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

Outline Motivation Motivation Topological spin density waves in correlated topological insulators Topological spin density waves in correlated topological insulators Quantum spin liquid states in correlated topological insulators Quantum spin liquid states in correlated topological insulators Conclusion Conclusion [1] Kou SP ， PHYS. REV. B 78, （ 2008 ）. [2] Sun GY and Kou SP, EPL, (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) [5] He J, Kou SP, Liang Y, Feng SP, PHYS. REV. B 83, (2011). [6] He J, Zong YH, Kou SP, Liang Y, Feng SP, PHYS. REV. B 84, (2011). [7] He J, Liang Y, Kou SP, PHYS. REV. B. 85, (2012). [8] He J, Wang B, Kou SP, PHYS. REV. B. submitted, arXiv: [9] Kou SP, “ Insulators: Types, Properties and Uses ” (Nova Science Publishers).

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

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

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

II. Topological spin density wave states in correlated topological insulators Instability of an interacting fermion system with topologically nontrivial band structure 1.Interacting spinful Haldane model 2.Interacting Kane-Mele model

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

The Kane-Mele model – T symmetry, no spin rotation symmetry Kane and Mele, Phys. Rev. Lett. 95, (2005)

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 (2010) QMC: M. Hohenadler, T. C. Lang, F. F. Assaad, Phys. Rev. Lett. 106, (2011) Dong Zheng, Congjun Wu and Guang-Ming Zhang, Phys. Rev. B 84, (2011) VCA : Shun-Li Yu, X.C. Xie, Jian-Xin Li, Phys. Rev. Lett. 107, (2011) DMF: Wei Wu, S. Rachel, Wu-Ming Liu, K. Le Hur, Phys. Rev. B 85, (2012)

Topological spin-density-wave states in interacting spinful Haldane model 1. Topological spin-density-wave states in interacting spinful Haldane model - spin rotation symmetry, no T symmetry He J, Zong YH, Kou SP, Liang Y, Feng SP, PHYS. REV. B 84, (2011) What is the ground state for the spinful Haldane model with the on-site interaction?

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

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

Low energy effective model

K-matrix formulation

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

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

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.

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

Possible fractional quanum hall states

1.What is the ground state for the correlated topological insulators in the flat- band limit? 2.What’s the dispersion of electrons and spin waves for correlated topological insulators in the flat-band limit?

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

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

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

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

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

Quantum spin-fluctuations in topological spin density wave states Transverse spin susceptibility is Spin coupling constant Spin wave velocity X. G. Wen, Quantum Field Theory of Many-Body Systems, (Oxford Univ. Press, Oxford, 2004) 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, (2004).

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

? ? ?

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

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

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

Slave-rotor approach

Mean field approach

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

Chiral spin order parameter

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

Effective Lagrangian from slave-rotor approach N = 4

? S=1/2, charge e fermion

Composite spin liquidspin liquid

g > g c g < g c S=1/2, charge e fermion ?

To be confirmed by QMC, … IV. Conclusion ?

Thanks for your attention!

Spin susceptility of spin order in metallic spin order

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

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

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

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

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

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

Quantum spin liquid from QMC Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad & A. Muramatsu, Nature 464, 847 (2010)

Results of the Hubbard Model on the Honeycomb Lattice from QMC of bigger size Sandro SorellaSandro Sorella, Yuichi Otsuka, Seiji Yunoki, arXiv: Yuichi OtsukaSeiji Yunoki