Composite 21

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1 Composite 21 A recap of the successes of the CF theory as it steps into adulthood, with emphasis on some aspects that are not widely known or appreciated. CF pairing at 5/2? Nature of FQHE for three body interactions?

2 FQHE 1.5 T ~ 35 mk R xx (kω) 1.0 5/2 3/2 1 3/4 2/3 1/2 2/5 1/3 2/7 1/ / /23 4/11 6/ MAGNETIC FIELD [T] FQHE provides a new paradigm for collective behavior in condensed matter physics. No BEC. No order parameter. No ODLRO. It is the quintessential example of a topological quantum liquid. A wealth of unprecedented concepts have been inspired by the FQHE.

3 CF theory: Intuitive physics Electrons capture 2p vortices to turn into composite fermions. Composite fermions behave as weakly interacting particles to a good approximation. The vortices produce Berry phases which effectively cancel part of the external magnetic field. Composite fermions experience a reduced effective magnetic field. Composite fermions form Λ levels of electrons) in the reduced magnetic field. levels (analogous to the Landau The FQHE of electrons is a manifestation of the IQHE of composite fermions. (This unification was the original motivation for postulating the formation of composite fermions.)

4 CF theory: Mathematical formulation ν = Ψ ν = P LLL Φ ν ν 2pν ± 1 (z j z k ) 2p j<k B = B 2pρφ 0 ( φ 0 = hc e ) The CF theory makes precise prediction for the structure of the low energy Hilbert space. It expresses the (unknown) FQHE wave functions for ground and low-energy excited states at arbitrary fillings in terms of the (known) IQHE wave functions at the effective magnetic field. The wave functions of incompressible FQHE states and their excitations contain no adjustable parameters. Their form is completely fixed once we postulate composite fermions. (Can they be taken seriously?) These enable a rigorous test of the CF theory, as well as calculation of numbers from a microscopic theory.

5 Too good to be true? The CF theory makes precise, falsifiable statements about both the microscopic solution and its physical meaning. Both have been subjected to extensive and decisive tests over the last 21 years.

6 Computer experiments Calculate the exact eigenfunctions and eigenenergies by a brute force numerical diagonalization. In an independent calculation, evaluate the eigenfunctions and eigenenergies from the CF theory. Compare the two sets of results. Comparisons are extremely powerful, because there are no fudge factors. The CF theory contains no adjustable parameters, thus precluding the possibility of ad hoc adjustments of CF results to match exact results.

7 Each comparison is nontrivial! Dev, Kamilla, Jain (CF energies) He, Xie, Zhang (exact diagonalization) L L

8 overlaps/energies ν n N overlap % agreement with no adjustable parameters! fully spin polarized FQHE partially spin polarized / spin singlet FQHE Wu, Dev, Jain, 1993

9 Away from the special fillings 1-4 CFs in the second Λ level 4 CFs in the second Λ level N=8, 2Q = 19 N=8, 2Q = 20 E [e 2 /!l] N=12, 2Q= L N=8, 2Q = 17 N=8, 2Q = 18 Scarola, Rezayi, Jain Jain and Kamilla L L

10 12.75 E (e 2 /") (E-bckg)/N (a)!=1/3 N=12, 2Q= (b)!=1/3 N=13, 2Q= E (e 2 /") (c)!=2/5 N=14, 2Q= (d)!=3/7 N=15, 2Q= L L E (e 2 /") 14.4 (E-bckg)/N (a)!=2/5 N=12, 2Q= (b)!=3/7 N=12, 2Q= L L Arkadiusz Wójs (Arek) Csaba Tőke

11 Literally more than a thousand comparisons have fully established the CF theory for the lowest LL FQHE. The CF theory gives a complete account of the low energy physics. It is also quantitatively accurate. (And this is only the zeroth level approximation -- the numbers can be improved by including Λ level mixing.) These studies also confirm the notion of effective magnetic field, Λ levels, and a deep connection between the IQHE and the FQHE. There is much more. Composite fermions exhibit many states and phenomena!

12 Many body states of composite fermions CF-IQHE (manifests as FQHE at 2pn ± 1 ) CF-FQHE (manifests as FQHE at 4/11 etc.) * CF Fermi sea (1/2 compressible state) Partially spin/valley polarized or unpolarized FQHE states Partially spin polarized CF Fermi sea Paired CF state (5/2) * CF Wigner crystal * n

13 Excitations Composite fermion theory gives a unified description of excitations. Charged excitations at are CF particles or CF holes. (The objects obeying fractional braid statistics are composite fermions.) Neutral excitations are CF excitons (with minima in the dispersion called rotons). The putative nonabelian quasiparticle at 5/2 is a majorana composite fermion, which is essentially half of a composite fermion. With spin, the excitations can be spin reversed CF particles, CF holes, or CF excitons.

14 Phenomena / quantum numbers of composite fermions effective magnetic field Λ levels semiclassical cyclotron orbits Shubnikov de Haas oscillations effective mass Fermi wave vector collective modes (excitons, rotons) thermopower

15 What s topological about the FQHE? Topological order is sometimes defined in terms of the absence of a Landau order parameter, or a genus dependent degeneracy of the ground state. In FQHE, topology enters most fundamentally through the very formation of composite fermions, which are topological particles (the vortex is a topological object). The topological character of composite fermions is responsible for the (experimentally confirmed) effective magnetic field, and thereby the FQHE and other phenomenology. All states of composite fermions are topological states of matter.

16 5/2: CF pairing and nonabelian braid statistics The proposal Moore, Read; Greiter, Wen, Wilczek; Read, Green; Das Sarma, Nayak, Freedman; Stern, Halperin Composite fermions form (approximately) a Fermi sea. However, the CF Fermi sea is unstable to a p_x+ip_y pairing of composite fermions. Gap opens to produce FQHE. The Abrikosov vortices of the paired CF state have zero mode solutions, which are majorana composite fermions. Majorana composite fermions obey nonabelian braid statistics. They can be used as nonlocal topological qubits, with braidings producing unitary rotations. Decoherence is suppressed due to topological protection (topological properties are robust to local disturbances).

17 A concrete realization of this physics is through the so-called Pfaffian wave functions, which are exact ground states of a model Hamiltonian with three body interaction (Moore Ψ Pf = Pf(M ij ) j<k(z [ and Read) ] j z k ) 2 exp 1 z k Ground state: ( M ij ) = [ z i z (paired CF state) j M ij = (z i η)(z j η ) + (i j) 2 quasiholes: (z i z j ) k (charge 1/4 for each) 2n quasiholes: M ij = n α=1 (z i η α )(z j η α+n ) + (i j) (z i z j ) Pf M ij = A(M 12 M 34 M N 1,N ) Ψ BCS = A[φ 0 (r 1, r 2 )φ 0 (r 3, r 4 ) φ 0 (r N 1, r N )] ( ) Does this describe the reality?

18 Pfaffian ground state comparison with the exact Coulomb 5/2 state: N overlap Scarola, Jain, Rezayi, PRL Overlaps not conclusive but decent.

19 Testing the Pfaffian quasihole wave functions N=10 2 quasiholes L = 1, 3, 5 respectively N=10 4 quasiholes This model predicts zero energy states a L = 0 2, 1 0, 2 4, 3 1, 4 4, 5 2, 6 3, 7 1, 8 2, 9 0, 10 1 No one-to-one correspondence he superscript between denotes the solutions the degeneracy) of the model and the Coulomb interactions. Lack of adiabatic connection.

20 Testing the Pfaffian quasihole wave functions N=10 2 quasiholes * * * L = 1, 3, 5 respectively N=10 4 quasiholes * * * * * * * * * ** * * This model predicts zero energy states a L = 0 2, 1 0, 2 4, 3 1, 4 4, 5 2, 6 3, 7 1, 8 2, 9 0, 10 1 No one-to-one correspondence he superscript between denotes the solutions the degeneracy) of the model and the Coulomb interactions. Lack of adiabatic connection. Toke, Regnault, and Jain, PRL

21 D E 2 E$19& (3) (3) C PfQH sector, for the.possibility that par- Ψ 0.6 as O which = Ψallows Φ The wave function 2 qh 0at or2 qh 2 qh $191 TABLE I: Overlaps between different the PfQH basiscombinations for two quasiticle braidings can produce linear bital angular momentum LCoulomb refers tointeraction the two quasihole eigen$1 holes and the lowest energy states for in of PfQH states, hence nonabelian statistics. (3) C Φ2 qh is two quasiholes state of H with quantum number Lz = L, and $89< the second Landau level at Q=2N-2. The overlaps are defined D the E E 2 energy state for the Coulomb interaction with the $89: L(3) lowest (3) C FIG. 1: Cha $8 $8 as O = Ψsame. The wave function Ψ at or 2 qh 2 qh Φ 2 qh quantum numbers. We note that for two quasiholes, the electrons at 2 (3) N = bital angular momentum L refers quasihole zero energy states of Hto the aretwo singly degenerate. C eigenthe presence (3) - quantum number Lz0.61 state ofnh= 10with = L, -and - Φ2 qh is The solid and the lowest the -Coulomb N =energy state - for interaction with the 1: sl thefig. ground same quantum the N = 14 numbers (3)We - note 0.13 that - for 0.39two- quasiholes, 0.27 respectively. for Ψpf 2 qh zero energy N states L =of0 H 2 are3 singly 4 degenerate (obtained by two delta between 0.54 quasiholes quasi the(d) - quasihole in-the four TABLE I: Overlaps the0.47 PfQH basis for two show th pole. 10 lowest 0.67 energy in - panel The holes and the states0.47 for Coulomb interaction split quasiholi (dotted the second Landau level at Q=2N-2. The overlaps are defined N L = 0D E E (3) (3) 2N 2. C as O = Ψ Φ. The wave function Ψ at or qh 2 qh 2 qh /3 (obtained bital0.67 angular momentum L0.21 refers to the two quasihole eigen the quasi (3) C state of HTABLE with quantum number Lz = L, Φ2 qh is -./.4 quasioverlaps between theand PfQH basis II: for four integrated the lowestholes energyand state for the Coulomb interaction with the the lowest energy states for Coulomb interaction in will exhib same quantum numbers. We note that for two quasiholes, the -./.3defined the second Landau level at Q=2N-1. The overlaps are (3) E 2 zero energy states of singly degenerate. PHN Dare(3) for four quasic TABLE II:asOverlaps the PfQH basis O = between Ψ Φ -./02 into 4 qh,j /N. We have take 4 qh,i i,j holes and the lowest energy for Coulomb interaction in have For two quasiholes, quasihole bandstates[6] analogous account the no N states degenerate forto each LWe value, Toke, Regnault, and Jain -./01 the Landau level atinq=2n-1. The overlapsspectrum are defined tion pote thesecond PfQH band is seen the exact Coulomb D E i and j refer to the corresponding degeneracy index. ThePoverlaps are notchigh, and 2 sometimes very low. (3) N as O = Φ Ψ /N. We have take into # # Overlaps

22 Landau level mixing appears to establish the Pf physics -- for ground state as well as quasiparticles and quasiholes Arkadiusz Wójs (Arek) Csaba Tőke

23 LL mixing not only renormalizes the two body interaction, but also induces a three body interaction. We use the Bishara-Nayak model for LL mixing. κ = e2 /ɛl ω c B = 2 15T κ = N=14, 2l= LL V 1 =0.04 Pf ground state + neutral exciton branch != L 0.0 Pfaffian L

24 N=12, 2l= != != Pf ground state + neutral exciton branch N=12, 2l= !=1.0!= L !=1.5 Pfaffian L squared overlap with Pf L= V L= ! 2

25 N=12, 2l= != != Two quasiparticles !=1.0!= L !=1.5 Pfaffian L squared overlap with Pf L=0 L=2 L= V 1 N=12, 2l= L=0 L=2 L= !

26 N=12, 2l= Two quasiholes !=0.0!= !=1.0!= L !=1.5 Pfaffian L squared overlap with Pf L= V 1 6 N=12, 2l= L= !

27 magnetic field (T) overlap overlap 2 with: p-h conj., Pf, APf dotted: 12+14@25 dashed: 14+16@29 solid: 16+18@33 overlap 2 with CF state overlap ! V 1 The Pf wins over the APf.

28 FQHE for purely three body interaction The Pfaffian model is exact for a short range three body interaction, which can be implemented in ultra-cold atoms / molecules. ARTICLES Three-body interactions with cold polar molecules H. P. BÜCHLER*, A. MICHELI AND P. ZOLLER Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria Institute for Quantum Optics and Quantum Information, 6020 Innsbruck, Austria * hanspeter.buechler@uibk.ac.at VOLUME 92, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S week ending 4 JUNE 2004 Exact Ground States of Rotating Bose Gases Close to a Feshbach Resonance N. R. Cooper T.C.M. Group, Cavendish Laboratory, Madingley Road, Cambridge, CB3 0HE, United Kingdom (Received 14 August 2003; published 4 June 2004) A=first pseudopotential B=second pseudopotential Arkadiusz Wójs (Arek) Csaba Tőke

29 Emergence of CFs for three body interaction

30

31 Phase diagram of FQHE for a purely three body interaction B/A /7 4/9 Pf CF Fermi sea!=3/5 paired CFs ! 2/3 weakly interacting CFs 5/7 7/9 4/5 Appearance of composite fermions for a three body interaction is surprising.

32 FQHE of composite fermions: 4/11 E [e 2 /!l] N=12, 2Q= L Try: exact Exact energy (Rezayi): (N=12) Energy of the CF w.f.: Overlap: 0.99 Chang and Jain The low energy Hilbert space is described in terms of composite fermions. The incompressible ground state is a 4/3 FQHE state of composite fermions.

33 What s the state at very low fillings? Filling (dimension) 1/3 (1,206) 1/5 (19,858) 1/7 (117,788) 1/9 (436,140) Laughlin Electron Wigner crystal Overlap^2 with the exact wave function for six particles

34 Composite fermion Wigner crystal Ψ CF WC = j<k (z j z k ) 2p Ψ WC Electrons bind fewer than the maximum available number of vortices, and use the remaining degree of freedom to form a crystal. The optimal value of 2p is determined by minimizing the energy. (For 1/5, 1/7 and 1/9, we have 2p=2, 4 and 6, respectively.)

35 Overlaps again Chang, Jeon, Jain, 2004 Filling (dimension) 1/3 (1,206) 1/5 (19,858) 1/7 (117,788) 1/9 (436,140) Laughlin Electron Wigner crystal CF Wigner crystal ?!! The CF Wigner crystal describes the actual crystal state remarkably accurately. (For 1/7 and 1/9, its energy is off by 0.016% and 0.006%.)

36 Composite Fermions Jainendra K. Jain Composite fermions are the fundamental building blocks of the FQHE. They are to FQHE what electrons are to the IQHE. The single principle of composite fermions explains most of the lowest LL phenomenolgy, makes nontrivial predictions. Composite fermions have been observed. Landau level mixing appears to establish the Pfaffian state at 5/2. FQHE for purely three body interaction has a rich phase diagram with a host of CF states, such as the n/(2pn+1) FQHE states; CF Fermi sea; paired state.