Week 10: Berry Phases, SSH Model, Bulk-Edge Correspondence

Concepts of Berry phases, SSH model, surface states, Chern number, phonon Hall effect, bulk-edge correspondence, and more. Dive into the SSH model as a topological insulator, special limits, scattering states vs. edge states, and the bulk-edge correspondence in winding numbers.

• Physics
• Topological Insulator
• SSH Model
• Berry Phases
• Winding Numbers

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1. Week 10, Berry phases, SSH Model Surface states, definition of Berry phase, Berry phase in SSH model, bulk-edge correspondence, Chern number, phonon Hall effect, etc

2. SSH model as a topological insulator ?1?2?1?2 0 1 2 3 4 . n = = + 0 2 t t u 1 0 0 2 t t u 2 0 = + = h.c. H 1 0 1 t c c 2 1 t c c 1 2 3 t c c c Hc 2 0 t 0 t t 1 0 t 0 t 1 2 = H 0 0 t 2 1 0 1

3. Special limit, ?1 0, ?2> 0, edge state ?1= 0?2?2 0 1 2 3 4 . n 0 0 0 0 0 0 0 t 0 t 0 0 0 0 t 0 0 0 t +?2 2 0 0 0 0 2 = = , H H 0 0 0 2 0 0 0 2 ?2 = = 0 (edge state, one-fold), (N-fold generate) t 2

4. Opposite limit, ?1> 0, ?2= 0 ?1?2= 0?1 0 1 2 3 4 . n 0 t 0 0 0 t 0 0 t 0 0 0 0 0 t t 1 0 0 0 0 +?1 1 0 0 0 0 1 = = , H H 0 0 1 0 t ?1 1 no edge state 1 = (N-fold generate) t 1

5. General cases = + + 2 1 2 2 2 cos(2 ) t t 1 2 t t ak ?1< ?2 ?1> ?2 gap = 2 ?1 ?2 ? ?1= ?2

6. Scattering states vs edge state near the end of SSH chain

7. Surface Greens function g00 ( focus on 0th column + + + + + ) + i = ( ) E I H g I g 0 j + = + + ( t g t g ) ( ( 1 t g t g E i g E E t g 00 1 ) ) 10 = = 0 0 i i g g 1 00 10 2 20 2 10 20 1 30 try g = = = n n , , 1, 0,1,2, A g B n 1,0 + 2 ,0 n 2 n + + i + i 2 1 2 2 2 ( ) t t E E = = + + = 2 , 1 0 g A 1 00 1 2 t t + 2 2 t 1 2 t t

8. Surface density of states, ? ? = 2 Im ?00 ?1= 1 > ?2= 1/2, ? = 0.02 ?1=1 ? = 0.001 2< ?2= 1, ?(?) ?1= ?2= 1, ? = 0.02 ?

9. SSH model in k space t k ( ) ( ) * c k c k 0 ( ) t k ( ) 0 A = = + = iak 2 ( ) ( ), ( ) c k , ( ) H c k t k t t e c k 1 2 k B 2 2 = + ( ) H k H k a Im ? ? 1 1 z = dz 2 i ?2 angle = 2?? As ? varies from 0 to 2?/(2?) the complex number ?(?) traces out a circle. The winding number is defined as how many times the origin (0,0) is enclosed. ?1 Re ? ?

10. Bulk-edge correspondence Winding number ? = number of edge states on site A minus the number of edge states on site B. ?1?2?1?2 A B A B A . For SSH model, there is no states on site B, and only on site A, so when ?1< ?2, ? = 1,??= 1 when ?1> ?2, ? = 0,??= 0

11. Overall phase and relative phases in quantum mechanics i e | 2> , , 1 2 3 | 1> = i | | e 21 1 2 1 2 | 3> The relative phase in a loop is invariant and has physical consequences. + + ( ) i | | | e 21 32 13 1 2 2 3 3 1

12. Berry phase in a loop of k space j+1 j 2 Na j = , k N j 3 2 1 i = | | e + , 1 j j + + 1 1 j j j j N N = + , 1 j j N-1 = 1 j = = i | | e k + + + 2 / k k k k k k k a k | | ( ) + = + + + = + k k k k k 2 2 ( ), 1 1 ( ) k O k k i O + k k k k k k k k + k k k 2 / a = i dk k k k 0

13. Compute the Zak phase for SSH model * 0 c c = = + = iq ( ) , , 2 , 0 2 H q c t t e q ak q 1 2 0 1 1 = eigenvectors c c 2 winding number = 2 2 * iq t c e c i dc c = = + + = c.c. c.c. i dq 2 dq + + 2 4 4 q 0 0

14. Berry curvature (need 2D parameter k space) Apply Stoke's theorem S ?? = k A A S d d ?? loop S A k k A k ( ) y = = = z A A k ( )| k ( ) k , ( ) Im | x z x y

15. Symmetries in SSH model = = N 1 In Fock space (2 dimensional): in single particle Hilbert space ( by matrices): in space (2 by 2 matrices): k S HS H c Hc = 1 N N S HS H = 1 ( ) k H k S k ( ) ( ) ( ) S H k What are the possible symmetries existing in SSH model? What are the operators S (or ? or ?(?)) that perform the symmetry transformations? Real space periodicity, space translation by one unit cell, j to j + 2 Time reversal, ?to ? Space inversion, ?to ? mod ? Chiral symmetry, exchange sublattice A B, 1? = ?

16. Hall effect steps ??? peaks ??? 1 ne h = = = = = , , 0 E j R B j R y yx x H z x H yx xx xx 2 e i Derive the Drude model result in magnetic field for the Hall effect (following Ashcroft/Mermin)

17. Quantum Hall effect, TKNN formula Thouless, Kohmoto, Nightingale, den Nijs (1982) 2 occ 1 e h = = 2 ( ) k k C C d yx n n n 2 n 1BZ k ( ): Berry curvature of band : Chern number n n C n

18. Phonon Hall effect First phonon Hall experiment on Tb3Ga5O12 at T=5.45 K, with kxy estimated to be 0.45 (mW/K-m) at B = 10 T. C. Strohm, G.L.J.A. Rikken, and P. Wyder, PRL 95, 155901 (2005). SrTiO3 Hall experimental result, X. Li, B. Fauqu , Z. Zhu, and K. Behnia, PRL 124, 105901 (2020).

19. Phonon Hall model u u u 1 x Positive definite 1 y 1 2 1 2 ( ) 2 = + = = T , H p Au u Ku u m r 2 x Velocity dependent force u Ny 2 d u dt du dt + + = 2 0 Ku A but non-dissipative 2 1 2 1 2 dE dt = + = = = 2 T T T , 2 0 E u u Ku u Au A A What is the physical original of A?

20. T. Qin et al formula 2 B Tk V q = ( ) ( i q ) q xy i , 0 i ( ) x 2 x = + + x x ( ) x 2 log 1 Li ( ) x e e 2 x 1 e 0 I = = = , 1 H i q eff i 2 ( ) A ( ) q q D x H q H q q q ( ) q q eff eff q q i j j i x y 0 I D u u x y = = = ( ) q Im , , i ( ) 2 0 j i q q i j