Triplon band splitting and topologically protected edge states in the dimerized antiferromagnet

Search for topological materials has been actively promoted in the field of condensed matter physics for their potential application in energy-efficient information transmission and processing. Recent studies have revealed that topologically invariant states, such as edge states in topological insulators, can emerge not only in a fermionic electron system but also in a bosonic system, enabling nondissipative propagation of quasiparticles. Here we report the topologically nontrivial triplon bands measured by inelastic neutron scattering on the spin-1/2 two-dimensional dimerized antiferromagnet Ba2CuSi2O6Cl2. The excitation spectrum exhibits two triplon bands that are clearly separated by a band gap due to a small alternation in interdimer exchange interaction, consistent with a refined crystal structure. By analytically modeling the triplon dispersion, we show that Ba2CuSi2O6Cl2 is the first bosonic realization of the coupled Su-Schrieffer-Heeger model, where the presence of topologically protected edge states is prompted by a bipartite nature of the lattice.


SUPPLEMENTARY NOTE 1. CRYSTAL STRUCTURE.
Supplementary Figure 1 shows the redetermined crystal structure of Ba 2 CuSi 2 O 6 Cl 2 . The structure is closely related to that of Ba 2 CoSi 2 O 6 Cl 2 [1]. The crystal structure has a CuO 4 Cl pyramid feature with a Cl − ion on an apex. The CuO 4 Cl pyramids are linked via SiO 4 tetrahedra in the ab plane. Magnetic spin-1/2 Cu 2+ is located at the center of the base composed of O 2− , which is parallel to the ab plane. Two neighboring CuO 4 Cl pyramids along the c axis are placed with their bases facing each other. The CuO 4 Cl pyramids are linked via SiO 4 tetrahedra in the ab plane. The atomic linkage in the ab plane is approximately the same as that of BaCuSi 2 O 6 [2,3].
It is natural to assume from the crystal structure that two Cu 2+ spins located on the bases of neighboring CuO 4 Cl pyramids along the c axis form an antiferromagnetic dimer, and the dimers are coupled by weak exchange interactions in the ab plane. In fact, the presented excitation spectrum supports this model. The exchange network of Ba 2 CuSi 2 O 6 Cl 2 is illustrated in Supplementary Fig. 1c. In the original crystal structure reported in Ref. [4], there is no alternation of the interdimer interactions along the a and b axes, while in the redetermined structure the interdimer interactions are alternate along the a axis. It is also close to a 2D exchange network in BaCuSi 2 O 6 [3,5]. However, it should be emphasized that all the dimers are symmetrically equivalent in Ba 2 CuSi 2 O 6 Cl 2 , while three inequivalent dimers are resolved in BaCuSi 2 O 6 owing to a structural transition [6].  To confirm that the decrease of the intensity centered at 2.6 meV is not an extrinsic effect, we checked the data measured at different E i values. Supplementary Figure 2 shows a color contour map measured at 2.5 K with an E i of 3.14 meV. The same dispersion relations as those measured with an E i of 5.9 meV are obtained, indicating that the gap between two triplons bands is intrinsic. In this section, we start from the model described in Fig. 1a and derive the dispersion relations of triplet excitations. The spin Hamiltonian is given by where H 0 represents intradimer exchange terms from J and H represents interdimer exchange terms from J ξ ij and J ξ ij (ξ = a, b). S mni is defined as the i-th Cu atom of the (m, n)-th dimer pair (see Fig. 1b in the main text). Dimers on the two different sublattices are distinguished by m and n: m + n becomes even for one sublattice and odd for the other. For each dimer pair (m, n), a spin operator S mn , T mn can be defined as Thus, H 0 is rewritten as since S 2 mn1 = S 2 mn2 = 3/4. In addition, H can be projected in a subspace constructed by the basis of H 0 as H = H tt + H ss , where , , Dispersion relations are obtained by applying a bond-operator approach [7][8][9] to Supplementary eqs. (3) and (4). Singlet and triplet creation operators are defined as so that they follow bosonic commutation relations. In this definition, the number of bosons per dimer is constrained to 1 as Then, the squared operator S 2 mn and α component (α = x, y, z) of S mn and T mn are given as where αβγ represents an antisymmetric tensor. At zero field, the ground state is a product of the singlet at each dimer, and thus, the triplon density is zero. Thus, a mean-field approximation that neglects the dynamics of singlet operators should be applicable. By replacing creation and annihilation operators by its expectation value, s † mn ∼ s mn ∼ 1, and neglecting high-order terms, Supplementary eqs. (3) and (4) become A k-dependent form is obtained by Fourier transformation defined at each sublattice as for m + n = even and for m + n = odd, where N describes the number of dimers. This procedure leads to the following quadratic form: where Supplementary eq. (12) can be described using a 4 × 4 matrix as which is the same as eq. (1) in the main text (except for the omitted constant term), where The dispersion relation can be obtained by Bogoliubov transformation, which is equivalent to a procedure determining a paraunitary matrix T k that satisfies Owing to orthogonality and completeness of the new basis, . Therefore, Supplementary eq. (16) is equivalent to the relation Thus, eigenenergies E +,k , E −,k , −E +,k , and −E −,k are obtained by diagonalizing ΣM k , leading to the dispersion relation given by eq. (3) in the main text.

SUPPLEMENTARY NOTE 4. CALCULATION OF BERRY CONNECTION
In this section, we start by determining T k and then derive the Berry connection of each subband from the Hamiltonian M k (Supplementary eq. (15)). By diagonalizing ΣM k , eigenvectors for each eigenenergy are determined as where A k = J + J 2 + 2J|d|, Thus, a paraunitary matrix can be constructed as T k = (t ++,k , t +−,k , t −+,−k , t −−,−k ). Note that this definition is not valid and a different gauge should be selected for d = (0, 0, −d)(d > 0). The following discussion can be also applied to eigenvectors with a different gauge.
The Berry connection can be defined by the following equation [10,11], where Γ j is a diagonal matrix, the j-th diagonal component of which is 1 while others are zero, and µ = x, y. From Supplementary eq. (20), the Berry connection of each subband for µ = x can also be rewritten as The first real term corresponds to the phase change of the eigenvector along the Brillouin zone, while the remaining of imaginary terms omitted in Supplementary eq. (22) are due to band deformation. For a one-dimensional system, the total phase change across the Brillouin zone corresponds to the Zak phase [12]: Under d z = 0, d can be represented by (|d| cos θ, −|d| sin θ, 0), leading to where the integer n represents the winding number. The exactly same form can be derived from M k = J1 + d · σ for an arbitrary gauge, indicating that topological properties are unchanged even if pair creation and annihilation terms are present. For triplon bands in Ba 2 CuSi 2 O 6 Cl 2 , the Berry connection can be obtained from d = (ReΛ k , −ImΛ k , 0) as which leads to the Zak phase quantized into γ ++ = −γ +− = −γ −+ = γ −− = ±π irrespective of k y .

SUPPLEMENTARY NOTE 5. CALCULATION OF AN ENERGY SPECTRUM
As discussed in the main text, edge states should appear at the end of the a-direction from an analogy with a coupled SSH model [13]. To confirm this, an energy spectrum of the present model is calculated by imposing open boundary conditions along the a-direction. For simplicity, Fourier-transformed operators are defined under periodic boundary conditions along the b-direction as where N b is the number of chains along b.
where m αky represents a 4N a (2N a ≡ N/N b ) component vector m αky ≡ (t 1, † αmky , t 1, † αmky , · · · , t Na, † αmky , t Na, † αmky , t 1 αm−ky , t 1 αm−ky , · · · , t Na αm−ky , t Na αm−ky ) T , and 1 is an N a × N a identity matrix. X ky is a 2N a × 2N a matrix defined as where J ky = 2J B cos(k y b/2). The energy spectrum shown in Figure 5 is obtained by diagonalizing the matrix with N a = 100 for each k y . Note that edge states exhibit a very weak dispersion, as shown in Supplementary Figure 3, which becomes even weaker with increasing N a . While pair creation and annihilation terms make the bulk energy spectrum asymmetric above and below energy J (Supplementary Figure 3a), they do not affect the dispersion of the edge modes (Supplementary Figure 3b).