Topological insulator (TI) Haldane model: QHE without Landau level Quantized spin Hall effect: 2D topological insulators: Kane-Mele model for graphene HgTe quantum well InAs/GaSb quantum well 3D topological insulators Outlook References: Topological insulators with inversion symmetry,, Liang Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007). Topological insulators and superconductors,, Xiao-Liang Qi and Shou-Cheng Zhang, Rev. Mod. Phys. 83, 1057 (2011).
,, 2 1.. B i i i A i i i ij j i i ij j i c c c c M h c c c e t c c t H 0 ) / (2 2 b a good quantum number a is still zero! unit cell is flux through a total numbers are real and 2 1 k t t
Phase diagram for the spinless Haldane model 2 xy e h Symmetries on the Haldane model Time reversal symmetry: Inversion symmetry:? Spin rotational symmetry:
Two-copy version of Haldane model spin dependent t2 terms, / 2 restore time reversal symmetry H ij tc i c j ij z it2 s c c h. c., 1 ij i j ij Symmetries on Kane-Mele model Time reversal symmetry: Inversion symmetry:? Spin rotational symmetry: Spin-orbit coupling
Edge states in Kane-Mele model The Dirac point originates from the topology of bulk energy band and is protected by time reversal symmetry
S.C. Zhang, Physics 1, 6 (2008)
Quantum spin Hall effect Spin-orbit coupling in graphene is too weak to realize it!
HgTe/CdTe first prediction of realistic materials first experiment
HgTe : band inversion picture Effective Hamiltonian: Atomic basis: M / B 0 M / B 0 B.A. Bernevig, T. L. Hughes and S.-C. Zhang, Science 314 1757(2006).
k y : Edge states in the BHZ model good quantum number y x k y 0 C D 2 x Neglecting C existing M B ia x and D E 0 2 x ia x M B solution 2 x 0( x) E ( x) particle - hole symmetry 0 General solution: ( x) ( ae 0 2 x M B ia ( x) e 2 ia M B 0 E x x x x 1 1 2 be ) ( ce de ), A A 2 Open bondary condition (0) 0 ( c d 0, ab 0) or ( a b 0, cd 1 2 1,2 2B 0) 4MB 0 Existence condition for edge states: Re 1,2 0 ( c d 0) or Re 1, 2 0 ( a b 0) These conditions can be satisfied only in the inverted regime when M / B 0.
InAs/GaSb quantum well gap locates away from Γ point C. X. Liu, T.L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang, PRL 100, 236601 (2008).
second quantum spin Hall insulator
Highly quantized conductance plateaus in Si-doped InAs/GaSb quantum well Lingjie Du, Ivan Knez, Gerard Sullivan and Rui-Rui Du, arxiv:1306.1925 (2013) <1% Si dopants serve as donors in InAs and acceptors in GaSb. Impurity concentration: 10 11 cm - 2. 11 cm
Evidences for in-gap localized states DOS: capacitance-gate voltage hybridization gap residual DOS Conductance in a Corbino disk
From 2D to 3D: a big challenge Layered 2D TIs can NOT form a 3D TI protected by topology! Two Dirac cones may merge each other to vanish!
From 2D to 3D : breakthrough
From 2D to 3D: strong and weak TIs
A suggested paper
prediction first experiment Bi0.9 Sb0.1
More compounds for 3D TIs Bi 2 Se 3 Bi 2 Te 3 etc.
Some exotic properties Odd number of Dirac cones No-Go theorem 2D and 3D, separated boundaries and connected surface Delocalization against nonmagnetic disorders C C, C 2 1 e i 1
Experimental evidence for Berry s phase
Outlook Interacting systems Fermionic systems: Kitaev s fermion chain Bosonic systems: Symmetry protected topological order Localization and disorder effect Anderson localization or other effect? Two-parameter scaling? Topological crystalline insulator Surface states protected by crystalline symmetry etc.