Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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BookID 160928 ChapID 16 Proof# 1 - 29/07/09

16.4 Numerical Studies 487

2Qφ

rrR=

Fig. 16.1. Haldane sphere of radius R with magnetic monopoles of strength 2Q

located at the center of the sphere

Here, l is the orbital angular momentum operator. The components of l satisfy 121

the usual commutation rules [l

]=i¯h

αβγ

, where the eigenvalues of l

122

and l

are, respectively, ¯h

l(l +1) and¯hm.

The single particle eigenstates 123

of (16.12) denoted by |Q, l, m are called monopole harmonics.Thestates124

|Q, l, m are eigenfunctions of l

and l

as well as of H

, the single particle 125

Hamiltonian, with eigenvalues 126

ε(Q, l, m)=

¯hω

[l(l +1)− Q

]. (16.13)

In writing (16.13), we noted that Λ·

R =

R·Λ = 0 and, hence, l·

R =

R·l =¯hQ.

127

Because this energy must be positive, the allowed values of l are given by 128

= Q + n,wheren =0,1,2,.... The lowest Landau level (or angular 129

momentum shell) occurs for l

= Q and has the energy ε

=¯hω

/2, which is 130

independent of m as long as m is a nonpositive integer. Therefore, the low- 131

est Landau level has (2Q + 1)-fold degeneracy. The nth excited Landau level 132

occurs for l

= Q + n with energy 133

¯hω

[(Q + n)(Q + n +1)−Q

]. (16.14)

134

An N-particle eigenfunction of the lowest Landau level can be written, in 135

general, as 136

, ···,m

 = c

†

···c

†

|0. (16.15)

We note that, in the presence of the magnetic ﬁeld, the total angular momentum

is given by Λ = r ×[−i¯h∇+ eA(r)] and that the eigenvalues of Λ

are not equal

to l(l +1)¯h

. Here, A is the vector potential and [Λ

]=i¯hε

ijk

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488 16 The Fractional Quantum Hall Eﬀect

Here, |m

|≤Q and c

†

creates an electron in state |l

.Sinceweare 137

concentrating on a partially ﬁlled lowest Landau level we have only 2Q +1 138

degenerate single particle states. The number of possible way of constructing 139

N-electron antisymmetric states from 2Q+ 1 single particle states or choosing 140

N distinct values of m out of the 2Q + 1 allowed values is given by 141



2Q +1



(2Q +1)!

N!(2Q +1− N)!

. (16.16)

Then, there are G

N-electron states in the Hilbert subspace of the lowest 142

Landau level. For the Laughlin ν =1/m state, we have 2Q

ν=1/m

= m(N −1). 143

For example, for the case of 2Q =9andN =4(ν =1/3 state for 4 electrons 144

in the lowest Landau level of degeneracy 2Q + 1 = 10), we have l =4.5and 145

there are G

= 10!/[4!(10 − 4)! = 210 of four-electron states in the Hilbert 146

space of the lowest Landau level. 147

Table 16.1 lists the values of the electron angular momentum 

,2Q +1 148

(the Landau level degeneracy), G

(the number of antisymmetric N-electron 149

states), L

MAX

(the largest possible angular momentum of the system), and 150

t1.1 Table 16.1. The angular momentum 

of an electron in the lowest Landau level;

2Q+1 the Landau level degeneracy; G

the dimension of N-electron Hilbert space;

Max

the maximum value of the total angular momentum L; the allowed L-values for

a system consisting of N electrons on the surface of a Haldane sphere. The exponent

of the allowed L-values indicates the number of times an L-multiplet appears and

the number in parenthesis denotes the total number of L-multiplets

t1.2 N

2Q +1 G

Max

Allowed L-values

t1.3 3 3.0 7 35 6 6 ⊕ 4 ⊕ 3 ⊕ 2 ⊕ 0(5)

t1.4 4 4.5 10 210 12

12 ⊕ 10 ⊕ 9 ⊕ 8

⊕ 7 ⊕ 6

⊕

5 ⊕ 4

⊕ 3 ⊕ 2

⊕ 0

(18)

t1.5 5 6.0 13 1,287 20

20 ⊕ 18 ⊕ 17 ⊕ 16

⊕ 15

⊕ 14

⊕

⊕ 12

⊕ 11

⊕ 10

⊕ 9

⊕

⊕ 6

⊕ 5

⊕ 4

⊕ 3

⊕

⊕ 1 ⊕ 0

(73)

t1.6 6 7.5 16 8,008 30

30 ⊕ 28 ⊕···

(338)

t1.7 7 9.0 19 50,382 42

42 ⊕ 40 ⊕···

(1, 656)

t1.8 8 10.5 22 319,770 56

56 ⊕ 54 ⊕···

(8, 512)

t1.9 9 12.0 25 2,042,975 72

72 ⊕ 70 ⊕···

(45, 207)

t1.10 10 13.5 28 13,123,110 90

90 ⊕ 88 ⊕···

(246, 448)

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BookID 160928 ChapID 16 Proof# 1 - 29/07/09

16.4 Numerical Studies 489

the allowed values of L (the total angular momentum) with a superscript 151

indicating how many times they appear. The number in parenthesis in the 152

allowed L-value column is the total number of diﬀerent L-multiplets that 153

appear. For three electrons there are ﬁve such states, all with diﬀerent L 154

values. For four electrons there are 18 states; L = 12, 10, 9, 7, 5, and 3 155

each appearing once, L = 8, 2, and 0 each appearing twice, and L =6156

and 4 each three times. For N =10andQ =13.5(ν =1/3 state of 10 157

electrons) G

=13, 123, 110 and there are 246,448 distinct L multiplets 158

with 0 ≤ L ≤ 90. 159

The numerical problem is to diagonalize the interaction Hamiltonian 160

int



i<j

V (|r

− r

|) (16.17)

in the G

dimensional space. The problem is facilitated by ﬁrst determining 161

the eigenfunctions |LM α of the total angular momentum. Here,

L =



, 162

M =



,andα is an additional label that accounts for distinct multiplets 163

with the same total angular momentum L (for example, for the ﬁve electron 164

system the seven L = 6 states correspond to seven diﬀerent values of α). The 165

210 four-electron states of four electrons give us 18 × 18 matrix that is block 166

diagonal with two 3 × 3blocks,three2× 2 blocks, and six 1 × 1blocks.For 167

small numbers of electrons these ﬁnite matrices can easily be diagonalized to 168

obtain the many-body eigenvalues and eigenfunctions.

169

In a plane geometry, the allowed values of m,thez-component of the sin- 170

gle particle angular momentum, are 0, 1, ···, N

− 1. M =



is the 171

total z−component of angular momentum, where the sum is over all occu- 172

pied states. It can be divided into the center-of-mass (CM) and relative (R) 173

contributions M

+ M

. The connection between the planar and spherical 174

geometries is as follows: 175

M = Nl + L

= Nl − L, M

= L + L

(16.18)

The interactions depend only on M

,so|M

 acts just like |L, L

. 176

The absence of boundary conditions and the complete rotational symmetry 177

make the spherical geometry attractive to theorists. Many experimentalists 178

prefer using the |M

 states of the planar geometry. The calculations 179

give the eigenenergies E as a function of the total angular momentum L. 180

Because H

int

is a scalar, the Wigner–Eckart theorem

L



int

|LMα = δ



L



int

|Lα

tells us that matrix elements of H

int

are independent of M and vanish unless L



L. This reduces the size of the matrix to be diagonalized enormously. For example,

for N =10andQ =27/2(ν =1/3 state of ten electrons) G

=13, 123, 110

and there are 246,448 distinct L multiplets with 0 ≤ L ≤ 90. However, the largest

matrix diagonalized is only 7069 by 7069.

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490 16 The Fractional Quantum Hall Eﬀect

2Q=27

Laughlin

ν=1/3 state

1QE+1QH

024681012

Fig. 16.2. The energy spectrum of 10 electrons in the lowest Landau level calculated

on a Haldane sphere with 2Q = 27. The open circle denotes the L = 0 ground state

The numerical results for the lowest Landau level always show one or more L 181

multiplets forming a low energy band. As an example, the numerical results 182

(E vs L) are shown in Figs. 16.2 and 16.3 for a system of 10 electrons with 183

values of 2Q between 25 and 29. It is clear that the states fall into a well 184

deﬁned low energy sector and slightly less well deﬁned excited sectors. The 185

Laughlin ν =1/3 state occurs at 2Q =3(N − 1) = 27 and the low energy 186

sector consists of a singlet L = 0 state as illustrated in Fig. 16.2. States with 187

larger values of Q contain one, two, or three quasiholes (2Q =28, 29, 30), and 188

states with smaller values of Q,suchas2Q = 25 or 26, contain quasielectrons 189

in the ground states. For 2Q = 26 the system is one single-particle state shy 190

of having the Laughlin ν =1/3 ﬁlling. In this case the low energy sector 191

corresponds to having a single Laughlin quasielectron of angular momentum 192

L =5. 193

16.5 Statistics of Identical Particles in Two Dimension 194

Let us consider a system consisting of two particles each of charge −e and mass 195

μ, conﬁned to a plane, in the presence of a perpendicular dc magnetic ﬁeld 196

B =(0, 0B)=∇×A(r). Since A(r) is linear in the coordinate r =(x, y), (for 197

example, A(r)=

B(−y, x) in a symmetric gauge), the Hamiltonian separates 198

into the center-of-mass and relative coordinate pieces R =

+ r

)and 199

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16.5 Statistics of Identical Particles in Two Dimension 491

1QE

(a) 2Q=28

1QH

024681012

(d) 2Q=25

2QE

024681012

(b) 2Q=29

2QH

Fig. 16.3. The energy spectra of 10 electrons in the lowest Landau level calculated

on a Haldane sphere with 2Q =28, 29, 26, 25. The open circles and solid lines mark

the lowest energy bands with the fewest composite fermion quasiparticles of n

for 2Q =28in(a), n

=2for2Q =29in(b), n

=1for2Q =26in(c), and

=2for2Q =25in(d)

r = r

− r

being the center-of-mass and relative coordinates, respectively. 200

The energy spectra for the center-of-mass and relative motion of the particles 201

are identical to that of a single particle of mass μ and charge −e.Wehave 202

seen that, as given by (16.6), for the lowest Landau level, the single particle 203

wavefunction is 204

)=N

−imφ

−r

/4l

For the relative motion φ is equal to φ

−φ

, and the interchange of the pair, 205

denoted by P Ψ(r

, r

)=Ψ(r

, r

), is accomplished by replacing φ by φ + π. 206

For two identical particles initially at positions r

and r

in a three dimen- 207

sional space, the amplitude for the path that takes the system from the initial 208

state (r

, r

) to the same ﬁnal state (r

, r

) depends on the angle of rotation 209

φ of the vector r

(=r

−r

). The end points represented by φ = π or 0 corre- 210

spond to exchange or non-exchange processes, and the angle φ is only deﬁned 211

modulo 2π. The angle of rotation φ is not a well-deﬁned quantity in three 212

dimension, but the statistics can not be arbitrary. Under the exchange of two 213

particles, the wavefunction picks up either a plus sign named bosonic statistics 214

or a minus sign named fermionic statistics with no other possibilities. Since 215

two consecutive interchanges must result in the original wavefunction, e

imπ

216

must be equal to either +1 (m is even; bosons) or −1(m is odd; fermions). 217

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492 16 The Fractional Quantum Hall Eﬀect

In two dimensions the angle φ is perfectly well-deﬁned for a given tra- 218

jectory. It is possible to keep track of how many times the angle φ winds 219

around the origin. Any two trajectories can not be deformed continuously 220

into one another since any two particles can not go through each other. The 221

space of particle trajectories falls into disconnected pieces that cannot be 222

deformed into one another if |r

| is not allowed to vanish. Each piece has a 223

deﬁnite winding number. Therefore, it is not enough to specify the initial and 224

ﬁnal conﬁgurations to characterize a given system completely. In constructing 225

path integrals, the weighting of trajectories can depend on a new parameter θ 226

(deﬁned modulo 2π) through a factor e

iθφ/π

.Forθ =0orθ = π we have the 227

conventional boson or fermion statistics. For the most general case, we have 228

Ψ(1, 2) = e

iθ

Ψ(1, 2). (16.19)

229

For arbitrary values of θ the particles are called anyons and satisfy a new 230

form of quantum statistics.

231

Let us consider a simple Lagrangian describing the relative motion of two 232

interacting particles, the relative position and reduced mass of which are 233

denoted by r[= (r, φ)] and μ, respectively. A simple way to realize anyon 234

statistics is to add a term ¯hβ

φ called Chern–Simons term to the Lagrangian, 235

where β(≡ θ/π)=qΦ/hc is the anyon parameter with 0 ≤ β ≤ 1. While q and 236

Φ are a ﬁctitious charge and ﬂux, θ is the numerical parameter of 0 ≤ θ ≤ 1. 237

For example, if 238

L =

μ(˙r

+ r

) − V (r)+¯hβ

φ (16.20)

the added (ﬁctitious charge–ﬂux) Chern–Simons term does not aﬀect the clas-

239

sical equations of motions because q and Φ are time independent. However, 240

the canonical angular momentum is given by p

∂L

∂

)=μr

φ +¯hβ. Because

241

2πip

/¯h

generates rotations of 2π,¯h

−1

must have integral eigenvalues . 242

However, the gauge invariant kinetic angular momentum, given by p

− ¯hβ, 243

can take on fractional values, which will result in fractional quantum statistics 244

for the particles. 245

16.6 Chern–Simons Gauge Field 246

Let us consider a two-dimensional system of particles satisfying some partic- 247

ular statistics and described by a Hamiltonian 248

H =

2μ





A(r

)





i>j

V (r

). (16.21)

A. Lerda, Anyons: Quantum Mechanics of Particles with Fractional Statistics,

Lecture Notes in Physics (Springer, Berlin, 1992) and F. Wilczek, Fractional

Statistics and Anyon Superconductivity (World Scientiﬁc, Singapore, 1990).

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16.6 Chern–Simons Gauge Field 493

Then, we can change the statistics by attaching to each particle a ﬁctitious 249

charge q and ﬂux tube carrying magnetic ﬂux Φ. The ﬁctitious vector potential 250

a(r

) at the position of the ith particle caused by ﬂux tubes, each carrying 251

ﬂux of Φ, on all the other particles at r

(= r

) is written as 252

a(r

)=Φ



j=i

ˆz × r

. (16.22)

The Chern–Simons gauge ﬁeld due to the gauge potential a(r

) becomes 253

b(r)=Φ



δ(r − r

)ˆz, (16.23)

254

where r

is the position of the ith particle carrying gauge potential a(r

). 255

Since no two electrons can occupy the same position and a given electron can 256

never sense the δ-function type gauge ﬁeld due to other electrons, b(r)hasno 257

eﬀect on the classical equations of motion. 258

In a quantum mechanical system, we rewrite the vector potential a(r)as 259

follows: 260

a(r

)=Φ



ˆz ×(r − r

)

|r −r

†

)ψ(r

). (16.24)

Here, ψ

†

)ψ(r

) denotes the density operator ρ(r

) for the electron liquid 261

and the gauge potential a(r) introduces a phase factor into the quantum 262

mechanical wavefunction. 263

Chern–Simons transformation is a singular gauge transformation which 264

transforms an electron creation operator ψ

†

(r) into a composite particle 265

creation operator ψ

†

(r) as follows: 266

†

(r)=ψ

†

(r)e

iα



arg(r−r



)ψ

†



)ψ



)

. (16.25)

Here, arg(r − r



) denotes the angle that the vector r − r



makes with the 267

x-axis and α is a gauge parameter. Then, the kinetic energy operator K

of 268

an electron is transformed into 269

2μ



rψ

†

(r)



−i¯h∇ +

A(r)+

a(r)



ψ(r), (16.26)

where a(r) is the total gauge potential formed at the position r due to the

270

Chern–Simons ﬂux attached to other particles. 271

a(r)=αφ





ˆz ×(r − r



)

|r −r



†



)ψ(r



Hence, the Chern–Simons transformation corresponds to a transformation

272

attaching to each particle a ﬂux tube of ﬁctitious magnetic ﬂux Φ(=αφ

) 273

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494 16 The Fractional Quantum Hall Eﬀect

and a ﬁctitious charge −e so that the particle could couple to the ﬂux tube 274

attached to other particles. 275

The new Hamiltonian, through Chern–Simons gauge transformation, is 276

obtained by simply replacing

A(r

) in (16.21) by

A(r

a(r

). 277

2μ



rψ

†

(r)



p +

A(r)+

a(r)



ψ(r)+



i>j

V (r

). (16.27)

278

The composite fermions obtained in this way carry both electric charge and 279

magnetic ﬂux. The Chern–Simons transformation is a gauge transformation 280

and hence the composite fermion energy spectrum is identical with the orig- 281

inal electron spectrum. Since attached ﬂuxes are localized on electrons and 282

the magnetic ﬁeld acting on each electron is unchanged, the classical Hamil- 283

tonian of the system is also unchanged. However, the quantum mechanical 284

Hamiltonian includes additional terms describing an additional charge–ﬂux 285

interaction, which arises from the Aharanov–Bohm phase attained when one 286

electron’s path encircles the ﬂux tube attached to another electron. 287

The net eﬀect of the additional Chern–Simons term is to replace the statis- 288

tics parameter θ describing the particle statistics in (16.19) with θ + πΦ

. 289

If Φ = p

when p is an integer, then θ → θ + πpq/e. For the case of q = e 290

and p =1,θ =0→ θ = π converting bosons to fermions and θ = π → θ =2π 291

converting fermions to bosons. For p = 2, the statistics would be unchanged 292

by the Chern–Simons terms. 293

The Hamiltonian H

contains terms proportional to a

(r)(n =0, 1, 2). 294

The a

(r) term gives rise to a standard two-body interaction. The a

(r)term 295

gives three-body interactions containing the operator 296

†

(r)Ψ(r)Ψ

†

)Ψ(r

)Ψ

†

)Ψ(r

The three-body terms are complicated, and they are frequently neglected.

297

The Chern–Simons Hamiltonian introduced via a gauge transformation is con- 298

siderably more complicated than the original Hamiltonian given by (16.21). 299

Simpliﬁcation results only when the mean-ﬁeld approximation is made. This 300

is accomplished by replacing the operator ρ(r) in the Chern–Simons vector 301

potential (16.24) by its mean-ﬁeld value n

, the uniform equilibrium elec- 302

tron density. The resulting mean-ﬁeld Hamiltonian is a sum of single particle 303

Hamiltonians in which, instead of the external ﬁeld B,aneﬀective magnetic 304

ﬁeld B

∗

= B + αφ

appears. 305

16.7 Composite Fermion Picture 306

The diﬃculty in trying to understand the fractionally ﬁlled Landau level in 307

two dimensional systems comes from the enormous degeneracy that is present 308

in the noninteracting many body states. The lowest Landau level contains N

309

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BookID 160928 ChapID 16 Proof# 1 - 29/07/09

16.7 Composite Fermion Picture 495

states and N

= BS/φ

, the number of ﬂux quanta threading the sample of 310

area S. Therefore, N

/N = ν

−1

is equal to the number of ﬂux quanta per 311

electron. Let us think of the ν =1/3 state as an example; it has three ﬂux 312

quanta per electron. If we attach to each electron a ﬁctitious charge q(= −e, 313

the electron charge) and a ﬁctitious ﬂux tube (carrying ﬂux Φ = 2pφ

directed 314

opposite to B,wherep is an integer and φ

the ﬂux quantum), the net eﬀect 315

is to give us the Hamiltonian described by Eqs.(16.21) and (16.22) and to 316

leave the statistical parameter θ unchanged. The electrons are converted into 317

composite fermions which interact through the gauge ﬁeld term as well as 318

through the Coulomb interaction. 319

Why does one want to make this transformation, which results in a much 320

more complicated Hamiltonian? The answer is simple if the gauge ﬁeld a(r

) 321

is replaced by its mean value, which simply introduces an eﬀective magnetic 322

ﬁeld B

∗

= B + b. Here, b is the average magnetic ﬁeld associated with the 323

ﬁctitious ﬂux. In the mean ﬁeld approach, the magnetic ﬁeld due to attached 324

ﬂux tubes is evenly spread over the occupied area S. The mean ﬁeld composite 325

fermions obtained in this way move in an eﬀective magnetic ﬁeld B

∗

. Since, 326

for ν =1/3 state, B corresponds to three ﬂux quanta per electron and b 327

corresponds to two ﬂux quanta per electron directed opposite to the original 328

magnetic ﬁeld B,weseethatB

∗

B. The eﬀective magnetic ﬁeld B

∗

acting 329

on the composite fermions gives a composite fermion Landau level contain- 330

ing

states, or exactly enough states to accommodate our N particles. 331

Therefore, the ν =1/3 electron Landau level is converted, by the composite 332

fermion transformation, to a ν

∗

= 1 composite fermion Landau level. Now, 333

the ground state is the antisymmetric product of single particle states con- 334

taining N composite fermions in exactly N states. The properties of a ﬁlled 335

(composite fermion) Landau level is well investigated in two dimension. The 336

ﬂuctuations about the mean ﬁeld can be treated by standard many body per- 337

turbation theory. The vector potential associated with ﬂuctuation beyond the 338

mean ﬁeld level is given by δa(r)=a(r) −a(r). The perturbation to the 339

mean ﬁeld Hamiltonian contains both linear and quadratic terms in δa(r), 340

resulting in both two body and three body interaction terms. 341

The idea of a composite fermion was introduced initially to represent an 342

electron with an attached ﬂux tube which carries an even number α (= 2p) 343

of ﬂux quanta. In the mean ﬁeld approximation the composite fermion ﬁlling 344

factor ν

∗

is given by the number of ﬂux quanta per electron of the dc ﬁeld 345

less the composite fermion ﬂux per electron, i.e. 346

∗

−1

= ν

−1

− α. (16.28)

347

We rememb er that ν

−1

is equal to the number of ﬂux quanta of the applied 348

magnetic ﬁeld B per electron, and α is the (even) number of Chern–Simons 349

ﬂux quanta (oriented oppositely to the applied magnetic ﬁeld B) attached 350

to each electron in the Chern–Simons transformation. Negative ν

∗

means the 351

eﬀective magnetic ﬁeld B

∗

seen by the composite fermions is oriented oppo- 352

site to the original magnetic ﬁeld B. Equation (16.28) implies that when 353

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496 16 The Fractional Quantum Hall Eﬀect

∗

= ±1, ±2,... and a nondegenerate mean ﬁeld composite fermion ground 354

state occurs, then 355

ν =

∗

1+αν

∗

(16.29)

generates, for α = 2, condensed states at ν =1/3, 2/5, 3/7,... and ν =

356

1, 2/3, 3/5,.... These are the most pronounced fractional quantum Hall states 357

observed in experiment. The ν

∗

= 1 states correspond to Laughlin ν =

1+α

358

states. If ν

∗

is not an integer, the low lying states contain a number of quasi- 359

particles (N

≤ N ) in the neighboring incompressible state with integral ν

∗

. 360

The mean ﬁeld Hamiltonian of noninteracting composite fermions is known 361

to give a good description of the low lying states of interacting electrons in 362

the lowest Landau level. 363

It is quite remarkable to note that the mean ﬁeld picture predicts not only 364

the Jain sequence of incompressible ground states, given by ν =

∗

1+2pν

∗

(with 365

integer p), but also the correct band of low energy states for any value of 366

the applied magnetic ﬁeld. This is illustrated very nicely for the case of N 367

electrons on a Haldane sphere. In the spherical geometry, one can introduce 368

an eﬀective monopole strength 2Q

∗

seen by one composite fermion. When 369

the monopole strength seen by an electron has the value 2Q,2Q

∗

is given, 370

since the α ﬂux quanta attached to every other composite fermion must be 371

subtracted from the original monopole strength 2Q,by 372

∗

=2Q − α(N − 1). (16.30)

This equation reﬂects the fact that a given composite fermion senses the vector

373

potential produced by the Chern–Simons ﬂux on all other particles, but not 374

its own Chern–Simons ﬂux. 375

Now, |Q

∗

| = l

∗

plays the role of the angular momentum of the lowest 376

composite fermion shell just as Q = l

was the angular momentum of the 377

lowest electron shell. When 2Q is equal to an odd integer (1+α)times(N −1), 378

the composite fermion shell l

∗

is completely ﬁlled (ν

∗

=1),andanL =0 379

incompressible Laughlin state at ﬁlling factor ν =(1+α)

−1

results. When 380

2|Q

∗

| + 1 is smaller than N, quasielectrons appear in the shell l

= l

∗

+1. 381

Similarly, when 2|Q

∗

| + 1 is larger than N , quasiholes appear in the shell 382

= l

∗

. The low-energy sector of the energy spectrum consists of the states 383

with the minimum number of quasiparticle excitations required by the value 384

of 2Q

∗

and N. The ﬁrst excited band of states will contain one additional 385

quasielectron – quasihole pair. The total angular momentum of these states in 386

the lowest energy sector can be predicted by addition of the angular momenta 387

or l

)ofthen

or n

quasiparticles treated as identical fermions. 388

In Table 16.2 we demonstrate how these allowed L values are found for a 389

ten electron system with 2Q in the range 29 ≥ 2Q ≥ 15. By comparing 390

with numerical results presented in Fig. 16.1, one can readily observe that the 391

total angular momentum multiplets appearing in the lowest energy sector are 392

correctly predicted by this simple mean ﬁeld Chern–Simons picture. 393