abolhassan)vaezi) cornell)university) · e e e e ⌫ = n e n = 1 3 n = 0(e) ba 2⇡ e ~c gsd n.i. =...

Universal quantum computa2on with topological phases (Part II)

Abolhassan Vaezi Cornell University

Cornell University, August 2015

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

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

Example 4

�e

�e�e

�e⌫ =

Ne

N�=

1

3

N� =�

�0(e)=

BA2⇡e ~c

GSDn.I. =

✓Ne

gLL

◆=

✓3N�

N�

◆Non-‐InteracAng picture:

InteracAng picture: GSDn.I. = finite

GSD depends on the topology of space, e.g. on sphere=1, and on torus=3

FracAonal Quantum Hall Effect

FracAonal Quantum Hall Effect

•  QuanAzed Hall conductance •  Protected gapless edge state •  Ground-‐state degeneracy (on torus) •  FracAonal charge •  Anyon staAsAcs

⌫ =Ne

N�

⌫ = 1/3

Filling fracAon

E

�e

�e�e

�e

FracAonal Quantum Hall Effect e/3

e/3 e/3 e ==

N�(q) =�

�0(q)=

BA2⇡q ~c

N�(e/3) = 1/3N�(e) Nfi = Ne

⌫q =Nq

N�(q)

⌫fi = 3⌫e = 1

fi fully occupies its LLL à w.f. = Slater determinant

ci = f1,if2,if3,i

fi =Y

i<j

(zi � zj)e� e

3B|zi|

2

4

e = f1 f2 f3 =Y

i<j

(zi � zj)3e�e

B|zi|2

4

⌫fi = 1 :z = x+ iy

Laughlin w.f. (On infinite plane)

FracAonal Hall conductance

IQH : �xy

(q) = nq2

h

�xy

(e) = �xy

(f1) + �xy

(f2) + �xy

(f3) =1

3

e2

h

�xy

(fi

) =(e/3)2

h

Ax

= �By, Ay

= 0

FQH states on Torus

lB = 1

2⇡

Lx

l2B

y

| |2

B = r⇥A

n=0m

(x, y) = e

�i( 2⇡L

x

m)xe

�(y� 2⇡m

L

x

)2/2

En,m = ~!c (n+ 1/2)

m = 0 m = 1 m = 2m = �1m = �2

Ur,s = (r2 � s2)e�2⇡2(r2+s2)/L2x

1/3 Laughlin state

Ideal Hamiltonian for 1/3 Laughlin state

V1 =X

i

X

r>s

Ur,sc†i+sc

†i+rci+r+sci

V1 Laughlin1/3 = 0

Thin torus limit: Lx

⌧ lB

: V1 'X

i

(U1,0ni

ni+1 + U2,0ni

ni+2)

Haldane, 1983

Bergholtz and Karlhede, JSM (2006);

|gi1 = |100100100100100100100...i ⌘ [100]

|gi2 = |010010010010010010010...i ⌘ [010]

|gi3 = |001001001001001001001...i ⌘ [001]

1/3 FQH: Thin torus limit

Effec2ve Hamiltonian:

Degenerate ground-‐states (CDW pa]erns)

lB = 1

2⇡

Lx

l2B

y

| |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)

Lx

⌧ Ly

: V1 'X

i

(U1,0nini+1 + U2,0nini+2)

!

[100100100100010010010010010100100100]

[100100100100|100100100100100|100100100]

ExcitaAons: domain-‐walls


![100100100100|100100100100100|100100100]

[100100100100010010010010010100100100]{ExcitaAons: domain-‐walls


![100100100100|100100100100100|100100100]

[100100100100010010010010010100100100]{



![100100100100|100100100100100|100100100]

[100100100100010010010010010100100100]{q⇤ = e/3



![100100100100|100100100100100|100100100]

[100100100100010010010010010100100100]{

q⇤ = �e/3


Energy cost = U2à bulk gap = U2


! !

i j

[100100100010010...010100100...] = V †1 (j)V1(i) |gi1

[100...100100001001001...] = V2(i)... |gi1!

q⇤ = ke/3 : |gia ! |gia+k%3


q⇤ = 2e/3


FracAonalizaAon

r

t = 0

r

t � 1

Eg

d � lB

e/3 e e/3

e/3

e/3 e/3 e/3

e

⇢(r)

⇢(r)

Charge distribuAon

�e

�e�e

�e

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

�2e/3

Aharanov-‐Bohom Effect: taking charge around flux (full braid):

! ei✓AB ✓AB = q��q

Topological spin

Similarly:

Rota2ng an anyons of charge & flux one around

itself amounts to

q =ne

m � =2⇡n

e

sn =✓n2⇡

=n2

2m

✓n = n2 ⇡

m

Topological spin

Exchanging 2 anyons of charge & flux (half braid) :

q =ne

m

� =2⇡n

e✓n = 2

q�

2= n2 ⇡

m q�

q�

FracAonal Topological Superconductors

with fracAonalized Majorana (parafermion) zero modes

Example 5

Frac2onal Topological Superconductor (FTSC)

v=1/m FQH

Superconductor

hc

2ehc

2eFTSC= FQH + SC

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 inserAon of two vorAces f.  Each vortex contributes to GSD. g.  Quantum dimension of vortex: dv =

p2m

p2m

Vaezi, 2012

Frac2onal Topological Superconductor (FTSC)

v=1/m FQH

Superconductor

hc

2ehc

2e

FTSC= FQH + SC

Vortex carries Parafermion (a.k.a fracAonalized Majorana) zero mode with quantum dimension dv =

p2m

v = 1/m

Z2m

m=1 case can be solved exactly and it is known that IQH+SC à TSC (p+ip)

Vaezi, 2012

Qi, Hughes, Zhang, 2010

v=1/m FQH

⇤⇤ SC SC FM

= parafermion zero mode (a.k.a frac2onalized Majorana zero mode)

⇤

Domain walls: Parafermion zero modes

Linder et al, (2012); Clark et al (2012), Cheng (2012) Barkeshli, Qi (2012); Barkeshli et al (2012)

v=1/m FQH

�2m1 = �2m

2 = 1 �†i = �2m�1

i

�1�2 = ei⇡/m�2�1

ei⇡

2m �†1�2 |qi = ei

⇡qm |qi |qi = �†

1

q|0i

Vaezi, PRB, 2013

Parafermions (fracAonalized Majorana fermions)

Braid staAsAcs

� ⇥ � = 1 + V1 + V2 + · · ·+ V2m�1

RotaAng V1 around itself CCW one round à phase change RotaAng V2 around itself CCW one round à phase change RotaAng Vn around itself CCW one round à phase change

ei⇡/m

ei4⇡/m

ein2⇡/m

Contains n anyons (Vn) |ni :

Exchanging vorAces CCW = rotaAng zero mode by 2⇡

B12 |ni = ein2⇡/m |ni

sn =n2

2m

Vaezi, PRB, 2013

Type I!Exchange!

time!

n12 n34

B1,2 |n12, n34i = ei⇡n2

12m |n12, n34i

B3,4 |n12, n34i = ei⇡n2

34m |n12, n34i

Braid StaAsAcs

Type II!Exchange!

time!

n12 n34

(1) Basis transformaAon to diagonalize (equivalently )

(2) transform back to the original basis

B2,3 |n12, n34i =?

Braid StaAsAcs

n23 �†3�2

Maximally entangled state

Spq =

eiqp/mp2m

Upq = �p,qe

i⇡q2

mB2,3 = S†US

B2,3 |n12, n34i =

time!

n12 n34

Type II!Exchange!

Braid StaAsAcs

v=2/3 Bilayer Fibonacci FQH state

Emergence of Fibonacci anyons

& Universal quantum computaAon

Example 6

Acknowledgement

Maissam Barkeshli Microsoe StaAon Q

Zhao Liu Princeton

Eun-‐Ah Kim Cornell

Kyungmin Lee Cornell

Outline for Example 6

•  FQH states at 2/3: Fibonacci phase

•  Thin torus limit

•  Parton construcAon

•  Numerical results

•  Experimental signatures

Experimental setup of perturbed 1/3+1/3 FQH

1/3 FQH

1/3 FQH B?

Bk

Charge distribuAon of 2DEG in a wide quantum well

Suen et al, PRL, 1994 Monaharan et al, PRL, 1996

FQH

1

3(") + 1

3(#)

Eisenstein et al, PRB, 1990

a) Layer index

b) Spin index c) Valley index: graphene

Du et al, Nature, 2009 BoloAn et al, Nature, 2009

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

Candidate states at 2/3 filling

Abelian states: McDonald & Haldane, PRB, 1996 Interlayer Pfaffian: Graedts, Zaletel, Papic & Mong, arXiv:1502.01340 Z4 RR state: Peterson, Wu, Cheng, Barkeshli, Wang & Das Sarma, arXiv:1502.02671 Bilayer Fibonacci: Liu, Vaezi, Lee, Kim, arXiv: 1502.05391; to appear in PRB(R)

A. Thin torus limit

Ax

= �By, Ay

= 0

FQH states on Torus

lB = 1

2⇡

Lx

l2B

y

| |2

B = r⇥A

n=0m

(x, y) = e

�i( 2⇡L

x

m)xe

�(y� 2⇡m

L

x

)2/2

En,m = ~!c (n+ 1/2)

m = 0 m = 1 m = 2m = �1m = �2

Ur,s = (r2 � s2)e�2⇡2(r2+s2)/L2x

1/3 Laughlin state

Ideal Hamiltonian for 1/3 Laughlin state

V1 =X

i

X

r>s

Ur,sc†i+sc

†i+rci+r+sci

V1 Laughlin1/3 = 0

Thin torus limit: Lx

⌧ lB

: V1 'X

i

(U1,0ni

ni+1 + U2,0ni

ni+2)

Haldane, 1983

Bergholtz and Karlhede, JSM (2006);

|gi1 = |100100100100100100100...i ⌘ [100]

|gi2 = |010010010010010010010...i ⌘ [010]

|gi3 = |001001001001001001001...i ⌘ [001]

1/3 FQH: Thin torus limit

Effec2ve Hamiltonian:

Degenerate ground-‐states (CDW pa]erns)

lB = 1

2⇡

Lx

l2B

y

| |2

Seidel et al, PRL (2005); Bergholtz and Karlhede, JSM (2006); Bergholtz and Karlhede, PRB (2008); Seidel and Yang, PRL (2008); Ardonne, PRL (2009)

Lx

⌧ Ly

: V1 'X

i

(U1,0nini+1 + U2,0nini+2)

Bilayer (330) state: Thin torus limit

|gi4 = [", #, 0, ", #, 0, ", #, 0, · · · ]|gi5 = [#, 0, ", #, 0, ", #, 0, ", · · · ]|gi6 = [0, ", #, 0, ", #, 0, ", #, · · · ]|gi7 = [#, ", 0, #, ", 0, #, ", 0, · · · ]|gi8 = [", 0, #, ", 0, #, ", 0, #, · · · ]|gi9 = [0, #, ", 0, #, ", 0, #, ", · · · ]

|gi1 = [2, 0, 0, 2, 0, 0, 2, 0, 0, · · · ]|gi2 = [0, 2, 0, 0, 2, 0, 0, 2, 0, · · · ]|gi3 = [0, 0, 2, 0, 0, 2, 0, 0, 2, · · · ]

GSD = 9 =3 x 3 “3” : Transla2on symmetry in Layer 1 (Abelian) “3” : Transla2on symmetry in Layer 2 (Abelian)

|gi1 = [2, 0, 0, 2, 0, 0, 2, 0, 0, · · · ]|gi2 = [0, 2, 0, 0, 2, 0, 0, 2, 0, · · · ]|gi3 = [0, 0, 2, 0, 0, 2, 0, 0, 2, · · · ]|gi4 = [1, 1, 0, 1, 1, 0, 1, 1, 0, · · · ]|gi5 = [1, 0, 1, 1, 0, 1, 1, 0, 1, · · · ]|gi6 = [0, 1, 1, 0, 1, 1, 0, 1, 1, · · · ]

GSD = 6 =3 x 2 “3” : Transla2on symmetry (Abelian) “2” : Non-‐Abelian sector: 1, X anyons

X ⇥X = 1 + nX

Bilayer Fibonacci state: : Thin torus limit

Vaezi, Barkeshli, PRL, 2014 Vaezi, PRX, 2014

Mong et al, PRX, 2014

1, 1 ⌘ |", #i � |#, "ip2

Quasi-‐hole excitaAons

[200...200110...110020...020011...011010110...]

[200...200110...110101...101011...011010110...]

[200...200110...110020...020011...011002002...]

[020]

[101]

[011]

[200]

[011]

[110][200]

[002]

[110]

[110]

[002]

[110]

q⇤ =

e

3

(mod e) :

1 1 2 3 5

GSD(n)=Fib(n) Vaezi, Barkeshli, PRL, 2014

[110][200]

[020]

[101][110]1st row: 4th row:

[020]

[101]

[011]

[110]

[200]

[002]Ae/3 =

0

BBBBBB@

0 0 0 1 0 00 0 0 0 0 10 0 0 0 1 00 1 0 0 1 01 0 0 0 0 10 0 1 1 0 0

1

CCCCCCA

[200] [020] [101] [011][110][002]

GSD(nqh) ' Tr (Anqh) ⇠ �nqh

1 de/3 = �1 =1 +

p5

2

GSD: Adjacency matrix

[020]

[101]

[011]

[110]

[200]

[002]

[200] [020] [101][011] [110][002]

Ae/3,�e/3 =

0

BBBBBB@

0 1 0 0 0 01 1 0 0 0 00 0 0 1 0 00 0 1 1 0 00 0 0 0 0 10 0 0 0 1 1

1

CCCCCCA

GSD(nqh) ' Tr (Anqh) ⇠ �nqh

1de/3,�e/3 = �1 =

1 +p5

2

[...011011011...][...200200200...]

[...011011011...] [...200200200...]

[...011011011...] [...011011011...]

Neutral excitaAons (Fibonacci anyons)

Vaezi, Barkeshli, PRL, 2014

Bilayer Fibonacci state

Operator Charge Topological spin Quantum dimension

1. 1 0 0 12. V1 2e/3 1/3 13. V2 e/3 1/3 14. ⌧ 0 2/5 1.618 · · ·5. V1⌧ 2e/3 11/15 1.618 · · ·6. V2⌧ e/3 11/15 1.618 · · ·

⌫ = 2/3 ctot

= 14/5

Va ⇥ Vb = Va+b ⌧ ⇥ ⌧ = 1 + ⌧

B. Parton construcAon of the bilayer Fibonacci state

Parton construc2on: 1/3 FQH

c = volume of the complex space of fa à SU(3) gauge invariance

X.-‐G. Wen, PRB (1999).

e/3

e/3 e/3 ci = f1,if2,if3,i⌫e = 1/3


X.-‐G. Wen, PRB (1999).

e/3

e/3 e/3

⌫e/3 = 1 e/3 =Y

i<j

(zi � zj)e� e

3B|zi|

2

4

ci = f1,if2,if3,i⌫e = 1/3



X.-‐G. Wen, PRB (1999).

e/3

e/3 e/3

⌫e/3 = 1 e/3 =Y

i<j

(zi � zj)e� e

3B|zi|

2

4

e =� e/3

�3=

Y

i<j

(zi � zj)3e�e

B|zi|2

4

ci = f1,if2,if3,i⌫e = 1/3


Bulk theory: Integra2ng out gapped fermions: SU(3)1 CS ac2on

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

X.-‐G. Wen, PRB (1999).

LCS =1

4⇡✏µ⌫⇢Tr

✓Aµ@⌫A⇢ +

2

3AµA⌫A⇢

◆1


t=0 à Gauge symmetry:

Bulk theory: Integra2ng out fermions: SU(3)1 x SU(3)1 CS ac2on

SU(3)" ⇥ SU(3)#

LCS =X

�=",#

1

4⇡✏µ⌫⇢Tr

✓A�

µ@⌫A�⇢ +

2

3A�

µA�⌫A

�⇢

◆

Vaezi & Barkeshli, PRL (2014)

Parton construc2on: 1/3+1/3 FQH


operator carries charge under à Higgs mechanism:

�ab = f†a,"fb,# A" �A#

A" = A# = A< �ab > 6= 0



SU(3)" ⇥ SU(3)#

LCS =X

�=",#

1

4⇡✏µ⌫⇢Tr

✓A�

µ@⌫A�⇢ +

2

3A�

µA�⌫A

�⇢

◆




SU(3)" ⇥ SU(3)#

LCS =X

�=",#

1

4⇡✏µ⌫⇢Tr

✓A�

µ@⌫A�⇢ +

2

3A�

µA�⌫A

�⇢

◆


operator carries charge under à Higgs mechanism:

�ab = f†a,"fb,# A" �A#

A" = A# = A< �ab > 6= 0

SU(3)2 CS LCS !4⇡

✏µ⌫⇢Tr

✓Aµ@⌫A⇢ +

2

3AµA⌫A⇢

◆2


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



C. Coupled wires construcAon

Coupled wires construcAon

Teo & Kane, PRB, 2014; Vaezi, PRX, 2014;

== FQH FQH

H0 =

X

I

vF1

4⇡

Zdx

h�@x

✓I�2

+

�@x

'I

�2i� gBS

Zdx cos

r2

⌫✓I

!

Coupled wires construcAon

Vaezi & Barkeshli, PRL (2014); Vaezi, PRX, 2014; Mong et al, PRX, 2014

He↵ =1

4⇡

Zdx

h(@

x

'

s

)2 + (@x

✓

s

)2i� u

Zdx

ht? cos(

p3's) + tk cos(

p3✓s)

i

2,L

2,R

1,R

1,L

Vaezi, (2014); Mong et al., (2014); Vaezi and Barkeshli, (2014)

III. Inter-‐wire coupling

c = 2 + 4/5

U(6)1/SU(3)2 = SU(2)3 ⇥ U(1)6

III. Parafermion edge state

Vaezi, (2014); Mong et al., (2014); Vaezi and Barkeshli, (2014)

D. Numerical results

Two-‐body interacAon

Papic & Regnault, 2009; Goerbig, Moessner & Doucot, 2006; Haldane, 1983; Haldane & Rezayi, 1988

Uee =e2

4⇡✏0r

PnLLUeePnLL =X

l

clVl

(a) (b)

(c)

(6)

(6)

(9)

(3)

(3)

(9)

(9)

H = V intra1 + U0V inter

0 + U1V inter1 +Ht

Model Hamiltonian 1

Liu, Vaezi, Lee, and Kim; arXiv:1502.05391 (to appear in PRB (R)

Model Hamiltonian 2

H = Hintra

coulomb

+Hinter

coulomb

+ U0

V inter

0

+ U1

V inter

1

U0=

�0.4

U1=

0.6


Entanglement measurements Level Cou

n2ng

(Even sector)

Finite size sc

aling of

energy gap

Entanglemen

t Entropy

Level Cou

n2ng (O

dd se

ctor)


ParAcle Entanglement Spectrum

Ne=8 NeA=4 Nqh=12

Ne=10 Ne

A=4 Nqh=18


Wave-‐funcAon overlap

6-‐fold degenerate state:

Negligible overlap with Z4 Read-‐Rezayi state Small overlap with interlayer Paffian à decreases with increasing Ne Small overlap with Abelian states

Experimental relevance

Papic & Regnault, 2009; Goerbig, Moessner & Doucot, 2006; Haldane, 1983; Haldane & Rezayi, 1988

Uee =e2

4⇡✏0r

PnLLUeePnLL =X

l

clVl

Experimental signatures

(1)  Detec2ng topological phase transi2on

Experimental probes of Bilayer Fibonacci state

Phase transiAon happens in the neutral sector Neutral Excitons carry electric dipole dipole-‐dipole correlaAon diverges at criAcal point Surface acous2c phonon measurment?

2/3 FQH 2/3 FQH

I / V 2gqh�1

xy

/T = c⇡2k2B3h

(2) Thermal Hall conduc2vity:

(3) Edge tunneling: (a) I-‐V curve (b) Zero bias conductance: �

xy

/ T 2gqh�2

(4) Interferometry experiments

Fib : gqh = 7/15 c = 14/5

Experimental probes of Bilayer Fibonacci state

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 V1 component à Bilayer Fibonacci state

a.  Large interlayer V1 component: Second Landau level in graphene-‐like systems?

Thanks for your apenAon

abolhassan)vaezi) cornell)university) · e e e e ⌫ = n e n = 1 3 n = 0(e) ba 2⇡ e ~c gsd n.i. =...

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