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

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Uncorrected Proof

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

16.7 Composite Fermion Picture 497

t2.1 Table 16.2. The eﬀective CF monopole strength 2Q

∗

, the number of CF quasipar-

ticles (quasiholes n

and quasielectrons n

), the quasiparticle angular momenta

and l

, the composite fermion and electron ﬁlling factors ν

∗

and ν,andthe

angular momenta L of the lowest lying band of multiplets for a ten electron system

at 2Q between 29 and 15

t2.2 2Q 29 28 27 26 25 24 23 22 21 15

t2.3 2Q

∗

11 10 9 8 7 6 5 4 3 −3

t2.4 n

2 1000 0 0 0 00

t2.5 n

0 0012 3 4 5 66

t2.6 l

5.5 5 4.5 4 3.5 3 2.5 2 1.5 1.5

t2.7 l

6.5 6 5.5 5 4.5 4 3.5 3 2.5 2.5

t2.8 ν

∗

12−2

t2.9 ν 1/3 2/5 2/3

t2.10 L 10,8,6, 5 0 5 8,6,4, 9,7,6,5, 8,6,5, 5,3,1 0 0

t2.11 4,2,0 2,0 4,3

,1 4

For example, the Laughlin L = 0 ground state at ν =1/3 occurs when 394

∗

= N −1, so that the N composite fermions ﬁll the lowest shell with angular 395

momentum l

∗

N−1

). The composite fermion quasielectron and quasihole 396

states occur at 2l

∗

= N − 1 ± 1 and have one too many (for quasielectron) 397

or one too few (for quasihole) quasiparticles to give integral ﬁlling. The single 398

quasiparticle states (n

= 1) occur at angular momentum N/2, for example, 399

at l

=5with2Q

∗

=8andl

=5with2Q

∗

=10forN = 10 as indicated 400

in Table 16.2. The two quasielectron or two quasihole states (n

= 2) occur 401

at 2l

∗

= N − 1 ∓ 2, and they have 2l

= N − 1and2l

= N +1. For 402

example, we expect that, for N = 10, l

=4.5with2Q

∗

=7andl

=5.5 403

with 2Q

∗

= 11 as indicated in Table 16.2, leading to low energy bands with 404

L =0⊕ +2 ⊕ +4 ⊕ +6 ⊕ +8 for two quasielectrons and L =0⊕ +2 ⊕ +4 ⊕ 405

+6 ⊕ +8 ⊕ 10 for two quasiholes. In the mean ﬁeld picture, which neglects 406

quasiparticle-quasiparticle interactions, these bands are degenerate. 407

We emphasize that the low lying excitations can be described in terms 408

of the number of quasiparticles n

and n

. The total angular momentum 409

can be obtained by addition of the individual quasiparticle angular momenta, 410

being careful to treat the quasielectron excitations as a set of fermions and 411

quasihole excitations as a set of fermions distinguishable from the quasielec- 412

tron excitations. The energy of the excited state would simply be the sum 413

of the individual quasiparticle energies if interactions between quasiparticles 414

were neglected. However, interactions partially remove the degeneracy of dif- 415

ferent states having the same values of n

and n

. Numerical results in 416

Fig. 16.3b and d illustrate that two quasiparticles with diﬀerent L have diﬀer- 417

ent energies. From this numerical data one can obtain the residual interaction 418



) of a quasiparticle pair as a function of the pair angular momen- 419

tum L



. In Fig. 16.2, in addition to the lowest energy band of multiplets, 420

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498 16 The Fractional Quantum Hall Eﬀect

the ﬁrst excited band containing one additional quasielectron-quasihole pair 421

can be observed. The “magnetoroton” band can be observed lying between 422

the L = 0 Laughlin ground state of incompressible quantum liquid and a 423

continuum of higher energy states. The band contains one quasihole with 424

=9/2 and one quasielectron with l

=11/2. By adding the angu- 425

lar momenta of these two distinguishable particles, a band comprising L of 426

1(=l

−l

) ≤ L ≤ 10(=l

+ l

) would be predicted. But, from Fig. 16.2 427

we conjecture that the state with L = 1 is either forbidden or pushed up by 428

interactions into the higher energy continuum above the magnetoroton band. 429

Furthermore, the states in the band are not degenerate indicating residual 430

interactions that depend on the angular momentum of the pair L



.Other 431

bands that are not quite so clearly deﬁned can also be observed in Fig. 16.3. 432

Although ﬂuctuations beyond the mean ﬁeld interact via both Coulomb 433

and Chern–Simons gauge interactions, the mean ﬁeld composite fermion pic- 434

ture is remarkably successful in predicting the low-energy multiplets in the 435

spectrum of N electrons on a Haldane sphere. It was suggested originally 436

that this success of the mean ﬁeld picture results from the cancellation of the 437

Coulomb and Chern–Simons gauge interactions among ﬂuctuations beyond 438

the mean ﬁeld level. It was conjectured that the composite fermion transfor- 439

mation converts a system of strongly interaction electrons into one of weakly 440

interacting composite fermions. The mean ﬁeld Chern–Simons picture intro- 441

duces a new energy scale ¯hω

∗

proportional to the eﬀective magnetic ﬁeld B

∗

, 442

in addition to the energy scale e

(∝

√

B) associated with the electron– 443

electron Coulomb interaction. The Chern–Simons gauge interactions convert 444

the electron system to the composite fermion system. The Coulomb interac- 445

tion lifts the degeneracy of the noninteracting electron bands. The low lying 446

multiplets of interacting electrons will be contained in a band of width e

447

about the lowest electron Landau level. The noninteracting composite fermion 448

spectrum contains a number of bands separated by ¯hω

∗

. However, for large 449

values of the applied magnetic ﬁeld B, the Coulomb energy can be made 450

arbitrarily small compared to the Chern–Simons energy ¯hω

∗

, resulting in the 451

former being too small to reproduce the separation of levels present in the 452

mean ﬁeld composite fermion spectrum. The new energy scale is very large 453

compared with the Coulomb scale, and it is totally irrelevant to the deter- 454

mination of the low energy spectrum. Despite the satisfactory description of 455

the allowed angular momentum multiplets, the magnitude of the mean ﬁeld 456

composite fermion energies is completely wrong. The structure of the low- 457

energy states is quite similar to that of the fully interacting electron system 458

but completely diﬀerent from that of the noninteracting system. The magne- 459

toroton energy does not occur at the eﬀective cyclotron energy ¯hω

∗

.What 460

is clear is that the success of the composite fermion picture does not result 461

from a cancellation between Chern–Simons gauge interactions and Coulomb 462

interactions. 463

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

16.8 Fermi Liquid Picture 499

16.8 Fermi Liquid Picture 464

The numerical result of the type displayed in Fig. 16.2 could be understood in 465

a very simple way within the composite fermion picture. For the 10 particle 466

system, the Laughlin ν =1/3 incompressible ground state at L = 0 occurs 467

for 2Q =3(N −1) = 27. The low lying excited states consist of a single quasi- 468

particle pair, with the quasielectron and quasihole having angular momentum 469

=11/2andl

=9/2.Themeanﬁeldcompositefermionpicturedoesnot 470

account for quasiparticle interactions and would give a magnetoroton band of 471

degenerate states with 1 ≤ L ≤ 10 at 2Q = 27. It also predicts the degenera- 472

cies of the bands of two identical quasielectron states at 2Q =25andoftwo 473

identical quasihole states at 2Q = 29. 474

The energy spectra of states containing more than one composite fermion 475

quasiparticle can be described in the following phenomenological Fermi liquid 476

model. The creation of an elementary excitation, quasielectron or quasihole, in 477

a Laughlin incompressible ground state requires a ﬁnite energy, ε

or ε

, 478

respectively. In a state containing more than one Laughlin quasiparticles, 479

quasiparticles interact with one another through appropriate quasiparticle- 480

quasiparticle pseudopotentials, V

QP−QP



. Here, V

QP−QP





) is deﬁned as the 481

interaction energy of a pair of electrons as a function of the total angular 482

momentum L



of the pair. 483

An estimate of the quasiparticle energies can be obtained by comparing 484

the energy of a single quasielectron (for example, for the 10 electron system, 485

the energy of the ground state at L = N/2=5for2Q =27− 1 = 26) or a 486

single quasihole (the L = N/2 = 5 ground state at 2Q =27+1=28forthe 487

10 electron system) with the Laughlin L = 0 ground state at 2Q = 27. There 488

can be ﬁnite size eﬀects, because the quasiparticle states occur at diﬀerent 489

values of 2Q from that of the ground state. But estimation of reliable ε

490

and ε

should be possible for a macroscopic system by using the correct 491

magnetic length l

= R/

√

Q (R is the radius of the Haldane sphere) in the 492

units of energy e

at each value of 2Q and by extrapolating the results as 493

a function of N

−1

to an inﬁnite system.

494

The quasiparticle pseudopotentials V

QP−QP



can be obtained by subtract- 495

ing from the energies of the two quasiparticle states obtained numerically, 496

for example, for the ten particle system at 2Q = 25(2QE state), 2Q = 497

27(QE − QH state), and 2Q = 29(2QH state) the energy of the Laughlin 498

ground state at 2Q = 27 and two energies of appropriate noninteracting quasi- 499

particles. As for the single quasiparticle, the energies calculated at diﬀerent 500

2Q must be taken in correct units of e

√

/R to avoid ﬁnite size 501

eﬀects. 502

P. Sitko, S.-N. Yi, K.-S. Yi, J.J. Quinn, Phys. Rev. Lett. 76, 3396 (1996).

Uncorrected Proof

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

500 16 The Fractional Quantum Hall Eﬀect

16.9 Pseudopotentials 503

Electron pair states in the spherical geometry are characterized by a pair 504

angular momentum L



(=L

). The Wigner–Eckart theorem tells us that the 505

interaction energy V



) depends only on L



and the Landau level index n. 506

The reason of the success of the mean ﬁeld Chern–Simons picture can be 507

seen by examining the behavior of the pseudopotential V

QP−QP





)ofapair 508

of particles. In the mean ﬁeld approximation the energy necessary to create 509

a quasielectron–quasihole pair is ¯hω

∗

. However, the quasiparticles will inter- 510

act with the Laughlin condensed state through the ﬂuctuation Hamiltonian. 511

The renormalized quasiparticle energy will include this self-energy, which is 512

diﬃcult to calculate. We can determine the quasiparticle energies phenomeno- 513

logically using exact numerical results as input data. The picture we are using 514

is very reminiscent of Fermi liquid theory. The ground state is the Laughlin 515

condensed state; it plays the role of a vacuum state. The elementary excita- 516

tions are quasielectrons and quasiholes. The total energy can be expressed as 517

518

E = E



QP,QP



QP−QP



(L)n



. (16.31)

519

The last term represents the interactions between pair of quasiparticles in 520

a state of angular momentum L. One can take the energy spectra of ﬁnite 521

systems, and compare the two quasiparticle states, such as |2QE, |2QH,or 522

|1QE + 1QH, with the composite fermion picture. The values of V

QP−QP



(L) 523

are obtained by subtracting the energies of the noninteracting quasiparticles 524

from the numerical values of E(L)forthe|1QP + 1QP



 states after the 525

appropriate positive background energy correction. It is worth noting that 526

the interaction energy for unlike quasiparticles depends on the total angular 527

momentum L while for like quasiparticles it depends on the relative angular 528

momentum R, which is deﬁned by R = L

Max

− L. One can understand it 529

by considering the motions in the two dimensional plane. Oppositely charged 530

quasiparticles form bound states, in which both charges drift in the direction 531

perpendicular to the line connecting them, and their spatial separation is 532

related to the total angular momentum L. Like charges repel one another 533

orbiting around one another due to the eﬀect of the dc magnetic ﬁeld. Their 534

separation is related to their relative angular momentum R.

535

If V

QP−QP





) is a “harmonic” pseudopotential of the form 536



)=A + BL



+1)

The angular momentum L

of a pair of identical fermions in an angular momen-

tum shell or a Landau level is quantized, and the convenient quantum number

to label the pair states is the relative angular momentum R =2l

− L

(on a

sphere) or relative angular momentum m (on a plane).

Uncorrected Proof

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

16.9 Pseudopotentials 501

every angular momentum multiplet having the same value of the total angu- 537

lar momentum L has the same energy. Here, A and B are constants and we 538

mean that the interactions, which couple only states with the same total 539

angular momentum, introduce no correlations. Any linear combination of 540

eigenstates with the same total angular momentum has the same energy. We 541

deﬁne V

QP−QP





) to be “superharmonic” (“subharmonic”) at L



=2l −R if 542

it increases approaching this value more quickly (slowly) than the harmonic 543

pseudopotential appropriate at L



− 2. For harmonic and subharmonic pseu- 544

dopotentials, Laughlin correlations do not occur. In Figs. 16.3b and d, it is 545

clear that residual quasiparticle–quasiparticle interactions are present. If they 546

were not present, then all of the 2QE states in frame (b) would be degenerate, 547

as would all of the 2QH states in frame (d). In fact, these frames give us the 548

pseudopotentials V

(R)andV

(R), up to an overall constant, describing 549

the interaction energy of pairs with angular momentum L



=2l −R. 550

Figure 16.4 gives a plot of V



)vsL



+1)for the n =0andn =1 551

Landau levels. For electrons in the lowest Landau level (n =0),V



)is 552

superharmonic at every value of L



. For excited Landau levels (n ≥ 1), V



) 553

is not superharmonic at all allowed values of L



. The allowed values of L



for a 554

pair of fermions each of angular momentum l are given by L



=2l −R,where 555

the relative angular momentum R must be an odd integer. We often write the 556

pseudopotential as V (R)sinceL



=2l−R. For the lowest Landau level V

(R) 557

is superharmonic everywhere. This is apparent for the largest values of L



in 558

0 100 200 300 400 500 600

L'(L'+1)

0.1

0.6

V (e

/ )

0 100 200 300 400 500 600

L'(L'+1)

(a) n=0 (b) n=1

=10 2

=15

=20 2

=25

Fig. 16.4. Pseudopotential V



) of the Coulomb interaction in the lowest (a)and

the ﬁrst excited Landau level (b) as a function of the eigenvalue of the squared pair

angular momentum L



+1). Here,n indicates the Landau level index. Squares

(l =5),triangles (l =15/2), diamonds (l = 10), and circles (l =25/2) indicate data

for diﬀerent values of Q = l + n

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

502 16 The Fractional Quantum Hall Eﬀect

Fig. 16.4. For the ﬁrst excited Landau level V

increases between L



=2l − 3 559

and L



=2l −1, but it increases either harmonically or more slowly, and hence 560

(R) is superharmonic only for R > 1. Generally, for higher Landau levels 561

(for example, n =2, 3, 4, ···) V



) increases more slowly or even decreases 562

at the largest values of L



. The reason for this is that the wavefunctions 563

of the higher Landau levels have one or more nodes giving structure to the 564

electron charge density. When the separation between the particles becomes 565

comparable to the scale of the structure, the repulsion is weaker than for 566

structureless particles.

567

When plotted as a function of R, the pseudopotentials calculated for 568

small systems containing diﬀerent number of electrons (hence for diﬀer- 569

ent values of quasiparticle angular momenta l

) behave similarly and, for 570

N →∞, i.e., 2Q →∞, they seem to converge to the limiting pseudopotentials 571

QP−QP



(R = m) describing an inﬁnite planar system. 572

The number of electrons required to have a system of quasiparticle pairs 573

of reasonable size is, in general, too large for exact diagonalization in terms 574

of electron states and the Coulomb pseudopotential. However, by restricting 575

our consideration to the quasiparticles in the partially ﬁeld composite fermion 576

shell and by using V

(R) obtained from numerical studies of small systems of 577

electrons, the numerical diagonalization can be reduced to manageable size.

578

Furthermore, because the important correlations and the nature of the ground 579

state are primarily determined by the short range part of the pseudopotential, 580

such as at small values of R or small quasiparticle–quasiparticle separa- 581

tions, the numerical results for small systems should describe the essential 582

correlations quite well for systems of any size. 583

In Fig. 16.5 we display V

(R)andV

(R) obtained from numerical diag- 584

onalization of N (6 ≤ N ≤ 11) electron systems appropriate to quasiparticles 585

of the ν =1/3andν =1/5 Laughlin incompressible quantum liquid states. 586

We note that the behavior of quasielectrons is similar for ν =1/3andν =1/5 587

states, and the same is true for quasiholes of the ν =1/3andν =1/5 Laughlin 588

states. Because V

(R =1)<V

(R =3)andV

(R =5)<V

(R =7), 589

we can readily ascertain that V

(R) is subharmonic at R =1andR =5. 590

Similarly, V

(R) is subharmonic at R = 3 and possibly at R =7. 591

There are clearly ﬁnite size eﬀects since V

(R) is diﬀerent for diﬀerent 592

values of the electron number N. However, V

(R) converges to a rather well 593

deﬁned limit when plotted as a function of N

−1

. The results are quite accurate 594

up to an overall constant, which is of no signiﬁcance when we are interested 595

As for a conduction electron and a valence hole pair in a semiconductor, the

motion of a quasielectron–quasihole pair, which does not carry a net electric

charge is not quantized in a magnetic ﬁeld. The appropriate quantum number to

label the states is the continuous wavevector k,whichisgivenbyk = L/R =

L/l

√

Q on a sphere.

The quasiparticle pseudopotentials determined in this way are quite accurate up

to an overall constant which has no eﬀect on the correlations.

Uncorrected Proof

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

16.9 Pseudopotentials 503

0.10

0.15

V (e

/ )

V (e

/ )

0.00

0.05

(a) QE's in ν=1/3 (b) QH's in ν=1/3

0.01

0.04

1357911 1357911

0.00

0.02

N=11 N=10

N= 9

N= 7 N= 6

N= 8

Fig. 16.5. Pseudopotentials V (R) of a pair of quasielectrons and quasiholes in

Laughlin ν =1/3andν =1/5 states, as a function of relative pair angular momen-

tum R. Diﬀerent symbols denote data obtained in the diagonalization of between

six and eleven electrons

only in the behavior of V

QP−QP

as a function of R. Once the quasiparticle– 596

quasiparticle pseudopotentials and the bare quasiparticle energies are known, 597

one can evaluate the energies of states containing three or more quasiparti- 598

cles. Figure 16.6 illustrates typical energy spectra of three quasiparticles in the 599

Laughlin incompressible ground states. The spectrum in frame (a) shows the 600

energy spectrum of three quasielectrons in the Laughlin ν =1/3 state of eleven 601

electrons. In frame (b) we show the energy spectrum of three quasiholes in 602

the nine electron system at the same ﬁlling. The results from exact numerical 603

diagonalization of the eleven and nine electron systems are represented by the 604

crosses and the Fermi liquid results are indicated by solid circles. The exact 605

energies above the dashed lines correspond to higher energy states containing 606

additional quasielectron–quasihole pairs. It should be noted that in the mean 607

ﬁeld composite fermion model, which neglects the quasiparticle–quasiparticle 608

interactions, all of the three quasiparticle states would be degenerate and 609

the energy gap separating the three quasiparticle states from higher energy 610

states would be equal to ¯hω

∗

=¯hω

/3. Although the ﬁt is not perfect in 611

Fig. 16.6, the agreement is quite good and it justiﬁes the use of the Fermi liq- 612

uid picture to describe non-Laughlin type compressible states at ﬁlling factor 613

ν =

2p+1

. 614

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

504 16 The Fractional Quantum Hall Eﬀect

0.15

0.25

−

1/3

0.0

0.1

ν=1/3+3QH

(a) N=11,2Q=27 (b) N=9,2Q=27

0 5 10 15

ν=1/3+3QE

Fig. 16.6. The energy spectra of three quasielectrons (a) and three quasiholes (b)

in the Laughlin ν =1/3 state. The solid circles denote the Fermi liquid calcula-

tion using pseudopotentials illustrated in Figs. 16.5a and b. The crosses correspond

to exact spectra obtained in full diagonalization of the Coulomb interaction for

electrons of N =11(a)andN =9(b)at2Q =27

16.10 Angular Momentum Eigenstates 615

We have already seen that a spin polarized shell containing N fermions each 616

with angular momentum l can be described by eigenfunctions of the total 617

angular momentum

L =



and its z-component M =



. We deﬁne 618

(N,l) as the number of multiplets of total angular momentum L that can 619

be formed from N fermions each with angular momentum l. We usually label 620

these multiplets as |l

; Lα, where it is understood that each multiplet con- 621

tains 2L + 1 states having −L ≤ M ≤ L,andα is the label that distinguishes 622

diﬀerent multiplets with the same value of L. We deﬁne

,the 623

angular momentum of the pair i, j each with angular momentum l. 624

We can write

625

; Lα =







Lα,L



)|l

; l

N−2



; L, (16.32)

where |l

N−2



 is the α



multiplet of total angular momentum L



of N −2 626

fermions each with angular momentum l.From|l

N−2



 and |l

 one 627

The following theorems are quite useful:

Theorem 1:

+ N(N − 2)



i,j

, where the sum is over all pairs.

Theorem 2: f

(N, l) ≥ f

(N,l

∗

), where l

∗

= l − (N − 1) and 2l ≥ N − 1.

Theorem 3: If b

(N,l)isthebosonequivalentoff

(N, l), then b

(N,l

), if l

= l

−

(N − 1).

The ﬁrst theorem can be proven using the deﬁnitions of

and



i,j

and

eliminating

from the pair of equations. The other two theorems are almost

obvious conjectures to a physicist, but there exist rigorous mathematical proofs

of their validity.

Uncorrected Proof

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

16.10 Angular Momentum Eigenstates 505

can construct an eigenfunction of total angular momentum L. Here, we mean 628

that the angular momentum multiplet |l

; Lα obtained from N fermions, 629

each with angular momentum l, can be formed from |l

N−2



 and a 630

pair wavefunction |l

 by using G

Lα,L



)thecoeﬃcient of fractional 631

grandparentage.TheG

Lα,L



) produces a totally antisymmetric eigen- 632

function |l

; Lα, even though |l

; l

N−2



; L is not antisymmetric 633

under exchange of particle 1 or 2 with any of the other particles. 634

Equation (16.32) is useful because of a very simple identity involving N 635

fermions: 636

+ N (N − 2)

−



i,j

=0, (16.33)

637

where

L is the total angular momentum operator,

, and the sum 638

is over all pairs. Taking the expectation value of this identity in the state 639

; Lα gives the following useful result: 640

L(L +1)+N(N − 2)l(l +1)=l

; Lα|



i,j

; Lα. (16.34)

Because (16.32) expresses the totally antisymmetric eigenfunction |l

; Lα as 641

a linear combination of states of well deﬁned pair angular momentum

,the 642

right hand side of Eq.(16.34) can be written as 643

N(N − 1)



Lα

+1). (16.35)

In this expression P

Lα

) is deﬁned by 644

Lα





Lα,L



, (16.36)

and is the probability that |l

; Lα contains pairs with pair angular momen- 645

tum L

. It is interesting to note that the expectation value of square of 646

the pair angular momentum summed over all pairs is totally independent 647

of the multiplet α. It depends only on the total angular momentum L. Because 648

the eigenfunctions |l

; Lα are orthonormal, one can show that 649





Lα,L



Lβ,L



)=δ

αβ

. (16.37)

From (16.34)–(16.37) we have two useful sum rules involving P

Lα

). They 650

are 651



Lα

) = 1 (16.38)

652

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

506 16 The Fractional Quantum Hall Eﬀect

and 653

N(N −1)



+1)P

Lα

)=L(L+1)+N(N −2)l(l+1). (16.39)

654

The energy of the multiplet |l

; Lα is given by 655

(L)=

N(N − 1)



Lα

)V (L

), (16.40)

656

where V (L

) is the pseudopotential describing the interaction energy of a 657

pair of fermions with pair angular momentum L

. 658

Equation (16.40) together with our sum rules, (16.38) and (16.39) on 659

Lα

) gives the remarkable result that, for a harmonic pseudopotential 660

), the energy E

(L) is totally independent of α and every multiplet 661

that has the same total angular momentum L has the same energy. This 662

proves that the harmonic pseudopotential introduces no correlations and the 663

degeneracy of multiplets of a given L remains under the interactions. Any 664

linear combination of the eigenstates of the total angular momentum having 665

the same eigenvalue L is an eigenstate of the harmonic pseudopotential. Only 666

the anharmonic part of the pseudopotential ΔV (R)=V (R) − V

(R)causes 667

correlations. 668

16.11 Correlations in Quantum Hall States 669

Since the harmonic pseudopotential introduces no correlations, only the 670

anharmonic part of the pseudopotential ΔV (R)=V (R) − V

(R) lifts the 671

degeneracy of the multiplets with a given L. The simplest anharmonic pseu- 672

dopotential is the case in which ΔV (R)=Uδ

R,1

.IfU is positive, the lowest 673

energy multiplet for each value of L is the one with the minimum value of 674

the probability P

(R = 1) for pairs with R = 1. That is, the state of the 675

lowest energy will tend to avoid pair states with R = 1 to the maximum 676

possible extent.

This is exactly what we mean by Laughlin correlations. 677

In fact, if U →∞, the only states with ﬁnite energy are those for which 678

(R = 1), the probability for pairs with R = 1, vanishes. This cannot occur 679

for 2Q<3(N −1), the value of 2Q at which the Laughlin L = 0 incompressible 680

quantum liquid state occurs. If U is negative, the lowest energy state will have 681

a maximum value of P

(R = 1). This would certainly lead to the formation 682

of clusters. However, this simple pseudopotential appears to be unrealistic, 683

with nothing to prevent compact droplet formation and charge separation. 684

Avoiding R = 1 is equivalent to avoiding pair states with m = 1 in the planar

geometry.