quantum numbers and collective phases of …cmp2008/lecturenotes/jain3.pdfcomposite fermions...

Quantum numbers and collective phases of composite fermions

Quantum numbers

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

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.

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

IQHE

FQHE

Source: Clark et al.; Tsui and Stormer

Comparing B and B*

Magnetic focusing (Goldman; Smet)

Antidot resonances (Kang) SAW (Willett, 1993)

B*

B

Cyclotron radius

e

CF

Fermion statistics; Fermi wave vector; Intrinsic charge

CF mass

CF exciton8.6 Excitations 273

!"#$!"%&'(!)!*+,-*./,012

&&&&'(!)!*+,3$12&-*,.1$"4!5*6212"4/3

'(!270,/$"

'(!.$/$"

'(!8*7$"

9&1

!

Fig. 8.7. Schematic dispersion of the composite fermion exciton. This exci-tation is obtained by promoting a composite fermion across one ! level, asshown in the inset. The typical energy dispersion contains several minima(maxima), called rotons (maxons), and approaches a constant at large wavevectors, where the excitation contains a far separated pair of CF-quasiparticleand CF-quasihole that can move independently and contribute to transport.

level. This is one of the signatures of the appearance of higher LL-likestructure within the lowest Landau level.

8.6 Excitations

Experiments have demonstrated an variety of excitations of the FQHEstate: independent quasiparticle quasihole pairs, rotons, bi-rotons, tri-ons, flavor altering excitons, spin waves, independent spin-flip quasiparticle-quasihole pairs, spin-flip rotons, and skyrmions. We consider spin con-serving excitations here; those involving spin are taken up in Chapter11.

8.6.1 The CF exciton

The principal excitation of an incompressible FQHE state is obtainedby promoting a composite fermion to an unoccupied ! level, as shownschematically in the inset of Fig. (8.7). This excitation amounts to creat-ing a particle hole pair of composite fermions, called a CF-exciton (Sub-sec. 5.9.5). The energy of a CF exciton has three contributions: the self-energy of the CF-quasiparticle, the self-energy of the CF-quasihole, andtheir interaction energy. When the CF-quasiparticle and CF-quasihole

Schematic dispersion of CF exciton

8

8.6 Excitations 275

0.0 1.0 2.0 3.0 4.0

kl

0.00

0.05

0.10

0.15

!ex

[e

2/"l]

1/3

2/5

1/5

1/71/9

Fig. 8.8. Theoretical dispersions of CF excitons at 1/3, 2/5, 1/5, 1/7 and 1/9for 20, 30, 25, 25, and 20 particles, respectively. Source: J.K. Jain and R.K.Kamilla, Chapter 1 in Composite Fermions, edited by Olle Heinonen (WorldScientific, New York, 1998). (Reprinted with permission.)

of the e!ective interaction are the most significant sources of error in thetheoretical predictions. Based on an estimation of the accuracy of thisapproach [501], the theoretical energies should not be trusted to betterthan 20%. The intrinsic error in wave functions has a relatively minore!ect.

A composite fermion can be excited across a " level by a variety ofexperimental means, the principal ones being thermal excitation, inelas-tic Raman scattering [530, 531, 114, 532, 533, 534, 335, 336, 133, 261,134, 135, 262], optical spectroscopy [371, 372, 373, 374, 48], and phononabsorption [453, 763, 122, 595, 596]. We summarize results from theseexperiments.

8.6.2 Transport Gaps

It had been appreciated since quite early on (e.g., Chang et al. [62]) thatthe longitudinal resistance exhibits Arrhenius behavior, !xx ! exp("#/2kBT ),in an intermediate range of temperatures†, as seen in Fig. (8.9). The

† At high temperatures the transport is not activated, whereas at very low temper-atures, it is dominated by variable range hopping [538].

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.

• 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.

• The composite fermion mass is defined by interpreting this energy as the effective cyclotron energy of composite fermions.

[!xx ! exp ("!/2kBT )]

B dependence of m*

! = Ce2

!"!"

B

! = !!!c = !eB!

m!c= ! eB

(2pn± 1)m!c

However:

m! !"

BTherefore, we must have:

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.

!xx ! exp!" !

2kBT

"

276 Incompressible ground states and their excitations

Fig. 8.9. Longitudinal resistance as a function of temperature at several fillingfactors of the sequence ! = n/(2n + 1). Source: R.R. Du, H.L. Stormer, D.C.Tsui, L.N. Pfei!er, and K.W. West, Phys. Rev. Lett. 70, 2944 (1993).(Reprinted with permission.)

activation energy !, referred to as the transport gap, is interpreted asthe energy required to create a far-separated pair of CF-quasiparticleand CF-quasihole, each of which can move independently and thus con-tribute to transport. Theoretically, the energy of such an excitationcan be obtained in two ways: by calculating the energies of the CF-quasiparticle (Fig. 5.4b) and the CF-quasihole (Fig. 5.4c) separately, orby determining the large wave vector limit of the CF exciton dispersion(Fig. 8.7). Activation gaps for two states related by particle-hole symme-try are the same when measured in units of e2/!" (exercise). (It shouldbe remembered, however, that the particle-hole symmetry is exact onlyin the absence of LL mixing.) Figure 8.10 shows typical dependence of

[!xx ! exp ("!/2kBT )]

Du et al.

CF exciton

theory

How do the actual numbers compare?

8.6 Excitations 277

0.00 0.04 0.08 0.12 0.16 0.20

1/N

0.03

0.04

0.05

0.06

0.07

0.08

Eg [

e2/!

l]

2/5

4/9

Fig. 8.10. N dependence of the gap for 2/5 and 4/9. Source: J.K. Jain andR.K. Kamilla, Int. J. Mod. Phys. B 11, 2621 (1997). (Reprinted withpermission.)

gaps on the number of particles N . For ! = 1/(2p + 1), the transportgap has been computed by Bonesteel [33].

Fig. 8.11 shows transport gaps for two high quality samples with dif-ferent densities for the sequence ! = n/(2n + 1) (Du, Stormer, Tsui,Pfei!er, and West [127]). Also shown are gaps predicted by the CFtheory (Park, Meskini, and Jain [516]), including modification of theinteraction nonzero thickness in an LDA scheme. The nonzero thick-ness reduces gaps from their pure 2D values, bringing them into betteragreement with experiment, but a discrepancy of ! 50% remains. Asmentioned earlier, LL mixing cannot account for the discrepancy. Theever-present disorder is suspected to cause a substantial suppression ofthe gap, but no reliable theoretical method can currently confirm thisquantitatively.

8.6.3 CF rotons

The dispersion of the CF-exciton has several minima, named “rotons”(or magnetorotons) by Girvin, MacDonald and Platzman [196], draw-ing on the analogy to the “roton” minimum in the dispersion of thephonon excitation of superfluid 4He. Extending the analogy, maxima inthe dispersion can be called “maxons.” In the case of superfluid 4He, thephonon, the maxon and the roton are part of a single excitation branch


!"!!

!"!#

!"$!%%&'()%*+,-./'00%1,2,*

%%34%*+')(5%6%789

%%':;'(,2'/*

!"! !"$ !"< !"= !">

$?@</6$A

!"!!

!"!#

!"$!

$?< #?$$

>?B

=?C

<?#

$?=

!

"D<"=E$!$$-2

!<

"D$"$E$!$$-2

!<#%F'<?$l 0G

Fig. 8.11. Comparison of theoretical and experimental gaps for two di!erentdensities shown on the figure. Squares are for a pure two-dimensional sys-tem, and circles for a heterojunction (with the nonzero width correction tointeraction evaluated in a local density approximation). Stars are gaps fromthe experiment of Du, Stormer, Tsui, Pfei!er, and West [127]. Source: K.Park, N. Meskini, and J.K. Jain, J. Phys. Condens. Matter 11, 7283 (1999).(Reprinted with permission.)

(as confirmed by neutron scattering), with no conceptual distinction be-tween them.† Rotons and maxons in the FQHE are similarly specialcases of the CF exciton. They occur simply because the internal struc-ture in the density profiles of the CF-quasiparticle and the CF-quasiholemakes their interaction non-monotonic at short distances, i.e., at smallwave vectors. Girvin, MacDonald and Platzman [196] have developed adensity-wave picture for neutral excitations of the FQHE states, known

† The name roton was originally coined by Landau [379], who envisioned it as anexcitation characterized by a rotational velocity flow, distinct from the phonon.It has been described colorfully as a “ghost of a vanishing vortex ring” (Feynman[166]) or a “ghost of a Bragg spot” (Nozieres [495]).

• 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.

Park, Scarola, Jain

A. Pinczuk, M. Kang, et al.

Comparison of the CF exciton energy in the q=0 limit

Better quantitative theory of neutral modes than charged excitations.

Splitting of the collective mode284 Incompressible ground states and their excitations

Fig. 8.14. Resonant Raman spectra of the low-lying mode at ! = 1/3 fromtwo samples as a function of the tilt angle in the backscattering geometry(see inset), which is related to the wave vector of the excitation k. (l0 isthe magnetic length.) The single peak splits into two as the wave vector isincreased. Source: C.F. Hirjibehedin, I. Dujovne, A. Pinczuk, B.S. Dennis,L.N. Pfei!er, and K.W. West, Phys. Rev. Lett. 95, 066803 (2005). (Reprintedwith permission.)

Tokatly and Vignale [662] have argued, using continuum elasticity the-ory (Section 12.7), that such mode splitting is a generic consequenceof classical hydrodynamic equations of motion for the incompressibleFQHE liquid.

8.6.5 Charged trions

Park [522] has shown, theoretically, that the CF roton can lower its en-ergy by forming a bound state with an already existing CF-quasiparticleor CF-quasihole. This bound state is known as a CF trion. He estimatesthe binding energy to be roughly 0.02e2/!". CF-trions may be observablein resonant inelastic light scattering experiments, since some localizedquasiparticles are likely to be always present in real experiments (in-duced by disorder). Hirjibehedin et al. [262] see additional excitationsat energies 0.005-0.01 e2/!" below the ordinary roton energy, which may

Hirjibehedin, Pinczuk et al.

0 0.5 1 1.5 2 2.5 30.05

0.1

0.15

0.2

Majumdar, Mandal, 2008

144 Foundations of the Composite Fermion Theory

ch!

*"en

ergy

quantum number

ener

gy

#"

quantum number

Fig. 5.2. The general structure of the energy spectrum of the many-bodysystem at an integral filling, !! = n. The ground state is non-degenerate,containing n Landau levels fully occupied (shown in Fig. 5.3). The excitedstates form bands separated by the cyclotron energy. The x-axis label is a con-venient quantum number (wave vector in the periodic geometry, or the orbitalangular momentum in the spherical geometry). The CF mean-field theorypredicts that the low-energy spectrum at fractional fillings ! = n/(2pn ± 1)has identical structure, except that states at ! are only quasi-degenerate andthe cyclotron gap evolves into a gap ! (the determination of which requires amicroscopic theory). Explicit calculations beyond the mean-field theory showthat the one-to-one correspondence displayed in this figure is valid for the low-est band, but the mean-field theory predicts spurious states in higher bandsat !.

5.8.2 From IQHE to FQHE: The mean-field approximation

This subsection walks us through the golden path connecting the integralquantum Hall e!ect to the fractional quantum Hall e!ect, following Ref.[281]. For illustration, we begin by considering the special filling factors!! = n. The connection is established through the following steps:

Step I: Let us consider non-interacting electrons at !! = n. Thesystem is incompressible, i.e., the ground state is non-degenerate, sepa-rated from the other eigenstates by a gap (equal to the cyclotron energy).The many-particle energy spectrum is shown in Fig. (5.2). The groundstate has n full Landau levels, shown schematically in the left columnof Fig. (5.3). The lowest energy excited state is a particle-hole pair, oran exciton, shown in the left column of Fig. (5.4 d) for !! = 3. Thesediagrams have precise wave functions associated with them. We denotethe magnetic field by B!, which can be either positive or negative. It isrelated to the filling factor by !! = "#0/|B!| = n.

The long range rigidity in the system at an integral filling factor (man-ifested by the presence of the gap) is caused solely by the Fermi statistics.Thinking in the standard Feynman path-integral language [168, 594] isuseful. The partition function gets contributions from all closed paths

Ener

gy

kl

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.

FQHE of composite fermions

Pan et al. 2003

• The new fractions cannot be understood in terms of a model of noninteracting composite fermions, which epxlains FQHE only at fractions of the form

• We need to consider interacting composite fermions.

• It is natural to suspect that the next generation fractions correspond to the “FQHE of composite fermions.”

! =n

2pn± 1 and ! = 1! n

2pn± 1

• 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?

CF diagonalization

• 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.

Mandal, Jain, 2002

Rezayi (exact); Mandal, Jain (CF diagonalization)

Exact energy (Rezayi): -0.4412

Energy of the CF w.f.: -0.4409

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.”

Microscopic confirmation

(N=12)

Chang and Jain, 2004

Try: exact

• 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.

One complicated structure on top of another:

It would be hard to take such a state seriously if it didn’t really occur.

Paired state of composite fermions

5/2 FQHE

Willett et al.Pan et al.Xia et al.

Pfaffian wave function

• The paired state is described by a “Pfaffian” wave function (Moore and Read):

7.4 Interacting composite fermions: new fractions 185

written earlier by Moore and Read [463]:

!Pf1/2 = Pf

!1

zi ! zj

" #

i<j

(zi ! zj)2 exp

$!1

4

%

k

|zk|2&

, (7.11)

where “Pf” stands for Pfa"an. Without the Pfa"an factor, the wave function still de-scribes a system at ! = 1/2, but has the wrong exchange symmetry. The Pfa"an of anantisymmetric matrix M is defined, apart from an overall normalization factor, as [419]

Pf Mij = A(M12M34 · · · MN!1,N ) , (7.12)

where A is the antisymmetrization operator. The Pfa"an factor makes !Pf1/2 properly

antisymmetric without altering the filling factor. In the spherical geometry, the Moore-Read wave function generalizes to

!Pf1/2 = #2

1 Pf M , (7.13)

with Mij = (uivj ! viuj)!1.While Pf Mij is a complicated function, its square is relatively simple [419]:

[Pf M ]2 = Det M . (7.14)

This property is useful in the calculation of the energy of the Moore-Read wave function,for which only the modulus of the squared wave function is needed.

The usual Bardeen-Cooper-Schrie$er wave function for fully polarized electrons can bewritten as [114]

!BCS = A["0(r1, r2)"0(r3, r4) · · · "0(rN!1, rN )] , (7.15)

which clarifies that it is a Pfa"an. Analogously, Pf'

1zi!zj

(describes a p-wave pairing of

electrons (p-wave because the system is fully spin-polarized), and !Pf1/2 is interpreted as the

p-wave paired state of composite fermions carrying two vortices.The feasibility of the concept of CF pairing has been investigated in several quantitative

studies. Park et al. [509] show that the Moore-Read wave function has a substantiallylower energy than the CF Fermi-sea wave function in the second Landau level (with reverseenergy ordering in the lowest Landau level). Several studies compare the Moore-Read wavefunction with the exact Coulomb ground state wave function at ! = 1/2 in the secondLandau level (Greiter, Wen and Wilczek [217, 219]; Morf [466]; Rezayi and Haldane [561];Scarola, Jain and Rezayi [583]). The overlaps of the Moore-Read wave function with theexact Coulomb ground state are given in Table 7.4.2 for several N . Morf [466] and Rezayiand Haldane [561] have shown that the overlaps can be improved by tweaking the form ofthe interaction for the exact state, and also by particle-hole symmetrizing the Moore-Readwave function. While not conclusive, these comparisons generally support the interpretationof the 5/2 state as a paired state of composite fermions. Unlike the BCS wave function forsuperconductors, the Moore-Read wave function does not have any variational freedom, andhow one can be introduced is not known; such freedom should prove useful in improving thewave function, as well as for determining if the pairing is in the strong or the weak couplinglimit.

Pairing is associated with attractive interactions. The original Hamiltonian only containsthe repulsive Coulomb interaction between electrons. How can pairing occur in spite of thestrong repulsion? In this context, it may help to recall that the objects that pair up arenot electrons but composite fermions, the interaction between which is di$erent from that



!Pf1/2 = Pf

!1

zi ! zj

" #

i<j

(zi ! zj)2 exp

$!1

4

%

k

|zk|2&

, (7.11)


Pf Mij = A(M12M34 · · · MN!1,N ) , (7.12)



!Pf1/2 = #2

1 Pf M , (7.13)


[Pf M ]2 = Det M . (7.14)



!BCS = A["0(r1, r2)"0(r3, r4) · · · "0(rN!1, rN )] , (7.15)


1zi!zj








!Pf1/2 = Pf

!1

zi ! zj

" #

i<j

(zi ! zj)2 exp

$!1

4

%

k

|zk|2&

, (7.11)


Pf Mij = A(M12M34 · · · MN!1,N ) , (7.12)



!Pf1/2 = #2

1 Pf M , (7.13)


[Pf M ]2 = Det M . (7.14)



!BCS = A["0(r1, r2)"0(r3, r4) · · · "0(rN!1, rN )] , (7.15)


1zi!zj






Compare with:

This is represents a paired state of composite fermions.

Understanding 5/2 FQHE without the Pfaffian wave function

• 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.

Toke and Jain, 2006

CF diagonalization

JAIN: “CHAP05” — 2006/12/28 — 16:53 — PAGE 130 — #26

130 5 Foundations of the composite fermion theory

!

ZZ

Fig. 5.10. CF diagonalization.

explicitly carried through for many nontrivial cases. Various multi-dimensional integralsrequired for this purpose do not reduce to known functions catalogued in standard books.But that is merely a technical point and should not be taken to indicate a gap in ourunderstanding. The integrals are well defined, and can be evaluated numerically, usuallyby the Metropolis Monte Carlo method, to produce energies to desired accuracy (typicallyfour or five significant figures) without making any approximations. Systems with as manyas 100 composite fermions have been studied. See Appendix L for further technical details.

5.8.5 ! levels and energy level diagrams

For each initial state at !! in the left column of Figs. 5.3 and 5.4, the corresponding finalstate at ! is depicted by the diagram in the right column. Each diagram on the right has aprecise wave function associated with it, obtained by composite-fermionization of the wavefunction of the corresponding state in the left column.

The CF theory suggests the following interpretation: We picture that composite fermionsform Landau-like levels in the reduced magnetic field B!. These are called “" levels.”15

While " levels are analogous to Landau levels of electrons at B!, the two are not the

15 They have also been called “composite fermion Landau levels,” “pseudo-Landau levels,” or “quasi-Landau levels.” Thesenames have been a source of confusion, however, because the term “Landau level” has the universally accepted meaning as

Lowest Landau level

8

! " # $ % &!'

!

!(!"

!(!#

!(!$

!(!%

!(&

!(&"

)*)!+,-".!/ 01

234&"

" # $ % &! &"'

234&#

" # $ % &! &"'

234&$

" # $ % &! &" &#'

234"!

!(!"

!(!#

!(!$

)*)!+,-".!/ 01

FIG. 10: Zeroth-order (top) and first-order (bottom) CF diagonalization excitation spectra for Nh = 12, 14, 16, 20 holes in thesecond Landau level.

! " # $ % &!'

!

!(!&

!(!"

!(!5

!(!#

)*)!+,-".!/ 01

234&"

" # $ % &! &"'

234&#

" # $ % &! &"'

234&$

" # $ % &! &" &#'

234"!

!(!!#

!(!!%

)*)!+,-".!/ 01

FIG. 11: Zeroth-order (top) and first-order (bottom) CF diagonalization excitation spectra for Nh = 12, 14, 16, 20 holes in thelowest Landau level.

Second Landau level 8

! " # $ % &!'

!

!(!"

!(!#

!(!$

!(!%

!(&

!(&"

)*)!+,-".!/ 01

234&"

" # $ % &! &"'

234&#

" # $ % &! &"'

234&$

" # $ % &! &" &#'

234"!

!(!"

!(!#

!(!$

)*)!+,-".!/ 01

FIG. 10: Zeroth-order (top) and first-order (bottom) CF diagonalization excitation spectra for Nh = 12, 14, 16, 20 holes in thesecond Landau level.

! " # $ % &!'

!

!(!&

!(!"

!(!5

!(!#

)*)!+,-".!/ 01

234&"

" # $ % &! &"'

234&#

" # $ % &! &"'

234&$

" # $ % &! &" &#'

234"!

!(!!#

!(!!%

)*)!+,-".!/ 01

FIG. 11: Zeroth-order (top) and first-order (bottom) CF diagonalization excitation spectra for Nh = 12, 14, 16, 20 holes in thelowest Landau level.

*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.

BCS wave function for composite fermions

Moller and Simon, 2007

!CF!BCS = PLLL!BCS

!

j<k

(zj ! zk)2

!BCS = Pf[g(!rj ! !rk)]

g(!rj ! !rk) =!

!k

g!k"!k(zj)"!!k(zk)

The parameters g_k are determined variationally.

• 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.

A topological quantum crystal of fermions

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)

Transition from CF liquid to WC268 Incompressible ground states and their excitations

0.0 0.1 0.2

1/N

-0.45

-0.44

-0.43

Egs

2/5

3/7

4/9

5/11

-0.43280(6)

-0.44228(6)

-0.44744(12)

-0.45080(18)

0.0 0.1 0.2

1/N

-0.350

-0.345

-0.340

Egs 2/9

3/13

-0.34278(4)

-0.34835(

2)

Fig. 8.2. Ground state energy as a function of 1/N for several incompressiblestates. Source: J.K. Jain and R.K. Kamilla, Int. J. Mod. Phys. B 11, 2621(1997). (Reprinted with permission.)

Figure 8.3 shows the ground state energy of the CF liquid along withthe best estimate for the energy of the Wigner crystal (Lam and Girvin[377]). It illustrates why the Wigner crystal is not observed as soon asthe kinetic energy is suppressed by forcing all electrons into the lowestLandau level. The Wigner crystal is preempted by the CF liquid for arange of filling factors in the lowest Landau level. As the system becomesmore classical with decreasing !, a crystal is eventually expected. Moreinformation on this topic can be found in Chapter 15.

Jain and Kamilla, 1998


0.0 0.1 0.2 0.3 0.4 0.5

!

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Eg

s -

Ecl [

e2/"l]

Composite Fermion Liquid (Jain & Kamilla)

Correlated Wigner Crystal (Lam & Girvin)

Ecl = -0.782133 !

!"#e2

/"l

Fig. 8.3. Energies per particle for the CF liquid and the WC as a functionof the filling factor. The WC energy is taken from Lam and Girvin [377].Energies are shown relative to Ecl = !0.782133

"! e2/"#, the energy of a

classical two-dimensional Wigner crystal with triangular symmetry. The opendiamond on the right vertical axis is the estimate of the CF sea energy at! = 1/2 (Egs,1/2 ! Ecl = 0.0877(2)e2/"#) obtained by an extrapolation ofthe solid diamonds. The energy of the CF liquid is shown only at the specialn/(2pn+1) filling factors; the full curve has cusps at these points. Source: J.K.Jain and R.K. Kamilla, Int. J. Mod. Phys. B 11, 2621 (1997). (Reprintedwith permission.)

Fig. 8.4. Density profiles ($(r)!$0)/$0 for the CF-quasiholes at ! = 1/3, 2/5,and 3/7. $0 is the average electron density. Source: K. Park, thesis (StateUniversity of New York, Stony Brook, 2000). (Reprinted with permission.)

Filling(dimension)

1/3(1,206)

1/5(19,858)

1/7(117,788)

1/9(436,140)

Laughlin 0.96 0.70 0.50 0.44

Electron crystal

0.65 0.72 0.74

Overlap^2 with the exact wave functionfor six particles

Composite fermion 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.)

Yi, Fertig, 1998

Narevich, Murthy, Fertig, 2001

Filling(dimension)

1/3(1,206)

1/5(19,858)

1/7(117,788)

1/9(436,140)

Laughlin 0.96 0.70 0.50 0.44

Electron crystal

0.65 0.72 0.74

CF crystal 0.89 0.99 0.99

Overlaps again

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%.)

Chang, Jeon, Jain, 2004

JAIN: “CHAP15” — 2006/12/28 — 17:00 — PAGE 451 — #10

15.3 Composite fermion crystal 451

Table 15.2. Electron crystal and CF crystal energies at lowfillings

! L Exact CFC WC Laughlin

1/3 45 2.8602 — 2.9163(9) 2.8643(3)1/5 75 2.2019 2.2042(5) 2.2196 2.2093(2)1/7 105 1.8533 1.8536(2) 1.8622 1.8617(2)1/9 135 1.6305 1.6306(1) 1.6361 1.6388(1)

Notes: Total interaction energies for exact ground state, CF crystal (CFC),Wigner crystal (WC), and Laughlin wave function for six particles atseveral filling factors.Uncertainty in last digit from Monte Carlo sampling is given in parentheses.Electron–background and background–background contributions are notincluded, but are expected to be the same for all states.Source: Chang, Jeon and Jain [68]. (Total energies were mislabeled asinteraction energies per particle in Ref. [68].)

of a large number of Slater determinant basis functions (see Table 15.1), involving D ! 1parameters. (ii) The CFC wave functions for ! = 1/7 and 1/9 are more accurate than theLaughlin wave function at ! = 1/3, in spite of the much larger Fock space dimensions at1/7 and 1/9. The CFC pair correlation function (not shown) is indistinguishable from exact[68]; the formation of composite fermions somewhat weakens the crystalline correlationsrelative to the Maki–Zotos uncorrelated Wigner crystal.

Chang et al. [68] consider states down to ! = 1/45, using a combination of exactdiagonalization and CF diagonalization for N = 6, and find similar accuracy at smaller !.The optimal value of 2p increases as the filling factor is reduced, with 2p = 38 producingthe best energy at ! = 1/45.

Finite size studies do not necessarily provide a reliable account of the thermodynamicstate. For example, for N = 6 the CFC gives a better description of the ! = 1/5 groundstate than the Laughlin liquid wave function, even though the thermodynamic state hereis known to be a liquid [205, 301]. (With increasing N , the Laughlin 1/5 wave functioneventually begins to have lower energy than the CFC [70].) Nonetheless, an extremelyprecise description of a finite N state (assuming N is not too small) gives a strong indicationfor the nature of the state in the thermodynamic limit. Even though the finite N study cannotpredict the precise ! value where a transition from liquid to crystal takes place, it makes acompelling case that whenever the thermodynamic state is a crystal, the crystal is made ofcomposite fermions.

The energy difference per particle, V CFC ! V WC, can be taken as a crude estimate ofthe temperature below which the quantum nature of the crystal should be robust to thermalfluctuations. From the energy differences in Ref. [68], for parameters appropriate for GaAs,the quantum crystal regime is estimated to be below 200 mK at ! = 1/9 at B = 25 T.

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.

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 non-perturbative 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.

CF crystal

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.

EZ = 2gµeBSz = gµeB = ! eB

mecIn vacuum:

In GaAs:

GaAs: effective mass differs differs by a factor of 14 and the g by a factor of 5.

Xie, Guo, Zhang, 1989

Early theoretical work (based on exact diagonalization studies)

Halperin (1983), Zhang and Chakraborty (1984), Maksym (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.

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)

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 ?

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.

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.

Quantitative tests of the microscopic wave functions for non-fully polarized FQHE states

Wu, Dev, Jain, 1993

Du et al. 1995

Transitions as a function of Zeeman energy

Landau level fan diagram for spinful composite fermions

Du et al. 1995

Consistent with free composite fermions with spin 1/2

Evidence for certain additional states that would require a consideration of the residual interaction between composite fermions.

Spin polarization from photoluminescence

Kukushkin et al. 1999

Park and Jain

• 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.

Quantitative estimates from CF theory

Park and Jain

Zeeman energy above which the state is fully spin polarized

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.

Kukushkin

Du

theory

The puzzle of spin polarization at 1/2

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.

Flaw: We are using the wrong mass!

“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.

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?

Melinte et al. 2000

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.

spin up spin downExcitations

• 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.

Mandal and Jain

Collective modes for spin reversed excitations

Spin roton

(curve d is for excitation out of unpolarized 2/5 state)

• 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.

quantum numbers and collective phases of …cmp2008/lecturenotes/jain3.pdfcomposite fermions...

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