Universal quantum computa2on with topological phases (Part II) Abolhassan Vaezi Cornell University

Transcription

1 Universal quantum computa2on with topological phases (Part II) Abolhassan Vaezi Cornell University Cornell University, August 2015

2 Outline of part II Ex. 4: Laughlin fracaonal quantum Hall states Ex. 5: FracAonal Topological Superconductors & parafermion zero modes Ex. 6: Bilayer Fibonacci state at v=2/3 filling

3 Example 4 FracAonal Quantum Hall Effect (Abelian v=1/3 Laughlin state)

4 FracAonal Quantum Hall Effect = N e N = 1 3 N = Non- InteracAng picture: 0(e) = BA 2 e ~c GSD n.i. = Ne g LL e e = e 3N N e InteracAng picture: GSD n.i. =finite GSD depends on the topology of space, e.g. on sphere=1, and on torus=3

5 FracAonal Quantum Hall Effect Filling fracaon = N e N e e e e QuanAzed Hall conductance Protected gapless edge state Ground- state degeneracy (on torus) FracAonal charge Anyon staasacs =1/3 E

6 FracAonal Quantum Hall Effect e = e/3 e/3 e/3 c i = f 1,i f 2,i f 3,i q = N q N (q) N (q) = 0(q) = BA 2 q ~c N (e/3) =1/3N (e) N fi = N e fi =3 e =1 f i fully occupies its LLL à w.f. = Slater determinant fi =1: f i = Y i<j(z i z j )e e 3 B z i 2 4 z = x + iy e = f1 f2 f3 = Y i<j (z i z j ) 3 e e B z i 2 4 Laughlin w.f. (On infinite plane)

7 FracAonal Hall conductance IQH : xy(q) =n q2 xy(e) = xy (f 1 )+ xy (f 2 )+ xy (f 3 )= 1 3 h xy(f i )= (e/3)2 h e 2 h

8 B = r A FQH states on Torus A x = By, A y =0 E n,m = ~! c (n +1/2) n=0 m (x, y) =e i ( 2 Lx m )x e ( y 2 L x l 2 B 2 2 m Lx )2 /2 m = 2 m = 1 l B =1 m =0 m =1 m =2 y

9 1/3 Laughlin state Ideal Hamiltonian for 1/3 Laughlin state V 1 = X i X r>s V 1 U r,s c i+s c i+r c i+r+sc i Laughlin 1/3 =0 U r,s =(r 2 s 2 )e 2 2 (r 2 +s 2 )/L 2 x Haldane, 1983 Thin torus limit: L x l B : V 1 ' X i (U 1,0 n i n i+1 + U 2,0 n i n i+2 ) Bergholtz and Karlhede, JSM (2006);

10 1/3 FQH: Thin torus limit 2 L x l 2 B 2 l B =1 Effec2ve Hamiltonian: L x L y : V 1 ' X (U 1,0 n i n i+1 + U 2,0 n i n i+2 ) i Degenerate ground- states (CDW pa]erns) gi 1 = i [100] gi 2 = i [010] gi 3 = i [001] y Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009), Bernevig & Haldan, PRLe (2008)

11 ExcitaAons: domain- walls [ ]! [ ] Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009), Bernevig & Haldan, PRLe (2008)

12 ExcitaAons: domain- walls [ ]! [ ] { Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009), Bernevig & Haldan, PRLe (2008)

13 ExcitaAons: domain- walls [ ]! [ ] { Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009), Bernevig & Haldan, PRLe (2008)

14 ExcitaAons: domain- walls [ ]! [ ] { Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009), Bernevig & Haldan, PRLe (2008)

15 ExcitaAons: domain- walls [ ]! [ ] { q = e/3 Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009), Bernevig & Haldan, PRLe (2008)

16 ExcitaAons: domain- walls [ ]! [ ] { q = e/3 Energy cost = U 2 à bulk gap = U 2 Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009), Bernevig & Haldan, PRLe (2008)

17 ExcitaAons: domain- walls [ ] =V 1 (j)v 1(i) gi 1! i! j [ ] =V 2 (i)... gi 1! q =2e/3 q = ke/3 : gi a! gi a+k%3 Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009), Bernevig & Haldan, PRLe (2008)

18 FracAonalizaAon e e/3 e/3 e/3 (r) e Charge distribuaon t =0 r (r) t 1 E g e/3 e/3 e/3 d l B r

19 FracAonal charge and staasacs 3 flux à 1 electron (q=e) 1 flux à anyon with q=e/3 2 flux à anyon with q=2e/3 e/3 e e e e 2e/3 Aharanov- Bohom Effect: taking charge q around flux (full braid):! e i AB AB = q

20 Exchanging 2 anyons of & flux (half braid) : = 2 n e Topological spin q = ne m charge n =2 q 2 = n2 m q q Similarly: Rota2ng an anyons of q = ne charge & flux one around m = 2 n itself amounts to e n = n 2 m s n = n 2 = n2 2m Topological spin

21 Example 5 FracAonal Topological Superconductors with fracaonalized Majorana (parafermion) zero modes

22 Frac2onal Topological Superconductor (FTSC) FTSC= FQH + SC hc 2e v=1/m FQH Superconductor a. Associated to every TWO voraces à Zero energy level (E=0) b. 2 electrons = 2m anyons of charge e/m c. Pauli exclusion: E=0 level can be occupied by 0,1,2,, (2m- 1) anyons d. E=0 level defines a 2m- dimensional Hilbert space e. GSD increases by 2m with pinseraon of two voraces f. Each vortex contributes 2m to GSD. g. Quantum dimension of vortex: d v = p 2m Vaezi, 2012 hc 2e

23 Frac2onal Topological Superconductor (FTSC) hc 2e hc 2e FTSC= FQH + SC v=1/m FQH Superconductor v =1/m Vortex carries Z 2m Parafermion (a.k.a fracaonalized Majorana) zero mode with d v = p 2m quantum dimension Vaezi, 2012 m=1 case can be solved exactly and it is known that IQH+SC à TSC (p+ip) Qi, Hughes, Zhang, 2010

24 Domain walls: Parafermion zero modes SC v=1/m FQH FM SC v=1/m FQH = parafermion zero mode (a.k.a frac2onalized Majorana zero mode) Linder et al, (2012); Clark et al (2012), Cheng (2012) Barkeshli, Qi (2012); Barkeshli et al (2012)

25 Parafermions (fracaonalized Majorana fermions) 2m 1 = 2 2m =1 i = 2m 1 i 1 2 = e i /m 2 1 e i 2m 1 2 qi = e i q m qi qi = 1q 0i Vaezi, PRB, 2013

26 Braid staasacs ni : =1+V 1 + V V 2m 1 Contains n anyons (V n ) s n = n2 2m Exchanging voraces CCW = rotaang zero mode by 2 RotaAng V 1 around itself CCW one round à RotaAng V 2 around itself CCW one round à RotaAng V n around itself CCW one round à e i /m e i4 /m e in2 /m phase change phase change phase change B 12 ni = e in2 /m ni Vaezi, PRB, 2013

27 Braid StaAsAcs Type I! Exchange! time! n 12 n 34 B 1,2 n 12,n 34 i = e i n B 3,4 n 12,n 34 i = e i n 2 12 m 2 34 m n12,n 34 i n12,n 34 i

28 Braid StaAsAcs Type II! Exchange! time! n 12 n 34 B 2,3 n 12,n 34 i =? n (1) Basis transformaaon to diagonalize (equivalently ) (2) transform back to the original basis

29 Braid StaAsAcs Type II! Exchange! time! B 2,3 = S US n 12 n 34 Uq p = p,q e i q2 m S p q = eiqp/m p 2m B 2,3 n 12,n 34 i = Maximally entangled state

30 Example 6 v=2/3 Bilayer Fibonacci FQH state Emergence of Fibonacci anyons & Universal quantum computaaon

31 Acknowledgement Maissam Barkeshli Microsoe StaAon Q Eun- Ah Kim Cornell Zhao Liu Princeton Kyungmin Lee Cornell

32 Outline for Example 6 FQH states at 2/3: Fibonacci phase Thin torus limit Parton construcaon Numerical results Experimental signatures

33 Experimental setup of perturbed 1/3+1/3 FQH a) Layer index 1/3 FQH B? 1/3 FQH Suen et al, PRL, 1994 Monaharan et al, PRL, 1996 B k Charge distribuaon of 2DEG in a wide quantum well b) Spin index c) Valley index: graphene 1 3 (")+1 3 (#) FQH Du et al, Nature, 2009 BoloAn et al, Nature, 2009 Eisenstein et al, PRB, 1990

34 Previous studies Fradkin, Nayak, Schoutens, Nuc. Phys. B, 1999 Wen, PRL, 2000 Ardonne & Schoutens, PRL, 2000 Cappelli et al, Nuc. Phys. B, 2001 Papic et al, PRB, 2010 Peterson & Das Sarma, PRB 2010 Wen, Rezayi & Read, arxiv, 2010 Barkeshli & Wen, PRB, 2010

35 Candidate states at 2/3 filling Abelian states: McDonald & Haldane, PRB, 1996 Interlayer Pfaffian: Graedts, Zaletel, Papic & Mong, arxiv: Z4 RR state: Peterson, Wu, Cheng, Barkeshli, Wang & Das Sarma, arxiv: Bilayer Fibonacci: Liu, Vaezi, Lee, Kim, arxiv: ; to appear in PRB(R)

36 A. Thin torus limit

37 B = r A FQH states on Torus A x = By, A y =0 E n,m = ~! c (n +1/2) n=0 m (x, y) =e i ( 2 Lx m )x e ( y 2 L x l 2 B 2 2 m Lx )2 /2 m = 2 m = 1 l B =1 m =0 m =1 m =2 y

38 1/3 Laughlin state Ideal Hamiltonian for 1/3 Laughlin state V 1 = X i X r>s V 1 U r,s c i+s c i+r c i+r+sc i Laughlin 1/3 =0 U r,s =(r 2 s 2 )e 2 2 (r 2 +s 2 )/L 2 x Haldane, 1983 Thin torus limit: L x l B : V 1 ' X i (U 1,0 n i n i+1 + U 2,0 n i n i+2 ) Bergholtz and Karlhede, JSM (2006);

39 1/3 FQH: Thin torus limit 2 L x l 2 B 2 l B =1 Effec2ve Hamiltonian: L x L y : V 1 ' X (U 1,0 n i n i+1 + U 2,0 n i n i+2 ) i Degenerate ground- states (CDW pa]erns) gi 1 = i [100] gi 2 = i [010] gi 3 = i [001] y Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009)

40 Bilayer (330) state: Thin torus limit gi 1 =[2, 0, 0, 2, 0, 0, 2, 0, 0, ] gi 2 =[0, 2, 0, 0, 2, 0, 0, 2, 0, ] gi 3 =[0, 0, 2, 0, 0, 2, 0, 0, 2, ] gi 4 =[", #, 0, ", #, 0, ", #, 0, ] gi 5 =[#, 0, ", #, 0, ", #, 0, ", ] gi 6 =[0, ", #, 0, ", #, 0, ", #, ] gi 7 =[#, ", 0, #, ", 0, #, ", 0, ] gi 8 =[", 0, #, ", 0, #, ", 0, #, ] gi 9 =[0, #, ", 0, #, ", 0, #, ", ] GSD = 9 =3 x 3 3 : Transla2on symmetry in Layer 1 (Abelian) 3 : Transla2on symmetry in Layer 2 (Abelian)

41 Bilayer Fibonacci state: : Thin torus limit gi 1 =[2, 0, 0, 2, 0, 0, 2, 0, 0, ] gi 2 =[0, 2, 0, 0, 2, 0, 0, 2, 0, ] gi 3 =[0, 0, 2, 0, 0, 2, 0, 0, 2, ] gi 4 =[1, 1, 0, 1, 1, 0, 1, 1, 0, ] gi 5 =[1, 0, 1, 1, 0, 1, 1, 0, 1, ] gi 6 =[0, 1, 1, 0, 1, 1, 0, 1, 1, ] 1, 1 ", #i #, "i p 2 GSD = 6 =3 x 2 3 : Transla2on symmetry (Abelian) 2 : Non- Abelian sector: 1, X anyons X X =1+nX Vaezi, Barkeshli, PRL, 2014 Vaezi, PRX, 2014 Mong et al, PRX, 2014

42 Quasi- hole excitaaons [ ] [ ] [ ] q = e 3 (mod e) : [200] [110] [020] [101] [011] [200] [011] [002] [110] [110] [002] [110] GSD(n)=Fib(n) Vaezi, Barkeshli, PRL, 2014

43 Quasi- hole excitaaons [ ] [ ] [ ] q = e 3 (mod e) : [200] [110] [020] [101] [011] [200] [011] [002] [110] [110] [002] [110] GSD(n)=Fib(n) Vaezi, Barkeshli, PRL, 2014

44 GSD: Adjacency matrix A e/3 = 0 [200] [020] [002] [110] [101] [011] C A [200] [020] [002] [110] [101] [011] [200] [110] 1 st row: 4 th row: [110] [020] [101] GSD(n qh ) ' Tr(A n qh ) n qh d 1 e/3 = 1 = 1+p 5 2

45 Neutral excitaaons (Fibonacci anyons) [ ] [ ] [ ] [ ] [ ] [ ] A e/3, e/3 = 0 [200] [011] [020] [101][002] [110] C A [200] [011] [020] [101] [002] [110] GSD(n qh ) ' Tr(A n qh ) n qh 1 d e/3, e/3 = 1 = 1+p 5 2 Vaezi, Barkeshli, PRL, 2014

46 Bilayer Fibonacci state =2/3 c tot = 14/5 Operator Charge Topological spin Quantum dimension V 1 2e/3 1/ V 2 e/3 1/ / V 1 2e/3 11/ V 2 e/3 11/ V a V b = V a+b =1+

47 B. Parton construcaon of the bilayer Fibonacci state

48 e =1/3 Parton construc2on: 1/3 FQH c i = f 1,i f 2,i f 3,i e/3 e/3 e/3 c = volume of the complex space of f a à SU(3) gauge invariance X.- G. Wen, PRB (1999).

49 e =1/3 Parton construc2on: 1/3 FQH c i = f 1,i f 2,i f 3,i e/3 e/3 e/3 c = volume of the complex space of f a à SU(3) gauge invariance e/3 =1 e/3 = Y i<j(z i z j )e e 3 B z i 2 4 X.- G. Wen, PRB (1999).

50 Parton construc2on: 1/3 FQH e =1/3 c i = f 1,i f 2,i f 3,i e/3 e/3 e/3 c = volume of the complex space of f a à SU(3) gauge invariance e/3 =1 e/3 = Y i<j(z i z j )e e 3 B z i 2 4 e = e/3 3 = Y i<j (z i z j ) 3 e e B z i 2 4 X.- G. Wen, PRB (1999).

51 Parton construc2on: 1/3 FQH Bulk theory: Integra2ng out gapped fermions: SU(3) 1 CS ac2on 1 L CS = 4 µ Tr A A + 23 A µa A X.- G. Wen, PRB (1999). Edge theory: 3 chiral fermions à U(3) 1 symmetry. SU(3) 1 subgroup is redundant (gauge symmetry) Edge CFT= U(3) 1 /SU(3) 1 =U(1) 3

52 Parton construc2on: 1/3+1/3 FQH t=0 à Gauge symmetry: SU(3) " SU(3) # Bulk theory: Integra2ng out fermions: SU(3) 1 x SU(3) 1 CS ac2on L CS = X =",# 1 4 µ Tr A A + 23 A µa A Vaezi & Barkeshli, PRL (2014)

53 Parton construc2on: 1/3+1/3 FQH t=0 à Gauge symmetry: SU(3) " SU(3) # Bulk theory: Integra2ng out fermions: SU(3) 1 x SU(3) 1 CS ac2on L CS = X =",# ab = f operator carries charge under a," f b,# A " A # < ab >6= µ Tr à Higgs mechanism: A A + 23 A µa A A " = A # = A Vaezi & Barkeshli, PRL (2014)

54 Parton construc2on: 1/3+1/3 FQH t=0 à Gauge symmetry: SU(3) " SU(3) # Bulk theory: Integra2ng out fermions: SU(3) 1 x SU(3) 1 CS ac2on L CS = X =",# 1 4 µ Tr A A + 23 A µa A ab = f operator carries charge under a," f b,# A " A # < ab >6= 0 à Higgs mechanism: A " = A # = A L CS! µ Tr A A A µa A 4 SU(3) 2 CS Vaezi & Barkeshli, PRL (2014)

55 Parton construc2on: 1/3+1/3 FQH Edge theory: 6 chiral fermions à U(6) 1 SU(3) 2 subgroup is redundant (gauge symmetry) Edge CFT= U(6) 1 /SU(3) 2 = SU(2) 3 x U(1) 6 Vaezi & Barkeshli, PRL (2014)

56 C. Coupled wires construcaon

57 Coupled wires construcaon FQH = FQH H 0 = X I v F 1 4 Z dx x I 2 x ' I 2i g BS Z dx cos r! 2 I Teo & Kane, PRB, 2014; Vaezi, PRX, 2014;

58 Coupled wires construcaon 1,L 2,L 1,R 2,R H e = 1 4 Z dx h (@ x ' s ) 2 +(@ x s ) 2i Z u dx h t? cos( p 3' s )+t k cos( p i 3 s ) Vaezi & Barkeshli, PRL (2014); Vaezi, PRX, 2014; Mong et al, PRX, 2014

59 III. Inter- wire coupling Vaezi, (2014); Mong et al., (2014); Vaezi and Barkeshli, (2014)

60 III. Parafermion edge state c =2+4/5 U(6) 1 /SU(3) 2 = SU(2) 3 U(1) 6 Vaezi, (2014); Mong et al., (2014); Vaezi and Barkeshli, (2014)

61 D. Numerical results

62 Two- body interacaon P nll U ee P nll = X l c l V l U ee = e2 4 0 r Papic & Regnault, 2009; Goerbig, Moessner & Doucot, 2006; Haldane, 1983; Haldane & Rezayi, 1988

63 Model Hamiltonian 1 (a) (b) (6) (3) (3) (9) (9) (c) (6) (9) H = V intra 1 + U 0 V inter 0 + U 1 V inter 1 + H t Liu, Vaezi, Lee, and Kim; arxiv: (to appear in PRB (R)

64 Model Hamiltonian 2 U0 = 0.4 U1 =0.6 H = Hcoulomb intra + Hcoulomb inter + U 0 V0 inter + U 1 V1 inter Liu, Vaezi, Lee, and Kim; arxiv: (to appear in PRB (R)

65 Entanglement measurements Finite size scaling of energy gap Entanglement Entropy Level Coun2ng (Even sector) Level Coun2ng (Odd sector) Liu, Vaezi, Lee, and Kim; arxiv: (to appear in PRB (R)

66 ParAcle Entanglement Spectrum N e =8 N ea =4 N qh =12 N e =10 N ea =4 N qh =18 Liu, Vaezi, Lee, and Kim; arxiv: (to appear in PRB (R)

67 Wave- funcaon overlap 6- fold degenerate state: Negligible overlap with Z4 Read- Rezayi state Small overlap with interlayer Paffian à decreases with increasing N e Small overlap with Abelian states

68 Experimental relevance P nll U ee P nll = X l c l V l U ee = e2 4 0 r Papic & Regnault, 2009; Goerbig, Moessner & Doucot, 2006; Haldane, 1983; Haldane & Rezayi, 1988

69

70 Experimental signatures

71 Experimental probes of Bilayer Fibonacci state (1) Detec2ng topological phase transi2on Phase transiaon happens in the neutral sector Neutral Excitons carry electric dipole dipole- dipole correlaaon diverges at criacal point Surface acous2c phonon measurment?

72 Experimental probes of Bilayer Fibonacci state (2) Thermal Hall conduc2vity: apple xy /T = c 2 k 2 B 3h (3) Edge tunneling: 2/3 FQH 2/3 FQH (a) I- V curve I / V 2g qh 1 (b) Zero bias conductance: xy / T 2g qh 2 Fib : g qh =7/15 c = 14/5 (4) Interferometry experiments

73 a. A novel NA state at 2/3 filling with Fibonacci anyons b. Fibonacci anyons à Universal quantum computaaon via braiding c. 2- body interacaon with large interlayer V 1 component à Bilayer Fibonacci state a. Large interlayer V 1 component: Second Landau level in graphene- like systems?

74 Thanks for your apenaon