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

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

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

Quantum exotic states in Quantum exotic states in correlated correlated topological insulatorstopological insulators

Su-Peng Kou (寇谡鹏 )

Beijing Normal University

Page 2: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

OutlineOutline MotivationMotivation Topological spin density waves in correlated Topological spin density waves in correlated

topological insulatorstopological insulators Quantum spin liquid states in correlated Quantum spin liquid states in correlated

topological insulatorstopological insulators ConclusionConclusion

[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. B 84, 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).

Page 3: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

I. Motivation: LookLook for quantum exotic states in for quantum exotic states in

correlated topological insulatorcorrelated topological insulator

X. G. Wen, Quantum Field Theory of Many-Body Systems

Page 4: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Spin liquid – emergent in physicsSpin liquid – emergent in physics

No broken symmetry

Deconfined spinons+

Spin liquid

Emergent gauge field+

Page 5: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Spin orders in strongly correlated electron systems

G. Misguich, arXiv:cond-mat/0310405

Page 6: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

II. Topological spin density wave states II. Topological spin density wave states in correlated topological insulatorsin correlated topological insulators

Instability of an interacting fermion system with topologically nontrivial band structure

1.1. Interacting spinful Haldane modelInteracting spinful Haldane model2. Interacting Kane-Mele model

Page 7: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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

Page 8: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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/

Page 9: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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)

VCA : Shun-Li Yu, X.C. Xie, Jian-Xin Li, Phys. Rev. Lett. 107, 010401 (2011)

DMF: Wei Wu, S. Rachel, Wu-Ming Liu, K. Le Hur, Phys. Rev. B 85, 205102 (2012)

Page 10: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

1. Topological spin-density-wave states Topological spin-density-wave states in interacting spinful Haldane model 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, 035127 (2011)

What is the ground state for the spinful Haldane model with the on-site interaction?

Page 11: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Mean field equation

where

Mean field approach

M is the staggered magnetization.

Page 12: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Phase diagramPhase diagram

C=2 topological insulator - QAH

Band insulator

Trivial AF-SDW order

B-type topologicalSDW order

A-type topologicalSDW order

Page 13: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

3.0 3.1 3.2 3.3 3.40.0

0.1

0.2

0.3

0.4

0.5

U0.0

0.5

1.0

AF

A-TSDW

QAHM

B-TSDW

Page 14: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Low energy effective modelLow energy effective model

Page 15: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

K-matrix formulationK-matrix formulation

Page 16: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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

Page 17: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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

Page 18: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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.

Page 19: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

2. Quantum spin orders in 2. Quantum spin orders in correlated topological insulator with correlated topological insulator with

flat-bandflat-band

Page 20: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Possible fractional quanum hall states

Page 21: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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?

Page 22: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Phase diagram : electrons on TFBPhase diagram : electrons on TFB

d is the hole concentration.

FM (topological) spin-density-wave

Page 23: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Dispersion of electrons in A-TSDW

Dispersion of spin-waves in

A-TSDW

A-TSDW : Half filling caseA-TSDW : Half filling case

qq )(

A-TSDW

AF-SDW

TFB

TFB

Page 24: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

FM (topological) spin-density-wave: quarter filling caseFM (topological) spin-density-wave: quarter filling case

Dispersion of electrons in FM order

Dispersion of spin wave in FM order

2)( qq

Page 25: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

FM order and AF order :FM order and AF order :d=0.3 filling cased=0.3 filling case

Dispersion of electrons

Order parameters

Page 26: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

III. Quantum spin liquids in III. Quantum spin liquids in interacting spinful Haldane modelinteracting spinful Haldane model

Short range A-type topological spin Short range A-type topological spin

density wave state: density wave state: chiral spin liquidchiral spin liquid

Short range B-type topological spin Short range B-type topological spin

density wave state : density wave state : composite spin composite spin

liquid liquid

Page 27: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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:

nz

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).

Page 28: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Spin coupling constant Spin coupling constant

t’=0.0228tt’=0.1t

Page 29: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

? ?

?

Page 30: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

What is the nature of the quantum What is the nature of the quantum disordered states for TSDWs? disordered states for TSDWs?

S. Chakravarty, et al., Phys. Rev. B 39, 2344 (1989).

Page 31: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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

Page 32: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Properties of chiral spin liquidProperties of chiral spin liquid

Spinon is semion with fractional statisticsSpinon is semion with fractional statistics Ground state degeneracy : 2 on torusGround state degeneracy : 2 on torus Chiral gapless edge states 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).

Page 33: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Slave-rotor approachSlave-rotor approach

Page 34: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Mean field approachMean field approach

Page 35: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

C=2 topological insulator Chiral spin liquid

Trvial AF order

A-TSDW

Page 36: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Chiral spin order parameterChiral spin order parameter

Page 37: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

π- vortex is semion

Statistics angle θ = π/2

With induced fermion numbWith induced fermion number , er , ππ-vortex beco-vortex becomes mes semionsemion. .

1fN

Page 38: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Effective Lagrangian from Effective Lagrangian from slave-rotor approach slave-rotor approach

N = 4

Page 39: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

?

S=1/2, charge S=1/2, charge e fermione fermion

Page 40: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Composite spin liquid spin liquid

Page 41: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

2.60 2.650.00

0.05

0.10E

nerg

y ga

p 2.55 2.60 2.65

0.0

0.2

0.4Electron's energy gap

U/t

E/t

U/t

B-TSDW

Charged skyrmion gap

Spin gap

g > gc g < gc

S=1/2, charge e fermionS=1/2, charge e fermion

?

Page 42: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

To be confirmed by QMC, …

IV. ConclusionIV. Conclusion

?

Page 43: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Thanks for your attention!

Page 44: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Spin susceptility of spin order Spin susceptility of spin order in metallic spin orderin metallic spin order

Page 45: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

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

tt ji ,ittttyx eiieii ,, ,

Page 46: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Quantum spin liquid near Mott transition of Quantum spin liquid near Mott transition of ππ--flux Hubbard modelflux 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).

Page 47: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Gapless Gapless Z2 topological spin liquid Z2 topological spin liquid

There are three types of quasi-particles : gapped fermionic spinons, gapped bosonic spinons and the gapped gauge field.

Page 48: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Nodal spin liquidNodal spin liquid

There are three types of quasi-particles : gapless fermionic spinons, gapped bosonic spinons and the roton-like gauge fie

ld.

Page 49: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Results from QMCResults from QMC

Chia-Chen Chang and Richard T. Scalettar, Phys. Rev. Lett. 109,

026404 (2012)

Page 50: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Global Phase diagram by spin-fluctuation theory

Sun GY and Kou SP, J. Phys. C. 23 (2011)

045603

Page 51: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Quantum spin liquid from QMCQuantum spin liquid from QMC

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

Page 52: Quantum exotic states in correlated topological insulators Su-Peng Kou ( 寇谡鹏 ) Beijing Normal University

Results of the Hubbard Model on the Results of the Hubbard Model on the Honeycomb Lattice from QMC of bigger sizeHoneycomb Lattice from QMC of bigger size

Sandro Sorella, Yuichi Otsuka, Seiji Yunoki, arXiv:1207.1783.