Quantum numbers and collective phases of composite fermions

Transcription

1 Quantum numbers and collective phases of composite fermions

2 Quantum numbers Effective magnetic field Mass Magnetic moment Charge Statistics Fermi wave vector Vorticity (vortex charge)

3 Effective magnetic field Composite fermions is that they experience an effective magnetic field, much reduced compared to the applied magnetic field. The effective field has a topological orgin: it is a direct measure of the vorticity of composite fermion. The effective magnetic field is responsible for many features of the FQHE liquid. The effective magnetic field is internal to composite fermions. Composite fermions themselves must be used to measure it. (An external magnetometer will measure the applied magnetic field.) The phenomenon has a quantum mechanical origin, and is very different from the Meissner effect in superconductors.

4 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 B and B*

5 Cyclotron radius CF e B B* Fermion statistics; Fermi wave vector; Intrinsic charge Antidot resonances (Kang) SAW (Willett, 1993) Magnetic focusing (Goldman; Smet)

6 CF mass

7 CF exciton! '(!8*7$" '(!.$/$" 1$"4!5*6212"4/3 '(!270,/$" 9&1!"#$!"%&'(!)!*+,-*./,012 &&&&'(!)!*+,3$12&-*,. Schematic dispersion of CF exciton

8 /3! ex [e 2 /"l] /9 1/5 1/7 2/ kl Jain and Kamilla The number of strong minima correlates with the numerator. Also, the energy at the roton minimum becomes negative for 1/9, suggesting an instability of the FQHE sate. 8

9 The large wave vector limit of the energy corresponds to a far separated particle hole pair of composite fermions, and is identified with the transport gap. [ρ xx exp ( /2k B T )] The composite fermion mass is defined by interpreting this energy as the effective cyclotron energy of composite fermions.

10 B dependence of m* ( ρ xx exp ) 2k B T = ωc = eb m c = eb (2pn ± 1)m c However: = C e2 ɛl B Therefore, we must have: m B Because the magnetic field does not change substantially along a FQHE sequence, m* can be taken to be approximately constant. That predicts a linear opening of the gap away from 1/2.

11 [ρ xx exp ( /2k B T )] Du et al.

12 CF exciton theory

13 How do the actual numbers compare?! E g [e 2 /!l] $?< #?$$ >?B =?C <?# $?= / /N #%F' <?$l 0 G!"$!!"!# 4/9!"!!!"$! "D<"=E$! $$ -2!< %%&'()%*+,-./'00%1,2,* %%34%*+')(5%6%789 %%':;'(,2'/* "D$"$E$! $$ -2!<!"!#!"!!!"!!"$!"<!"=!"> $?@</6$A

14 We must remember that while the CF mass is a useful way of parametrizing the energy gaps, it is not a fundamental property of composite fermions, and should not be overinterpreted. It is very sensitive to various parameters, such as magnetic field, disorder etc. That follows from the fact that there is no bare mass for composite fermions; the entire mass is generated from interactions alone.

15 Comparison of the CF exciton energy in the q=0 limit A. Pinczuk, M. Kang, et al. Park, Scarola, Jain Better quantitative theory of neutral modes than charged excitations.

16 Splitting of the collective mode Hirjibehedin, Pinczuk et al.

17 energy h! c "* energy # " quantum number quantum number 0.2 Energy kl Majumdar, Mandal, 2008

18 Condensed states of composite fermions IQHE states Fermi sea FQHE states Paired states Quantum crystals Partially spin-polarized FQHE states and Fermi sea States of quantum dots, rotating BEC, graphene (numerical experiments) Some of these states are explicable in terms of noninteracting composite fermions, whereas the others are produced by the weak residual interaction between composite fermions.

19 FQHE of composite fermions

20 Pan et al. 2003

21 The new fractions cannot be understood in terms of a model of noninteracting composite fermions, which epxlains FQHE only at fractions of the form ν = n 2pn ± 1 and ν = 1 n 2pn ± 1 We need to consider interacting composite fermions. It is natural to suspect that the next generation fractions correspond to the FQHE of composite fermions.

22 The residual interaction between composite fermions is a remnant of the Coulomb interaction between electrons. As true for any bound states, the inter-cf interaction is very complex. It is also very weak. (The temperature scale for 4/11 is much smaller than for the nearby 1/3 or 2/5 state.) Q: Do composite fermions provide a precise enough understanding of this strongly correlated state to capture the subtle physics of the next generation FQHE?

23 CF diagonalization Mandal, Jain, 2002 Construct a basis of all states of composite fermions with the lowest kinetic energy. Diagonalize (numerically) the Coulomb Hamiltonian in that basis to obtain the spectrum. The method produces very accurate results. Rezayi (exact); Mandal, Jain (CF diagonalization)

24 Microscopic confirmation Try: Chang and Jain, 2004 exact Exact energy (Rezayi): Energy of the CF w.f.: (N=12) Overlap: 0.99 Remarkable accuracy (0.07%) for a 12 particle FQHE state. Establishes the physics of the next generation states as the FQHE of composite fermions.

25 One complicated structure on top of another: First, electrons capture two flux quanta to turn into composite fermions, with the lowest quasi-landau level fully occupied and the second partially occupied. The composite fermions in the second level capture two more flux quanta to turn into higher order composite fermions. These fill their own quasi-landau levels to produce new FQHE. It would be hard to take such a state seriously if it didn t really occur.

26 Paired state of composite fermions

27 5/2 FQHE Willett et al. Pan et al. Xia et al.

28 Pfaffian wave function ( ) [ The paired state is described by a Pfaffian wave function (Moore and Read): ( ) 1 Ψ Pf 1/2 = Pf (z i z j ) 2 exp z i z j i<j Pf M ij = A(M 12 M 34 M N 1,N ) ds for Pfaffian. Compare with: [ 1 4 ] z k 2 Without the Pfaffian factor, the wav Ψ BCS = A[φ 0 (r 1, r 2 )φ 0 (r 3, r 4 ) φ 0 (r N 1, r N )] ( ) This is represents a paired state of composite fermions. k

29 Understanding 5/2 FQHE without the Pfaffian wave function Toke and Jain, 2006 The Pfaffian wave function does not contain any adjustable parameters. It does not produce the CF Fermi sea as one limit. We begin with noninteracting composite fermions and ask if the residual interaction between composite fermions opens up a gap.

30 CF diagonalization α Z Z

31 Lowest Landau level )*)! +,- ".!/ 0 1!(!!%!(!!# 2 3 4&" 2 3 4&# 2 3 4&$ 2 3 4"!!(!#!(!5 )*)! +,- ".!/ 0 1!(!"!(!&!! " # $ % &! ' " # $ % &! &" ' " # $ % &! &" ' " # $ % &! &" &# '

32 Second Landau level )*)! +,- ".!/ 0 1!(!$!(!#!(!"!(&"!(& 2 3 4&$ 2 3 4"! )*)! +,- ".!/ 0 1!(!%!(!$!(!#!(!" 2 3 4&" 2 3 4&#!! " # $ % &! ' " # $ % &! &" ' " # $ % &! &" ' " # $ % &! &" &# ' *Residual interactions between composite fermions open a gap at 5/2. The size of the gap (~0.02) is consistent with Morf s estimates from exact diagonalization studies. *Nonabelian statistics does not appear naturally in this picture.

33 BCS wave function for composite fermions Moller and Simon, 2007 Ψ CF BCS = P LLL Ψ BCS (z j z k ) 2 j<k Ψ BCS = Pf[g( r j r k )] g( r j r k ) = k g k φ k (z j )φ k (z k ) The parameters g_k are determined variationally.

34 The earlier Pfaffian wave function can be well approximated by this form, indicating that it belongs to a more general class of wave functions. The CF-BCS wave function also reduces to the CF Fermi sea in one limit. The adiabatic connectivity of the excitations has not yet been established, however.

35 A topological quantum crystal of fermions

36 At very low fillings, electrons are far from one another, so one may expect them to behave classically and form a Wigner crystal. An insulating state is observed at low fillings, which is interpreted as a pinned crystal. The most natural wave function for the electron crystal (EC) is a Hartree Fock wave function, which can be projected into a definite angular momentum state and compared to the exact state. (Maki and Zotos; Yannouleas and Landman)

37 Transition from CF liquid to WC (6) Egs (6) (12) (18) 2/5 3/7 4/9 5/ /N E gs - E cl [e 2 /"l] Composite Fermion Liquid (Jain & Kamilla) Correlated Wigner Crystal (Lam & Girvin) E cl = !!"# e 2 /"l Egs (4) (2) 2/9 3/ ! /N Jain and Kamilla, 1998

38 Overlap^2 with the exact wave function for six particles Filling 1/3 1/5 1/7 1/9 (dimension) (1,206) (19,858) (117,788) (436,140) Laughlin Electron crystal

39 Composite fermion crystal Yi, Fertig, 1998 Narevich, Murthy, Fertig, 2001 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.)

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

41 ν L Exact CFC WC Laughlin 1/ (9) (3) 1/ (5) (2) 1/ (2) (2) 1/ (1) (1)

42 The CF crystal remains valid to arbitrarily low fillings. The quantum regime is valid at available temperatures. E.g., below 0.2 K for 1/9 at 25 T.

43 CF crystal The lowest Landau level crystal is not a simple Wigner crystal of electrons, but an inherently quantum mechanical crystal of composite fermions with long range quantum coherence. The binding of electrons and quantized vortices is a nonperturbative quantum effect. Because each composite fermions sees vortices on every other composite fermion, there is long range quantum coherence in the CF crystal state. Just as the CF liquid behaves qualitatively differently from the electrons liquid, the CF crystal can be expected to have properties distinct from the electron crystal. First fermion crystal with non-trivial quantum behavior.

44 FQHE in a spin In the very large B limit, the spin degree of freedom is frozen. FQHE typically occurs in high magnetic fields. However, the Zeeman splitting in GaAs is quite small: Therefore, it may sometimes be energetically favorable for electrons to reverse their spin if they can gain more in interaction energy. In vacuum: E Z = 2gµ e BS z = gµ e B = eb m e c GaAs: effective mass differs differs by a factor of 14 and the g by a factor of 5. In GaAs:

45 Early theoretical work (based on exact diagonalization studies) Xie, Guo, Zhang, 1989 The excitations of 1/3 involve spin reversal at small Zeeman energies. Many FQHE states are not fully polarized at small Zeeman energies. Generalization of Laughlin s w.f. to spin singlet states at 2/5, 2/9, etc. Halperin (1983), Zhang and Chakraborty (1984), Maksym (1989)

46 Early experimental work The Zeeman energy can be changed continuously in tilted field experiments. The FQHE states at 8/5, 4/3, 2/3, 3/5 were found not to be maximally polarized for a range of parameters. Clark et al. (1989) Eisenstein et al. (1989)

47 Questions What are the possible spin polarizations for various FQHE states as the Zeeman energy is varied from zero to infinity? Ground state wave functions? Excitations? Quantitative theory? Spin phase diagram of the FQHE? Polarization of compressible states, e.g. at?

48 Composite fermions with a spin CF theory can be generalized straightforwardly to include the spin. Consider both spins for composite fermions at the effective magnetic field. The CF theory makes a prediction for the possible spin polarizations for all filling factors.

49 Schematic evolution of the CF state with Zeeman energy Example: 4/9 state (4 filled CF-LLs) Three distinct spin polarizations predicted at 4/9. Similar analysis can be carried out for other fractions. The state at zero Zeeman energy is unpolarized for even numerators, and partially polarized otherwise.

50 Quantitative tests of the microscopic wave functions for non-fully polarized FQHE states Wu, Dev, Jain, 1993

51 Transitions as a function of Zeeman energy Du et al. 1995

52 Landau level fan diagram for spinful composite fermions Consistent with free composite fermions with spin 1/2 Du et al. 1995

53 Spin polarization from photoluminescence Evidence for certain additional states that would require a consideration of the residual interaction between composite fermions. Kukushkin et al. 1999

54 Quantitative estimates from CF theory The interaction energies of variously polarized FQHE states can be determined from the CF theory. The energy ordering is consistent with the Hund s rule. The energy differences are very small. From the energy differences, the critical Zeeman energies where transitions take place can be determined. Park and Jain

55 Park and Jain

56 Zeeman energy above which the state is fully spin polarized Du theory Kukushkin Satisfactory agreement is obtained considering that the relevant energy differences are extremely small, there are no free parameters, and finite thickness and disorder have not been incorporated.

57 The puzzle of spin polarization at 1/2

58 The puzzle If we treat the ½ state as a Fermi sea of free fermions with an effective mass derived from the activation gaps, the Fermi sea is close to being unpolarized. At B=9T, 45% spins are reversed. However, the experimental Fermi wave vector is consistent with a close to fully spin polarized Fermi sea.

59 Flaw: We are using the wrong mass!

60 Polarization mass of composite fermions The effective mass model works well. The spin polarization is governed by a different mass than the excitation gap. The polarization mass is much bigger than the activation mass. These aspects are fully confirmed by experiment. The value of either mass is still off by a factor of two, presumably because of several effects left out in the theory.

61 The resolution The correct mass for determining the spin polarization is the polarization mass. Then, at B=9T, only 14% spins are reversed at 1/2, giving a 7% correction to the Fermi wave vector. This brings consistency with experiment. Two different (large and small) Fermi seas?

62 Optically pumped NMR determination of 2D electron spin polarization for the ½ Fermi sea (from the hyperfine shift of the Ga nuclei) The temperature dependence of the spin polarization (at zero tilt) provides support to the non-interacting CF model, and the value of the effective mass is in excellent agreement with theory. Melinte et al. 2000

63 Excitations Excitations are understood as excitations of composite fermions across CF-LLs. Several kinds of excitations are possible, involving change in spin or/and CF-LL index. spin up spin down

64 Collective modes for spin reversed excitations Spin roton (curve d is for excitation out of unpolarized 2/5 state) Mandal and Jain

65 The spin physics of composite fermions / FQHE is extremely rich. (Richer than the IQHE physics of electrons.) The reason is that the Zeeman energy happens to be of the same order as the effective cycloron energy of composite fermions. The qualitative features of experiment are in agreement with the predictions of the simple model of non-interacting composite fermions. A satisfactory semi-quantitative understanding has been achieved with the help of microscopic wave functions.