Building Frac-onal Topological Insulators. Collaborators: Michael Levin Maciej Kosh- Janusz Ady Stern

Transcription

1 Building Frac-onal Topological Insulators Collaborators: Michael Levin Maciej Kosh- Janusz Ady Stern

2 The program Background: Topological insulators Frac-onaliza-on Exactly solvable Hamiltonians for frac-onal topological insulators When are they really frac-onal topological insulators?

3 Ordinary Insulators all states are local OR Can be deformed into a product of local states without going through a phase transi-on (Images from Kane and Mele: Science 15, 1692)

4 Chern insulators Cannot localize all states without a phase transi-on Gapless edge modes

5 Topological insulators Cannot localize all states without a phase transi-on OR breaking T Time- reversal protected gapless edge modes

6 Topological insulators 2D: quantum spin Hall effect (HgTe; Bernevig Hughes Zhang; Konig et al) 3D: single surface Dirac cone (Moore Balents; Fu Kane; Roy; Hsieh et al) Moore, Nature physics 2009 (Hsieh et al, Nature, 2009; Bi2Se3 )

7 What s the big deal? Even without interac-ons, there are different kinds of insulators Bulk proper-es are all ordinary (gapped, incompressible states) Dis-nguished by proper-es of boundary (Quan-zed transport of charge, spin)

8 Frac-onaliza-on What? Excita-ons carry a frac-on of quantum numbers of cons-tuent par-cles. How? Strong interac-ons Example Frac-onal QHE

9 Non- interac-ng electrons Insulators Ordinary Topological insulators Chern insulators (IQHE) FTI Strongly interac-ng electrons gapped Frac-onalized FQHE

10 Frac-onal topological insulators Can you make a model with a single surface Dirac cone of frac-onally charged fermions? Exactly solvable laece model Frac-onally charged fermionic excita-ons Band structure. Is it a topological insulator? Time reversal protected edge modes

11 Solvable models with frac-onally charged fermions Strongly interac-ng charged bosons ( ) (RVB; Toric code; Senthil & Motrunich) Charge conserving, exactly solvable q B p Couple to fermions q f = e + k q B p Band structure

12 Solvable models with frac-onally charged fermions n s (quantum dot) n f n ss (hard core boson) V t

13 Frac-onally charged bosons V = Q 2 s Q s = s n ss +2n s Q s =0 1 Q s =

14 Frac-onally charged bosons V = Q 2 s Q s = s n ss +2n s Q s =0 Q s =

15 U ss = b s b ss + b ss b s B P = U 12 U 23 U 34 U 41 H = V s Q 2 s u 2 P [Q s,b P ]=0 Equal amplitude superposi-on of all link occupa-on numbers compa-ble with Q s B P + B P

16 Q s = links : 2 ; sites : - 1 Q s = links : 2 ; sites : 1-2

17 Frac-onal Charge N S Qs =1 N s Qs =0 = 1 p

18 Frac-onally charged fermions Q s = Q s kn f,s Q s = k q f = (p +2k)e p n f =1

19 H = V s Q 2 s u 2 BP + B p + µ n f,s P s Q s =0 n f,s =0 B P =1 u, V µ Low- energy excita-ons: composite fermions (frac-onally charged)

20 Band structure H kin = ss t ss σσ c s σ c sσ U k ss Hops one fermion and k frac-onally charged bosonic excita-ons Q s = k Q s =0 U k ss Q s =0 Q s = k

21 H = V s Q 2 s u 2 B p + B P + H kin P Q s =0 [H kin, Q s ]=0 n f,s =0 [H kin,b P ]=0 B P =1

22 H = t ss σσ d s σ d sσ n f,s =0

23 Frac-onal topological insulators We can choose any band structure Choose TI band structure Frac-onally charged surface states

24 A few interes-ng proper-es Topological order p 2 p 3 (2D) (3D) 1

25 A few interes-ng proper-es Magnetoelectric effect Break T on the surface ½ integer Hall conduc-vity TI σ xy = 1 2 e 2 h

26 A few interes-ng proper-es Frac-onal magnetoelectric effect Break T on the surface ½ integer Hall conduc-vity TI σ xy = 1 2 q 2 f h

27 Frac-onal topological insulators Can you make a model with a single surface Dirac cone of frac-onally charged fermions? Exactly solvable laece model Frac-onally charged fermionic excita-ons Band structure. Is it a topological insulator? Time reversal protected edge modes

28 Time- reversal protected gapless boundary modes Kramers theorem: if T : k k k T 2 = 1 then states must be degenerate. k =0, π

29 Topological insulators: odd number of Kramers pairs in the edge spectrum E k E k ky =0 ky = π k y =0 k y = π Ordinary insulator Topological insulator

30 Interac-ng equivalent: Φ Φ =0= π E k E k ky =0 ky = π k y =0 k y = π (Fu and Kane 06)

31 Interac-ng equivalent: Φ (Fu and Kane 06) Φ =0, π : Time- reversal invariant Kramers degeneracies: TI Non-TI Φ =0 Yes No Φ = π No No Φ =0 No Yes Φ = π Yes Yes Odd Even

32 Construc-ng low- lying excited states Φ = π Φ =0 E π π

33 Flux inser-on in frac-onal insulators Levin and Stern 09 q f e Φ 0 e 2 Φ 0 Inser-ng 2 : a different ground state q f odd: Kramers No Kramers e q f e even: No change q f e = p +2k (p odd) 1 2 (p +2k) (p even)

34 2D Frac-onal topological insulators q f : protected edge modes e odd q f : Not protected! e even Can add interac-ons that preserve T and gap the edge

35 Frac-onal 3D insulators e Φ 0 e 2 : same ground state sector q f : changes Kramers degeneracy if odd : protected surface modes e odd q f :? e even q f e

36 Conclusions Models of frac-onal insulators Frac-onally charged fermions Exactly solvable: band structure Gapless edge/ surface states Frac-onal topological insulators? q f e odd q f e even : protected (2D & 3D) : Not protected (2D) :? (3D)