Quantum exotic states in correlated topological insulators

1. Quantum exotic states in correlated topological insulators Su-Peng Kou (寇谡鹏) Beijing Normal University

2. Outline Motivation Topological spin density waves in correlated topological insulators Quantum spin liquid states in correlated topological insulators Conclusion [1] Kou SP，PHYS. REV. B 78, 233104（2008）. [2] Sun GY and Kou SP, EPL, 87 67002 (2009). [3] Kou SP, and Liu LF，EUR. PHYS. J. B. 81, 165 (2011) . [4] Sun GY and Kou SP, J. Phys. C 23 (2011) 045603. [5] He J, Kou SP, Liang Y, Feng SP, PHYS. REV. B 83, 205116 (2011) . [6] He J, Zong YH, Kou SP, Liang Y, Feng SP, PHYS. REV. B84, 035127 (2011) . [7] He J, Liang Y, Kou SP, PHYS. REV. B.85, 205107 (2012). [8] He J, Wang B, Kou SP, PHYS. REV. B. submitted, arXiv:1204.4766. [9] Kou SP, “Insulators: Types, Properties and Uses” (Nova Science Publishers).

3. I.Motivation:Lookfor quantum exotic states in correlated topological insulator X. G. Wen, Quantum Field Theory of Many-Body Systems

4. Spin liquid – emergent in physics No broken symmetry + Deconfined spinons + Emergent gauge field Spin liquid

5. Spin orders in strongly correlated electron systems G. Misguich, arXiv:cond-mat/0310405

6. II. Topological spin density wave states in correlated topologicalinsulators Instability of an interacting fermion system with topologically nontrivial band structure • Interacting spinful Haldane model • Interacting Kane-Mele model

7. The spinful Haldane model– spin rotation symmetry, no T symmetry

8. The Kane-Mele model – T symmetry, no spin rotation symmetry Kane and Mele, Phys. Rev. Lett. 95, 146802 (2005) http://www.physics.upenn.edu/~kane/

9. Possible quantum spin liquid in the interacting Kane-Mele model – T symmetry, no spin rotation symmetry Slave-rotor theory: Stephan Rachel and Karyn Le Hury, Phys. Rev. B 82. 075106 (2010) QMC: M. Hohenadler, T. C. Lang, F. F. Assaad, Phys. Rev. Lett. 106, 100403 (2011) Dong Zheng, Congjun Wu and Guang-Ming Zhang, Phys. Rev. B 84, 205121 (2011) DMF: Wei Wu, S. Rachel, Wu-Ming Liu, K. Le Hur, Phys. Rev. B 85, 205102 (2012) VCA : Shun-Li Yu, X.C. Xie, Jian-Xin Li, Phys. Rev. Lett. 107, 010401 (2011)

10. 1. Topological spin-density-wave states in interacting spinful Haldane model - spin rotation symmetry, no T symmetry What is the ground state for the spinful Haldane model with the on-site interaction? He J, Zong YH, Kou SP, Liang Y, Feng SP, PHYS. REV. B84, 035127 (2011)

11. Mean field approach M is the staggered magnetization. Mean field equation where

12. Phase diagram C=2 topological insulator - QAH A-type topological SDW order Band insulator B-type topological SDW order Trivial AF-SDW order

14. Low energy effective model

15. K-matrix formulation

16. Spin-charge separated charge-flux binding effect in A-TSDW

17. spin-charge synchronization charge-flux binding effect in B-TSDW

18. Different spin-density-wave states in correlated topological insulators with the same local order parameter may have different topological properties, including the induced quantum numbers on topological objects, the edge states, the quantum Hall effects.

19. 2. Quantum spin orders in correlated topological insulator with flat-band

20. Possible fractional quanum hall states

21. What is the ground state for the correlated topological insulators in the flat-band limit? • What’s the dispersion of electrons and spin waves for correlated topological insulators in the flat-band limit?

22. Phase diagram : electrons on TFB FM (topological) spin-density-wave d is the hole concentration.

23. A-TSDW : Half filling case Dispersion of electrons in A-TSDW A-TSDW Dispersion of spin-waves in A-TSDW TFB TFB AF-SDW

24. FM (topological) spin-density-wave: quarter filling case Dispersion of electrons in FM order Dispersion of spin wave in FM order

25. FM order and AF order :d=0.3 filling case Order parameters Dispersion of electrons

26. III. Quantum spin liquids in interacting spinful Haldane model • Short range A-type topological spin density wave state: chiral spin liquid • Short range B-type topological spin density wave state : composite spin liquid

27. Quantum spin-fluctuations in topological spin density wave states Transverse spin susceptibility is Spin coupling constant X. G. Wen, Quantum Field Theory of Many-Body Systems, (Oxford Univ. Press, Oxford, 2004) Spin wave velocity One obtains spin stiffness and the transverse spin susceptibility: H.J. Schulz, in The hubbard Model, edited by D. Baeriswyl(Plenum, New York, 1995). Z. Y. Weng, C. S. Ting, and T. K. Lee, Phys. Rev. B43, 3790 (1991). K. Borejsza, N. Dupuis, Euro Phys. Lett. 63, 722 (2003); Phys. Rev. B 69, 085119 (2004).

28. Spin coupling constant t’=0.1t t’=0.0228t

29. ? ? ?

30. S. Chakravarty, et al., Phys. Rev. B 39, 2344 (1989). What is the nature of the quantum disordered states for TSDWs?

31. He J, Liang Y, Kou SP, PHYS. REV. B.85, 205107 (2012).

32. Properties of chiral spin liquid • Spinon is semion with fractional statistics • Ground state degeneracy : 2 on torus • Chiral gapless edge states He J, Liang Y, Kou SP, PHYS. REV. B.85, 205107 (2012). X. G.Wen, F.Wilczek, and A. Zee, Phys. Rev. B 39, 11413 (1989).

33. Slave-rotor approach

34. Mean field approach

35. C=2 topological insulator Chiral spin liquid A-TSDW Trvial AF order

36. Chiral spin order parameter

37. π- vortex issemion • Statistics angle θ= π/2 With induced fermion number , π-vortex becomes semion.

38. Effective Lagrangian from slave-rotor approach N = 4

39. S=1/2, charge e fermion ?

40. Composite spin liquid spin liquid

41. S=1/2, charge e fermion ? g > gc g < gc

42. IV. Conclusion ? To be confirmed by QMC, …

43. Thanks for your attention!

44. Spin susceptility of spin order in metallic spin order

45. 1. Spin liquid in the π-flux Hubbard model and the Hubbard model on honeycomb lattice

46. Quantum spin liquid near Mott transition of π-flux Hubbard model Sun GY and Kou SP, EPL, 87 67002 (2009). Kou SP, Liu LF, He J, Wu YJ，EUR. PHYS. J. B. 81, 165 (2011).

47. Gapless Z2 topological spin liquid There are three types of quasi-particles : gapped fermionic spinons, gapped bosonic spinons and the gapped gauge field.

48. Nodal spin liquid There are three types of quasi-particles : gapless fermionic spinons, gapped bosonic spinons and the roton-like gauge field.

49. Results from QMC Chia-Chen Chang and Richard T. Scalettar, Phys. Rev. Lett. 109, 026404 (2012)

50. Global Phase diagram by spin-fluctuation theory Sun GY and Kou SP, J. Phys. C. 23 (2011) 045603