Hall effect of triplons in a dimerized quantum magnet

The celebrated Shastry Sutherland model has a gapped dimer singlet ground state. The material SrCu$_2$(BO$_3$)$_2$ serves as a good realization of this model, upto small anisotropies arising from Dzyaloshinskii-Moriya (DM) interactions. The DM interactions admix a triplet component into the singlet ground state and give rise to weakly dispersing triplon bands. We show that an applied magnetic field splits the triplon modes and opens band gaps. Surprisingly, we are left with topological bands with Chern numbers $\pm 2$. SrCu$_2$(BO$_3$)$_2$ thus supports topologically protected triplonic edge modes and is a magnetic analogue of the integer quantum Hall effect. At a critical value of the magnetic field set by the strength of DM interactions, the three triplon bands touch once again in a spin-1 generalization of a Dirac cone, and lose their topological character. We predict a strong thermal Hall signature in the topological regime.

The celebrated Shastry Sutherland model has a gapped dimer singlet ground state. The material SrCu2(BO3)2 serves as a good realization of this model, upto small anisotropies arising from Dzyaloshinskii Moriya (DM) interactions. The DM interactions admix a triplet component into the singlet ground state and give rise to weakly dispersing triplon bands. We show that an applied magnetic field splits the triplon modes and opens band gaps. Surprisingly, we are left with topological bands with Chern numbers ±2. SrCu2(BO3)2 thus supports topologically protected triplonic edge modes and is a magnetic analogue of the integer quantum Hall effect. At a critical value of the magnetic field set by the strength of DM interactions, the three triplon bands touch once again in a spin-1 generalization of a Dirac cone, and lose their topological character. We predict a strong thermal Hall signature in the topological regime. 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, etc.) 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 magnons [1][2][3][4][5], phonons [6][7][8] and skyrmionic textures [9], with the thermal Hall effect [10] as the experimental probe of choice. We present a manifestation of this physics in the prototypical quantum magnet SrCu 2 (BO 3 ) 2 , well known as a realization of the Shastry Sutherland model [11,12]. SrCu 2 (BO 3 ) 2 is a layered material consisting of Cu S = 1/2 moments arranged in orthogonal dimers [13,14].
To a very good approximation, this arrangement conforms to the Shastry Sutherland model with spins on each dimer forming a singlet. Low energy excitations correspond to 'triplons' wherein a singlet is broken and replaced with a triplet. 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 [15]. However, electron spin resonance (ESR) [16], infrared absorption (IR) [17], neutron scattering [18] and Raman scattering [19] measurements show a weak dispersion that has been attributed to small Dzyaloshinskii-Moriya (DM) couplings [20,21]. NMR measurements also support the presence of DM couplings [22]. Fig. 1(a) illustrates the lattice geometry and the interactions between the spins. The resulting Hamiltonian is given by We include a small magnetic field h z , perpendicular to the SrCu 2 (BO 3 ) 2 plane. The intra-dimer coupling D is allowed by symmetry below a structural phase transition at T ∼ 395 K[23, 24]. We only keep the zcomponent of the inter-dimer DM interaction D ′ ; ab initio estimates[25] and previous fits to ESR, IR and neutron data indicate that its in-plane components are small. As seen in Fig. 1(a), the D ′ ⊥ couplings encode a sense of clockwise rotation; this ultimately drives a Hall effect of triplon excitations as we report below.
In a given dimer, the Hilbert space is spanned by a singlet |s = (| ↑↓ − | ↓↑ )/ √ 2 and three triplets: In the pure Shastry-Sutherland model, the ground state is a direct product of singlets |s over the dimers as long as J ′ 0.675J[12, 27, 28]. Since the DM interactions are small compared to J, we assume that the ground state remains a product wavefunction. Minimizing the overall energy, we find that the ground state has the wavefunction |s h ∼ |s h +α|t y h and |s v ∼ |s v − α|t x v on horizontal and vertical dimers, respectively; the direction of D on each bond determines whether |t y or |t x is admixed. The triplet admixture is proportional to the intra-dimer DM coupling D with α ≈ D/2J ≪ 1. Here, as in the rest of this article, we only retain terms up to linear order in D, D ′ , h z which are small compared to the J ′ s. We use the values J = 722 GHz, J ′ = 468 GHz, |D| = 60 GHz, and D ′ ⊥ = −21 GHz and g z = 2.28, which reproduce the ESR data in Ref. [16].
On each dimer, we choose a new Hilbert space by ro- (2) on horizontal and vertical dimers respectively. In the ground state, each dimer is in the |s state given by the first row in the corresponding W matrix. We have three local excitations given by the mutually orthogonal 'triplon' states |t x , |t y and |t z .
At low magnetic fields, the low-energy excitations are spanned by single-triplon states with their dynamics captured by hopping processes of the form i t α |H|t β j . Introducing a bosonic representation for triplons, we obtain a Hamiltonian with purely hopping-like terms. The definition of W v encodes a gauge choice under which the Hamiltonian takes on a convenient form, viz., the two dimers in the unit cell are rendered equivalent. We may henceforth drop v/h indices and work with the reduced unit cell in Fig. 1(b). In momentum space, the Brillouin zone (BZ) is enlarged as shown in Fig.2 For a more complete treatment, we may include pairing-like terms (t † i,αt † j,β ) within a bond operator formalism as in Ref. [21]. 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 below.
Dirac cone physics generalized to spin-1. In momentum space, we obtain the Hamiltonian where the Hamiltonian matrix is given by where γ 1 (k) = sin k x , γ 2 (k) = sin k y , and γ 3 (k) = 1 2 (cos k x + cos k y ). This 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 ] = iε ξηζ L ζ SU(2) algebra. Thus, in momentum space, the triplons behave as (pseudo)spin-1 objects coupled to a pseudomagnetic field We now draw an analogy with the usual two-band physics wherein the 2 × 2 Hamiltonian takes the same form as Eq. (5) but with spin-1/2 Pauli matrices instead of spin-1 L matrices. There, we obtain two bands corresponding to eigenvalues J ± d(k)/2 (we denote d(k) = |d(k)|). If d(k) is non-zero throughout the BZ, we obtain two well separated bands whose Chern numbers are ±N s , where N s is the number of skyrmions in the d(k) field over the BZ. The d(k) field contains all the information about the band structure; its skyrmion count determines the topological structure of the bands.
Likewise, in our spin-1 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) becomes zero at some k, all three bands touch and give rise to a generalized Dirac cone. If d(k) is non-zero throughout the BZ, the spectrum consists of three wellseparated triplon bands with well-defined Chern numbers Magnetic field tuned topological transitions: The magnetic field h z acts as a useful handle to tune topological transitions in SrCu 2 (BO 3 ) 2 , as shown in Fig. 2. When h z = 0, the three bands touch at (0, π) and (π, 0), the edge centres of the BZ (corresponding to the corner in the structural BZ). A small non-zero field opens a nontrivial band gap, allowing for three well-separated bands with Chern numbers {−2, 0, +2} or {+2, 0, −2}, depending on the sign of h z . When the field reaches a critical strength h c = 2D ′ ⊥ /g z , the three bands touch at the BZ corner (π, π) (corresponding to Γ in the structural BZ). Indeed, this band touching has already been seen in ESR [16] and infrared absorption [17] spectra at h z ≈ 1.4 T; however, its significance as a spin-1 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 2D BZ (an S 1 × S 1 torus), we assign the 3D vector d(k): this gives us a closed 2D surface embedded in 3 dimensions. If the bands are to remain well-separated, the surface cannot touch the origin, i.e., 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 N s ∈ Z, 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 z-component of d(k). As shown in Fig. 3, the BZ maps to a closed surface of width DJ ′ /J and height 4D ′ ⊥ , 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 [ Figs. 3(a) and (d)]. When −h c < h z < 0, the origin lies inside the upper chamber [ Fig. 3(b)], 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. 3(c)], 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 the relativistic spin-orbit coupling. The critical magnetic field h c is proportional to the coupling D ′ ⊥ . 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 that connect the Chern bands appear within the bulk band gap. Fig. 4(a) shows the triplon energies for a strip with periodic boundary conditions in the horizontal (x) direction and finite width in the y direction. We recover the bulk bands and in addition, we clearly see four edge states consistent with bulk boundary correspondence[30] for Chern numbers ±2. The edges constitute two 'right-movers' and two 'left-movers' (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. 4

(b)]
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 ther- mal Hall effect provides an alternative. Semi-classical analysis shows that a wave packet in a Chern band also undergoes rotational motion[31, 32]. 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 imbalanced, 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. [1]. Subsequently, Matsumoto et al. [3] showed that there is an extra contribution from the orbital motion of excitations. Fig. 5(a) shows the thermal Hall conductivity as a function of external magnetic field calculated using the expression in Ref. [3]. SrCu 2 (BO 3 ) 2 is quasi-two-dimensional 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 three dimensional sample. As the magnetic field is tuned away from h = 0, a non-zero Hall signal develops with the sign of κ xy depending on the direction of magnetic field. When the critical magnetic field strength h c is reached, the topological nature of triplon bands is lost and the Hall signal is diminished. Fig. 5(b) 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 interaction terms. We expect this assumption to hold atleast until ∼ 10 K[33]. The required magnetic field strength is of a few Tesla, which can be easily realized in experiments.
Discussion: Magnetic analogues of the integer quantum Hall effect have been proposed in ferromagnets, wherein the DM interaction allows magnon bands to acquire Chern numbers. Our calculation presents the first quantum realization of this physics in a spin dimer system. The triplon Hamiltonian in SrCu 2 (BO 3 ) 2 gives a spin-1 generalization of Dirac cones. Such a feature with threefold band touching has previously been seen in various contexts[34-38]. Our study elucidates its implications for band structure topology; the spin-1 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 [39, 40] with non-zero DM couplings, and possibly in ZnCu 3 (OH) 6 Cl 2 (Herbertsmithite) [41].
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 low magnetic fields ( 2T). It should be possible to see the band gap opening and closing transitions at h z = 0 and h z = h c . It may even be possible to directly probe the edge states using precise low-angle scattering measurements.
We thank R.  where n = −L, . . . , L and d(k) = |d(k)|. As k is continuously varied over the BZ, each of these eigenvalues can be taken to form a band. We have (2L + 1) such bands, with the spacing between each pair of bands given by d(k). If the amplitude of d is never zero over the BZ, no two bands touch, they are well-separated that can be indexed by n. Before we evaluate the Chern numbers of these bands, we define the skyrmion number of the d field: The Berry curvature of a band n is given by We use f (k) and g(k) as a shorthand notation for the ∂d/∂k x and ∂d/∂k y , respectively. To evaluate the matrix elements in Eq. (S.4), we define a local coordinate system so that the local z axis points along d(k). We use the form of H, Eq. (S.1), and Eq. (S.5) to write The operators L z , L + , and L − are spin operators in the local basis, i.e., with z along d(k). They have the usual matrix elements for spin operators. Furthermore, f ± = (f x ± if y )/2 and g ± = (g x ± ig y )/2. In the expression for the Berry curvature, the only intermediate states that contribute are the immediately higher and lower states m = n ± 1: In the expressions above, we have used the fact that the energy difference ω n − ω n±1 is simply ∓d, the amplitude of the d(k) vector. Next, we use the commutation relation L + L − − L − L + = 2L z and The quantity f x g y − f y g x is the z-component of g × f in the local coordinate system which has z along d(k). This can be rewritten as In this section, we apply the expression for thermal conductivity κ xy = 1 β n BZ d 2 k c 2 (ρ n )2Im ∂ kx ψ n |∂ ky ψ n = 1 β n BZ d 2 k c 2 (ρ n ) (S.17) Since the band dispersions and splittings are much smaller than the gap between the bands and the ground state, ω +1 − ω −1 ≪ ω n , we expand Eq. (S.17) in d/ω 0 : where the difference of Bose occupation numbers is .