B.Spivak

University of Washington

with S. Kivelson, S. Sondhi,

S. Parameswaran

A typology of quantum Hall liquids.Weakly coupled Pfaffian state as a type 1

quantum Hall fluid

integerarenm,,2

xy e

n

m

Integer quantum Hall effect

integerisn,2

xy en

Fractional quantum Hall effect

I will discuss the cases m/n=1/2, 5/2, ….

Spectrum of electrons in two dimensions in magnetic field B

c

mc

eBn cc ;12

Density of states on each Landau level:LH is the magnetic length

2

1

2 Hc Lc

eB

Filling factor:c

J. Jain, R. Laughlin, S. Girvin, A. McDonnald,S. Kivelson, S.C. Zheng, E. Fradkin, F. Wilczek,P. Lee, N. Read, G. Moore, B. Halperin, D. Haldane

Aharonov-Bohm effect

quantumfluxtheis2 00 e

cdl

Φ

Φ

Φ

Composite fermions

0k

e = fermion if k=2nboson if k=n

n is an integer

The statistical phase can be interpreted as an Aharonov-Bohm effect: when charge is moving around the flux (it acquires a phase

time

space

Chern-Simons theory of the quantum Hall effect (Fermion version k=2)

)ln(Im,

,

,

0

kjjkkj

jkj

ii

zze

ck

k

c

ei

c

ei

a

ABrrarb

aAA

B and b are the magnetic field and statistical magnetic field A and a are the vector potential and statistical vector potential

0k

e = composite fermion

Halperin- Lee-Read (HLR) state: “Fermi liquid” of composite Fermions, k=2

At the filling factor =½ the statistical and external magnetic fields cancel each other:

2

222 1,,

2 H

HF

F

HpotF

Le

mLp

m

p

L

eEE

What are the effective mass and the Fermi energyof composite fermions?

at the mean field level the system is in a Fermi liquid state in a zero effective magnetic field!

B + b =0

jze c

k 0

Mean field electrodynamics of HLR state

eEj CFOhm’s law for composite Fermions:

,fieldeffectivethein

movefermionscompositeif

2/10

2/1

kbB

bothxxandxyare not quantized !

Experiments supporting HLR theory

bB

bB

cc

as diverge fermions composite of r radius cyclotron

Fv

cr

mc

ec

|| bBbB

0bB

J.P. Eisenstein, R.L. Willet, H.L. Stormer, L.N. Pffiffer, K.W. West

Superconductivity of composite fermions

),(')()( r'r,rrr'rr'r, ΔΠΔ g

Chern-Simons Superconducting order parameter

222Fyx

yx

ppp

ipp

pΔ

P-wave (triplet) order parameter

the system has an isotropic gap

Moore-Read Pfaffian 5/2 QH state, weakly coupled (BCS) p-wave superconductivity of composite fermions

)(

2

2,

0

0

eEv

aAvvj

jez

ab

s

et

m

c

e

meN

k

Nk

s

ss

z is a unit vector perpendicular to the plane, at T=0 Ns=N

42

1||

.......,2

5,

2

1

,2

0)(

0

00

2*

22

0

e

k

ed

kdee

ee

et

m

xy

s

rar

ezj

eEeEv

Ψ

Correspondence between the perfect conductivity of the superconductors and the quantization of the Hall conductance:

Meissner effect incompressibility

Quantized vortices fractionally charged quasiparticles

gaptheisE

constantstructurefinetheis

ionconcentratelectrontheis

radiusBohrtheis

||

1,1

1

F

3/1

2/1

2/1

2

2

Δ

Δ

e

B

F

eLs

BeeL

n

a

v

nrif

anen

mc

Two types of conventional superconductors

jBc

curl4

lengthmagnetictheis

field"magneticlstatisticathe"oflengthnPenetratio.2

||lengthCoherence.1

H

H

F

L

L

v

Δ

Two characteristic lengths in the Pfaffian state at T=0

Two characteristic energy scales

2

2

energyFermiThe

||gapThe

HF mLE

a) Type 2 QH fluids where roughly In this case the surface energy between HLR and Pfaffian states is negative. Consequently density deviations are accommodated by the introduction of single quasiparticles/vortices

b) Type I QH state: , (or EF >> In this case the surface energy between is positive. Quasiparticles (vortices) agglomerate and form multi-particle bound states

electronic microemulsions

Two possible types of quantum Hall fluids

1||

1

vorticestwoofmergingwithassociatedenergyCoulomb

)1

(votexaofenergyonCondensati

2/1

2/12/1

2

2

22

Δ

ΔΔ

e

EN

Ee

E

Fc

F

F

If vortices agglomerate into big bubbles

Nc is the number of electrons in the bubble

If Nb ~1 the system is in “electronic microemulsion phase”which can be visualized as a mixture of HLR and Pfaffian on mesoscopic scale. Nb is the bubble concentration

Schematic phase diagram

Bosonic Chern-Simons theory.At Bogomolni’s point vortexes do not interact

!interactnotdovortexes

point)s(Bogomolnilinearisequationthe,If

D

casestatic

rr

02

'2

0||2/2

,0||22

||,2

1,

2

0

20

2

22

4int

0int0

m

D

m/kΦλ

mkm

D

iDDD

AaiDm

ddtSaak

ddtSSSSS

yx

kjiijkcscs

Φ

Φ

1.Numerical simulations: H. Lu, S. das Sarma, K. Park, cond-mat. 1008.1587; P. Rondson, A. E. Feiguin, C. Nayak, cond. mat. 1008.4173; G. Moller, A. Woijs, N. Cooper, cond-mat. 1009.4956 e2/LHEF ~10-30, /EF ~12.a) Activation energy in transport experiments is approximately two orders of magnitude smaller than EF , and sometimes decreases further as a function of gate voltage and parallel magnetic field. b) the characteristic temperature where the 5/2 plateau of QHE disappears is much smaller then EF

Do we know that in the Pfaffian state

An exapmple: superfluid 3He:

310~;40~

FF

pot

EE

U Δ

Existing experiments on measuring the effective vortex charge near 5/2filling fraction cannot distinguish between the first and second type of quantum Hall states.

They only prove that the elementary building blocks for any charged structure(either vortices, or bubbles, or more complex objects) have charge e/4.

KEF 100

Willet’s experiments measure the totalnumber of vortices of charge e/4 in a sample

R. L. Willett, L. N. Pfeiffer, and K. W. West, Phys. Rev.B 82, 205301 2010

Edge states

c

In Heiblum’s group experiments the edge state carrier charge is inferred from shot noise measurements. Edge states exist even exactly at 5/2 filling fraction.

J. Nuebler, V. Umansky, R. Morf, M. Heiblum, K. von Klitzing, and J. Smet, Phys. Rev. B 81, 035316 (2010)

The Yacoby group’s experiments are based on the fact that samples are disordered and there are puddles of HLR states embeddedinto the Pfaffian state. The charge of big HLR puddles grows in steps e/4 as a functionof the gate voltage

Pfaffian HLR

Vivek Venkatachalam, Amir Yacoby, Loren Pfeiffer, Ken West, Nature 469, 185, 2011

Experiments on the activation energy of xx

The longitudinal resistance exists due to motion of vortices. The activation energy is determined by the pinning of vortices.Thus these experiments do not provide direct information about the value of the gap

In pure samples the value of the “critical temperature” is directly related to the value of the gap.However in disordered samples the value of the “critical temperature” may be determined by weak links between superconducting droplets.The situation is quite similar to that in granular superconductors.

disorder

T

Pfaffian Pfaffian glass HLR

Since the Jij have random sign, near the critical point the system is Pfaffian (p-wave superconducting) glass

An effective model of Joshepson junctions

energykineticquantumcos

jiji

ijJ Σ

Conclusion:Weakly coupled Pfaffian state is equivalent toType 1 p+ip superconducting state. In this state vortices attract each other and agglomerate into big bubbles.

There is a quantum phase transition between HLR and Pfaffian states as a function of disorder

Depending on interaction, conventional QHfractions can be type 1 as well.