emil j. bergholtz- one-dimensional theory of the quantum hall system

8/3/2019 Emil J. Bergholtz- One-dimensional theory of the quantum Hall system

1/29

One-dimensional theory ofthe quantum Hall system

Anders Karlhede (Stockholm)

Thors Hans Hansson (Stockholm)Janik Kailasvuori (Berlin)Emma Wikberg (Stockholm)Maria Hermanns (Stockholm)Eddy Ardonne (Caltech/Nordita)

Juha Suorsa (Helsinki/Oslo)Susanne Viefers (Oslo/Stockholm)

with:

Seminar atENS, ParisNovember 2008

Emil J. Bergholtz(MPI-PKS, Dresden)


2/29

I will:

1. Introduce the quantum Hall effect (QHE)

2. Solve it exactlyin a simple well defined limit

3. Convince you that this limit is relevant(I hope.......)

a) Hierarchyb) Gapless states

c) Non-abelian states


3/29

I

Vx

Vy

2 dimensional electron gas

x

y

B

T 0

The basic experiment

1. Sendcurrent I through sample

Vx Vyand2. Measure

Rxx = Vx/IRxy = Vy/I3.Plot and as functions ofB

Low temperature T - near 0KStrong magnetic field BClean sample


4/29

(Willett et al1987)

Rxx

(h /e2)

Rxy

B (incompressiblequantum liquids)

Quantum Hall states

Rxy =1

s

h

e2Rxx = 0

h Plancks constanteelectron charge

Metallic states

(compressiblequantum liquids)

s =1

2,1

4, . . .s 1/B

1/21234s

s = 1, 2, 3,... Integer QHE

von Klitzing, Dorda, Pepper 1980

Fractional QHE

Tsui, Strmer, Gossard 1982

s =

1

3 ,

2

5 ,

3

7 . . .

Non-abelian

Fractional QHE?

Willet et al 1987

s =5

2, . . .


5/29

Problem to understand:

What are they?

Groundstate and low energy excitationsdetermine everything.

Cold electrons moving on a surface in a perpendicularmagnetic field.

(+disorder)

Is the system gapped?


6/29

ONE ELECTRON IN A MAGNETIC FIELD B(two dimensions)

En = (n + 12)hc n = 0, 1, 2, 3...

c =eB

mc

Kinetic energy quantized

Landau levels

0

3

2

1

E/hc

spin

= number of filled LL (filling factor)Define

s 1/B

Each Landau level (LL) is highly degenerate.

One state per flux quantum 0 = hc/e

# of states B

The crucial parameter!

(within a Landau level)

Quantized area of electron state 10 nm

c

eB =

257AB

1 teslaA = 2 ;


7/29

= 1IQHE

unique state

empty one electron state

filled one electron state

Fixed area per state 0 = hc/e(one state per flux quantum )

?

incompressible liquid

!

= 1/3FQHE

673132974506580171230064

degenerate states

10 nm

(Real samples much bigger)


8/29

Standard theory - brief review

e = e/(2m + 1)incompressible quantum liquid with particles = 1/(2m+ 1)

The fractionally charged qps condense and form an incompressible liquid just aselectrons condensed at 1/(2m+1). Iterating this gives all odd. = p q, qHierarchy:

(Haldane 83, Halperin 83)

(Jain 89)

Composite

fermions:

and/or

Electrons form new particles, composite fermions,by absorbing magnetic flux. FQHE is IQHE of these composite fermions.Gives directly. = p/(2mp + 1)

= p/qBut other fractions less clear. = p/q, q(All odd, experimentally similar.)

Many-body wave functions

(Moore and Read 91)Non-abelions Appear (?) in higher Landau levels (and/or in rotating condensates)Motivated by conformal field theory (CFT-FQHE correspondence)

...and it goes on.....

(Laughlin 83)

Gapless states (Halperin, Lee and Read 93)Half filled Landau level; free composite fermions


9/29

= 1/3

(By Kwon Park)

composite fermion

Composite fermions at

Nice pictures, but (how) does it work?

(cf. ) = 1

incompressible liquid

!

Supported by:

Numerical tests of wavefunctions

Mean field theory and exp. near = 1/2

But:

Microscopic understanding

is lacking... (my opinion)


10/29

L1

L2

Each electron state has fixed area

(any shape is OK)

choose

(Landau gauge)

Consider sample with lenghts L1, L2

L2

L1Two dim electron gas

Our approach: a one-dimensional view(and a solvable limit)


11/29

L1

L2

One-dimensional lattice model

Each box is either empty 0 or filled 1

A possible state at = 1/31 0 0 1 1 0 0 0 1 0 0 0 0 1 0

1..0.....0..1 0..1.....1..0 (all ee-terms that preserve position of CM)

Hamiltonian (ee-interaction) (egCoulomb V(r)=e /r)2 No kinetic energy!

k+m k-m

Vk,m

(electrostatic repulsion)including

1......1 Vk,0k

x

y k eik 2L1

ye(xk 2

L1)2/2

(Landau gauge)

Exact mapping of a single Landau level!


12/29

Idea: physics as varies L2 fixed and largeL1

L1

Experimental situation

Complicated 1D interaction

L1 L

2 large

L2

Smooth development (we claim)

2/L1(Lattice constant: )

Simple 1D interaction

Solvable limit

smallL1


13/29

Exact solution

Hopping 1..0.....0..1 0..1.....1..0 makes ground state complicated.

L1 0But, when

hopping vanishes and only electrostatic repulsion remains: 1......1

This is a simple classical electrostatics problem!

States with electrons in fixed positions are the energy eigenstates -

groundstate obtained by separating the electrons as much as possible:

= 2/5

1 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 0

= 1/3

Unit cell

1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0

(Tao-Thouless (TT)states)

At ground state is TT-state with p electrons in unit cell of length q. = p q

gapped crystal


14/29

We claim:

Motivation is to come.

* Same qualitative properties

The TT-state at any odd, is a QH-state!1. = p/q, q

The stability of the state decreases when q increases.2.

(More stable states are seen at higher disorder.)

L* Develops smoothly as no phase transition

The QH hierarchy


15/29

100100100100100100100100100 groundstate

10010010100100100100100100 -e/3(domain wall separating degenerate groundstates)

(At , quasiparticles with charge ) = p/q e = e/q

Charge determined by Su-Schrieffers counting argument:

100100100100100100100100100100100100100100100100100100

100101001001001001010010010010010010100100100100100100

-2e-3e

Excited states: fractional charge as L 0

= 1/3

Quasielectrons with charge -e/3 are obtained by inserting a 10:

e = e/3


16/29

Stability of states ~ 1/q

Disorder destroys a QH-state - the more energy it takes

to disturb a state, the more stable it is.

-e/q +e/q

The smallest energy disturbance is creation of a pair:e/q

Stability ~ gap ~ 1/q

(This argument is correct in spirit, not in detail.

However, the result is correct.)


17/29

Parents and daughters - the hierarchy

Add many quasielectrons 10 to 1/3, lowest energy states:

100101001010010100101001010010100101001010010100101001010010.... 10210, = 2/5

10010010010100100100100101001001001001010010010010100100100100.... (102)310, = 4/11

100100100100100100100100100100100100100100100100100100.... 100 = 102, = 1/3

100100100100100101001001001001001001010010010010010010010.... (102)510, = 6/17

Condensation of quasielectrons in give states with

unit cells - the ground states at .(102)2m110 = 2m/(6m 1) = 1/3

1/q

p/q

Stability

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

!!p/q

"#q

Fractal structure


18/29

Phase transitions:

Two assumptions:

1) Condensation of quasielectrons or quasiholes

gives phase transition from one QH state to

another

2) More stable states are seen at higher disorder

leads to......


19/29

......the phase diagram

0.25 0.3 0.35 0.4 0.45 0.50

0.05

0.1

0.15

0.2

0.25

0.3

1/3

2/5

2/7 3/7

3/11

4/9

4/11

4/13 5/135/17 6/17

6/19

7/177/19

8/19

!=p/q

1/q

disorder

1/4 1/2Shown here for

- all regions look the same.

Experiment, varying B

.....at lower disorder


20/29

The observed states:

Stability ~ 1/q agrees with experiment

0.2 0.4 0.6 0.8 1

p

q

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1

q

range of experiment

stability

CF states

non-CF states

States observed by

Pan et al (2003)


21/29

TT-states 100, ... develop smoothly into bulk QH-states Surprising - but true!

Same qualitative properties (gap, quasiparticles, ...)

New unique bulk wave function for any state that is obtained by

successive condensation of quasielectrons. (conformal field theory construction)

Gives TT-states as L 0

Gives Laughlin/Jain wave functions where these exist

Going to the bulk L the crucial question

Numerical studies (exact diagonalization and DMRG)

Supported by numerics for simplest non-L/J state = 4/11

Overall picture. More to come...

Same structure from a different perspective

Proven for Laughlin states and pseudopotential interaction


22/29

Gapless states =1/2

But: metallic, ie gapless state as (experiment & theory)L ?!

This solution develops smoothly as L (numerical evidence)

TT-state 10101010.... ground state when (has a gap )L 0

Phase transition to gapless state at finite L

Interacting electrons in magnetic field free neutralparticles in 1 dim

composite fermions!?

(valid at small but finite L)- exact solution for gapless state


23/29

Exact solution at = 1/2

Hamiltonian: V211001 0110 V10 = 2V20and (rest =0)

(Good approximation for small L)

Ground state is 1d Fermi sea of free neutral dipoles

composite fermions!?

Free particles not affected by magnetic field

The TT state melts at finite L

(Note the contrast to the1/3 state 100100100...)

101010... is annihilated by the shortest range hopping 1001 0110Competition between electrostatics and hopping

Luttinger liquid!Corrections

= V21

i

(s+is

i+1+H.c.)

10 , 01

Low energy sectorcontained in spin space


24/29

Luttinger liquid(s) describe the bulk at half-filling!?

Towards the bulk...L( )

9.08.2 8.4 10.6

10101010....

0.993 0.996 0.995 0.998

L5.3

(5,2) (6,4) (8,3) (0,0)

1

Solvable!Luttinger liquid

Solvable!

Smooth transitions

Overlap1.000

Exact diagonalization:

Bulk state believed to be described by the Rezayi-Read wave function

free 2d composite fermions


25/29

Non-trivial groundstate degeneracy

Non-abelian QH statesL 0(as )

Domain wall excitations with non-trivial chargeand degeneracy.

Crucial for non-abelian statistics!

(Examples to come....)

Consistent with CFT results where these exists (and

gives clues on how to construct new theories).

Appear (?) in higher Landau levels (2D electrons and/orin rotating condensates)

Interaction (matrixelements) changes


26/29

The Moore-Read pfaffian(the example....)

Six ground states

101010101010101010101010

110011001100110011001100

Three-body interaction ---never three electrons on four sites

Half quasiholes as domain walls

101010101010101010101010101010101010101010101010

101010100110011001100101010101001100110011001010 e

Supposedly describes the state = 5/2 ( 1/2)

e/4 e/4 e/4 e/4

e

= e/4


27/29

Quasihole degeneracy

10101010100110011001100

100110011001100or Two possibilities!1010101010

Create a pair of qhs

Leads to 3x2n/2 degenerate states for n qh s at fixed positions.

Generalizes to quasiparticles by particle hole symmetry.


28/29

Total degeneracy for n quasiholes is given by the number of periodic paths...

This can be generalized..... (here for the k=4, M=0 bosonic Read-Rezayi state)

These domain walls encode the fusion rules of the pertinent CFT

[04]

[13]

[22]

[31]

[40]

ddd

ddd

ddd

ddd

ddd

. . .

. . .

ddd

ddd

ddd

. . .

. . .

n = 0 1 2 3 4 5

Bratteli diagram


29/29

Conclusion

The quantum Hall problem is exactly solvable in a limitthat accommodate the rich structure of the system.

The solution is smoothly connected to the experimental

regime.

References:

Half-filling: Hierarchy: Non-abelian states:

Bergholtz and Karlhede,PRL 94, 026802, (2005)

Bergholtz and Karlhede,J. Stat. Mech. L04001, (2006)

Bergholtz, Hermanns, Hansson,and Karlhede,

PRL 99, 256803 (2007)

Bergholtz and Karlhede,PRB 77, 155308 (2008)

Bergholtz, Hermanns, Hansson,

Karlhede and Viefers

Bergholtz, Kailasvuori, Wikberg,Hansson, and Karlhede,

PRB 74, 081308(R) (2006)

Ardonne, Bergholtz, Kailasvuori, Wikberg,J. Stat. Mech. P04016, (2008)

emil j. bergholtz- one-dimensional theory of the quantum hall system

Documents