained here to that of the Hal- dane model for electrons (with in- In Fig.3 we plot C˜D as a function of κT/JS, for side the gap) computed in [46], arising also from elastic a DM interaction of D = J/10. At zero temperature, gauge fields. In the electronic case, the Hall viscosity H 2 2 a Bose-Einstein condensate forms at zero energy, where is ηelec = 4~β /9πa , whereas for magnons we obtained ˜ H 2 ˜ 2 the Berry curvature vanishes and consequently CD = 0. ηmag = 4~β CD/9πa . Even if they are material de- As the temperature increases, states with non zero Berry pendent,− we can reasonably assume β and a to be roughly H curvature are populated, and C˜D, and by extension η , of the same order in magnon and electronic systems, so 6 we have length, lB, whereas in the Hall viscosity coming from elastic gauge fields the magnetic length is substituted by ηH mag ˜ the lattice spacing a [46]. For magnetic fields of the order H CD. (34) ηelec ∼ − of 10 Tesla, and for the lattice constant of, say, graphene ˚ 2 2 3 (a 2.5 A), we have lB/a 10 , so the value of the Hall If we take our Dirac magnon system to be a monolayer of viscosity≈ from elastic gauge∼ fields is three orders of mag- CrI3, as we estimated above our calculations are valid up nitude bigger than the conventional one. Therefore, even to temperatures of 4.5 K, for which we obtain a maximum if the magnon Hall viscosity computed here is (at best) value of C˜D 0.1. Then we get one order of magnitude down on its electronic counter- ∼ part, it is still (at best) two orders of magnitude bigger ηH than the conventional Hall viscosity of electrons under mag 0.1, (35) H magnetic fields. ηelec ∼ which means that, at best, the magnon Hall viscosity is one order of magnitude down on its electronic counter- ACKNOWLEDGMENTS part. It is important to remark that, in electronic systems, We thank Hector Ochoa and Alberto Cortijo for en- the Hall viscosity arising from elastic gauge fields is sev- lightening and fruitful discussions. Y. F. acknowledges eral orders of magnitude bigger than that coming from support from the ERC Starting Grant No. 679722. conventional or metric deformations [46], which The work of M. V. has been supported by Span- greatly increases the chances of experimental detection. ish MECD grant FIS2014-57432-P, the Comunidad de Basically, the conventional Hall viscosity under magnetic Madrid MAD2D-CM Program (S2013/MIT-3007), and fields is inversely proportional to the squared magnetic by the PIC2016FR6.

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