Composite Fermions

Jainendra Jain

The Pennsylvania State University

Klaus v. Klitzing Horst L. Stormer

Edwin H. Hall Robert B. Laughlin

Quantum Hall Effect

Daniel C. Tsui

H R

B

2D

Quantum Hall Effect

R H

B

Hall Effect

3D

2HhRie

= IQHE

•  Accuracy of quantization: 3 parts in 10 billion.

•  Universal effect, independent of sample type, geometry, or disorder.

Landau levels (2D)

0

degeneracy Bρφρ

ν = =

00

BLandau level degeneracy per unit area = , hce

φφ

=

Fermi sea Landau levels

chω

FE

Filling factor =

B

Origin of the IQHE

•  The IQHE is a manifestation of the Landau level quantization of the electron kinetic energy.

•  It can be understood in terms of non-interacting electrons. (Some disorder is also needed for plateaus.)

2ν =

(Laughlin, ‘81)

chω

Landau levels gap at integer fillings IQHE

1/3 plateau ! FQHE

( ) 213

HhRe

=

21/

1( ) exp[ | | ]4

mm j k l

lj k

z z z<

Ψ = − − ∑∏

j j jz x iy= −

Theory of 1/m

•  Gap at FQHE

•  Fractional charge (e*=e/m)

ν =1m

1m

•  1/3 plateau is indicative of a correlated state.

Goldman, Su de Picciotto et al. Saminadayar et al. Reznikov et al.

•  1/3 was just the beginning of an avalanche! •  Evidence now exists for more than 50 fractions in the lowest Landau level. Some examples are:

2HhRfe

=

1 2 3 4 2 3 6 2 10, , , , , , , , ,...3 5 5 5 7 11 23 9 21

f =

•  The denominator is an odd integer (exception: 5/2).

•  The longitudinal resistance vanishes as T 0.

Willett, Stormer, Tsui et al.

211/ 4( ) exp[ | | ]mm j k l

lj k

z z z<

Ψ = − − ∑∏

•  In Laughlin’s wave function, the exponent m must be an odd integer. It therefore only applies to 1/3, 1/5 ...

•  What about the other fractions? (Haldane, 1983; Halperin, 1984)

The FQHE Mystery

•  What physical principle describes the physics of the broad range of phenomenon?

•  What is the nature of the correlations? Why do they manifest themselves in such a rich, yet stunningly simple fashion?

•  Why does a gap open up at certain fractional fillings?

•  What fractions are possible? What is the order of their stability?

•  Why is the denominator odd? (Why is 5/2 an exception?)

•  What is the nature of the state at even denominator fractions, e.g. at ½?

•  What are the excitations?

•  What is the role of spin?

•  What is the quantitative theory?

GOAL ?? Given particles + strong interactions

= new particles + weak interactions

•  If such an identification can be made, mysterious phenomena that were impossible to understand in terms of the old particles become straightforwardly comprehensible as properties of nearly free particles.

•  The new particles are the “true particles” of the phenomenon. Only weakly interacting objects have a sufficiently well defined identity to deserve the title “particle.”

•  The original particles were the particles of the problem, the new particles are the particles of the solution. They embody the profound reorganization of the strongly correlated system.

non-perturbative perturbative

Q: What are the true particles of the FQHE? Not electrons.

Q: What principle unifies the fractional and the integral quantum Hall effects?

•  True particles of the FQHE are composite fermions.

•  Unification of FQHE and IQHE: FQHE is the IQHE of composite fermions.

COMPOSITE FERMION

What is a composite fermion?

Composite fermion = electron + 2p vortices

Since a vortex is similar to a flux quantum, it is intuitively useful to view the composite fermion as:

Composite fermion = electron + 2p flux quanta

2CF 4CF 6CF

The composite fermion theory •  Electrons transform into composite fermions by capturing 2p “flux quanta” of the external field.

•  Composite fermions experience a much reduced effective magnetic field.

*0 02 ( / )B B p hc eρφ φ= − =

*

*2 1pν

νν

=±

Interacting electrons

Non-interacting electrons

Wave functions for ground and excited states at arbitrary filling factors.

Laughlin-Jastrow factor

*2( ) pj k

j k

z zν ν<

Ψ = − Φ∏

*2

1/(2 1) 1( ) pp j k

j k

z zν ν= + =<

Ψ = − Φ∏

Special case: * 11 = 2p+1

ν ν= ⇔

2 214( ) ( ) exp( | | )p

j k j k llj k j k

z z z z z< <

⎡ ⎤= − − −⎢ ⎥

⎣ ⎦∑∏ ∏

2 1 21( ) exp[ | | ]4

pj k l

lj k

z z z+

<

= − − ∑∏

Laughlin’s wave function recovered with 2 1m p= +

It is interpreted as one filled composite-fermion Landau level.

13

ν =

Why composite fermions form

*2( ) pj k

j k

z zν ν<

Ψ = − Φ∏

The probability of two composite fermions coming close to one another vanishes as (as opposed to the typical for electrons).

Electrons can keep apart efficiently by putting on vortices and transforming into composite fermions.

2(2 1)pr +

2r

Are composite fermions fact or fantasy?

Comparing IQHE and FQHE

The dynamics of interacting electrons at B resembles that of non-interacting fermions at B*.

IQHE

FQHE

Source: Clark et al.; Tsui and Stormer

Effective magnetic field

EXPLANATION OF THE FQHE IQHE of composite fermions occurs at * nν =

Here, *

*2 1 2 1n

p pnν

νν

= =± ±

These are precisely the prominently observed fractions!

1 2 3 10, , ,..., 2 1 3 5 7 21nn

ν = =+

2 3 5 10, , ,..., 2 1 3 5 7 19nn

ν = =−

1 2 6, ,..., 4 1 5 9 25nn

ν = =+

2 3 6, ,..., 4 1 7 11 23nn

ν = =−

CF Landau levels IQHE of composite fermions

' 1ν ν= −

1 13⇔

2 25⇔

3 37⇔

Electrons + B composite fermions CF-LLs FQHE

*2( ) pj k

j k

z zν ν<

Ψ = − Φ∏

•  “Minimal” theory: The wave functions contain no adjustable parameters. The only variational aspect is the form of the wave function, which is fixed by the composite fermion physics. •  Too restrictive? The lack of adjustability may appear to be a serious shortcoming of these wave functions.

Microscopic theory

CF exciton Two filled CF Landau levels

25

ν =

Comparison with “computer experiments”

•  Exact: Exact eigenfunctions and eigenvalues can be obtained by brute force diagonalization of systems with up to ~10-15 electrons. The energies of the (projected) wave functions for composite fermions can also be calculated “exactly.”

•  No free parameters: The CF wave functions and the exact diagonalization do not involve any adjustable parameters -- no possibility of any ad hoc, case-by-case adjustment.

•  The comparisons are rigorous, unbiased and substantial.

21 2 j k jk

eHr≠

= ∑ Model •  lowest Landau level •  spin frozen •  no mass parameter

13

ν =

25

ν =

37

ν =

L L Dev, Kamilla, Jain (CF energies) He, Xie, Zhang (exact diagonalization)

0.05% accuracy with no adjustable parameters! •  The lack of free parameters turns out to be one of the strongest assets. •  Composite fermions not only provide a “physical picture,” but an accurate “microscopic theory.”

( CF particle and hole at maximum separation)

½ puzzle

•  At the half filled Landau level, composite fermions absorb all magnetic flux, giving B*=0.

•  Composite fermions form a Fermi sea here.

•  No gap no FQHE. Non-trivial physics behind the lack of FQHE at even denominator fractions.

Halperin, Lee, Read Kalmeyer, Zhang

CF Fermi sea

•  Experiments have established that the radius of the semiclassical cyclotron orbit near is governed by B* (rather than B):

**FhkR

eB=

•  Composite fermions exist in the vicinity of ½.

•  Fermi wave vector measured. Fermi sea confirmed.

•  Grand Unification: Composite fermions unify not only all fractions, the FQHE and the IQHE, but also describe states that do not exhibit FQHE.

Willett et al. Kang et al. Goldman et al. Smet et al.

CF Fermi wave vector

12

ν =

CF Fermi sea

The observation of composite fermions near ½ demonstrates that they are more general than the FQHE, with other manifestations.

Electrons + B

CF Landau levels

FQHE

Du et al. Manoharan et al. Pan et al. Coleridge et al. Leadley et al. Halperin, Lee, Read Murthy, Shankar Stern, Halperin Park, Meskini, Jain Bonesteel Morf et al. Scarola, Jain

CF mass

CF exciton

*chω

CF SdH oscillations

Leadley et al. Du et al. Manoharan et al.

CF thermopower

Zeitler et al. Ying et al. Tieke et al. Bayot et al.

Landau level Fan diagram is consistent with spin ½ for composite fermions.

Du et al. Kukushkin et al. Pan et al. Kang et al. Nicholas et al. Leadley et al. Melinte et al. Dementyev et al. Pinczuk et al. Yeh et al. Pan et al. Chughtai et al. Wu, Dev, Jain Park, Jain Lopez, Fradkin Murthy, Shankar Mandal, Jain Nakajima, Aoki

CF Spin

CF g-factor

Raman spectrum at 2/5

Pinczuk et al. Kukushkin et al. Davies et al. Mellor et al. Zeitler et al. Devitt et al. Kang et al. Dujovne et al. Girvin,MacDonald,Platzman Simon, Halperin Lopez, Fradkin Jain, Kamilla Xie Scarola, Park, Jain Murthy Mandal, Jain Quinn et al.

CF collective modes

Microwave absorption

(2002)

Why not a Fermi sea here?

There is growing evidence that the CF Fermi sea is unstable here to pairing of composite fermions, which opens up a gap, producing FQHE.

Willett et al. Pan et al. Moore, Read Greiter, Wen, Wilczek Bonesteel Rezayi, Haldane Morf Park et al. Scarola et al.

CF pairing

half filled second Landau level 5 122 2= + =

Why are composite fermions interesting?

The composite fermion is a remarkable particle.

•  Collective particle: The vortices – a part of the composite fermion – are made up of all other particles

•  Quantum particle: One of its constituents, the vortices, are quantum mechanical objects. While all particles are described by quantum mechanics, the very definition of the composite fermion requires quantum mechanics. The composite fermion could not exist in a purely classical world.

•  It is astonishing that composite fermions behave as ordinary fermions to a great extent.

A new particle in condensed matter

Composite fermions provide a dramatic example of “emergent behavior:” Fundamentally new and unexpected conceptual structures emerge when a large number of particles are put together.

Emergent Physics

“More Is Different” – P.W. Anderson (1972)

A new paradigm for a strongly correlated system

•  A string of dramatic experimental clues has led us to an understanding of the new state of matter. •  The single principle of composite fermion gives a simple, unified, and accurate understanding of the new state of matter, without requiring any parameters!

Super simple solution!

•  All states degenerate in the absence of interactions. For 100 electrons, say at 2/5, there are degenerate ground states! •  On purely theoretical grounds we have no clue where to start. The problem appears hopeless.

7210:

Super non-perturbative problem:

Macroscopic Quantum Phenomenon

•  Single-valuedness of the wave function requires that the vorticity (2p) of the composite fermion be an integer. This exact, topological quantization of the vorticity lies at the root of the exactness of the quantization of the Hall resistance.

•  Antisymmetry under exchange requires that the vorticity be an even integer, which results in the odd denominator rule.

*2( ) pj k

j k

z zν ν<

Ψ = − Φ∏

Microscopic principles of quantum mechanics have dramatic macroscopic manifestations.

Other Noteworthy Aspects •  Mass generation Massive particles appear in a theory that had no mass to begin with.

•  Quantum mechanical renormalization of the vector potential Part of the Aharonov-Bohm phase is canceled by the phases generated by the vortices of composite fermions.

•  Quantized screening / fractional charge Screening takes place through binding of precisely 2p vortices.

•  Pairing from repulsive interactions Even though electrons repel, composite fermions can attract!

Thanks! Phil Allen Nick Bonesteel Sankar Das Sarma Fred Goldhaber Vladimir Goldman Steve Kivelson Noureddine Meskini Ganpathy Murthy Nick Read Ed Rezayi R. Shankar Doug Stone Nandini Trivedi Xincheng Xie Fuchun Zhang

Lotfi Belkhir Gautam Dev Ken Graham Rajiv Kamilla Tetsuo Kawamura Seung-Yeop Lee Sudhansu Mandal Kwon Park Vito Scarola Xiao-Gang Wu Lizeng Zhang

Congratulations!