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

_ _ _ _ _ _

++ + + ++I

−I= nνHal

Integral QHE

Magnetic Field

longitudinal resistance

2D electrons

Historical Summary:

1980 : Discovery of theIQHE (v. Klitzing)

1983 : Discovery of theFQHE (Tsui, Störmer);

Laughlin :

incompressible quantum

liquids

électrons 2D

_ _ _ _ _ _

++ + + ++I

−I= nν

= p/(2ps+1)ν

Integral QHE

Magnetic Field

Fractional QHE

3/54/7

4/93/7

2D electrons

Historical Summary:

Laughlin :

liquids

électrons 2D

_ _ _ _ _ _

++ + + ++I

−I= nν

= p/(2ps+1)ν

Integral QHE

Magnetic Field

Fractional QHE

3/54/7

4/93/7

2D electrons

Historical Summary:

Laughlin :

liquids

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

4.03.83.63.43.2Magnetic Field (Tesla)

7/2 3+1/5

Reentrant IQHE

Magnetic field

3/54/7

4/93/7

Historical Summary:

Laughlin :

liquids

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

4.03.83.63.43.2Magnetic Field (Tesla)

7/2 3+1/5

Reentrant IQHE

Magnetic field

Self−similarity of the Hall curve

3/54/7

4/93/7

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 :

� � ��

IQHE: single-particle localisation

class.

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

R xyxxR

h/e n2

class.

(n+1)LL

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

R xyxxR

h/e n2

class.

(n+1)LL

electrons in partially filled LL trapped by impurities ( )

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

R R xyxyxx xxRR

h/e n2

class.

(n+1)LL

ε ε ε

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

h/e n2

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

0.8 v (r)n

r/10lB20 40 60 80 100

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

universal function

2 length scales:

� and

(interparticle separation)

Wigner Crystal and Bubbles

A. Wigner Crystal(WC):

� �

Quasi-classical limit

B. FQHE:

C. Bubbles (super-WC):

Not in LLL,

d’ dd’ 2d−d’>2Rc

~energy = u

energy = 2 u

� �

B. FQHE:� � �

Not in LLL,

~energy = u

energy = 2 u

� �

B. FQHE:� � �

� ��

Not in LLL,

� ��

~energy = u

energy = 2 u

Energy of competing ground states

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)

Rlong RHall

5 10 15 20 25

Partial filling of the last level

−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

M=2 (electrons per site)

quantum liquids

impurities

800400

4.0 3.8 3.6 3.4 3.2

Magnetic Field (Tesla)

0.350.30

7/23+1/5

� � � ��

0.1 0.2 0.3 0.4 0.5

1/31/5

crystal liquid crystal liquid crystal

filling of the last level

Rlong RHall

5 10 15 20 25

−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

quantum liquids

impurities

800400

4.0 3.8 3.6 3.4 3.2

0.350.30

7/23+1/5

� � � ��

0.1 0.2 0.3 0.4 0.5

1/31/5

1/3.5[h

Rlong RHall

5 10 15 20 25

−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

quantum liquids

impurities

800400

4.0 3.8 3.6 3.4 3.2

0.350.30

7/23+1/5

� � � ��

0.1 0.2 0.3 0.4 0.5

1/31/5

1/3.5[h

Rlong RHall

5 10 15 20 25

−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

quantum liquids

impurities

800400

4.0 3.8 3.6 3.4 3.2

0.350.30

7/23+1/5

� � � ��

0.1 0.2 0.3 0.4 0.5

1/31/5

1/3.5[h

Rlong RHall

5 10 15 20 25

−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

quantum liquids

impurities

800400

4.0 3.8 3.6 3.4 3.2

0.350.30

7/23+1/5

� � � ��

0.1 0.2 0.3 0.4 0.5

1/31/5

1/3.5[h

Rlong RHall

5 10 15 20 25

−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

quantum liquids

impurities

800400

4.0 3.8 3.6 3.4 3.2

0.350.30

7/23+1/5

� � � ��

0.1 0.2 0.3 0.4 0.5

1/31/5

1/3.5[h

5 10 15 20 25

-0.175

-0.125

-0.075

-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

Partial filling factor

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

partial filling factor

mixed phase

Mixed phase

Wigner crystal/Bubble

Pinning mode at

0.15 0.26M=1

mixed phase

Mixed phase

Wigner crystal/ Bubble

Pinning mode at

��

0.15 0.26M=1

mixed phase

Mixed phase

Pinning mode at

��

2.01.51.00.5 f(GHz)

ν=4.26 Data fit peak 1 peak 2

ν=4.21

ν=4.16

ν=4.12

] (µS

ν=4.18

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

0.15 0.26M=1

mixed phase

Mixed phase

Pinning mode at

��

f pk (

4.354.304.254.204.154.10

Discovery of a new FQHE at � � � � �

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

= p/(2ps+1)ν

magnetic field

3/54/7

4/93/7

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

magnetic field

3/54/7

4/93/7

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

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

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

filling 1/3 of first excited CF level

theory

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

theory

1filled CF level

Explanation of

� ��

state (??):IQHE of C

�Fs � � � ��

ν∗ = 1+1/3

theory

1filled CF level

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

ν∗ = 1+1/3

1.st generation CF

intra-level

ν∗ = 1+1/3

1.st generation CF

intra-level

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

0.0075

0.0125

0.0175

2 3 4 5 6 71

(ν=6/17)(ν=7/19)(ν=11/31)

4/11(ν= )

width in units of

s=1, p=1

s=1, p=2

s=2, p=1

s=2, p=2

s=1 ~ ~

2 4 6 8 10 12 14

1 2 3 4 5 6 7

(ν=4/19)(ν=6/29)(ν=7/33)(ν=11/53)

lB width in units of

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

-0.015

-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

Electronic filling factor

partial CF filling factor

quant. liquids (FC )

(CF Wigner crystal)

(CF bubbles)

CF stripes

FC state stable at

Reentrance in the

Reentrant FQHE

5 10 15 20 25

-0.015

-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

(CF Wigner crystal)

(CF bubbles)

CF stripes

state stable at

� � � ��

Reentrance in the

Reentrant FQHE

5 10 15 20 25

-0.015

-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

(CF Wigner crystal)

(CF bubbles)

CF stripes

state stable at

� � � ��

Reentrance in the

Conclusions I: Self-similarity of QHE

= p/(2ps+1)ν

Integral QHE

Magnetic Field

Fractional QHE= IQHE of CFs 4/11

Self−Similarity

of the Hall Curve

3/54/7

4/93/7

� �� 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)

= p/(2ps+1)ν

Integral QHE

Magnetic Field

Self−Similarity

of the Hall Curve

3/54/7

4/93/7

theory of theFQHE

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

= p/(2ps+1)ν

Integral QHE

Magnetic Field

Self−Similarity

of the Hall Curve

3/54/7

4/93/7

theory of theFQHE

Model reveals self-similarity of the FQHE

� new hierarchy scheme (higher CF generations)

Conclusions II: Phase Diagram

−electroncrystals

� � � � �

0.0 0.1 0.2 0.3 0.4 0.5

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

Perspectives

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

Quantum Hall Ferromagnet at � � �

m=N −1φm=0.....

m=1 m=2

Perspectives

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

Magneto-excitons � bosons

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

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

+ : impureté: électron

0.0 0.1 0.2 0.3 0.4 0.5

��

� � � � � � � � � �

� � � � � � � � �

aggrandissement :

auto−similarité

îlots à électrons

(EHQF)

phases de FC

5/2, 7/2EHQF

� � � � �

1/3 2/5

remplissage él.

remplissage FC

4/11 5/13p=1

isolantsFCs

rubans de FCniveau

� � �

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

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

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

Generation:

(q=3)p =1i

encircled states are stable

(non-Laughlin)

Continued fraction:

� �

� �� $ � ��

� � : positive integer

� : odd integer

��

2/5 2/7

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

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

Generation:

(q=3)p =1i

(non-Laughlin)

Continued fraction:

� �

� �� $ � ��

� � : positive integer

� : odd integer

��

2/5 2/7

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

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

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

modular group

Continued fraction:Fixed point ( series,

-th CF generation(

� � *

��

� � : C

� modular group

��

� �

Continued fraction:Fixed point ( series,

-th CF generation(

� � *

��

� � : C

� modular group

��

� �

Continued fraction:

� �

� � � �

� � $ �

��

Fixed point ( series,):

-th CF generation(

� � *

��

� � : C

� modular group

��

� �

Continued fraction:

� �

� � � �

� � $ �

��

Fixed point (

� � �

series,

��

� � � � � � � � �

CF Interaction Potential

Haldane’s pseudopotential expansion

� � �� $ � ��

��

� � � �

�� $ � � � � � ��

2 4 6 8 10

1 5 1197 13 15 17 19 21m

−0.01

p=1p=2

CF Phases in

� � � � �

5 10 15 20 25

-0.004

-0.003

-0.002

-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

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

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

electronic filling factor

s=1p=2

Quantum liquid (C

F) ceases to be ground state at

� � � � � � �

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quantum hall effect - ucsb...

exact fractional quantum hall states

anomalous quantum hall effect- an incompressible quantum...