Composite Fermions. Jainendra Jain. The Pennsylvania State University

Transcription

1 Composite Fermions Jainendra Jain The Pennsylvania State University

2 Quantum Hall Effect Klaus v. Klitzing Horst L. Stormer Edwin H. Hall Daniel C. Tsui Robert B. Laughlin

3 3D Hall Effect 2D Quantum Hall Effect R H R H B B

4 R H h = IQHE 2 ie Accuracy of quantization: 3 parts in 10 billion. Universal effect, independent of sample type, geometry, or disorder.

5 Landau levels (2D) E F B hω c Fermi sea Landau levels B Landau level degeneracy per unit area =, φ 0 φ 0 = hc e ρ Filling factor = ν = = degeneracy ρφ B 0

6 Origin of the IQHE (Laughlin, 81) Landau levels gap at integer fillings IQHE hω c ν = 2 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.)

7 FQHE 1/3 plateau! R H = ( 1 ) e 2 3 h

8 Theory of 1/m 1/3 plateau is indicative of a correlated state. Ψ = m 2 1/ m ( zj zk) exp[ zl ] j< k 4 l Gap at ν = 1 1 FQHE m m Fractional charge (e*=e/m) 1 zj = xj iyj Goldman, Su de Picciotto et al. Saminadayar et al. Reznikov et al.

9 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: R H = h fe 2 f = ,,,,,,,,, The denominator is an odd integer (exception: 5/2). The longitudinal resistance vanishes as T 0. Willett, Stormer, Tsui et al.

10 Ψ = 1 2 1/ ( ) m m zj zk exp[ 4 zl ] j< k l 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)

11 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?

12 GOAL?? Given particles + strong interactions non-perturbative = new particles + weak interactions perturbative 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.

13 Q: What are the true particles of the FQHE? Not electrons. Q: What principle unifies the fractional and the integral quantum Hall effects?

14 COMPOSITE FERMION True particles of the FQHE are composite fermions. Unification of FQHE and IQHE: FQHE is the IQHE of composite fermions.

15 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 2 CF 4 CF 6 CF

16 The composite fermion theory Electrons transform into composite fermions by capturing 2p flux quanta of the external field. * B = B pρφ0 φ0 = hc e * ν = Interacting electrons 2 ( / ) ν * 2pν ± 1 Ψ = ( z z ) 2 p Φ ν j k * ν j< k Composite fermions experience a much reduced effective magnetic field. Laughlin-Jastrow factor Non-interacting electrons Wave functions for ground and excited states at arbitrary filling factors.

17 Special case: ν = 1 ν = 2p+1 * 1 Ψ = ( z z ) Φ 2 p ν = + = j< k 1/(2 p 1) j k * ν = ( z ) p j zk ( zj zk) exp( 4 zl ) j< k j< k l = j< k 2p+ 1 2 ( zj zk) exp[ zl ] 4 l Laughlin s wave function recovered with m= 2p+ 1 1 It is interpreted as one filled composite-fermion Landau level. ν = 1 3

18 Why composite fermions form 2 p Ψ ν = ( zj zk) Φ * ν j< k The probability of two composite fermions coming 2(2 p+ 1) close to one another vanishes as (as opposed to the typical r 2 r for electrons). Electrons can keep apart efficiently by putting on vortices and transforming into composite fermions.

19 Are composite fermions fact or fantasy?

20 Effective magnetic field IQHE The dynamics of interacting electrons at B resembles that of non-interacting fermions at B*. FQHE Source: Clark et al.; Tsui and Stormer Comparing IQHE and FQHE

21 EXPLANATION OF THE FQHE IQHE of composite fermions occurs at Here, * ν ν = = n * 2pν ± 1 2pn± 1 * ν = n These are precisely the prominently observed fractions! CF Landau levels IQHE of composite fermions n ν = =,,,..., 2n n ν = =,,,..., 2n n ν = =,,..., 4n n ν = =,,..., 4n ν ' = 1 ν

22 Electrons + B composite fermions CF-LLs FQHE

23 Microscopic theory Ψ = ( z z ) 2 p Φ ν j k ν j< k * ν = 2 5 Two filled CF Landau levels CF exciton 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.

24 Comparison with computer experiments H 1 = e 2 2 j k rjk Model lowest Landau level spin frozen no mass parameter 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.

25 ν = 1 3 ν = 2 5 ν = 3 7 Dev, Kamilla, Jain (CF energies) He, Xie, Zhang (exact diagonalization) L L

26 ( CF particle and hole at maximum separation) 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.

27 ½ puzzle

28 CF Fermi sea Halperin, Lee, Read Kalmeyer, Zhang 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.

29 CF Fermi wave vector Experiments have established that the radius of 1 the semiclassical cyclotron orbit near ν = is governed by B* (rather than B): 2 R * = hk F * eb Willett et al. Kang et al. Goldman et al. Smet et al. 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.

30 Electrons + B CF Fermi sea CF Landau levels The observation of composite fermions near ½ demonstrates that they are more general than the FQHE, with other manifestations. FQHE

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32 CF mass * hω c CF exciton 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

33 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.

34 CF Spin Landau level Fan diagram is consistent with spin ½ for composite fermions. CF g-factor 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

35 CF collective modes 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.

36 Microwave absorption (2002)

37 CF pairing 5 1 = 2 + = half filled second Landau level 2 2 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.

38 Why are composite fermions interesting?

39 A new particle in condensed matter 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.

40 Emergent Physics More Is Different P.W. Anderson (1972) 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.

41 A new paradigm for a strongly correlated system Super non-perturbative problem: 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. Super simple solution! 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!

42 Macroscopic Quantum Phenomenon Microscopic principles of quantum mechanics have dramatic macroscopic manifestations. 2 p Ψ ν = ( zj zk) Φ * ν j< k 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.

43 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!

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