Topological Insulator

Transcription

1 Topologial Insulator Topologial insulator : Quantum Spin Hall state Classial to Quantum 2D TNG 3D

2 Classial to Quantum Quantum Spin Hall Effet Quantum Hall Effet without magneti field Quantum Hall Effet with time-reersal inariane QHE B TR is broken

3 Quantum Spin Hall Effet Classial to Quantum Quantum Hall Effet without magneti field Quantum Hall Effet with time-reersal inariane No external net magneti field QSHE B TR inariant spin up spin down B

4 Quantum Spin Hall Effet Classial to Quantum Quantum Hall Effet without magneti field Quantum Hall Effet with time-reersal inariane No external net magneti field QSHE B TR inariant spin up spin down B Quantum effets! No lassial orrespondent! Think different Following the QHE

5 so-alled Topologial Insulator Classial to Quantum Topologial insulator : Quantum Spin Hall state Need to undestand! Time Reersal Kramers degeneray TNG Let me explain! Spin Hall ondutane is not quantized Spin is not onsered (spin-orbit)

6 Classial to Quantum Time-Reersal (TR) symmetry & Kramers degeneray TR: Anti-Unitary : i H i H ij j TR inariane [, H] H J H * J -- a t b d d a b a d, b t i i#! JH J H i# i J i & omplex onjugate J i y d {H} ij H ij a b d b a t : Spin independent hopping a : Spin-orbit, Rashba term, et. H b d

7 Classial to Quantum Time-Reersal (TR) symmetry & Kramers degeneray TR: Anti-Unitary : i H i H ij j TR inariane [, H] H J H * J -- a t b d d a b a d, b t i i#! JH J H & hermitiity i# i J i & omplex onjugate J i y d {H} ij H ij a b d b a t t a hermite H b d! e anti-symmetri

8 Classial to Quantum Time-Reersal (TR) symmetry & Kramers degeneray Shrödinger Equation t u t E u tu + Eu t + ( u )E u + t E + t ( u )E( u )

9 Classial to Quantum Time-Reersal (TR) symmetry & Kramers degeneray Shrödinger Equation t u t E u tu + Eu u + t E t + ( u )E + t ( u )E( u ) Ah! H u E u, u u & u u is also an eigen state with the same energy : the same energy, degenerate?

10 Classial to Quantum Time-Reersal (TR) symmetry & Kramers degeneray Shrödinger Equation t u t E u tu + Eu u + t E t + ( u )E + t ( u )E( u ) Ah! H u E u, u u & u u is also an eigen state with the same energy : the same energy, degenerate? Not yet! the same state??

11 Classial to Quantum Time-Reersal (TR) symmetry & Kramers degeneray Shrödinger Equation t u t E u tu + Eu u + t E t + ( u )E + t ( u )E( u ) Ah! H u E u Not yet! the same state??, u u & u u : the same energy, degenerate? Orthogonal! u is also an eigen state with the same energy OK! surely different! u u + ( u )

12 Classial to Quantum Time-Reersal (TR) symmetry & Kramers degeneray Shrödinger Equation t u t E u Kramers degeneray Any one partile state is doubly degenerate tu + Eu t + ( u )E u + t E + t ( u )E( u ) Ah! H u E u Not yet! the same state??, u u & u u : the same energy, degenerate? Orthogonal! u is also an eigen state with the same energy OK! surely different! u u + ( u )

13 Time Reersal & Quaternions Classial to Quantum F.J.Dyson 6-- Quaternion omplex number real number R C H No magi, neither razy just Pauli matries t Hamilton t (Ret)I 2 +(Imt) i z +(Re ) i y +(Im ) i x (Ret)+(Imt)i H +(Re ) j H +(Im ) k H : Quaternion ( ) i H i z,j H i y,k H i z i 2 H k 2 H k 2 H i H j H k H Quaternion 2 2 Matrix, Yang Monopole & quantization: YH, NJP2, 654 (2)

14 Time-Reersal, Spins & Spinors spin Ŝ x y z Ŝ spinor A S, S, 2 S S JS J i i i i Classial to Quantum # J x y z S S Magneti field B S B S Zeeman term breaks TR

15 n # Rotation: Spin & Spinor spinor # U() # Classial to Quantum U 2 SU(2) n # U( )e is n os 2 in sin 2 det U e i(tr S) n e S S spin A USU Q( ) A S : hermite Tr S TrSU U Expand by Pauli matries with real oeffiients Tr S S TrS S 2 Tr S S Q Q Tr S S Q e Q E 3 Q 2 SO(3) 2 Q Q 2 (Q e Q) ontiniously onneted to E 3

16 n Rotation: Spin & Spinor A USU Q( ) A n S S spin Classial to Quantum # spinor # U() U( )e is n os 2 in sin 2 # #

17 n Rotation: Spin & Spinor A USU Q( ) A n S S spin Classial to Quantum # spinor # U() U( )e is n os 2 in sin 2 # # Spin Spinor goes bak by 2 rotation does not go Q( +2 ) U( +2 ) + -- Q() U()

18 n Rotation: Spin & Spinor A USU Q( ) A n S S spin Classial to Quantum # spinor # U() U( )e is n os 2 in sin 2 # # Spin Spinor goes bak by 2 rotation does not go Q( +2 ) U( +2 ) + -- Q() U() 4 is always OK : ontinuously deformed to rotation.f. 2D

19 n Rotation: Spin & Spinor A USU Q( ) A n S S spin Classial to Quantum # spinor # U() U( )e is n os 2 in sin 2 # # Spin Spinor goes bak by 2 rotation does not go Q( +2 ) U( +2 ) + -- Q() U() 4 is always OK : ontinuously deformed to rotation.f. 2D

20 n Rotation: Spin & Spinor A USU Q( ) A n S S spin Classial to Quantum # spinor # U() U( )e is n os 2 in sin 2 # # Spin Spinor goes bak by 2 rotation does not go Q( +2 ) U( +2 ) + -- Q() U() H. Weyl Penrose-Rindler 4 is always OK : ontinuously deformed to rotation.f. 2D

21 n Dira sissors Rotation: Spin & Spinor A USU Q( ) A n S S spin Classial to Quantum # spinor # U() U( )e is n os 2 in sin 2 # # Spin Spinor goes bak by 2 rotation does not go Q( +2 ) U( +2 ) + -- Q() U() H. Weyl Penrose-Rindler 4 is always OK : ontinuously deformed to rotation.f. 2D

22 Classial to Quantum Quantum Spin Hall effet?? Topologial insulator : Quantum Spin Hall state

23 Classial to Quantum Quantum Spin Hall effet?? Topologial insulator : Quantum Spinor Hall state

24 Classial to Quantum Quantum Spin Hall effet?? Topologial insulator : Quantum Spinor Hall state Purely quantum mehanial!