Topological insulator (TI)Topological insulator (TI)•• Haldane model: QHE without Landau levelHaldane model: QHE without Landau level•• Quantized spin Hall effect: 2D topological Quantized spin Hall effect: 2D topological

insulators: insulators: –– KaneKane--MeleMele model for graphenemodel for graphene–– HgTeHgTe quantum wellquantum well–– InAs/GaSbInAs/GaSb quantum wellquantum well

•• 3D topological insulators3D topological insulators•• OutlookOutlookReferences:References:

““Topological insulators with inversion symmetryTopological insulators with inversion symmetry””, Liang Fu and C. L. Kane, , Liang Fu and C. L. Kane,

Phys. Rev. B 76, 045302 (2007).Phys. Rev. B 76, 045302 (2007).

““Topological insulators and superconductorsTopological insulators and superconductors””, Xiao, Xiao--Liang Liang QiQi and and ShouShou--Cheng Cheng

Zhang, Rev. Mod. Phys. 83, 1057 (2011).Zhang, Rev. Mod. Phys. 83, 1057 (2011).

,,

21 ..

Biii

Aiii

ijji

i

ijji

ccccM

chccetcctH

0/)2(2 ba

number quantum good a still is

! zero is cellunit agh flux throu total numbers real are and 21

k

tt

Phase diagram for the Phase diagram for the spinlessspinless Haldane modelHaldane model

hexy

2

Symmetries on the Haldane modelSymmetries on the Haldane model

Time reversal symmetry: Time reversal symmetry: ××

Inversion symmetry: ?Inversion symmetry: ?

Spin rotational symmetry: Spin rotational symmetry: √√

1.,.2

ij

ijji

zij

ijji chccsitcctH

TwoTwo--copy version of Haldane modelcopy version of Haldane model

symmetry reversal timerestore 2/ terms, dependent spin 2 t

Symmetries on KaneSymmetries on Kane--MeleMele modelmodel

Time reversal symmetry: Time reversal symmetry: √√

Inversion symmetry: ?Inversion symmetry: ?

Spin rotational symmetry: Spin rotational symmetry: ××

SpinSpin--orbit couplingorbit coupling

The Dirac point originates from the

topology of bulk energy band and is protected by time reversal symmetry

Edge states in KaneEdge states in Kane--MeleMele modelmodel

S.C. Zhang, Physics 1, 6 (2008)

Quantum spin Hall effectQuantum spin Hall effect

SpinSpin--orbit coupling in graphene is too weak to realize it !orbit coupling in graphene is too weak to realize it !

first prediction of realistic materials

first experiment

HgTe/CdTe

HgTeHgTe : band inversion picture: band inversion picture

B.A. Bernevig, T. L. Hughes and S.-C. Zhang, Science 314 1757(2006).

Effective Hamiltonian:Effective Hamiltonian:

0/ BM0/ BM

Atomic basis:Atomic basis:

Edge states in the BHZ modelEdge states in the BHZ model

x

y)()(

0

002

22 xEx

BMiAiABM

DC

k

xx

xxx

y

number quantum good : yk

solution 0 existingsymmetry hole-particle and Neglecting

EDC

0)(2

2

0

E

BMiAiABM

ex x

MBAAB

decebeaex xxxx 421,)()()( 2

2,102121

General solution:

)0,0(or )0,0(0)0( condition bondary Open cdbaabdc

)0(0Reor )0(0Re 2,12,1 badc

These conditions can be satisfied only in the inverted regime whThese conditions can be satisfied only in the inverted regime whenen .0/ BM

Existence condition for edge states:

InAs/GaSbInAs/GaSb quantum well quantum well

C. X. Liu, T.L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang, PRL 100, 236601 (2008).

gap locates away from gap locates away from ΓΓ pointpoint

second quantum spin Hall insulator

Highly quantized conductance plateaus Highly quantized conductance plateaus in in SiSi--dopeddoped InAs/GaSbInAs/GaSb quantum well quantum well

Si Si dopantsdopants serve as donors in serve as donors in InAsInAs

and acceptors in and acceptors in GaSbGaSb. .

Impurity concentration: 10Impurity concentration: 101111cmcm--22..

Lingjie Du, Ivan Knez, Gerard Sullivan and Rui-Rui Du, arXiv:1306.1925 (2013)

<1%<1%

DOSDOS: capacitancecapacitance--gate voltagegate voltage

hybridization gaphybridization gap

residual DOSresidual DOS

Conductance in a Conductance in a CorbinoCorbino disk disk

Evidences for inEvidences for in--gap localized statesgap localized states

From 2D to 3D: a big challengeFrom 2D to 3D: a big challenge

Two Dirac cones may merge each other to vanish !

Layered 2D TIs can NOT form a 3D TI protected by topology !

From 2D to 3D : breakthroughFrom 2D to 3D : breakthrough

From 2D to 3D: From 2D to 3D: strong and weak strong and weak TIsTIs

A suggested paperA suggested paper

Bi0.9 Sb0.1

prediction

first experiment

More compounds for 3D More compounds for 3D TIsTIs

•• BiBi22 SeSe33

•• BiBi22 TeTe33

•• etc.etc.

Some exotic propertiesSome exotic properties

Odd number of Dirac conesOdd number of Dirac cones

NoNo--Go theoremGo theorem

2D and 3D, separated boundaries and connected surface2D and 3D, separated boundaries and connected surface

Delocalization against nonmagnetic disordersDelocalization against nonmagnetic disorders

1, 2 CCC

1

ie

Experimental evidence for Berry’s phase

OutlookOutlook•• Interacting systemsInteracting systems

–– Fermionic systems: Fermionic systems: KitaevKitaev’’ss fermion chainfermion chain–– BosonicBosonic systems: Symmetry protected systems: Symmetry protected

topological ordertopological order•• Localization and disorder effectLocalization and disorder effect

–– Anderson localization or other effect?Anderson localization or other effect?–– TwoTwo--parameter scaling?parameter scaling?

•• Topological crystalline insulatorTopological crystalline insulator–– Surface states protected by crystalline symmetrySurface states protected by crystalline symmetry

•• etc.etc.