Abstract
SrCu_{2}(BO_{3})_{2} is the archetypal quantum magnet with a gapped dimersinglet ground state and triplon excitations. It serves as an excellent realization of the Shastry–Sutherland model, up to small anisotropies arising from Dzyaloshinskii–Moriya interactions. Here we demonstrate that these anisotropies, in fact, give rise to topological character in the triplon band structure. The triplons form a new kind of Dirac cone with three bands touching at a single point, a spin1 generalization of graphene. An applied magnetic field opens band gaps resulting in topological bands with Chern numbers ±2. SrCu_{2}(BO_{3})_{2} thus provides a magnetic analogue of the integer quantum Hall effect and supports topologically protected edge modes. At a threshold value of the magnetic field set by the Dzyaloshinskii–Moriya interactions, the three triplon bands touch once again in a spin1 Dirac cone, and lose their topological character. We predict a strong thermal Hall signature in the topological regime.
Introduction
Topological phases of bosons have steadily gained interest, driven by the goal of realizing protected edge states that do not suffer from dissipation. As bosonic carriers (phonons, magnons and so on) are electrically neutral, they are weakly interacting and show good coherent transport. As a first step in this direction, analogues of the integer quantum Hall effect have been proposed using photons^{1,2,3,4}, magnons^{5,6,7,8,9}, phonons^{10,11,12} and skyrmionic textures^{13}, with the thermal Hall effect^{14} as the experimental probe of choice. It would be very interesting to observe this phenomenon in quantum magnets, where the quantization of spin produces a new class of bosonic excitations called triplons.
The archetypal quantum magnet is SrCu_{2}(BO_{3})_{2}, a layered material consisting of Cu S=1/2 moments arranged in orthogonal dimers^{15,16}. To a very good approximation, this arrangement conforms to the Shastry–Sutherland model^{17,18}. Lowenergy excitations correspond to breaking a singlet to form a triplet. Such excitations are called triplons and can be thought of as spin1 bosonic particles^{19}. Indeed, triplons undergo Bose condensation in many systems^{20,21,22}. If SrCu_{2}(BO_{3})_{2} were an exact realization of the Shastry–Sutherland model, the triplons would be local excitations forming a threefolddegenerate flat band^{23}. However, electron spin resonance (ESR)^{24}, infrared absorption^{25}, neutron scattering^{26} and Raman scattering^{27} measurements show a weak dispersion that has been attributed to small Dzyaloshinskii–Moriya (DM) anisotropies^{28,29,30}. Nuclear magnetic resonance measurements also support the presence of DM couplings^{31}.
Here, we show that DM couplings give rise to topological character in the triplon energy spectrum of SrCu_{2}(BO_{3})_{2}. A small magnetic field suffices to induce phases with Chern bands and topologically protected edge states. As triplons carry energy but no electrical charge, we predict a thermal Hall effect of triplons.
Results
Microscopic model
Figure 1a illustrates the lattice geometry and the interactions between the spins in SrCu_{2}(BO_{3})_{2}. The Hamiltonian is given by
J is the strength of the exchange coupling on intradimer bonds. The intradimer DM coupling D is allowed by symmetry below a structural phase transition at T∼395 K (refs 32, 33; see Supplementary Figs 1 and 2, Supplementary Table 1 and Supplementary Note 1). We have included a small magnetic field h^{z} perpendicular to the SrCu_{2}(BO_{3})_{2} plane, with the gfactor denoted as g_{z}. On the interdimer bonds, J′ and D′ are the exchange and DM couplings, respectively, with the dominant DM component being outofplane. As seen in Fig. 1a, the outofplane couplings encode a sense of clockwise rotation; this ultimately drives a Hall effect of triplon excitations as we report below.
Triplon description
A thorough bond operator treatment of the Hamiltonian in equation 1 has been presented in ref. 30. We present a simplified treatment suitable for SrCu_{2}(BO_{3})_{2} in a weak magnetic field. Previous studies have largely focused on plateau phases at high fields(ref. 34 and references therein). In contrast, we show that the lowfield regime has exotic topological properties.
In a given dimer, the Hilbert space is spanned by a singlet and three triplets: and . In the pure Shastry–Sutherland model, the ground state is a direct product of singlets s〉 over the dimers as long as J′≲0.675 J (refs 18, 35, 36). In SrCu_{2}(BO_{3})_{2}, as the DM anisotropies are small compared with J, we assume that the ground state remains a product wavefunction. Minimizing the overall energy, we find that the ground state has the wavefunction and on horizontal and vertical dimers, respectively; the direction of D on each dimer determines whether t_{y}〉 or t_{x}〉 is admixed. The triplet admixture is proportional to the intradimer DM coupling D with . Here, as in the rest of this article, we only retain terms up to linear order in D,D′, and h^{z}, which are small compared with the J′s.
On each dimer, we choose a new Hilbert space by rotating to using
on horizontal and vertical dimers, respectively. In the ground state, each dimer is in the state given by the first row in the corresponding W matrix. We have three local excitations given by the mutually orthogonal triplon states , and .
At lowmagnetic fields, the lowenergy excitations are spanned by singletriplon states with their dynamics captured by hopping processes of the form . Introducing a bosonic representation for triplons, we obtain a Hamiltonian with purely hoppinglike terms. By defining W_{v} as above with complex entries, the Hamiltonian takes on a convenient form, viz., the two dimers in the unit cell become equivalent (see Supplementary Note 2 for details). We may henceforth drop v/h indices and work with the reduced unit cell in Fig. 1b. In momentum space, the Brillouin zone (BZ) is enlarged as shown in Fig. 2b.
For a more complete treatment, we may include pairinglike terms () within a bond operator formalism as in ref. 30. We ignore such terms as they do not change the triplon energies to linear order in D, D′ and h^{z}; we have checked that their inclusion does not alter the results presented here.
Spin1 Dirac cone physics
The triplon Hamiltonian in momentum space is given by
where the Hamiltonian matrix is given by
with γ_{1}(k)=sin k_{x}, γ_{2}(k)=sin k_{y} and (see Supplementary Note 2 for details). Only two components of the interdimer DM coupling enter the Hamiltonian, viz., the outofplane component and the ‘staggered’ component shown in Fig. 1a. A third nonstaggered component is allowed by symmetry, but does not appear at this level (see Supplementary Fig. 2 and Supplementary Note 1). Intradimer D and inplane interdimer act in consonance so that only the linear combination appears in the Hamiltonian similar to the analysis in ref. 29. In the following, we use the values J=722 GHz, , and g_{z}=2.28 in the M(k) matrix, which reproduce the ESR data in ref. 24. The parameter J is not the microscopic exchange strength, but rather the measured spin gap, which determines the effective coupling in the presence of quantum fluctuations.
The M(k) matrix is of the form
where 1 is the 3 × 3 identity matrix and
is a vector of 3 × 3 matrices satisfying the [L^{ξ}, L^{η}]=iɛ_{ξηζ} L^{ζ} SU(2) algebra. Thus, in momentum space, the triplons behave as (pseudo)spin1 objects coupled to a pseudomagnetic field
We now draw an analogy with the usual twoband physics wherein the 2 × 2 Hamiltonian takes the same form as equation (5) but with spin1/2 Pauli matrices instead of spin1 L matrices. There, we obtain two bands corresponding to eigenvalues J±d(k)/2 (we denote d(k)=d(k)). If d(k) is nonzero throughout the BZ, we obtain two wellseparated bands whose Chern numbers are ±N_{s}, where N_{s} is the number of skyrmions in the d(k) field over the BZ^{37}. The d(k) field contains all information about the band structure; its skyrmion count determines the topological character of bands. We emphasize here that topological properties will not change with small corrections to the Hamiltonian such as nextnearest neighbour hopping (see Supplementary Figs 3 and 4, and Supplementary Note 3).
Likewise, in our spin1 realization, we read off the eigenvalues as {J+d(k),J,J−d(k)}. Note that the band in the middle is always flat with energy J, irrespective of the value of d(k). If the pseudomagnetic field d(k) vanishes at some k, all three bands touch in a ‘spin1 Dirac cone’, resembling graphene but with an additional flat band passing through the band touching point. If d(k) is nonzero throughout the BZ, the spectrum consists of three wellseparated triplon bands with welldefined Chern numbers {−2N_{s},0,+2N_{s}}, where N_{s} is again the skyrmion number. More generally, for the arbitrary spinS generalization of equation (5), we have (2S+1) bands with Chern numbers {−2SN_{s},−2(S−1)N_{s},…, 2(S−1), 2SN_{s}} (see Supplementary Table 2 and Supplementary Note 4).
Magnetic fieldtuned topological transitions
The magnetic field h^{z} provides a handle to tune topological transitions in SrCu_{2}(BO_{3})_{2}, as shown in Fig. 2. With small magnetic fields, even though the ground state remains a product of dimer singlets, the band structure of excitations shows topological transitions. When h^{z}=0, the three bands touch at the edge centres of the BZ (corresponding to corners in the structural BZ). A small applied field opens a nontrivial band gap, allowing for three wellseparated bands with Chern numbers {−2,0,+2} or {+2,0,−2}, depending on the sign of h^{z}. When the field reaches a threshold strength , the three bands touch at the Γ point. Indeed, this band touching has already been seen in ESR^{24} and infrared absorption^{25} spectra at h^{z}≈1.4 T; however, its significance as a spin1 Dirac point was not appreciated. As h^{z} is increased further, a trivial band gap opens with all three Chern numbers being zero.
The topology of triplon bands can be understood in terms of the d(k) field. To every point in the twodimensional (2D) BZ (an torus), we assign the 3D vector d(k): this gives us a closed 2D surface embedded in three dimensions. If the bands are to remain wellseparated, the surface cannot touch the origin, that is, d(k)≠0 anywhere in the BZ. The origin is thus special and acts as a monopole for Berry phase. The topology of the band structure reduces to whether or not the 2D surface encloses the origin; if it does, how many times does it wrap around the origin? This defines a skyrmion number , that is related to the Chern number.
To see the role of h^{z}, we note that it enters solely as an additive contribution in the zcomponent of d(k). As shown in Fig. 3, the BZ maps to a closed surface of width and height proportional to , which is composed of an upper and a lower chamber. The chambers are disconnected, but touch along line nodes. The surface is orientable: the outer surface of the lower chamber smoothly connects to the inner surface of the upper chamber and vice versa. When h^{z}>h_{c}, neither chamber encloses the origin; we have N_{s}=0 with all Chern numbers zero (Fig. 3a,d). When −h_{c}<h^{z}<0, the origin lies inside the upper chamber (Fig. 3b), the net Berry flux is positive and Chern numbers are {+2,0,−2}. When 0<h^{z}<h_{c}, the origin lies inside the lower chamber (Fig. 3c), the Berry flux is negative and Chern numbers are {−2,0,+2}.
The key ingredient that gives rise to topological properties is the DM interaction that originates from relativistic spinorbit coupling. The threshold magnetic field h_{c} is proportional to the coupling . The intradimer DM coupling D also plays a role: we do not find any Chern bands upon setting D=0, as is appropriate for T>395 K, above a structural transition in SrCu_{2}(BO_{3})_{2}.
Edge states
The topological character of bands is revealed when edges are introduced. For 0<h^{z}<h_{c} (and for −h_{c}<h^{z}<0), edge states connecting the Chern bands appear within the bulk band gap, as shown in Fig. 4a for a strip geometry. Apart from recovering the bulk bands, we clearly see four edge states consistent with bulk boundary correspondence^{38} for Chern numbers ±2. The edge states constitute two ‘rightmovers’ and two ‘leftmovers’ (with group velocity pointing right/left), localized on the opposite edges of the strip. The wave functions of the edge states decay exponentially into the bulk, as shown in Fig. 4b.
Thermal Hall effect
Chern bands in electronic systems can be easily probed by doping the system so that the Fermi level lies in the band gap. This gives a transverse electrical conductivity quantized to integer values. In bosonic systems where this is not possible, the thermal Hall effect provides an alternative. Semiclassical analysis shows that a wave packet in a Chern band undergoes rotational motion^{39,40}. To exploit this, a temperature gradient is used to populate the band differently at the system’s edges. The rotational motion of the triplons is then unbalanced, leading to a transverse triplon current. As triplons carry energy, this leads to a measurable transverse thermal current.
An expression for thermal Hall conductivity was derived using the Kubo formula in ref. 5. Subsequently, Matsumoto et al.^{7} showed that there is an extra contribution from the orbital motion of excitations. Figure 5a shows the thermal Hall conductivity as a function of external magnetic field calculated using the expression in ref. 7. SrCu_{2}(BO_{3})_{2} is quasi2D and the Hall response in each layer is in the same direction. Therefore, we add the contribution from each layer to get κ^{xy} for a threedimensional sample (see Supplementary Note 5). As the magnetic field is tuned away from h^{z}=0, a nonzero Hall signal develops with the sign of κ^{xy} depending on the direction of magnetic field. When the threshold magnetic field strength h_{c} is reached, the topological nature of triplon bands is lost and the Hall signal is diminished. Figure 5b shows the peak thermal Hall conductivity increasing monotonically with background temperature. Our calculation assumes that the temperature is low enough that the triplon bands are weakly populated, allowing us to neglect triplon–triplon interactions. We expect this assumption to hold atleast until ∼5 K where the filling of bosons is ∼0.2%. Neutron scattering data show that the intensity of the single triplet excitations is essentially unchanged up to 5 K showing no damping^{26}.
Discussion
We have demonstrated that SrCu_{2}(BO_{3})_{2} hosts a Hall effect of triplons. A small external magnetic field of the order of a few Tesla suffices to tune topological transitions in the band structure. The triplons form novel spin1 Dirac cones with threefold band touching. Such a feature has been seen in various contexts^{41,42,43,44,45}. Our study elucidates its implications for band structure topology; the spin1 structure naturally gives Chern numbers ±2 instead of the more common ±1. Similar topological phases could exist in dimer compounds such as Rb_{2}Cu_{3}SnF_{12} (refs 46, 47) with nonzero DM couplings, and possibly in ZnCu_{3}(OH)_{6}Cl_{2} (Herbertsmithite)^{48}.
We predict a thermal Hall signature in SrCu_{2}(BO_{3})_{2} that can be verified by transport measurements. We also suggest neutron scattering experiments to study the evolution of band structure in lowmagnetic fields (≲2 T). Such measurements can see the spin1 Dirac cone features at h^{z}=0 and h^{z}=h_{c}. It may even be possible to directly probe the edge states using precise lowangle scattering measurements.
Additional information
How to cite this article: Romhányi, J. et al. Hall effect of triplons in a dimerized quantum magnet. Nat. Commun. 6:6805 doi: 10.1038/ncomms7805 (2015).
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Acknowledgements
We thank R. Shankar (Chennai), A. Paramekanti and M. Daghofer for useful discussions. This work was supported by the Hungarian OTKA Grant No. 106047.
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Romhányi, J., Penc, K. & Ganesh, R. Hall effect of triplons in a dimerized quantum magnet. Nat Commun 6, 6805 (2015). https://doi.org/10.1038/ncomms7805
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