zooming in on the quantum hall effect - capri school · zooming in on the quantum hall effect...

Zooming in on the Quantum HallEffect

Cristiane MORAIS SMITH

Institute for Theoretical Physics, Utrecht University, The Netherlands

Capri Spring School – p.1/31

Experimental Motivation

électrons 2D

B

RL RH

_ _ _ _ _ _

++ + + ++I

−I= nνHal

l Res

ista

nce

Integral QHE

Magnetic Field

longitudinal resistance

1

2

34

5

2D electrons

Historical Summary:

1980 : Discovery of theIQHE (v. Klitzing)

1983 : Discovery of theFQHE (Tsui, Störmer);

Laughlin :

incompressible quantum

liquids



électrons 2D

B

RL RH

_ _ _ _ _ _

++ + + ++I

−I= nν

= p/(2ps+1)ν

Integral QHE

Hal

l Res

ista

nce

Magnetic Field


Fractional QHE

2/3

3/54/7

4/93/7

2/5

1/3

1/2

1

2

34

5

2D electrons

Historical Summary:



Laughlin :


liquids



électrons 2D

B

RL RH

_ _ _ _ _ _

++ + + ++I

−I= nν

= p/(2ps+1)ν

Integral QHE

Hal

l Res

ista

nce

Magnetic Field


Fractional QHE

2/3

3/54/7

4/93/7

2/5

1/3

1/2

1

2

34

5

2D electrons

Historical Summary:



Laughlin :


liquids

1989 : composite fermions(Jain, Read, Lopez/Fradkin,...)



800

400

0

Rxx

(O

hms)

4.03.83.63.43.2Magnetic Field (Tesla)

b)

0.35

0.30

0.25

Rxy

(h

/e2 )

h/4e2

h/3e2

7/2 3+1/5

3+4/5

a)

Reentrant IQHE


Hal

l R

esis

tanc

e

Magnetic field

2/3

3/54/7

4/93/7

2/5

1/3

1/2

1

2

34

5

Historical Summary:



Laughlin :


liquids

2002 : discovery of Reentrant IQHE (Eisenstein et al.)



800

400

0

Rxx

(O

hms)

4.03.83.63.43.2Magnetic Field (Tesla)

b)

0.35

0.30

0.25

Rxy

(h

/e2 )

h/4e2

h/3e2

7/2 3+1/5

3+4/5

a)

Reentrant IQHE


Hal

l R

esis

tanc

e

Magnetic field

4/11

5/13

3/8

Self−similarity of the Hall curve

2/3

3/54/7

4/93/7

2/5

1/3

1/2

1

2

34

5

Historical Summary:



Laughlin :


liquids

2002 : discovery of Reentrant IQHE (Eisenstein et al.)2003 : discovery of the

� ��

FQHE (Pan et al.)Capri Spring School – p.2/31

Theoretical Model

2D electrons in a perpendicular magnetic field

spin !no

1−particle Hamiltonian Coulomb interactions

�� : energy quantization (Landau levels)

�� : impurity potential (pinning)

�� : FQHE, electron-solid phases


IQHE: single particle picture

one electron in

�� :

��

��

degenerate Landaulevels (LLs)

Density of states perLL: ��

filling factor :

� � ��

heB/m

m

1

3

2

4

n = 0

Land

au L

evel

s


IQHE: single-particle localisation

class.

n

ε

ν

electrons in full LLs: one quantum of conductance e /h per LL2

R xyxxR

B=n

n

h/e n2



class.

n

(n+1)LL

th

ε

ν

electrons in full LLs: one quantum of conductance e /h per LL2

++ +

R xyxxR

B=n

n

h/e n2



class.

n

(n+1)LL

th

ε ε

ν

electrons in partially filled LL trapped by impurities ( )

electrons in full LLs: "inert" background (c.f. noble gases, full shells)

++ +

+

R R xyxyxx xxRR

BB=n

n n

h/e n2

+

+ +



class.

n

(n+1)LL

th

ε ε ε

ν

electrons in partially filled LL trapped by impurities ( )

electrons in full LLs: "inert" background (c.f. noble gases, full shells)

++ +

+

RR R xy xyxyxx xxRR

BBB

Rxx

=n

h/e n2

n n n

h/e (n+1)2h/e n2

+

+ +

++ +


When Coulomb becomes essential

(

�� )

= n ν < 1ν < 1ν= nν



(

�� )

= n ν < 1ν < 1ν= nν

Hamiltonian in the �th LL

� ��

� ��

projected density:

� � � � � ��



(

�� )

= n ν < 1ν < 1ν= nν

Hamiltonian in the �th LL

��

� ��


Effective interaction potential

M.Goerbig and C.M.S., Europhys. Lett. 63, 736 (2003)

��

� ��

� � � � ��

� � �

20 40 60 80 100

0.2

0.4

0.6

0.8 v (r)n

r/10lB20 40 60 80 100

0.25

0.5

0.75

1

1.25

1.5

BR = l (2n+1)c 1/2

~2Rc

v (r)

r/10R

~

c

universal function

2 length scales:

� and

�

(interparticle separation)


Wigner Crystal and Bubbles

A. Wigner Crystal(WC):

� �

�

Quasi-classical limit

d

Rc

B. FQHE:

C. Bubbles (super-WC):

Not in LLL,

v (r)

r

2Rc

n

d’ dd’ 2d−d’>2Rc

2)u

d d

1)

~

~energy = u

energy = 2 u




� �

�


d

Rc

B. FQHE:� � �

�


Not in LLL,

v (r)

r

2Rc

n


2)u

d d

1)

~

~energy = u

energy = 2 u




� �

�


d

Rc

B. FQHE:� � �

�


� ��

�

�

Not in LLL,

� ��

v (r)

r

2Rc

n


2)u

d d

1)

~

~energy = u

energy = 2 u


Energy of competing ground states

M=7

Bubble crystal Stripe phase Wigner crystal

Wigner crystal and Bubble: Hartree-Fock + impurities

� � � ��

� � � � ��

� � ��

� � � ��

Excitations of the quantum liquid: Hamiltonian TheoryMurthy and Shankar, Rev. Mod. Phys. 75, 1101 (2003)

� ��


Results for � � �

Goerbig, Lederer, CMS, PRB 68, 241302(R) (2003)

I I

Rlong RHall

+++

+

5 10 15 20 25

-0.2

-0.15

-0.1

-0.05

Partial filling of the last level

n=1

−0.15

−0.20

−0.10

−0.051/9 1/7 1/5 1/3

0.1 0.2 0.3 0.4 0.5

M=1Ene

rgy

M=2 (electrons per site)

quantum liquids

impurities

800400

0

Rxx

(O

hms)

4.0 3.8 3.6 3.4 3.2

Magnetic Field (Tesla)

b)

0.350.30

0.25

Rxy

(h

/e2)

h/4e2

h/3e2

7/23+1/5

3+4/5

a)

� � � ��

??

0.1 0.2 0.3 0.4 0.5

1/2

1/31/5

1/3

1/3.5

[h/e

]2

Hal

l re

sist

ance

crystal liquid crystal liquid crystal

filling of the last level



Goerbig, Lederer, CMS, PRB 68, 241302(R) (2003)

I I

Rlong RHall

+++

+

5 10 15 20 25

-0.2

-0.15

-0.1

-0.05

Partial filling of the last level

n=1

−0.15

−0.20

−0.10

−0.051/9 1/7 1/5 1/3

0.1 0.2 0.3 0.4 0.5

M=1Ene

rgy

M=2 (electrons per site)

quantum liquids

impurities

800400

0

Rxx

(O

hms)

4.0 3.8 3.6 3.4 3.2

Magnetic Field (Tesla)

b)

0.350.30

0.25

Rxy

(h

/e2)

h/4e2

h/3e2

7/23+1/5

3+4/5

a)

� � � ��

� � � ��

??

0.1 0.2 0.3 0.4 0.5

1/2

1/31/5

1/3

1/3.5[h

/e ]2

H

all

resi

stan

ce

crystal liquid crystal liquid crystal

filling of the last level



5 10 15 20 25

-0.175

-0.15

-0.125

-0.1

-0.075

-0.05

-0.025

impurities

quantum liquid

stripes

0.20.1 0.3 0.4 0.5

−0.15

−0.10

−0.051/9 1/31/7 1/5

M=1

Ene

rgy

Partial filling factor

M=2

M. O. Goerbig, P. Lederer, C. M. S., PRB 69, 115327 (2004)

No FQHE at� � � � ��

, but indications at

� � � � ��


Phase TransitionsGoerbig, Lederer, and CMS, PRB 69, 115327 (2004)

0.15 0.26M=1

M=2

n = 2

partial filling factor

ener

gy (

a.u.

)

mixed phase

Mixed phase

Wigner crystal/Bubble

Pinning mode at



0.15 0.26M=1

M=2

n = 2


ener

gy (

a.u.

)

mixed phase

Mixed phase

Wigner crystal/ Bubble

Pinning mode at

��



0.15 0.26M=1

M=2

n = 2


ener

gy (

a.u.

)

mixed phase

Mixed phase


Pinning mode at

��

15

10

5

0

2.01.51.00.5 f(GHz)

ν=4.26 Data fit peak 1 peak 2

9

6

3

0

ν=4.21

6

4

2

0

ν=4.16

4.0

2.0

0

ν=4.12

Re

[σxx

] (µS

) 6

4

2

0

ν=4.18

Lewis et al., PRL 93, 176808 (04)



0.15 0.26M=1

M=2

n = 2


ener

gy (

a.u.

)

mixed phase

Mixed phase


Pinning mode at

��

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

f pk (

GH

z)

4.354.304.254.204.154.10

ν

fp1

fp2


Discovery of a new FQHE at � � � � �

CF theory : classified all the known FQHE plateaus ...

= p/(2ps+1)ν

= nν

IQHE

FQHE

Hal

l Res

ista

nce

magnetic field


2/3

3/54/7

4/93/7

2/5

1/3

1/2

1

2

34

5

Self-similarity ofthe Hall curve


Discovery of a new FQHE at � � � � �

CF theory : classified all the known FQHE plateaus ...... until 2003 : new class of states (Pan et al.)

7/1910/27

= p/(2ps+1)ν

8/2111/29

= nν

IQHE

FQHE

Hal

l Res

ista

nce

magnetic field


4/11

2/3

3/54/7

4/93/7

2/5

1/3

1/2

1

2

34

3/8

5

5/13Self-similarity ofthe Hall curve


Composite FermionsIdea: interpret strongly correlated electrons in terms ofquasi-particles (CF) with negligible interactions

ν = 1/3

pseudo−vortex

electronic filling 1/3theory

CF

1 filled CF level

electron

"free" flux quantum

(with 2 flux quanta)

composite fermion (CF)

At ,FQHE of electrons IQHE of CFs


Composite FermionsIdea: interpret strongly correlated electrons in terms ofquasi-particles (CF) with negligible interactions

ν = 1/3

ν = 2/5

pseudo−vortex

theory

CF

2 filled CF levels

electron

"free" flux quantum

(with 2 flux quanta)

composite fermion (CF)

electronic filling 1/3 1 filled CF level

At � � � � ��

,FQHE of electrons � IQHE of CFs


Hamiltonian Theory of the FQHE

– Treat pseudo-vortex as “new” particle (charge � �):

�� "! �# � � � � $ �

� �&% (' �� ) �� *

Murthy/Shankar, Pasquier/Haldane, Read

Constraint :

(charge −c )2

electron (charge 1)

vortexpseudo−

– Preferred combination (CF density):

– At (non degenerate state):completely filled CF levels




�� "! �# � � � � $ �

� �&% (' �� ) �� *


Constraint :

�&% (' � � � ��

� * (charge −c )2

electron (charge 1)

vortexpseudo−






�� "! �# � � � � $ �

� �&% (' �� ) �� *


Constraint :

�&% (' � � � ��


electron (charge 1)

vortexpseudo−


� � � (' � � � (' � � � �&% (' � � � � � � � � � ��





�� "! �# � � � � $ �

� �&% (' �� ) �� *


Constraint :

�&% (' � � � ��


electron (charge 1)

vortexpseudo−


� � � (' � � � (' � � � �&% (' � � � � � � � � � ��

– At � � � �� (non degenerate state):

� completely filled CF levels � � � � � � ��


Second Generation of CFs

At � � � � � : CF levels are degenerate � CF interactions

=

ν∗ = 1+1/3

filling 1/3 of first excited CF levelCF of first generation (with 2 flux quanta)

Explanation of state (??):IQHE of C Fs

New hierarchy scheme of states




=

=

ν∗ = 1+1/3

+

2

2

2

filling 1/3 of first excited CF level

theory

C F

CF of first generation (with 2 flux quanta)

1 CF + CF vortex(with 2 additional flux quanta)

1filled CF level

1 filled C F level(in first excited CF level)

CF of second generation (C F)

Explanation of state (??):IQHE of C Fs





=

=

ν∗ = 1+1/3

+

2

2

2

filling 1/3 of first excited CF level

theory

C F

CF of first generation (with 2 flux quanta)

1 CF + CF vortex(with 2 additional flux quanta)

1filled CF level

1 filled C F level(in first excited CF level)

CF of second generation (C F)

Explanation of

� ��

state (??):IQHE of C

�Fs � � � ��



Interacting CFs at ��

ν∗ = 1+1/3

1.st generation CF

Low energyexcitations of CFs :

intra-level

- Wave functions : numerical calculations (finite size)Problem : ambiguous results (no thermodynamical limit)[Mandal and Jain, PRB 66, 155302 (2002);

Chang and Jain, PRL 92, 196806 (2004)]

- Hamiltonian theory : simple analytical frame


Model for interacting CFs at ��

Goerbig, Lederer, C.M.S., Europhys. Lett. 68, 72 (2004)

��

��

�� ' � � (' �

Density restricted to level � ,� � � �

� � � � �

�

��

:

Interaction potential

Effective Hamiltonian:

Similarity with original model Self-similarity of FQHE




��

��

�� ' � � (' �


� � � � �

�

��

:

Interaction potential � � ��

�

��

�


��

��

� � �� ' � �� ' �

Similarity with original model Self-similarity of FQHE




��

��

�� ' � � (' �


� � � � �

�

��

:

Interaction potential � � ��

�

��

�


��

��

� � �� ' � �� ' �

Similarity with original model

�

Self-similarity of FQHE


Activation Gaps of C

�

F StatesGoerbig, Lederer, C.M.S., PRB 69, 155324 (2004)Inter-level excitations � screened interaction (RPA)Finite width

�

: � � � �

��

��

2 4 6 8 10 12 14

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0.02

2 3 4 5 6 71

0.02

0.01

(a)

(ν=6/17)(ν=7/19)(ν=11/31)

lB

4/11(ν= )

width in units of

Act

ivat

ion

gaps

s=1, p=1

s=1, p=2

s=2, p=1

s=2, p=2

s=1 ~ ~

~ ~

~ ~

~ ~

p=1

2 4 6 8 10 12 14

0.001

0.002

0.003

0.004

0.005

0.006

0.002

0.004

0.006

1 2 3 4 5 6 7

(b)

(ν=4/19)(ν=6/29)(ν=7/33)(ν=11/53)

lB width in units of

Act

ivat

ion

gaps

s=1, p=1

s=1, p=2

s=2, p=1

s=2, p=2

~ ~

~ ~

~ ~

~ ~

s=2p=1

one order of magnitude smaller than for CF states !


Reentrant FQHE

Goerbig, Lederer, C.M.S., PRL 93, 216802 (2004)Self-similarity � same approach as for electrons

5 10 15 20 25

-0.02

-0.015

-0.01

-0.005

0.005

1/5 1/3

0.20.1 0.3 0.4 0.5

1/3 3/86/17

− 0.01

− 0.02s=1

p=1

M=2

M=1

2

4/11

Ene

rgy

Electronic filling factor

partial CF filling factor

quant. liquids (FC )

(CF Wigner crystal)

(CF bubbles)

CF stripes

FC state stable at

Reentrance in the

FQHE


Reentrant FQHE

Goerbig, Lederer, C.M.S., PRL 93, 216802 (2004)Self-similarity � same approach as for electrons

5 10 15 20 25

-0.02

-0.015

-0.01

-0.005

0.005

1/5 1/3

0.20.1 0.3 0.4 0.5

1/3 3/86/17

− 0.01

− 0.02s=1

p=1

M=2

M=1

2

4/11

Ene

rgy

Electronic filling factor


quant. liquids (FC )

(CF Wigner crystal)

(CF bubbles)

CF stripes

FC

�

state stable at

� � � ��

Reentrance in the

FQHE


Conclusions I: Self-similarity of QHE

= nν

= p/(2ps+1)ν

Integral QHE

Hal

l Res

ista

nce

Magnetic Field

Fractional QHE= IQHE of CFs 4/11

5/13

3/8

Self−Similarity

of the Hall Curve

2/3

3/54/7

4/93/7

2/5

1/3

1/2

1

2

34

5

� �� state due to

residual CFinteractions

Interacting CFmodel derived inthe Hamiltonian

theory of theFQHE

Model reveals self-similarity of the FQHEnew hierarchy scheme (higher CF generations)


Conclusions I: Self-similarity of QHE

= nν

= p/(2ps+1)ν

Integral QHE

Hal

l Res

ista

nce

Magnetic Field

Fractional QHE= IQHE of CFs 4/11

5/13

3/8

Self−Similarity

of the Hall Curve

2/3

3/54/7

4/93/7

2/5

1/3

1/2

1

2

34

5

� �� state due to

residual CFinteractions

Interacting CFmodel derived inthe Hamiltonian

theory of theFQHE

Model reveals self-similarity of the FQHE

� new hierarchy scheme (higher CF generations)


Conclusions II: Phase Diagram

−electroncrystals

M

1

1 1

1

1

2

2 3

� � � � �

� � � � �

� � � � �

� � � � �

� � � � �

� � � � �

0.0 0.1 0.2 0.3 0.4 0.5

n=0

n=1

n=2

n=3

ν

Loca

lisat

ion

LL

partial filling of the last LL

electronic filling1/3 2/5

1 2CF filling

��

�

��

�

��

��

��

��

��

��

��

��

��

��

��

�

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

C F C F2 2

4/11 5/13CF stripes

CFs CF levelp=1

insulating

Zoom in

CF phases

quantum liquids

strip

es


Perspectives

What about SPIN?Each Landau Level splits into two levels (Zeemanenergy)

Quantum Hall Ferromagnet at � � �

:

n=0 g

m=N −1φm=0.....

.....

m=1 m=2


Perspectives

What about SPIN?Each Landau Level splits into two levels (Zeemanenergy)

Magneto-excitons � bosons

n=0 g

m=N −1φm=0.....

.....

m=1 m=2


Bosonization theory: 2DES at � � �Doretto, Caldeira, Girvin, PRB 71, 45339 (2005)

Interacting 2DEG at

non-interacting bosons RPA

interaction term Skyrmion/anti-Skyrmion pair



Interacting 2DEG at � � �






� � �

� ��

� � ��

� � �

� �� ) �� ) �� ) �






� � �

� � ��

��

��

��

� � �� ) ��

� � ) � ' � � � � �� $��

� ��

��






� � �

� � ��

��

��

��

� � �� ) ��

� � ) � ' � � � � �� $��

� ��

��

non-interacting bosons � RPA

interaction term � Skyrmion/anti-Skyrmion pair


Bosonization theory: 2DES at � � � �

Doretto et al. PRB 72, 35341 (2005)

� � � ��

?- include SPIN in Hamiltonian theory- use Bosonization theory for � � � �

Spin-excitations of the QH FM of composite fermions

What aboutPSEUDOSPIN?Bosonization theoryfor QH bilayers:next talk


Conclusion

1

1 1

1

1

2

2 3

M

+

+

+

+

+

+

++

+

+

+

+ : impureté: électron

n

0.0 0.1 0.2 0.3 0.4 0.5

n=0

n=1

n=2

n=3

ν

��

��

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

� � � � � � � � �

aggrandissement :

auto−similarité

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

M=3

îlots à électrons

ruba

ns

(EHQF)

loca

lisat

ion

à un

e pa

rtic

ule

phases de FC

5/2, 7/2EHQF

� � � � �

� � � � �

� � � � �

� � � � �

� � � � �

1/3 2/5

1 2

remplissage él.

remplissage FC

2 2

4/11 5/13p=1

isolantsFCs

rubans de FCniveau

FC FC

� � �

� � �

� � �

� � �

� � �

� � �


Construction of Low-Energy Model at ��

Restriction to a singleCF level �

ν∗ = 1+1/3

Restriction of CF density:

Because of factorisation :

independent of !




ν∗ = 1+1/3

Restriction of CF density:

� � � (' � ��

� � ��

� � � ��


independent of !




ν∗ = 1+1/3

Restriction of CF density:� � � � (' � � ��

� � ��

��

�

�� (' �


independent of !




ν∗ = 1+1/3


� � ��

��

�

�� (' �

Because of factorisation � � ��

� � � � � � � � � � ��

:

independent of !




ν∗ = 1+1/3


� � ��

��

�

�� (' �

Because of factorisation � � ��

� � � � � � � � � � ��

:

� �� (' ��

�� ) � � � �� (' ) ��

� ��

independent of � !

� � � �

� � ��


Hierarchical States, Haldane/Halperin (1983)

Motivation: FQHE e.g. at � � � ��

(non-Laughlin)

� QP form a Laughlin state due to QP interactions ?

Continued fraction:

: positive integer: odd integer

2/5 2/7

1/3

4/9 8/19 12/31 8/21 8/27 12/41 8/29 4/15

3/7 5/13 5/17 3/11

(I)

(II)

(III)

(IV)

Generation:

(q=3)p =1i

encircled states are stable




(non-Laughlin)


Continued fraction:

� �

�

� �� $ � ��

� � : positive integer

� : odd integer

��

*

2/5 2/7

1/3

4/9 8/19 12/31 8/21 8/27 12/41 8/29 4/15

3/7 5/13 5/17 3/11

(I)

(II)

(III)

(IV)

Generation:

(q=3)p =1i





(non-Laughlin)


Continued fraction:

� �

�

� �� $ � ��

� � : positive integer

� : odd integer

��

*

2/5 2/7

1/3

4/9 8/19 12/31 8/21 8/27 12/41 8/29 4/15

3/7 5/13 5/17 3/11

(I)

(II)

(III)

(IV)

Generation:

(q=3)p =1i


Stable states at � � � � ��


Self-Similarity – Hierarchy of States

Recursion formula for possible states of

�

-th CF generation(

� � *

: electrons): � ��

��

��

� � : C

�

F level filling factor; � � : number of filled CF levels

�� : number of “attached” flux pairs in C

�

F

modular group

Continued fraction:Fixed point ( series,

):




�

-th CF generation(

� � *


��

��

� � : C

�



�

F

� modular group

��

� �

Continued fraction:Fixed point ( series,

):




�

-th CF generation(

� � *


��

��

� � : C

�



�

F

� modular group

��

� �

Continued fraction:

� �

�

� � � �

� � $ �

��

Fixed point ( series,):




�

-th CF generation(

� � *


��

��

� � : C

�



�

F

� modular group

��

� �

Continued fraction:

� �

�

� � � �

� � $ �

��

Fixed point (

� � �

series,

��

):

� � � � � � � � �


CF Interaction Potential

Haldane’s pseudopotential expansion

� � �� $ � ��

��

� � � �

�� $ � � � � � ��

2 4 6 8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

CF

V

s=1

mp

m

1 5 1197 13 15 17 19 21m

0.01

−0.01

0.0

0.02

pseu

dopo

tent

ials

V

p=1p=2

p=3

3


CF Phases in

� � � � �

5 10 15 20 25

-0.004

-0.003

-0.002

-0.001

0.001

quantum liquid (C F)2stripeCF

1/31/5

M=1M=2

0.1 0.2 0.3 0.4 0.5

11/27 7/17 5/12

− 0.001

− 0.002

− 0.003

− 0.004


cohe

sive

ene

rgy

2/5

electronic filling factor

s=1p=2

Quantum liquid (C F) ceases to be ground state at


CF Phases in

� � � � �

5 10 15 20 25

-0.004

-0.003

-0.002

-0.001

0.001

quantum liquid (C F)2stripeCF

1/31/5

M=1M=2

0.1 0.2 0.3 0.4 0.5

11/27 7/17 5/12

− 0.001

− 0.002

− 0.003

− 0.004


cohe

sive

ene

rgy

2/5

electronic filling factor

s=1p=2

Quantum liquid (C

�

F) ceases to be ground state at

� � � � � � �


zooming in on the quantum hall effect - capri school · zooming in on the quantum hall effect...

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