Abolhassan)Vaezi) Cornell)University) · e e e e ⌫ = N e N = 1 3 N = 0(e) BA 2⇡ e ~c GSD n.I. =...
Transcript of 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
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)
![100100100100|100100100100100|100100100]
[100100100100010010010010010100100100]{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)
![100100100100|100100100100100|100100100]
[100100100100010010010010010100100100]{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)
![100100100100|100100100100100|100100100]
[100100100100010010010010010100100100]{
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)
![100100100100|100100100100100|100100100]
[100100100100010010010010010100100100]{q⇤ = e/3
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)
![100100100100|100100100100100|100100100]
[100100100100010010010010010100100100]{
q⇤ = �e/3
ExcitaAons: domain-‐walls
Energy cost = U2à bulk gap = U2
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)
! !
i j
[100100100010010...010100100...] = V †1 (j)V1(i) |gi1
[100...100100001001001...] = V2(i)... |gi1!
q⇤ = ke/3 : |gia ! |gia+k%3
ExcitaAons: domain-‐walls
q⇤ = 2e/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)
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
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
c = volume of the complex space of fa à SU(3) gauge invariance
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
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
⌫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
Parton construc2on: 1/3 FQH
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
Parton construc2on: 1/3 FQH
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
Vaezi & Barkeshli, PRL (2014)
operator carries charge under à Higgs mechanism:
�ab = f†a,"fb,# A" �A#
A" = A# = A< �ab > 6= 0
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
�⇢
◆
Parton construc2on: 1/3+1/3 FQH
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)
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
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)
Parton construc2on: 1/3+1/3 FQH
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
Liu, Vaezi, Lee, and Kim; arXiv:1502.05391 (to appear in PRB (R)
Entanglement measurements Level Cou
n2ng
(Even sector)
Finite size sc
aling of
energy gap
Entanglemen
t Entropy
Level Cou
n2ng (O
dd se
ctor)
Liu, Vaezi, Lee, and Kim; arXiv:1502.05391 (to appear in PRB (R)
ParAcle Entanglement Spectrum
Ne=8 NeA=4 Nqh=12
Ne=10 Ne
A=4 Nqh=18
Liu, Vaezi, Lee, and Kim; arXiv:1502.05391 (to appear in PRB (R)
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