Flat band and localized excitations in the magnetic spectrum of the fully frustrated dimerized magnet Ba 2 CoSi 2 O 6 Cl 2

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1 Flat band and localized excitations in the magnetic spectrum of the fully frustrated dimerized magnet Ba 2 CoSi 2 O 6 Cl 2 γ 1 tr φ θ φ θ i Nobuo Furukawa Dept. of Physics, Aoyama Gakuin Univ.

2 Collaborators Theory: Daisuke Yamamoto, Takuya Kanesaka (Aoyama Gakuin) Shin Miyahara (Fukuoka) Isao Maruyama(Fukuoka Inst. Tech.) Experiment: Nobuyuki Kurita, Hidekazu Tanaka (Tokyo Inst. Tech) Seiko Ohira-Kawamura, Kenji Nakajima (J-PARC) Supported by KAKENHI: Theory of magnetic-field-induced topological phase transition in 2d frustrated quantum spin systems

3 Geometrical frustration and localizations? ϕ=0 J J t t ϕ=+ϕ 0 ϕ=-ϕ 0 Spin frustrations degeneracies, Particle propagations localization,

4 Flat band and magnetic crystal (plateau) in Kagome-lattice Mike Zhitomirsky, Hiro Tsunetsugu (2005) magnon wavefunction on a hexagonal plaquette Localized flat band Wigner crystal of magnons Magnetic plateau W

5 Flat band and magnetic crystal (plateau) in a frustrated ladder (J 1 -J 2 dimers) Honecker-Mila-Troyer(2000) Singlets and triplets are localized J 0 >> J 1 =J 2 Frustrated Ladder Coupled dimer limit Ground state: Dimer-singlets s 1 triplon excitation: flat bands (localized triplon) t s s s s s t s t t t t t H=0 H>H s

6 Ba 2 CoSi 2 O 6 Cl 2 : crystal structure H. Tanaka et al.: J. Phys. Soc. Jpn 83, (2014) Found as a by-product of Ba 3 CoSb 2 O 9 (S=1/2 triangular-lattice Heisenberg AFM) Co 2+ in a CoO 4 Cl pyramid Orbital triplet 4 T 1 x Kramers doublet Kramers doublet ground state S eff =1/2 with anisotropy (XY-like XXZ model) S=1/2 Bi-layer square lattice

7 Ba2CoSi2O6Cl2: Magnetization Susceptibilities dimer gap Fractional Magnetization Plateau g ab ~4, g c ~2 S=1/2 s s s s t s t t t S=1/2 Possibly, a coupled-dimer system s s s s s s t s t s t s t t t t t t

8 Ba 2 CoSi 2 O 6 Cl 2 : Exchange paths Detailed crystal structure around Co 2+ ions Intra-dimer exchange Through a SiO 4 tetrahedon Inter-dimer exchange Through two SiO 4 tetraheda Almost Perfect Frustration almost decoupled Dimers J 0 >> J 1 J 2

9 Ba2CoSi2O6Cl2: Magnetic excitations A crucial test for the coupled dimer model: Can it explain the magnetic excitation Spectrum? Neutron inelastic spectrum (many) flat bands Strongest signal: 5.8 mev peak is dispersionless, Intensity is Q-independent. Local s> t> excitation? (other peaks will be shown later) arxiv:

10 XY-like Dimer model Single dimer: Energy Levels Triplet 0 Triplet 1,-1 singlet With Inter-dimer couplings

11 Inter-dimer frustrated model XY-like dimers with Nearly perfect inter-dimer frustration Schwinger bosons No exchange between a singlet and a triplet if J 11 +J 22 =J 12 +J 22 s t t s See also Oleg Derzhko et al. PRB82 Hikihara et al. (2017)

12 Triplet excitation spectra Flat band excitations from the dimer-singlet ground state (nearly) flat band for (J 11 +J 22 )-(J 12 +J 22 ) << J Coupled dimer model can explain the flat band at 5.8 mev. Q-independent intensity suggests a local excitation.

13 Ba 2 CoSi 2 O 6 Cl 2 : Many flat band excitations More than 3 flat bands observed in neutron spectra??? arxiv: Main peak at 5.8 mev: intensity is Q independent Side peaks at 4.8, 6.7 mev: typical Q dependent intensity

14 Bulk-Dimer + Spin-vacancy model Lone spin Spin vacancy (lattice defect, ion substitution, Co valence change, ) XY dimer XY dimer+ lone spin Note: One vacancy affects 9 neighboring sites Vacancy effects are multiplied by an order.

15 Side peaks in magnetic excitations Spectrum Summary Isolated dimer Dimer + lone spin Fitting parameters

16 Ba 2 CoSi 2 O 6 Cl 2 : Neutron intensities Neutron scattering intensities are Q dependent for side peaks. Inelastic scattering cross section:,,,, in Nearest neighbor spin correlations give Q-dependences

17 Ba2CoSi2O6Cl2: Neutron spectra Q=(Q a, Q b ) dependent Neutron intensities at side peaks Neutron Model Peaks at Half-integer wavenumbers Peaks at integer wavenumbers Spin vacancies act as probes to confirm coupled-dimer model

18 Topological aspects: Localized dimer singlets/triplets and Z 2 Berry Phase Honecker-Mila-Troyer(2000) 1D U(θ) Maruyama-Miyahara (arxiv: ) γ 2 =π γ 2 =0 γ 2 =π 2D γ 1 i tr φ θ φ θ γ 2 =0 D=1 γ 2 =π D=1 γ 2 =π D=2 U(θ) Maruyama (unpublished)

19 Summary Ba 2 CoSi 2 O 6 Cl 2 : a perfectly frustrated coupled-dimer Magnetic Plateaus due to Flat bands. J 0 >> J 1 J 2 Spin vacancies act as probes to further confirm the model Quantum spin dimer singlets Topological Z 2 berry phase