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