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

Quantum exotic states in Quantum exotic states in correlated correlated topological insulatorstopological 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).

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

Spin liquid – emergent in physicsSpin 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/0310405

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

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, 146802 (2005)

http://www.physics.upenn.edu/~kane/

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)

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?

Mean field equation

where

Mean field approach

M is the staggered magnetization.

Phase diagramPhase diagram

C=2 topological insulator - QAH

Band insulator

Trivial AF-SDW order

B-type topologicalSDW order

A-type topologicalSDW order

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

Low energy effective modelLow energy effective model

K-matrix formulationK-matrix formulation

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

spin-charge synchronization charge-spin-charge synchronization charge-flux binding effect in B-TSDWflux 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 2. Quantum spin orders in correlated topological insulator with correlated topological insulator with

flat-bandflat-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 TFBPhase 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 caseA-TSDW : Half filling case

qq )(

A-TSDW

AF-SDW

TFB

TFB

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

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

Dispersion of electrons

Order parameters

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

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).

Spin coupling constant Spin coupling constant

t’=0.0228tt’=0.1t

? ?

?

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).

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

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).

Slave-rotor approachSlave-rotor approach

Mean field approachMean field approach

C=2 topological insulator Chiral spin liquid

Trvial AF order

A-TSDW

Chiral spin order parameterChiral spin order parameter

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

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

N = 4

?

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

Composite spin liquid spin liquid

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

?

To be confirmed by QMC, …

IV. ConclusionIV. Conclusion

?

Thanks for your attention!

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

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

tt ji ,ittttyx eiieii ,, ,

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).

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.

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.

Results from QMCResults from QMC

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

026404 (2012)

Global Phase diagram by spin-fluctuation theory

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

045603

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)

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.