/21. Tsuneya Yoshida. Collaborators: Robert Peters, Satoshi Fujimoto, and N. Kawakami 2013/6/07 (EQPCM) 1. Kyoto Univ.

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1 2013/6/07 (EQPCM) 1 /21 Tsuneya Yoshida Kyoto Univ. Collaborators: Robert Peters, Satoshi Fujimoto, and N. Kawakami T.Y., Satoshi Fujimoto, and Norio Kawakami Phys. Rev. B 85, (2012)

2 Outline 2 /21 1. Introduction Topological phase in d-,f- electron systems Several studies of correlated TBI Mott vs. TBI 2. Purpose 3. Model and Method 4. Numerical Results 5. Summary (DMFT study of BHZ+U model) Related studies: (If time allows.)

3 1.Introduction ~Properties of topological insulators~ 3/21 Topological insulators Gapless edge states (robust against non-magnetic perturbations) C. L. Kane et al. PRL Non-trivial band structure (Bulk) Characteristic magnetoelectric response Quantized spin Hall conductivity. (QSH ins.) Topological magnetoelectric effect. (3D strong-tbi)

4 ~Topological phase in d,f electron systems ~ 4/21 Na2IrO3 (Iridium oxide) LuPtSb etc. (Heusler compounds) CeOs4Sb12 filled skutterudite Ir O A.Shitade et al. PRL (2009) Lu Pt Sb S. Chadov et al. Nature Materials 9, 541 (2010) H. Lin et al. Nat. Mat (2010) B. Yan et al. arxiv: Band inversion is observed only at the Γ point. Parity :odd Lattice distortion opens the bulk gap. even (DFT) :probability of s-orbital occupation of Sb sites. Heusler compounds (e.g. LuPtSb) can be topological insulator.

5 Topological phase in d,f electron systems 5/21 Correlated topological insulators Exotic phases! Symmetry protected phases correlation + topological structure Intrinsic topological phase (FQHE etc.) Symmetry protected topological phase (Haldane phase in S=1 Heisenberg chain) Topological phases induced by Coulomb interaction Phase competition : [Topological phase] vs. [ordered phases] magnetic phase, charge density wave phase etc

6 Topological phases induced by interactions 6/21 S. Raghu et al. PRL 100, Coulomb interactions Spin-orbit interaction Such phases are also reported in pyrochlore and diamond lattice.

7 Phase competition : Topological phase v.s. magnetic phase ( Kane-Mele+U ) (Auxiliary field QMC) M. Hohenadler et al. PRL /21 Spin configuration (in-plane) (with VCA) S. Yu et al. PRL 107,

8 Outline 8 /21 1. Introduction 2. Purpose 3. Model and Method 4. Numerical Results DMFT+CT-QMC Relation between spin Hall conductivity and spin Chern number spin Hall conductivity, spectral function, magnetic instability 5. Summary and Outlook

9 2.Purpose 9 /21 Correlated topological insulators are extensively studied! Weakly correlated Strongly correlated Topological ins. Mott ins. Understand the phase competition with non-perturbative method. Bernevig-Hughes-Zhang model+u Dynamical Mean field theory + Continuous Time-Quantum Monte Carlo simulation

10 10 /21 1. Introduction 2. Purpose 3. Model and Method Model ~ BHZ+U model ~ Method ~DMFT+CT-QMC ~ How to detect the topological property ~Relation between spin Hall conductivity and spin Chern number ~ 4. Numerical Results 5. Summary and Outlook

11 3.Model and Method 11 /21 Model (BHZ model + U) Y X Non-interacting case 5 4 Orbital 2 Orbital 1 1 k gap-closing BHZ+U DMFT+CT-QMC

12 Method : Dynamical Mean Field Theory (DMFT+CT-QMC) 12/21 schematic picture CT-QMC Advantage DMFT has had a great success describing Mott transitions CT-QMC provides numerically exact solutions.

13 ~ How to detect topological property ~ 13/21 Even in : Kubo formula K. Ishikawa et al. Nucl. Phys. B Genetic diagram contributing to Spin Chern number Ward-identity [Vertex] = Three external lines. ( is included.) is anti-symmetric tensor Periodic boundary condition.

14 14 /21 1. Introduction 2. Purpose 3. Model and Method 4. Numerical Results Phase competition Topological ins. Mott ins. (i). Spin Hall conductivity (ii). Spectral function (iii). Magnetic instability (at finite temperature) 5. Summary and Outlook

15 ~(i) Spin Hall conductivity ~ : Quantized at T=0 Orbital 2 Orbital 1 TBI 15/21 [gap]~ discrepancy of : Temp. effect Gap renormalization Increase of effective temperature ([Temp.]/[Gap size]) Discontinuous change hysteresis 1 st order transition Mott [gap size] Trivial Mott ins.

16 ~(ii) Spectral function ~ :TBI :MI 16/21 Behavior of the gap and Topological structure non-interacting case gap Non-trivial trivial Mott transition gap coexisting region Non-trivial trivial Change of Topological structure without gap closing (Mott trans. is 1 st order)

17 ~The gap renormalization~ the renormalization depends on origin of the gap in U=0. Two orbitals +U+ local hybridization V c f V V (gap) is enhanced! :TBI :MI 17/21 A(ω) R. Sato et al. (2004) U However, the gap renormalization is generic behavior of TBIs. Gap Spin orbit interaction (contributes to kinetic term) (should be renormalized) Renormalization of the gap at generic behavior of TBIs.

18 Magnetic moment ~(iii) Magnetic instability ~ 18/21 Y X Mott transition is masked by AF. Observation of Mott trans. in geometrically frustrated TBI

19 Summary 19/21 Phase competition between TBI and MI are studied. (in BHZ+U with DMFT+CT-QMC) Spin Hall conductivity Double occupancy Topological ins. (topologically trivial) Mott ins. 1 st order transition Change of Top. # without gap-closing Magnetic instability AF phase masks the Mott transition geometrically frustrated topological insulators : pyrochlore lattice D.A. Pesin et al. Nat. Phys

20 Related studies 20/21 Topological antiferromagnetic insulators para AF topological antiferromagnetic phase T.Y, R. Peters, S. Fujimoto, and N. Kawakami PRB 87, (2013) U/t Topological Kondo insulators -RKKY and Kondo effect in TBI - topological phase in metal T.Y, R. Peters, S. Fujimoto, and N. Kawakami PRB 87, (2013)

21 21/21 Thank you for your attention!