Non-Abelian Anyons in the Quantum Hall Effect · Non-Abelian Anyons in the Quantum Hall Effect...
Transcript of Non-Abelian Anyons in the Quantum Hall Effect · Non-Abelian Anyons in the Quantum Hall Effect...
Non-Abelian AnyonsNon-Abelian Anyonsin the Quantum Hall Effectin the Quantum Hall Effect
● Incompressible Hall fluids: bulk & edge excitations● CFT description● Partition function● Signatures of non-Abelian statistics:
– Coulomb blockade & thermopower
Andrea Cappelli(INFN and Physics Dept., Florence)
with L. Georgiev (Sofia), G. Zemba (Buenos Aires), G. Viola (Florence)
OutlineOutline
Quantum Hall EffectQuantum Hall Effect● 2 dim electron gas at low temperature T ~ 10 mK
�
and high magnetic field B ~ 10 Tesla
BBB
● Conductance tensor● Plateaux: no Ohmic conduction gap
�
● High precision & universality● Uniform density ground state:
Incompressible fluid
x
yB
ExJy
Ji = ¾ijEj ; ¾ij = R¡1ij ; i; j = x; y
¾xx = 0; Rxx = 0
¾xy = R¡1xy =e2
hº; º = 1(§10¡8); 2; 3; : : : 1
3;2
5; : : : ;
5
2;
½o =eB
hcº
Laughlin's quantum incompressible fluidLaughlin's quantum incompressible fluid
Electrons form a droplet of fluid: incompressible = gap fluid =
½ ½
Rq
Nºx
y
©o = hce
# degenerate orbitals = # quantum fluxes,
filling fraction: density for quantum mech.
º = 1 º = 13
½(x; y) = ½o = const:
DA = BA=©o;
º =N
DA= 1; 2; : : :
1
3;1
5; : : :
Laughlin's wave functionLaughlin's wave function
● filled Landau level: obvious gap ● non-perturbative gap due to Coulomb interaction
n
effective theories● quasi-hole = elementary vortex
q
fractional charge & statistics
&
! = eBmc À kTº = 1
º = 13
Q = e2k+1
µ¼ = 1
2k+1
Anyons vortices with long-range topological correlations
º = 12k+1 = 1; 13 ;
15 ; : : :
ª´ =Q
i (´ ¡ zi)ªgs
ª´1;´2 = (´1 ¡ ´2)1
2k+1Q
i (´1 ¡ zi) (´2 ¡ zi)ªgs
ªgs (z1; z2; : : : ; zN ) =Y
i<j
(zi ¡ zj)2k+1 e¡P jzij2=2
Chern-Simons gauge theoryChern-Simons gauge theory
C
Special facts of 2+1 dimensions:● matter current gauge field:
m
● low-energy effective action, P, T:
J¹ = "¹º½@ºA½
J¹ = (½; Ji); @¹J¹ = 0; hJ¹i = 0
ext. source
eq. of motion no local degrees of freedom
exp
µi
I
z2
A¶= ei2¼=k
F¹º =2¼
k"¹º½s
½; B =2¼
k±(2)(z ¡ z2)
Aharonov-Bohm phase
SCS =k
4¼
Z"¹º½A¹@ºA½ +A¹s
¹ +1
MF2¹º
Conformal field theory of edge excitationsConformal field theory of edge excitations
● edge ~ Fermi surface: linearize energy● relativistic field theory in 1+1 dimensions, chiral (X.G.Wen '89)
(
chiral compactified c=1 CFT (chiral Luttinger liquid)
The edge of the droplet can fluctuate: edge waves are massless
t
µ
½V
Fermi surface
R / J
"(k) = vR (k ¡ kF ); k = 0; 1; : : :
CFT descriptions of QHE: bulk & edgeCFT descriptions of QHE: bulk & edge
● same function by analytic continuation from the circle:– both equivalent to Chern-Simons theory in 2+1 dim (Witten '89, X.G.Wen '89)
● simplest theory for is chiral Luttinger liquid (U(1) CFT):– wavefunctions: spectrum of anyons and braiding– edge correlators: physics of conduction experiments
plane (bulk excit)
µ
r
¿cylinder (edge excit)
(z1 ¡ z2)®2
anyon wavefunction
edge-excit. correlator
z = reiµ
³ = e¿ eiµ
(³1 ¡ ³2)®2
h Á1 Á2 iCFT
º = 1=p
Measure of fractional chargeMeasure of fractional charge
● electron fluid squeezed at one point: L & R edge excitations interact● fluctuation of the scattered current: Shot Noise (T=0)
– low current tunnelling of weakly interacting carriers
�
Poisson statistics
P
● CFT description & integrable massive interaction: (Fendley, Ludwig, Saleur)
S
universality & “anomalous” scaling
I = G ¢V¡ +
IT
IB
L
R
+VG
+VG
G =e2
h
1
3F
µVGT 2=3
¶
SI = hj±I(!)j2i!!0 =e
3IB
IB ¿ I
(Glattli et al '97)
Non-Abelian fractional statisticsNon-Abelian fractional statistics● described by Moore-Read “Pfaffian state” ~ Ising CFT x U(1)● Ising fields: identity, Majorana = electron, spin = anyon● fusion rules:
– 2 electrons fuse into a bosonic bound state– 2 channels of fusion = 2 conformal blocks2
hypergeometrich
state of 4 anyons is two-fold degenerate (Moore, Read '91)
● statistics of anyons ~ analytic continuation 2x2 matrixs
º = 52
I Ã
à ¢ à = I
h¾(0)¾(z)¾(1)¾(1)i = a1F1(z) + a2F2(z)
00 z 1 1
00 z 1 1
(all CFT redone for Q. Computation: M. Freedman, Kitaev, Nayak, Slingerland,...., 00'-10' )
¾
¾ ¢ ¾ = I + Ã
µF1F2
¶¡zei2¼
¢=
µ1 00 ¡1
¶µF1F2
¶(z)
µF1F2
¶¡(z ¡ 1)ei2¼
¢=
µ0 11 0
¶µF1F2
¶(z)
Topological quantum computationTopological quantum computation● qubit = two-state system ● QC: perform unitary transformations in qubit Hilbert space● Proposal: (Kitaev; M. Freedman; Nayak; Simon; Das Sarma '06)
((
use non-Abelian anyons for qubits and operate by braiding
u
4-spin system is 1 qubit (2n-spin has dim )● anyons topologically protected from decoherence (local perturbations)● more stable but more difficult to create and manipulate
great opportunity
new experiments and model building
2n¡1®jF1i+ ¯jF2i
U(2n) n
jÂi = ®j0i+ ¯j1i
Models of non-Abelian statisticsModels of non-Abelian statistics● Study Rational CFTs with non-Abelian excitations:
– best candidate: Pfaffian & its generalization, the Read-Rezayi states
b
– alternatives: other (cosets of) non-Abelian affine groups
● Identify their N sectors of fractional charge and statistics – Abelian (electron) & non-Abelian (quasi-particles)
A
● Compute physical quantities that could be signatures of non-Abelian statistics:– Coulomb blockade conductance peaks
– thermopower & entropy
º = 2 +k
k + 2;n
k = 2; 3; : : :M = 1 U(1)k+2 £
SU(2)kU(1)2k
U(1)£ GH
use partition functionuse partition function
● quantity defining Rational CFT (Cardy '86; many people)● complete inventory of states (bulk & edge)● modular invariance as building principle:
– S matrix and fusion rules– further modular conditions for charge spectrum– straightforward solution for any non-Abelian state– useful to compute physical quantities
● Inputs:– non-Abelian RCFT (i.e. )– Abelian field representing the electron “simple current”
● Output is unique
G
H
U(1)£ GH
Annulus partition functionAnnulus partition function ¯
R
modular invariance conditions geometrical properties & physical interpretation (A. C., Zemba, '97)
S matrix
half-integer spin excitations globally
completeness
integer charge excitations globally
add one flux: spectral flow
T 2 : Z(¿ + 2; ³) = Z(¿; ³);
S : Z
µ¡1¿;¡³¿
¶= Z(¿; ³);
U : Z(¿; ³ + 1) = Z(¿; ³);
V : Z(¿; ³ + ¿) = Z(¿; ³);
µ¸(³ + ¿) » µ¸+1(¿)
i2¼³ = ¯(¡Vo + i¹)
L0 ¡ L0 =n
2
Q¡Q = n
¢Q = º
i2¼ ¿ = ¡¯ vR
+ it; ¯ =1
kBT
Zannulus =
pX
¸=1
jµ¸(¿; ³)j2; µ¸(¿; ³) = TrH(¸)
hei2¼¿(L0¡c=24)+i2¼³Q
i
µ¸
µ¡1¿
¶=X
¸0
S¸¸0µ¸0(¿)
Disk partition functionDisk partition function ¯
R
Annulus -> Disk (w. bulk q-hole ) ¸
p
● sectors with charge
s
basic quasiparticle + n electrons● : electrons have half-integer dimension (=J),
:
and integer relative statistics with all excitations● # sectors = Wen's topological order
##
we recover phenomenological conditions on the spectrum
w
¹Q = ¸p
Q = ¸p + n
p = dim (S¸¸0)
T 2
U : Q¡Q = n
µ¸(¿; ³) = K¸(¿; ³; p) =1
´
X
n
ei2¼
·¿ (np+¸)2
2p +³ np+¸p
¸
; º =1
p; c = 1
Zannulus ! Zdisk; ¸ = µ¸(¿; ³)
Pfaffian & Read-Rezayi states Pfaffian & Read-Rezayi states
● parafermion sectors and charactersp
● electron is Abelian ● + electron:+
parity rule
● sectors labeled by
Ø
● # sectors = topological order
º = 2 +k
k + 2;n
k = 2; 3; : : :M = 1 U(1)k+2 £
SU(2)kU(1)2k
µ`a =k¡1X
¯=0
Ka+¯(k+2) Â`a+2¯
Z3
I
6
3
0
¡3
3
m
`
Ã1
Ã2
¾1
¾2
"
Ã2
Ã1
Zk
a = 0; : : : ; k + 1; ` = 0; : : : ; k; a = ` mod 2
(k + 2)£ k(k+1)2
£ 1k= (k+2)(k+1)
2
(a; `)
"
¾1
¾2
(q;m; `) ! (q + p;m+ 2; `)
q = m mod kp = 2 + k ;
Â`m; ` = 0; 1; : : : ; k; m mod 2k
(`;m)
ªe = ei®'Ã1; (`;m) = (0; 2)
Q =q
p
Ex: Pfaffian (k=2)
EEEE
● charge parts● Ising parts (Majorana fermion)
– 6 sectors– also Abelian excitations
ZRRannulus =
kX
`=0
p̂¡1X
a=0
¯̄µ`a(¿; ³)
¯̄2; ZRR
disk = µ`a =k¡1X
¯=0
Ka+¯(k+2) Â`a+2¯
ZPfa±anannulus = jK0I +K4Ãj2 + jK0Ã +K4Ij2 + j(K1 +K¡3)¾j2
+ jK2I +K¡2Ãj2 + jK2Ã +K¡2Ij2 + j(K3 +K¡1)¾j2
non-Abelian quasiparticle
ground state + electrons
K¸ Q =¸
4+ 2n
I; Ã; ¾
Q = 0;§1
2
(Milavanovich, Read '96; AC, Zemba '97)
● charge and neutral q. #'s are coupled by “parity rule”● but S-matrix for is factorized:
b
● generalization to other N-A models: (A.C, G. Viola, '10)
– Wen's non-Abelian Fluids– Anti-Read-Rezayi– Bonderson-Slingerland– N-A Spin Singlet state
NN
unique result once N-A CFT and electron field have been chosen
U(1)£ Ising £ SU(n)1
U(1)£ SU(2)k
U(1)£ SU(2)k
U(1)q £ U(1)s £SU(3)kU(1)2
Sa`;a0`0 » ei2¼aa0N=M s``0µ`a
ZRRannulus =
kX
`=0
p̂¡1X
a=0
¯̄µ`a(¿; ³)
¯̄2; ZRR
disk = µ`a =k¡1X
¯=0
Ka+¯(k+2) Â`a+2¯
Experiments on non-Abelian statisticsExperiments on non-Abelian statistics
● (a) interference of edge waves
Aharonov-Bohm phase, checks fractional statistics– experiment is hard
● (b) electron tunneling into the droplet
Coulomb blockade conductance peaks– check quasi-particle sectors
● Thermopower
(Goldman et al. '05; Willett et al '09)
(Chamon et al. '97; Kitaev et al 06)
(a) (b)
(Stern, Halperin '06)
(Ilan, Grosfeld, Schoutens, Stern '08 )
(Stern et al.; A.C. et al. '09 - '10)
e3
e3
(Cooper, Stern; Yang, Halperin '09; Chickering et al. '10)
ThermopowerThermopower● fusion of -type quasiparticles:
– multiplicity● Entropy
E
● put temperature and potential gradients between two edges● at equilibrium: ● thermopower
t
● entropy from Z:
e
● it could be observable by varying B off the plateau center
(Chickering et al '10)
¢T ¢Vo
Q = ¡¢Vo¢T
=S
Q
(quantum dimension)
Q =
¯̄¯̄B ¡Bo
e¤B0
¯̄¯̄ log(d1)
¯
`
n `
d = ¡SdT ¡QdVo = 0
2¼R
S(T = 0) » n log(d`); d` =s`0s00> 1
(Cooper,Stern;Yang, Halperin '09)
» (d`)n; n!1
S =
µ1¡ ¿ d
d¿
¶logµ`a(¿ +¢¿; ³ +¢³)
µ00(¿; ³)» log
s`0s00; ¿ » ¯
R! 0
Coulomb blockadeCoulomb blockade● Droplet capacity stops the electron● Bias & T ~ 0: needs energy matching
BB
current peak● energy deformation by
equidistant peaks
e
modulated pattern
¢S
¸e3
¢S » ¢Qbkg
¢¾ = B¢S©o
= 1º ; ¢S = e
no
¾
¢¾
E¾(n)
U(1)
U(1)£ GH
¢¾`m =1
º+vnv
¡h`m+2 ¡ 2h`m + h`m¡2
¢ vnv» 1
10
E(n+ 1; S) = E(n; S)
E(n; S) =v
R
(¸+ pn¡ ¾)22p
/ (Q¡Qbkg)2
● compares states in the same sector● spectroscopy of lowest CFT states● : cannot distinguish NA state from “parent” Abelian state
::
corrections
c
● two scales:
tt
multiplicity of neutral states
m
in (331) & Anti-Pfaff, not in Pfaff
i
matrix of non-Abelian part
mm
test non-Abelian part of disk partition function
T > 0
T = 0
T < Tn : ¢¾`m = ¢ ¢ ¢+ T
Tchlog
µ(d`m)2
d`m+2 d`m¡2
¶;
Tn < T < Tch : ¢ ¢ ¢+ / T
Tche¡h
11T=Tn
s`1s`0; S
(Bonderson et al. '10)
(Stern et al., Georgiev, AC et al. '09, '10)
µ`a =k¡1X
¯=0
Ka+¯(k+2) Â`a+2¯
d`m
hQiT » @@Vo
log µ`a
0 < Tn < Tch; Tn =vnR; Tch =
v
R» 10 Tn
ConclusionsConclusions
● non-Abelian anyons could be seen ● partition function:
– it is simple enough – it defines the CFT, its sectors, fusion rules etc.– it is useful to compute observables – it can be the basis for further model building