With this extra vector potential, the Hamiltonian is

2

e È0

1 e

^

H°qÞ ¼ pþ AÀq y:

2m* c c Ly

Here we have explicitly indicated the parametric dependence on q. The cur-

rent operator can then be written

!

eLy @H°qÞ

e e qh

^^

Jy °r; qÞ ¼ À pþ AÀ y Áy¼ : °10:2:5Þ

@q

m* c Ly h

We now make an interesting observation. According to Eq. (10.2.5) we can

evaluate the current in any state by forming the expectation value of the

derivative of the Hamiltonian with respect to a ¬ctitious vector potential in

that state. If a state carries any current, this derivative must obviously be

nonzero, and the eigenvalue spectrum must also depend on this ¬ctitious

vector potential. But this added vector potential is ˜˜pure gauge,™™ which is

to say that it does not correspond to any physical magnetic ¬eld and can be

completely removed by a gauge transformation. Therefore, it should have no

e¬ect whatsoever on the spectrum of the system. The solution to this paradox

lies in the fact that the vector potential a adds a phase to the electron wave-

Ð

function. This is the so-called Aharonov“Bohm phase AB ¼ Àe=0c a Á dr.

In the present case we integrate along the y-direction and obtain AB ¼ 2q.

The phase of a single-particle electron wavefunction thus advances by 2q as

349

10.2 Two-dimensional electron gas and the integer quantum Hall effect

it travels around the circumference Ly in the y-direction. This is precisely

what happens if there is a ¬‚ux È ¼ ÀqÈ0 piercing the center of the ring. In

addition to this Aharonov“Bohm phase, the phase of the wavefunction also

advances by kLy , where k is the wavenumber along the y-direction. If q is an

integer, q ¼ 0; Æ1; Æ2; . . . ; nothing new is added and the standard wavenum-

bers ki ¼ 2i=Ly with i ¼ 0; Æ1; Æ2; . . . satisfy the condition that the wave-

functions be single-valued. But if q is not an integer, the phase added to the

wavefunction due to a as we go around the ring is not an integer times 2.

The wavenumbers ki would then make the wavefunction multiple-valued. In

order to avoid this, we have to adjust the wavenumbers so that ki Ly carries

an extra phase that precisely cancels the phase due to the vector potential a,

and the allowed wavenumbers are now ki ¼ 2°i À qÞ=Ly . In other words, the

presence of the vector potential a changes the boundary conditions, unless a

corresponds to an integer number of ¬‚ux quanta piercing the system. Note

that this e¬ect hinges on the phase coherence of the wavefunction extending

around the ring. If the localization length is much smaller than the circum-

ference, the wavefunction will not run the risk of being multiple-valued. For

example, the wavefunction can be localized at some position y0 and decay

exponentially with a decay length ˜ ( Ly away from y ¼ y0 . The same wave-

function is then single-valued as we go around the ring no matter what q is

(except for some exponentially small corrections that we can ignore). The

spectrum of H°qÞ then has no dependence on q and the wavefunctions cannot

carry any current, which must obviously be the situation if the wavefunctions

are localized. This is the case for disordered insulators.

We now apply this kind of argument speci¬cally to a two-dimensional

electron gas on a ˜˜ribbon™™ of width Lx along the x-axis and having a cir-

^

cumference Ly along the y-axis. We apply a ¬eld E ¼ Ex x across the width of

the ribbon and then calculate the Hall resistivity of the system. We use our

trick from the previous paragraph of adding a fraction q of a ¬‚ux quantum

piercing the system to calculate the current density. The single-particle

Schrodinger equation in the Landau gauge and in the absence of disorder

¨

is then

& !2 '

1 e qÈ0

^

H°qÞ °x; yÞ ¼ pþ B xÀ y þ eEx x °x; yÞ

2m* c BLy

¼ E °x; yÞ; °10:2:6Þ

where represents an enumeration of the eigenstates. It shows what happens

as we slowly add a fraction q of a ¬‚ux quantum through the system “ the

350 The quantum Hall effect

electrons ˜˜march™™ to the right, moving their center points from xki ¼ Àki ˜2 ¼

B

À°2i=Ly Þ˜B to xki þ qÈ0 =BLy . As we complete the addition of one unit of

2

¬‚ux quantum through the system, the set of center points becomes mapped

back onto itself. If there were no electric ¬eld present, the single-particle

eigenstates for di¬erent values of q would all be equivalent, there would be

no dependence of the spectrum of H on q, and thus no current. But the

presence of the electric ¬eld changes this. There will now be a dependence

of the spectrum of eigenvalues on q. We make this explicit by inserting single-

particle states kn;q °xÞeiky into Eq. (10.2.6) and completing the squares:

& !2

121 qÈ0 vd

H°qÞkn;q °xÞeiky ¼ px þ m*!2 x þ xk À þ

c

BLy !c

2m* 2

'

1 Ex qÈ0

À m*v2 À 0kvd þ e kn;q °xÞeiky ;

d

2 B Ly

where vd is the classical drift velocity vd ¼ cE=B. The electric ¬eld introduces

a dependence of the single-particle energies on wavenumber k, and hence on

the center point xk . This means that as we now add a fraction q of a ¬‚ux

quantum, the energy of the system changes. By virtue of the relation between

single-particle energies and currents, the states carry currents in the presence

of the electric ¬eld. Since the total energy of the system changes as we insert

some ¬‚ux, this apparently means that we must do work on the system in

order to insert ¬‚ux. Clearly, the work that we do in this process must be

related to how the electron single-particle states march to the right and

increase their energies. Imagine that we slowly insert precisely one ¬‚ux quan-

tum into the system. The single-particle states and their energies are all the

same before and after inserting the ¬‚ux quantum. But in the process, all

occupied states moved one step over to the right, so that at the end of the

process, we have transferred precisely one electron per occupied Landau level

across the width Lx of the system. The cost in energy of this process is clearly

ÁE ¼ neV ¼ neEx Lx , where n is the number of occupied Landau levels.

We can work this out in more detail. Let the resistivity tensor of the system

be q. If there is no dissipation, then the diagonal components of the resistivity

tensor must vanish, and only the o¬-diagonal components are nonzero. Now

we consider the electric ¬eld in the y-direction rather than the x-direction,

and use Faraday™s law to write

° ° °

1 dÈ 1 dB

¼ dS Á ¼ d< Á Ey ¼ d˜yx jx ;

c dt c dt C C

351

10.2 Two-dimensional electron gas and the integer quantum Hall effect

where C is a contour enclosing the ¬‚ux quantum and jx is the current density,

equal to Jx =Ly , in the x-direction. If we now integrate this equation from

t ¼ À1 to t ¼ 1, we can relate the change in ¬‚ux ÁÈ to a transfer of charge

along the x-axis:

° ° °

1

ÁÈ ¼ yx d˜ dt jx ¼ yx dt Jx : °10:2:7Þ

c C

Ð

Now choose ÁÈ ¼ È0 . Then the net charge transferred ( dt Jx ) is Àne, where

n is the number of occupied Landau levels. Thus

1

È ¼ Àyx ne;

c0

so, using È0 ¼ hc=e, we obtain

h

yx ¼ À : °10:2:8Þ

ne2

While we have indeed derived a quantized Hall resistance for an ideal

ribbon-like system using a rather sophisticated gauge-invariance argument,

we could for the simple system above have taken a much simpler approach.

We could have calculated the current carried by each single-particle state,

summed up the result to get the total current, calculated the energy di¬erence

between the left-most and right-most occupied states, and we would have

arrived at the same result. So why did we go through all this e¬ort to calcu-

late something we could have derived using very elementary techniques? The

reason is that real systems are not ideal, but are composed of interacting

electrons in the presence of disorder. The simple methods cannot be used in

those cases. The gauge-invariance argument, on the other hand, is very

powerful, and is independent of the details of the system. It allows us to

turn now to a ˜˜real™™ system with disorder.

First, we summarize the main ingredients of the gauge-invariance argument

that we use in this case: (a) only states that are extended through the system

respond to the ¬‚ux inserted; (b) there is a mobility gap, i.e., there is a ¬nite

energy gap separating the bands of current-carrying states from each other,

so that the system remains dissipationless and the diagonal part of the resis-

tivity tensor vanishes; and (c) if we add precisely one ¬‚ux quantum, the

eigenstates of the system before and after the ¬‚ux quantum is inserted are

equivalent. Therefore, if there is a change in energy as we slowly add one ¬‚ux

quantum, this change must be due to a di¬erent occupation of single-particle

states within the same Landau level. It cannot be due to exciting electrons to

higher Landau levels, since such processes must overcome the cyclotron

352 The quantum Hall effect

energy and cannot occur adiabatically. It must be due to transferring n

electrons from one side of the system to the other.

The remaining issues are how to relate n to the number of ¬lled Landau

levels, and the origin of the mobility gap. For the ideal noninteracting system

in the absence of an external electric ¬eld, the density of states consists of a

series of delta-functions at the energies E ¼ °n þ 1Þ0!c , each of which has a

2

weight A=°2˜B Þ. As we add electrons to the system, the chemical potential

2

will always be at one of these energies, except when an integer number of

Landau levels are completely ¬lled. Hence, there are almost always extended

states just above and below the Fermi energy, without any energy gap separ-

ating them. As we add impurities to the system, extended states will start to

mix due to scattering o¬ the impurities. This both introduces dissipation (due

to a ¬nite probability of an incident electron being back-scattered) and also

broadens each Landau level into a band. We will here assume that the band-

width is smaller than the cyclotron energy so that each broadened Landau

level is separated from the neighboring ones (see Fig. 10.2.4). It is generally

then assumed that at the center of each broadened Landau level, there

remains a small number of extended states that can carry current, while

the states on each side of the center of the Landau level are localized. This

is a crucial assumption. According to the localization theory of noninteract-

ing particles, all electrons in two dimensions in the presence of any disorder

should be localized, which seems contrary to the assumption we just made.

What makes the di¬erence is the presence of the magnetic ¬eld. There is

strong theoretical and experimental evidence that the localization length of

the localized states diverges as the center of the Landau level is approached.

In a more pragmatic vein, we can also argue that it is an experimental fact

that these systems do carry current, and so there must be some extended

states present.

Figure 10.2.4. In the presence of disorder, the density of states shows that the

Landau levels have broadened into bands. The shaded areas represent localized

states that carry no current.

353

10.3 Edge states

As most of the states in the original Landau levels become localized and

shifted in energy away from the center of the Landau level, this gives us the

mobility gap that we need “ an energy range through which we can sweep the

Fermi energy while an energy gap separates the occupied extended states

from the unoccupied ones. While the Fermi energy lies in a band of localized

states, the transport is dissipationless, since dissipation is due to scattering

between current-carrying occupied and unoccupied states at the Fermi

energy. Here, all occupied current-carrying states are well below the Fermi

energy and cannot scatter to unoccupied states without a ¬nite increase in

energy.

Next, let us for simplicity assume that all localized states are in a region

ÀLx =2 þ x < x < Lx =2 À x and that the extended states occupy the regions

jxj > Lx =2 À x. As we now adiabatically insert a ¬‚ux quantum through the

system, the extended states in both regions march one step to the right. But

that means that we must have e¬ectively transferred one electron for each

Landau level with its extended states below the Fermi energy across the

region of localized states and across the width of the system “ since all states

in the localized region are initially occupied, there is no empty state to move

into from the extended-state region at ÀLx =2 < x < ÀLx =2 þ x unless the

net e¬ect is to transfer one electron across the band of localized states for

each such Landau level. Therefore, for this example, the integer n in the

gauge argument above is equal to the number of ¬lled Landau levels even

in the presence of (moderate) disorder. We should, however, point out that

while this argument can still be strengthened a little bit, there is no general

proof that n must be equal to the number of ¬lled Landau levels, or, for that

matter, nonzero. For example, in a strip of ¬nite width, all the energy levels

are discrete, and it is therefore impossible to move a charge adiabatically

across the system without adding any energy during this process. Only in the

limit of very wide strips do the energy levels form a continuum and make it

possible to move charges adiabatically. Another approach to quantization

will be presented in the next section, in which the current response to changes

in the chemical potential is studied using the Landauer“Buttiker formalism.

¨

That approach has the advantage of being closer to real experiments in which

the current response is measured in systems that are real, disordered, and of

¬nite size.

10.3 Edge states

In the previous section, we implicitly attached a special signi¬cance to the

edges in the physics of the quantized Hall conductance. For the ideal system

354 The quantum Hall effect

with an electric ¬eld, we transferred one electron per Landau level from

current-carrying edge states on one side of the system to current-carrying

edge states on the other side. Furthermore, while there is an excitation gap in

the bulk of the system, the edge states provided gapless excitations. It turns

out that because of the strong magnetic ¬eld, there will always be gapless

excitations of current-carrying states ¬‚owing along the perimeter of the sys-

tem. In this section, we examine these edge states more closely. We will ¬nd

that there is a very natural interpretation of the quantized Hall resistance

using these edge states in a Landauer“Buttiker formalism.

¨

We start by ¬rst giving a simple argument, due to Allan MacDonald, which

demonstrates that in a bounded system, there must always be gapless excita-

tions at the boundaries of the system. Consider a ¬nite system with a density

* at which the bulk is incompressible with a ¬lling factor *. The chemical

potential then lies in the bulk excitation gap, i.e., we have to pay the price

of the energy gap in order to add particles to the bulk of the system. We now

imagine that we increase the chemical potential by an in¬nitesimal amount

. In the bulk, the current density cannot change since is in¬nitesimal

and cannot overcome the mobility gap in the bulk. It follows that if there is a

change in the current density as a response to , this change must be at the

edges of the system. Charge conservation also requires that if there is a

resulting change in the current along the edge, this change must be uniform

along the edge. We can relate the change in current I to the change in orbital

magnetization density through

c

I ¼ M; °10:3:1Þ

A

with A the total area of the system. This relation is nothing but the equation

for the magnetic moment of a current loop. But we can write M in terms of

using a Maxwell relation:

@M @N

¼ :

M ¼ °10:3:2Þ

@B

@ B

By combining Eqs. (10.3.1) and (10.3.2) we arrive at

@

I

¼c : °10:3:3Þ

@B

When the ¬lling factor is locked at a particular value * then changing the

magnetic ¬eld at ¬xed necessarily changes the density, since

355

10.3 Edge states

@*=@B ¼ °@=@BÞ°*=2˜2 Þ ¼ *e=20c. Then Eq. (10.3.3) shows that there is

B

a corresponding current response to a change in the chemical potential. We

conclude that: (a) there must be gapless excitations in the system (since there

were states into which we could put more particles at an in¬nitesimal cost in

energy); and (b) these excitations must be located at the edges of the system.

Since all real systems are ¬nite and inhomogeneous, the low-energy proper-

ties probed by experiments such as transport measurements must be deter-

mined by the gapless edge excitations.

Next, we discuss in more detail the origin of these gapless edge states. First

of all, we may quite generally assume that there is some con¬ning potential

Vext °rÞ that keeps the electrons in the system. This potential is caused by

electron“ion interactions and electron“electron interactions, but for simpli-

city we assume that we have noninteracting electrons con¬ned by a potential

Vext °xÞ. In the center of the system the potential is ¬‚at, and we can here set

Vext °xÞ ¼ 0, but as we approach the edges of the system, the potential bends

upwards, providing a well that con¬nes the electrons to the interior of the

system. The nonzero gradient of the con¬ning potential also causes the states

near the edges to carry a ¬nite current. From our earlier discussion about

gauge invariance we related the derivative of the Hamiltonian with respect to

a ¬ctitious ¬‚ux to the current operator:

@H

/ jy :

@q

We now take the expectation value of this relation in one of the eigenstates of

H. The result is

0Ly

@E nk

¼À i;

e nk

@k

where ink is the net current carried by the state jnki. This equation just relates

the group velocity (@E=@k) to the current carried. In the interior, where the

con¬ning potential is ¬‚at, the eigenvalues are constant with respect to k and

these states carry no net current. Near the edges, where the con¬ning poten-

tial slopes upward, the eigenvalues change with k, giving rise to a ¬nite

velocity of the eigenstates, and hence a ¬nite current carried by each state.

Along one edge the current ¬‚ows in the positive y-direction, and along the

other edge, the current ¬‚ows in the negative y-direction, since the gradient of

Vext °xÞ has opposite signs at the two edges. The eigenvalues change because

of the relation xk ¼ À˜2 k between the center points xk of the states and the

B

356 The quantum Hall effect

wavenumber k. For example, in the limit of a very slowly rising potential

˜B dVext °xÞ

( 1;

0!c dx

and the eigenvalues are approximately E nk % °n þ 1Þ0!c þ Vext °xk Þ, so

2

dE nk =dk % ˜B dVext °xk Þ=dx. We brie¬‚y described these current-carrying edge

2

states in terms of semiclassical skipping orbits in Section 1.8.

The theorem we discussed at the beginning of this section stated that the

current-carrying states must be located at the edges. This lateral localization

is due to the strong external magnetic ¬eld. For a con¬ning potential Vext °xÞ

that preserves translational invariance along the y-axis, the single-particle

eigenstates can be labeled by the y-momentum k and can be constructed

from basis states that are a product of eiky and a function of x À xk . All

these basis states are localized in the x-direction about xk on a scale given

by ˜B . In the presence of the potential at the edge, linear combinations of

these will form new energy eigenstates, which will also be localized in the x-

direction. The strong magnetic ¬eld also prevents mixing of edge states at

opposite edges, provided the separation between the edges is large compared

with the magnetic length ˜B . Consider the e¬ect of a local potential V°x; yÞ on

two well separated edge states j1i and j2i. In perturbation theory, the mixing

of the edge states depends on the matrix element h1jVj2i, which falls o¬

roughly as exp °Àd 2 =˜2 Þ, where d is the separation between the edges.

B

In the language of the Landauer“Buttiker formalism, the lack of mixing

¨

between current-carrying edge states at opposite edges makes the transmis-

sion probabilities for edge states unity. To be more precise, it can be shown

that the matrix element for back-scattering across a system in the presence of

disorder is of order exp °ÀLx =Þ, where is a disorder-dependent length

characterizing the extent of the edge state in the direction across the system.

The suppression of back-scattering makes the Landauer“Buttiker formalism

¨

particularly well suited for systems in the integer quantum Hall regime, and

also provides these systems with a very convenient framework for interpre-

tation. Consider ¬rst a system with the Fermi energy midway between the

centers of two Landau levels. In discussing bulk systems we argued that this

places the Fermi energy in the mobility gap of the bulk states, so that in the

bulk there are no current-carrying states at the Fermi energy. On the edge,

however, there will be current-carrying states at the Fermi energy. We now

connect the system to a source and a drain and apply an in¬nitesimal elec-

trochemical potential energy di¬erence between them. The current-carrying

edge states injected at one terminal i cannot back-scatter but ¬‚ow along their

357

10.4 The fractional quantum Hall effect

respective edges until they encounter the next terminal j along the edge with

transmission probability Tji ¼ 1 and all other transmission probabilities zero.

It is then a matter of simple algebra to conclude that: (a) the resistance

between source and drain is h=e2 N, with N the number of ¬lled Landau levels;

(b) the resistance between any two terminals along the same edge is zero; and

(c) the resistance between any two terminals on opposite edges is h=e2 N.

As we increase the Fermi energy, it will eventually approach the center of

the next Landau level. There are now extended states all across the system

that are mixed by impurity scattering. Back-scattering is therefore no longer

prohibited, and the longitudinal resistance between terminals along the same

edge attains a ¬nite, nonzero value. At the same time, we are beginning to

add edge states belonging to a new Landau level, and the Hall resistance and

resistance between source and drain decreases. As soon as the Fermi energy

has swept past this new Landau level, the bulk states are in a new mobility

gap, there is no back-scattering, and the longitudinal resistance vanishes. At

the same time, we have populated the edge states originating from this new

Landau level, and the Hall resistance attains a new quantized value

h=e2 °N þ 1Þ.

As with all transport phenomena, the simple linear theory of the integer

quantum Hall e¬ect fails at su¬ciently large currents. The transport at a

quantized plateau then ceases to be dissipationless, while the Hall resistance

may or may not change appreciably from its quantized value. This can

happen through a variety of mechanisms. When one increases the electro-

chemical potential di¬erence Á between source and drain it is observed that

at some value of Á the longitudinal resistance starts to increase dramati-

cally, and eventually becomes Ohmic, and thus linear in Á. One loss

mechanism involves coupling to phonons. As soon as the drift velocity

exceeds the sound velocity, electrons can emit phonons, and dissipation

occurs even in a system with no other source of disorder.

10.4 The fractional quantum Hall effect

As we stated earlier, the fractional quantum Hall e¬ect is observed at very

large magnetic ¬elds in very clean systems. Here, the energy gap is caused by

electron“electron interactions. In order to observe resistance plateaus of ¬nite

width, there must be some degree of disorder present in order to provide a

mobility gap, but too much disorder has the contrary e¬ect of quenching this

necessary energy gap. The ¬rst theoretical evidence for an energy gap caused

by electron“electron interactions came from numerical diagonalizations of

358 The quantum Hall effect

small systems, which showed a downward dip in the ground state energy per

particle near ¼ 1. As the magnetic ¬elds considered are of the order of 10 T,

3

it is a good ¬rst approximation to assume that the cyclotron energy is

much larger than any other energy scale. This means that we can restrict

the basis states for electrons in the bulk of the two-dimensional sample to

only the lowest Landau level. Hence, in the absence of external potentials, all

single-particle states are completely degenerate. The problem becomes one of

¬nding the ground state and elementary excitations of this system in

which many electrons of equal unperturbed energy interact through a

Coulomb potential that is screened only by the static dielectric constant of

the material.

Almost all our understanding of the fractional quantum Hall e¬ect comes

from a bold variational trial wavefunction ¬rst proposed by Laughlin in

1983. He demonstrated that this wavefunction is incompressible at ¬lling

factors ¼ 1=p ¼ 1=°2m þ 1Þ with m an integer and that the quasiparticles

at these ¬llings have fractional charge Æ e=p ¼ Æ e=°2m þ 1Þ. Subsequent

theoretical advances based on Laughlin™s suggestion helped to establish

why his wavefunction gives a good description of the ground state. This

work showed that there is a low-energy branch of collective modes called

magneto-rotons (named in analogy with rotons in liquid helium), and estab-

lished that there is a hidden, so-called o¬-diagonal long-range order in the

Laughlin ground state. This latter insight led to the development of e¬ective

¬eld theories in which this order parameter and long-wavelength deviations

from it are the central quantities. Subsequent pioneering work by Jain, also

based on the Laughlin wavefunction and the o¬-diagonal long-range order it

contains, showed that the fractional quantum Hall e¬ect can be described as

an integer quantum Hall e¬ect of composite particles consisting of electrons

bound to an even number of ¬‚ux quanta. These entities are called composite

fermions. Finally, it was pointed out that the spin degree of freedom in GaAs

systems is very important, and leads to a new class of excitations with spin

textures. This is at ¬rst counter-intuitive, since one is inclined to assume that

in strong magnetic ¬elds, the spin degree of freedom is frozen out. However,

in GaAs, atomic and band-structure properties conspire to drive the e¬ective

´

Lande g-factor close to zero, rendering the spin contribution to the energy

per electron much smaller than any other energy.

We will for now ignore the spin degree of freedom and consider N spin-

polarized electrons in a magnetic ¬eld strong enough that we need only

consider single-particle basis states in the lowest Landau level. It is conveni-

ent to work in the symmetric gauge, in which A ¼ 1 °B ‚ rÞ, since then the

2

system is rotationally invariant about the z-axis, and the z-component of

359

10.4 The fractional quantum Hall effect

total angular momentum, Lz , commutes with the Hamiltonian. Our task is

then to ¬nd the best choice for the ground state of the degenerate system of

electrons in eigenstates of Lz when the Coulomb interaction is turned on. It is

conventional (even though it is not a little confusing!) to use the complex

notation zj xj À iyj for the coordinates of the jth electron. In the lowest

Landau level, the single-particle basis functions in the symmetric gauge are

then written as

m

zj

1 2 2

eÀjzj j =4˜B :

m °zj Þ ¼

˜B

°2˜2 2m m!Þ1=2

B

The probability densities of these states form circles about the origin with the

p¬¬¬¬¬¬¬

peak density occurring at r ™ ˜B 2m. One can verify that m °zj Þ is an eigen-

state of Lz with eigenvalue 0m.

The set of all N-particle Slater determinants composed of the lowest-

Landau-level single-particle wavefunctions forms a basis in which we can

expand the N-particle wavefunctions. For the special case of ¼ 1, we can

write down the wavefunction by inspection. It is, except for a trivial normal-

ization factor,

Y Y 1

É1 ¼ °zi À zj Þ exp À 2 jzk j :

2

°10:4:1Þ

4˜B

i<j k

This clearly satis¬es the requirement imposed by the Pauli principle that the

wavefunction vanish as zi ! zj for any pair i; j. The fact that it vanishes

linearly with the separation of the particles stems from its nature as a

Slater determinant of N wavefunctions. As we saw in Chapter 2, this also

has the e¬ect of lowering the repulsive Coulomb energy. If we now turn to a

fractional quantum Hall state at ¼ 1=p ¼ 1=°2m þ 1Þ, we must impose the

following conditions on a candidate for the ground-state wavefunction: (a)

the wavefunction must be odd under interchange of the positions of

P

two electrons; (b) it must contain a factor exp°À i jzi j2 =4˜2 Þ; and (c) the

B

wavefunction must be an analytic function that is an eigenstate of the total

angular momentum. The second of these comes from the fact that all

single-particle lowest-Landau-level wavefunctions contain the factor

exp °Àjzj2 =4˜2 Þ, and so must any Slater determinant constructed from

B

them. The wavefunction must be analytic because it can be expanded in

Slater determinants of lowest-Landau-level single-particle wavefunctions,

each of which is a polynomial in the zi . The simplest possible wavefunctions

360 The quantum Hall effect

that satisfy these requirements are

Y Y 1

Ép ¼ °zi À zj Þp exp À 2 jzk j2 ;

4˜B

i<j k

with p an odd integer. These wavefunctions are of the same form as (10.4.1),

except for the fact that the wavefunction now vanishes as an odd power p

with the separation of two particles. This gives the particles even higher

impetus to stay apart and avoid the Coulomb repulsion. They are also eigen-

functions of the z-component of total angular momentum, with eigenvalue

2 0N°N À 1Þp.

1

This wavefunction is not easy to visualize, or even to make use of in

calculating the energy of the system analytically or numerically. Instead,

we shall avail ourselves of a neat trick to study the particle density described

by these wavefunctions. The probability distribution of the electrons in Ép is

Y Y 1

jÉp j2 ¼ jzi À zj j2p exp À 2 jzk j2

2˜B

i<j k

X

1X

jzi j ¼ eÀHp ;

¼ exp 2p ln jzi À zj j À 2 2

2˜B

i<j

where we have de¬ned

X 1X

Hp ¼ À2p ln jzi À zj j þ jzi j2 :

2

2˜B

i<j

The ¬ctitious Hamiltonian Hp actually describes a real, albeit classical, sys-

tem. Consider a two-dimensional plasma, consisting of charges Àq at posi-

tions ri in the plane, interacting with one another and with a neutralizing

positive background charge density of density . The total interaction

between the particles is given by

X

Vp ¼ Àq 2

ln rij ; °10:4:2Þ

i>j

with rij the distance between the charges. The logarithmic interaction comes

from the fact that the charges are in¬nitely long rods. The interaction energy

with the neutralizing background charge density can be shown to be

2X 2

Vbackground ¼ q ri : °10:4:3Þ

2 i

361

10.5 Quasiparticle excitations from the Laughlin state

The system described by Eqs. (10.4.2) and (10.4.3) is called a one-component

classical two-dimensional plasma. In order to minimize its energy, the parti-

cles will spread out uniformly to reach the density of the neutralizing

background charge. In comparing Hp and Eqs. (10.4.2) and (10.4.3), it is

clear that the probability density of Ép corresponds to that of a classical one-

component two-dimensional plasma with density p ¼ 1=°2˜2 pÞ and charges

B

q ¼ 2p. We can therefore infer that the wavefunction Ép corresponds to a

2

state with uniform electron charge density p ¼ 1=°2˜2 pÞ. This is the electron

B

density at the fractional quantum Hall ¬llings ¼ 1=p. Detailed calculations

of the electron“electron correlation functions have veri¬ed that Ép describes

a translationally invariant liquid, and not a solid. This is signi¬cant, because

another contender for a fractional quantum Hall state is the so-called Wigner

crystal, in which the electrons arrange themselves on a regular lattice in order

to minimize the Coulomb repulsion. The Wigner crystal is, as the name

implies, a crystal and not a liquid. However, numerical calculations have

shown that the Laughlin wavefunction Ép has the lower energy for ¬lling

factors greater than about 1.7

Numerical calculations have also veri¬ed that the Laughlin wavefunction is

remarkably accurate. This was done by calculating numerically the ground-

state energy by exact diagonalization and comparing that with the energy

expectation value of the Laughlin wavefunction, and by also calculating the

overlap of the numerically obtained ground state with the Laughlin state.

The reason for this accuracy (in spite of the apparent simplicity of wavefunc-

tion) is very deep and is intimately connected with the extra powers with

which the wavefunction vanishes as two electrons are brought together.

10.5 Quasiparticle excitations from the Laughlin state

The elementary excitations from the Laughlin state have the remarkable

property that they have fractional charge. At ¬rst, this may seem really

bizarre, and may lead one to suspect that quarks are somehow involved.

This is of course not the case. The fractionally charged elementary excitations

are not single particles in the sense that they can exist alone, but are displace-

ments of the electron charge density such that the total local de¬cit or excess

of charge adds up to a fraction of an electron charge. The local charge

density is made up of a complicated correlated motion of the real electrons

in the system, and in order to create a quasihole, we need to create a local

charge de¬cit. Let us ¬rst consider how a ¬lled Landau level ( ¼ 1) responds

to a ¬‚ux tube carrying a total ¬‚ux È inserted adiabatically through the center

of the system. In the symmetric gauge, the single-particle wavefunctions in

362 The quantum Hall effect

the lowest Landau level are

m

jzj2

z

m °zÞ ¼ exp À 2 ;

˜B 4˜B

where z is again equal to x À iy and not the out-of-plane Cartesian coordinate,

and we have omitted an uninteresting normalization constant. The extra ¬‚ux È

adds an Aharonov“Bohm phase of ¼ 2È=È0 to each single-particle state

as we encircle the origin. In order to preserve single-valuedness of the wave-

functions, we must then add a compensating phase, and the single-particle

state now becomes

mþ

jzj2

z

m; °zÞ ¼ exp À 2 :

˜B 4˜B

If the ¬‚ux È is precisely one unit of ¬‚ux quantum, È ¼ È0 , the mth single-

particle wavefunction becomes the °m þ 1Þth single-particle wavefunction.

This is precisely analogous to our discussion earlier in the Landau gauge.

If we started with a full Landau level with uniform charge density, then by

inserting one unit of ¬‚ux quantum we have expelled precisely one electron

from the center of the system by pushing all electron charge uniformly out to

the edges.

We use these ideas to create a charge de¬cit “ a quasihole “ in the Laughlin

state. Since the change in the single-particle wavefunctions depends on funda-

mental principles such as gauge invariance and minimal coupling, the Laughlin

wavefunction must respond in a very similar way by shifting zm ! zmþ1 if the

i i

center of the system is pierced by a ¬‚ux quantum. The result will be some

de¬cit of charge. Laughlin used this observation to propose the following

Ansatz wavefunction for a quasihole at the position z0 :

Y Y Y jzj j2

Éþ ¼ °zi À z0 Þ °zi À zj Þ exp À 2 :

p

p

4˜B

j

i i<j

One can verify that this wavefunction has a component of angular momentum

in the direction of the applied magnetic ¬eld equal to 0½1 N°N À 1Þp þ N. There

2

is now an extra zero at zi ¼ z0 for each electron, so that all electrons are

pushed away from this point, at which there is then a local de¬cit of charge

relative to the ground state. We can calculate this charge de¬cit by using the

plasma analogy again. Then

þ

jÉþ j2 ¼ eÀHp ;

p

363

10.5 Quasiparticle excitations from the Laughlin state

with

X X

X jzi j2

1

þ

Hp ¼ À2p ln jzi À z0 j þ ln jzi À zj j þ :

2˜2

pi B

i<j i

Again with the identi¬cation 2p ! e2 we see that Hþ corresponds to a clas-

p

sical two-dimensional one-component plasma with an extra repulsive phan-

tom charge e* that is ¬xed at z ¼ z0 , and which is smaller in magnitude by a

factor of 1=p than the other charges. Since this charge is repulsive, the plasma

will respond by depleting charge Àe=p from around z ¼ z0 . This charge is

expelled to the edges of the system. Another way to understand that Éþ must

p

have electrons depleted from z ¼ z0 is the fact that we have inserted into the

wavefunction a factor that vanishes as any electron approaches z ¼ z0 . There

must therefore be a charge de¬ciency near z ¼ z0 . Since there is a de¬ciency of

Àe=p near z ¼ z0 , this means that there is a net extra charge of þe=p relative

to the ground state Ép near z ¼ z0 , so this must be a quasihole.

The quasiparticle, which has charge Àe=p, can be constructed in an analo-

gous manner. Instead of adding a ¬‚ux tube at z ¼ z0 with extra ¬‚ux, which

pushes electrons away from the ¬‚ux tube, we add a ¬‚ux tube that depletes the

¬‚ux at z ¼ z0 by one ¬‚ux quantum. Technically, this is a little more compli-

cated than creating the quasihole. We need to construct an operation that

locally removes one ¬‚ux quantum at z ¼ z0 , thereby decreasing the z-compo-

nent of total angular momentum by 0N by moving all single-particle states in

one step towards z0 , which we can without loss of generality take to be the

origin. We can then phrase the task at hand in the following way: given the

Laughlin function Ép , how can we construct an operator that transforms all

the polynomial factors zm to zmÀ1 ? A candidate for this operator is

Y 2 @

N

Sy À z* ;

2˜B 0

m

@zi

i

where it is understood that this operator acts only on the polynomial part of

Ép , leaving the exponential factor intact. Note that the single-particle state

with zero angular momentum about z0 is lifted to the next Landau level by

this operator. Since we demand that our states have no components in

Landau levels other than the lowest one, we interpret this as an annihilation.

The same line of argument as for the quasihole can then be used to show that

ÉÀ ¼ Sy Ép

p m

364 The quantum Hall effect

corresponds to an excess charge Àe=p near z ¼ z0 in an otherwise uniform

state. The extra charge is removed from the edges and moved to z ¼ z0 by

˜˜contracting™™ all single particle states towards z ¼ z0 .

The elementary excitations in the Laughlin state are not only fractionally

charged “ they also can be thought of as having fractional statistics. A good

way to start to clarify this concept is to consider regular particles “ bosons or

fermions “ in three dimensions, where the spin-statistics theorem tells us that

these are the only two classes of particles allowed. The bosons are described

by a wavefunction that is symmetric as two particles exchange positions, and

fermions are described by one that is antisymmetric under this exchange. It is

useful to cast this in terms of an ˜˜exchange phase.™™ We imagine a many-body

system in three dimensions consisting of identical bosons or fermions, and

keep all but two of the particles ¬xed at their positions. We then move one of

the remaining two adiabatically in a counterclockwise rotation of about the

other, and perform a translation that puts one particle at the original position

of the other, and vice versa. The net result is then to have interchanged the

positions of two particles. During the operation, the phase of the wavefunc-

tion changes. This change in phase depends directly only on the positions of

the two particles that we move. At the end of the operation, the change in

phase is an even integer times if the particles are bosons, and an odd integer

times if the particles are fermions. These are the only possibilities in three

dimensions. In two dimensions, on the other hand, particles can be chosen to

be fermions or bosons or anything in between (with respect to the statistics of

the particles), so long as the change in statistics, the change in exchange phase,

is compensated for by including some interaction between the particles.

Let us now try to apply this kind of reasoning to two quasiholes in a Ép

Laughlin state. First, we have to construct a viable candidate for a two-

quasihole wavefunction. We might think that something like

Y Y Y jzi j2

g

Éþ2 ¼ °zi À uÞ°zj À wÞ °zi À zj Þ exp À 2 ; °10:5:1Þ

p

p

2˜B

i i<j i

where u and w are the positions of the quasiholes of charge e=p, might do the

trick. However, this wavefunction has a serious de¬ciency. Imagine moving

one quasihole in a circle about the other, keeping all other charges

(including the other quasihole) ¬xed. Since we are moving a charge e=p

around a ¬‚ux tube (the location of the other quasihole) carrying a unit ¬‚ux

quantum, the many-body wavefunction must pick up an Aharonov“Bohm

phase of 2=p. However, the wavefunction given in Eq. (10.5.1) clearly picks

up a phase that is only an integer times 2, and so we must add something to

365

10.5 Quasiparticle excitations from the Laughlin state

the wavefunction to ¬x this di¬culty. In order for the wavefunction to pick

up the correct phase, it must be a function only of the di¬erence of the

coordinates of the quasiholes. This is also more generally required by the

fact that the system must be translationally invariant. In addition, we must

require that the wavefunction remain in the lowest Landau level. A trial

wavefunction that satis¬es these conditions is

Y Y

É2þ ¼ °u À wÞ 1=p

°zi À uÞ°zj À wÞ °zi À zj Þp

p

i; j i<j

! !

Y jzk j2 1

‚ exp À 2 exp À °juj þ jwj Þ :

2 2

°10:5:2Þ

4p˜2

4˜B B

k

By using the by now familiar plasma analogy we can see that this wavefunc-

tion indeed corresponds to a uniform plasma interacting with two positive

phantom charges e=p located at u and w, respectively, plus a term corre-

sponding to the interaction between the two phantom charges themselves.

We are now in a position to examine the exchange phase and the statistics

of the quasiholes. We note that the factor °u À wÞ1=p in Eq. (10.5.2) makes the

wavefunction É2þ multi-valued in the parameters u and w, and so we are

p

prepared to believe that something odd can happen to the phase of the

wavefunction. Indeed, if we now perform the exchange operation on the

two quasiholes at u and w, we ¬nd that the phase of the wavefunction

changes by =p. Not only is the charge of the quasiholes a fraction set by

the denominator in the ¬lling factor, so is the exchange phase! The possibility

of exchange phases and therefore particle statistics other than and 2 is

unique to two dimensions, and has led to use of the term anyons to describe

particles of arbitrary exchange phase and statistics.

Just as the quasiholes obey fractional statistics, so too do the quasiparti-

cles. The same arguments about translational invariance, Aharonov“Bohm

phase, and analyticity can be applied to a two-quasiparticle wavefunction,

but the algebra is a little more involved.

There is another way, due to Arovas and coworkers, to arrive at the

exchange phase. Imagine that we start with a Laughlin state Ép , and we

then add a quasihole of charge e* ¼ e=p at ¬xed external magnetic ¬eld.

This means that we cannot easily write down the wavefunction for this

state, since the quasiholes or quasiparticles we constructed previously

involved adding or subtracting ¬‚ux quanta, which necessarily changes the

total external magnetic ¬eld. Next, we drag the quasihole around some closed

contour C enclosing an area A. This gives an Aharonov“Bohm phase change

366 The quantum Hall effect

to the total wavefunction, with

þ

e* È

e*

¼À A Á d< ¼ À2 ;

0c e È0

C

where È is the total ¬‚ux enclosed by the contour. Since there are p ¬‚ux

quanta per electron, we also have È ¼ À2pNenc , where Nenc ¼ ÀA=e is

the number of electrons enclosed by the contour. So the Aharonov“Bohm

phase counts the charge enclosed by the contour. Next, we add another

quasihole of charge e* inside the contour, still keeping the total magnetic

¬eld ¬xed. As we now drag our ¬rst quasihole around the contour C, there is

a net charge of ÀeNenc þ e*, so the phase change of the wavefunction is now

0 ¼ À2½Nenc À 1=p. That is, there is a phase contribution of À2=p due to

the one quasiparticle encircling the other quasiparticle inside the contour.

Since the exchange phase of two quasiholes is half of this, we conclude that

the exchange phase is of magnitude =p. The same argument can also be

made for quasiparticles.

To show that the Hall resistivity is quantized in a Laughlin state, we can

apply a gauge argument similar to the one we used in the integer quantum Hall

e¬ect. The di¬erence is that we have to apply p ¬‚ux quanta in order for the

ground state to return to itself, if we consider a fractional quantum Hall state at

¼ 1=p on a ribbon. This gives a Hall resistivity of Àph=e2 (cf. Eq. (10.2.8)).

Perhaps we can feel intuitively that we must have H ¼ Àh=°e2 Þ for the

fractional quantum Hall e¬ect, too. One factor of e in the denominator

comes from the minimal coupling to the vector potential, and a factor

of e comes from the charge of the quasiparticles. The ¬nite width of the

plateaus comes again from disorder, which creates bands of localized states

(in this case, quasiparticles and quasiholes, which are created as the ¬lling

factor is moved away slightly from 1=p), in which we can pin the Fermi

energy. The transport is dissipationless because of the excitation gap for

extended states.

We have now seen that the Laughlin wavefunction gives a very good

description of the ground state and its elementary excitations at ¬lling factors

¼ 1=p, with p an odd integer. What about other quantum Hall fractions,

such as ¼ 2=5? One can attempt to construct sequences of Laughlin states

to describe these. For example, we can imagine that we start with the ¼ 1=3

state. We then further increase the external magnetic ¬eld. This will move the

system to a lower ¬lling, which we can attempt to describe as a Laughlin state

with a relatively high density of quasiholes. Eventually, these holes may

condense into a Laughlin state of quasiholes. However, the technical details

367

10.6 Collective excitations above the Laughlin state

of this sort of description rapidly become intractable. In addition, the energy

gaps predicted by a simple application of this approach do not bear a very

great resemblance to those deduced from experimental observations of the

relative prominence of the various plateaus. It turns out that there is a much

more convenient and simple description, based on composite fermions, which

not only correctly describes the sequence of energy gaps of the ground states,

but also the elementary and collective excitations above these. We discuss

composite fermions in the last section of this chapter.

10.6 Collective excitations above the Laughlin state

Suppose we move the overall ¬lling factor slightly o¬ a Laughlin ¬lling factor

¼ 1=p, for example by increasing or decreasing the strength of the external

magnetic ¬eld. It is energetically favorable for the system to respond by

creating quasiholes or quasiparticles, while keeping most of the electron

density ¬xed at ¼ 1=p. These excitation energies, E þ and E À , consequently

give us the change in slope of the ground state energy at a ¬lling factor

¼ 1=p. Equivalently, the lowest-energy way to add or remove electrons

from the system is to have the density excess or de¬cit break into quasipar-

ticles or quasiholes, leaving most of the system unchanged at ¼ 1=p. The

quasiholes and quasiparticles are charged excitations. We can also consider

neutral excitations, which occur when the particle number and magnetic ¬eld

are kept ¬xed. We can construct these from quasiholes and quasiparticles as

long as we maintain charge neutrality by creating only quasihole“quasi-

particle pairs. We can imagine that we create such a pair, and then move

the localized quasihole and quasiparticle far apart. The situation is now

reminiscent of the one we encountered in Section 2.7, when we considered

the e¬ect of creating an electron“hole pair in the three-dimensional electron

gas. There we found that the operator cy cp created a satisfactory excitation

pþq

when q was large, but that when q was small we needed to form the linear

combination of the operators cy cp that created density ¬‚uctuations. The

pþq

same circumstances arise in considering excitations above the Laughlin

ground state. If we consider excitation energy as a function of wavevector

k, this quasihole“quasiparticle pair corresponds to the excitation energy at

large wavevectors, k˜B ) 1. The limit k˜B ! 0, on the other hand, gives the

energy of a very long-wavelength density ¬‚uctuation, which must be made up

of a linear combination of many quasihole“quasiparticle pairs. Since the

system is incompressible, there must also be an energy gap at k˜B ! 0. We

have already noted that at ¬lling factors smaller than about 1=7, the ground

state of the system is a Wigner crystal. This means that as we lower the ¬lling

368 The quantum Hall effect

factor down towards 1=7, the electron liquid of the Laughlin state must

somehow freeze and transform into a crystal structure with a length scale

of about ˜B . The analogy we make now is with the Peierls transition discussed

in Section 6.3, where a phonon mode was softened until its frequency van-

ished, whereupon a permanent distortion occurred. We may expect a similar

phenomenon here, and look for an excitation mode at k $ 1=˜B to become

increasingly soft as the ¬lling factor is reduced, and for its excitation energy

to reach zero. When this happens, the translationally invariant liquid is no

longer the ground state.

It turns out that there is in fact a minimum in the excitation energy of the

Laughlin state as a function of wavevector at about k % 1=˜B . This minimum

is very well described by a theory analogous to the theory that produced the

so-called roton minimum in liquid 4 He shown in Fig. 3.4.2, and the excita-

tions at this minimum are therefore appropriately called magneto-rotons.

The core of the theory is a calculation of the energy expectation value of a

variational Ansatz, or trial, wavefunction for an excited state corresponding

to a density wave of wavevector q. The obvious choice is

Éq ¼ N À1=2 q jÉ0 i;

where q is the Fourier transform of the density operator and jÉ0 i the ground

state whose energy is E 0 . The factor of N À1=2 , with N the particle number, has

been inserted for convenience. The norm of this state, which is required to

evaluate the expectation energy, is

s°qÞ ¼ N À1 hÉ0 jy q jÉ0 i;

q

which is also known as the static structure factor of the ground state jÉ0 i.

This quantity can be measured directly by, for example, neutron scattering.

The expectation value of the excitation energy of the state Éq is then

‚ Ã

hÉq jH À E 0 jÉq i N À1 hÉ0 jy H; q jÉ0 i f °qÞ

q

Á°qÞ ¼ ¼ ¼ ;

s°qÞ s°qÞ s°qÞ

where f °qÞ is called the oscillator strength. It is a measure of how much of the

phase space available to excitations is ¬lled by the mode under consideration.

For the case of liquid helium, it is easy to derive the result

02 q2

f °qÞ ¼ ;

2m

369

10.6 Collective excitations above the Laughlin state

since the potential energy and the density operators commute with one another,

while the kinetic energy and the density operators do not. This leads to the

so-called Feynman“Bijl formula for the liquid He excitation energy:

02 q2

Á°qÞ ¼ :

2ms°qÞ

This equation says that the excitations are essentially free particle excitations

renormalized by the structure factor s°qÞ due to correlations between particles.

One can construct a very similar theory for collective excitations in the

fractional quantum Hall e¬ect. One complication, which makes the algebra

too lengthy to reproduce here, is that we need to project all actions of

operators on wavefunctions onto the lowest Landau level. At the end of

the day, one arrives at an equation very similar to the Feynman“Bijl formula,

except that the oscillator strength and the structure factor have to be replaced

by quantities projected onto the lowest Landau level, f"°qÞ and s°qÞ. The

"

corresponding equation for the collective mode excitation energy is then

f"°qÞ

Á°qÞ ¼ :

"

s°qÞ

One important di¬erence between liquid helium and the fractional quantum

Hall e¬ect is that the latter has an excitation gap at q ! 0, while, as we saw

in Fig. 3.4.2, the former does not. This implies that limq!0 Á°qÞ is ¬nite. One

can show that f"°q ! 0Þ $ jqj4 , which means that we must have s°q ! 0Þ $

"

jqj in order to have a ¬nite gap at q ! 0. Detailed calculation shows that

4

this is indeed the case for any liquid ground state in the lowest Landau level,

not just the Laughlin functions. For ¬nite q the excitation energies Á°qÞ

for Laughlin states can be found by numerically evaluating the projected

structure factor for the Laughlin wavefunction. The result is in very good

agreement with direct numerical diagonalizations for the lowest-lying excited

mode, and with experimental observations.

This approach to ¬nding the magneto-roton collective modes is called a

single-mode approximation, because it assumes that there is a single excita-

tion mode for each q. The reason that this theory works for liquid helium lies

in the fact that the symmetry of the wavefunction for bosons only allows for

low-lying collective density modes. But this is not the case for a system of

fermions, for which there can in principle be a continuum of single-particle

excitations and, for the case of the quantum Hall e¬ect, intra-Landau level

excitations. This is where the existence of a gap comes in and saves the day.

370 The quantum Hall effect

The excitation gap in the fractional quantum Hall e¬ect quenches out single-

particle-like excitations and leaves only the low-lying collective modes.

10.7 Spins

So far, we have limited our discussion to a fully spin-polarized system. This

seems reasonable, since in the strong magnetic ¬elds used in experiments on

quantum Hall systems one might expect the Zeeman spin splitting gB B,

where g is the e¬ective Lande factor and B the Bohr magneton, to be

´

large enough that the high-energy spin direction would be energetically inac-

cessible. However, two factors conspire to make the Zeeman splitting very

low in GaAs, which is the material from which many quantum Hall devices

are constructed. First of all, spin“orbit coupling in the GaAs conduction

band e¬ectively lowers the Lande factor to g % 0:44. Second, the low e¬ective

´

mass, m* % 0:067me , further reduces the ratio of spin-splitting energy to

cyclotron energy to about 0.02, compared with its value of unity for free

electrons. For magnetic ¬elds of about 1“10 T, the Coulomb energy scale of

the electron“electron interactions in GaAs is e2 =°˜B Þ, with the static dielec-

tric constant being about 12.4, and is of the same order as the cyclotron

energy. As a ¬rst approximation, one should then set the Zeeman energy to

zero, rather than in¬nity, since it is two orders of magnitude smaller than the

other energy scales. As a consequence, the spin degree of freedom is governed

by the electron“electron interactions, rather than by the Zeeman energy. This

dramatically changes the nature of the low-energy bulk single-particle excita-

tions near ¬lling factors ¼ 1=p, with p odd, from single-particle spin-¬‚ips to

charge-spin textures. In these objects, loosely called skyrmions, the spin den-

sity varies smoothly over a distance of several magnetic lengths, so that the

system can locally take advantage of the exchange energy by having spins

roughly parallel over distances of the order of a magnetic length. The mag-

netization has a ¬nite winding number n, which is to say that if we encircle a

skyrmion along some closed path, the magnetization direction will change by

2n, where n is an integer. The kind of spin texture excitations that make up

skyrmions have been known to exist in other models of magnets, but what is

remarkable about the skyrmions in the quantum Hall e¬ect is that they carry

charge, and the charge is equal to ne=p, with 1=p being the ¬lling factor .

This coupling between charge and spin is a direct consequence of the fact

that these are two-dimensional systems in the presence of a strong external

magnetic ¬eld. Let us suppose that the (bulk) ¬lling factor is initially unity.

If there is a region in space where the spin is slowly varying spatially,

^

that changes the e¬ective magnetic ¬eld Beff °rÞ ¼ Bz þ 4M°rÞ, where M°rÞ

371

10.7 Spins

is the magnetization density due to the varying spin density. But that would

result in a concomitant change in local e¬ective ¬lling factor away

from unity. The system desperately wants to maintain a ¬lling factor of

unity, so it responds by locally transferring some charge into the region of

varying spin density to maintain an e¬ective ¬lling factor of unity. The net

e¬ect is a local accumulation (or de¬cit) in charge relative to the ground

state.

We can make this argument more formal in the following way. Let us

assume that the spin density varies slowly on the scale of ˜B for a quantum

Hall system at bulk ¬lling factor ¼ 1=p. The spin Sj of electron j sees an

e¬ective exchange ¬eld b°rÞ due to the spin density of the other electrons.

This exchange ¬eld just expresses the fact that the electrons gain exchange

energy by keeping their spins parallel. Formally, it is de¬ned as the change

in exchange-correlation energy as we change the direction of one electron™s

spin, while keeping the others ¬xed. In a mean-¬eld approximation, we

do not distinguish between the exchange ¬elds of di¬erent electrons, but

take the ¬eld at r to be a suitable average over the exchange ¬elds in

some neighborhood about r. A model Hamiltonian expressing this coupling

would be

X

N

Heff ¼ À b°rÞ Á Sj :

j

Imagine that we move a single electron adiabatically around a closed path C

in real space, keeping all other electrons (and their spins) ¬xed. The electron

will keep its spin aligned with the exchange ¬eld as we move it along C and

trace out some path ! in spin-space. This path is the path drawn by a unit

vector on the unit sphere as the vector moves through the same angles °; Þ

as the spin along C. This means that the electron wavefunction will acquire an

extra phase, a so-called Berry™s phase, which is analogous to the Aharonov“

Bohm phase that a charge acquires as it moves through a region with a ¬nite

vector potential. The Berry™s phase is =2, where is the solid angle sub-

tended by the path !. This is illustrated in Fig. 10.7.1.

We must not forget that the electron also acquires an Aharonov“Bohm

phase, since it is charged and moves through a region with a ¬nite vector

potential. There are thus two contributions to the added phase, the Berry™s

phase, and the Aharonov“Bohm phase. The electron cannot tell where the

contributions to the phase come from, and we might as well replace the

Berry™s phase by adding some extra ¬‚ux ÁÈ in the region enclosed by C,

such that the Aharonov“Bohm phase due to this ¬‚ux equals the Berry™s

372 The quantum Hall effect

Figure 10.7.1. As an electron moves along a path C in real space with an inhomo-

geneous magnetic ¬eld, the direction of the electron™s spin traces a path ! on the unit

sphere in spin space. The solid angle subtended by the path ! is .

phase. This means that

ÁÈ ¼ È;

4 0

where È0 is the ¬‚ux quantum. But adding extra ¬‚ux in the region enclosed by

C will add extra charge, for we remember that the quasiholes and quasipar-

ticles were generated by adding ¬‚ux at the positions of the elementary excita-

tions. A simple application of the ¬‚ux argument from Section 10.5 shows that

the extra charge induced is

ÁÈ

ÁQ ¼ Àe ¼ Àe ;

È0 4

if the bulk Hall conductivity xy ¼ Àe2 =h.

We can construct simple trial functions for skyrmions of charge Æe for the

case of ¼ 1. It is convenient to work in the symmetric gauge, and to consider

the wavefunction of a skyrmion centered at the origin. The basic idea is to

start with a ¬lled, spin-polarized Landau level, which we write as

jÉ0 i ¼ …m cy j0i, where j0i is the vacuum state, and cy creates an up-spin

m m

electron of z-component of orbital angular momentum 0m in the lowest

Landau level. To construct a skyrmion hole we ¬rst remove the spin-up

electron in the m ¼ 0 state by use of the operator c0 , and then replace part

of the amplitude of the other spin-up states with a component with spin down

and with a value of m reduced to m À 1. We do this by operating with a

product of terms of the form °vm þ um by cmþ1 Þ where by creates a down-spin

m m

y

electron of angular momentum 0m. The term °v0 þ u0 b 0 c1 Þ, for example,

replaces part of the spin-up electron amplitude in the m ¼ 1 state by a con-

tribution with spin down in the m ¼ 0 state. It thus has the double e¬ect of

increasing the amount of spin-down component in the wavefunction and

bringing closer to the origin some of the charge. This reduces somewhat the

373

10.7 Spins

charge de¬cit near the origin due to the initial destruction of the spin-up

electron with m ¼ 0. The resulting wavefunction will then be

Y

1

½um by þ vm cy j0i:

j Ài ¼ °10:7:1Þ

m mþ1

m¼0

In a similar way we can create a skyrmion particle by ¬rst adding a down-

y

spin electron in the m ¼ 0 state by use of the operator b0 , and then operating

with terms like °vm À um by cm Þ. This has the e¬ect of replacing part of

mþ1

the amplitude of the spin-up states with a component with spin down

and with a value of m increased to m þ 1. It yields the skyrmion particle

wavefunction

Y

1

y

½Àum by þ vm cy b 0 j0i:

j þi ¼ °10:7:2Þ

mþ1 m

m¼0

In this case some of the charge is repelled from the origin by the increase in

m, which reduces the increase of charge near the origin due to the initial

creation of a spin-down electron with m ¼ 0. The set of numbers um and vm

are to be determined by a minimization of the energy in a procedure similar

to the one we used in the BCS theory of superconductivity in Section 7.3,

subject to the normalizing constraint that jum j2 þ jvm j2 ¼ 1. The signs of um

and vm are chosen to make j þ i and j À i orthogonal. The spin texture of a

skyrmion is illustrated in Fig. 10.7.2.

What we now want to do is to demonstrate that j þ i and j À i are approx-

imate energy eigenstates when we make the right choices of um and vm with

u0 6¼ 0 and um decaying as m increases so that the original spin-polarized

Landau level is restored far away from the skyrmion. We must also verify

that the energies of these states are lower than the single-electron quasi-

electron or quasihole energies. It is a good exercise to verify that the spin

Figure 10.7.2. The spin orientation due to a skyrmion located at the origin.

374 The quantum Hall effect

densities of j þ i and j À i point downwards at the origin and upwards as

m ! 1, and that the projection of the spin polarization on the xy-plane

rotates by Æ2 along any path encircling the origin.

We start with the Coulomb Hamiltonian for the lowest Landau level, but

include the Zeeman energy. We have

1X

Vm1 m2 m3 m4 : ½by 1 bm2 þ cy 1 cm2 À m1 m2 ½by 3 bm4 þ cy 3 cm4 À m3 m4 :

H¼ m m m m

2 m1 ;m2

m3 ;m4

X

½cy cm À by bm :

À gB B m m

m

Here Vm1 m2 m3 m4 are the matrix elements of the Coulomb interaction between

angular-momentum single-particle states,

°

e2

Vm1 m2 m3 m4 ¼ d r1 d r2 * 1 °r1 Þ* 2 °r2 Þ

2 2

°r Þ °r Þ;

jr1 À r2 j m3 2 m4 1

m m

and : ½. . . : indicates normal ordering, which keeps all the creation

operators to the left of the annihilation operators. We have not included a

uniform positive background charge density, since that is not important

here.

Next, we proceed with the Hartree“Fock reduction of the terms with four

electron creation and annihilation operators. When we were studying the

uniform electron gas in Section 2.4, our procedure was to group together

pairs of electron creation and annihilation operators to make number opera-

tors, which we then replaced with their expectation values. In the present

case, our wavefunction is no longer a simple Slater determinant, but is of the

form given by Eqs. (10.7.1) and (10.7.2). Thus we must allow for the fact that

not only terms like hby bm i and hcy cm i but also those like hby cmÆ1 i will

m m m

contribute. (If we had been considering skyrmions with winding numbers

larger than unity, there could also in principle be other combinations, such

as hby cmÆn i with n > 1.)

m

The minimization of the expectation value of H proceeds in a manner very

close to that used in the BCS theory to arrive at Eq. (7.3.12), which related

the coe¬cient xk to the gap parameter Á through the relation

Ek

xk ¼ Æ q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ ;

2 E 2 þ Á2

k k

375

10.7 Spins

but with Ák self-consistently dependent on the interaction V and on xk itself.

For the skyrmion problem we have the very similar results

UÆ °mÞ

sk

um ¼ q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ ;

E 2 þ jUÆ °mÞj2 sk

m

°10:7:3Þ

Em

vm ¼ q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬ ;

E 2 þ jUÆ °mÞj2 sk

m

but with rather more complicated de¬nitions of the components. Here

E m ¼ 1 fE b °mÞ À E c °mÞ þ ½°E b °mÞ À E c °mÞÞ2 þ 4jUÆ °mÞj2 1=2 g;

sk

°10:7:4Þ

2

and, with a standing for either b or c,

E a °mÞ ¼ sa gB B þ U H °mÞ þ Ua °mÞ;

ex

°10:7:5Þ

with sa ¼ 1 for a ¼ b, and sa ¼ À1 for a ¼ c. The interaction-energy terms U

are

X

Vmm1 mm1 ½hby 1 bm1 i þ hcy 1 cm1 i À 1

U °mÞ ¼

H

m m

m1

X

Vmm1 m1 m hay 1 am1 i

Ua °mÞ ¼

ex

°10:7:6Þ

m

m1

X

Vm;m1 ;m1 Æ1;mÆ1 hby 1 cm1 Æ1 i;

UÆ °mÞ ¼

sk

m

m1

with U H °mÞ the direct (Hartree) term, U ex °mÞ the exchange (Fock) term, and

UÆ °mÞ the skyrmion-speci¬c terms that describe a spin-¬‚ip together with a

sk

change in angular momentum. The order parameters, i.e., the expectation

values that contribute to the energy, are given by the relations

hby bm i ¼ jum j2 ;

m

hcy cm i ¼ jvmÇ1 j2 ; °10:7:7Þ

m

hby cmÆ1 i ¼ u* vm :

m m

The self-consistent equations (10.7.3) to (10.7.7) must be solved numeri-

cally. The result is that for any value of g, the hole-skyrmion and the particle-

skyrmion have lower energies than the simple quasihole or quasiparticle, in

376 The quantum Hall effect

which a regular hole or an electron with opposite spin is introduced at the

origin in the spin-polarized lowest Landau level. The energy di¬erences are

largest for g ! 0 and vanish as g becomes very large. Physically, what hap-

pens when the Zeeman energy is increased by increasing g is that the region of

the skyrmion in which the spin is varying shrinks in size, since it will cost

increasingly more energy not to align the spin density with the external ¬eld.

In the limit of zero size, the skyrmions become identical to quasihole and

quasiparticle excitations.

The fact that skyrmions have lower energy than quasiparticle and quasi-

hole excitations near ¼ 1 has experimental consequences. If skyrmions did

not exist, or had higher energies than quasiparticles and quasiholes, the

ground state would remain spin-polarized as the ¬lling factor is reduced

below unity. Electrons would simply be removed from the lowest Landau

level (or new, empty states added) and the remaining ones would all be spin-

polarized. Also, as the ¬lling factor is increased above unity, the spin polar-

ization would decrease at the rate at which new spin-reversed electrons are

added, assuming still that the spin splitting of Landau levels is smaller than

the cyclotron energy. However, it is experimentally observed that the ground-

state polarization is rapidly destroyed when the ¬lling factor is either

increased or decreased away from unity. The reason for this is that it is

cheaper for the system to add skyrmion particles or skyrmion holes than

quasiparticles or quasiholes. The net spin of skyrmions, obtained by integrat-

ing the spin density, is about 70=2, compared with the 0=2 found for quasi-

particles and quasiholes. This means that considerable spin is added with

each skyrmion, and there is a resulting rapid destruction of the ground-

state polarization.

10.8 Composite fermions

We have so far, within the context of Laughlin™s wavefunction, only dis-

cussed the ˜primary™ fractional quantum Hall states with ¬lling factors

¼ 1=p, with p odd. In Section 10.5 we indicated that one may try to con-

struct Laughlin-type wavefunctions for quasiparticles or quasiholes, which

could conceivably form a strongly correlated liquid on top of the underlying

basic fractional quantum Hall state. Then this correlated ˜˜Laughlin liquid™™

of quasiparticles or quasiholes would admit fractionally charged excitations,

which in turn could condense and form a new correlated state, and so on. The

problem with this picture, apart from its rapidly increasing and open-ended

complexity, is that it does not adequately explain the sequence of energy gaps

observed in fractional quantum Hall states. One would expect qualitatively

377

10.8 Composite fermions

that the stability of states, as re¬‚ected in the magnitude of the energy gaps,

should rapidly decrease as one climbs to higher levels in this hierarchical

picture. This is generally not the case. For example, the states ¼ 1=3 and

¼ 2=5 are in general more stable than the ¼ 1=5 state. Also, the quasi-

particles and quasiholes in one hierarchical level have a ¬nite size, and one

would rapidly reach a situation in which the number of quasiparticles or

quasiholes needed to condense at one level to form a new one would be so

great that these elementary excitations would overlap substantially, in which

case the notion of elementary quasiparticles and quasiholes ceases to be

meaningful. Finally, the Laughlin wavefunction provides no connection

between the integer quantum Hall states and the fractional ones, treating

the two as fundamentally very di¬erent.

Let us go back and consider the sequence of fractions at which the frac-

tional quantum Hall e¬ect is observed. We can group the sequences in the

following manner:

1234

n

¼ ¼ ; ; ; ;...

2n þ 1 3 5 7 9

2345

n

¼ ¼ ; ; ; ;...

2n À 1 3 5 7 9

12 3

n

¼ ¼ ; ; ;...

4n þ 1 5 9 13

23

n

¼ ¼ ; ;...

4n À 1 7 11

47

n

¼1À ¼ ; ;...

4n þ 1 5 9

By inspection, we see that we can in general write these fractions as

n

¼ ; °10:8:1Þ

2pn Æ 1

and

n

¼1À : °10:8:2Þ

2pn Æ 1

For given p and n, the fractions in Eqs. (10.8.1) and (10.8.2) are related by an

electron“hole symmetry that is exact if we ignore the presence of any other

than the lowest Landau level. In a real system, the energy gap “ the cyclotron

energy “ separating di¬erent Landau levels is ¬nite, with the consequence

378 The quantum Hall effect

that the real wavefunction will have some admixture of higher Landau levels.

However, this admixture is small enough that the electron“hole symmetry is

almost exactly preserved.

The composite fermion picture, originated by Jainendra Jain, provides a

simple and natural picture within which these sequences can be viewed. It

also attempts to tie together the physics of the integer and fractional quan-

tum Hall e¬ects. In addition, it provides a powerful computational method

with which to study general fractional quantum Hall ground states, as well as

their quasiparticle and collective excitations. While the origins of the com-

posite fermion picture are empirical, numerical calculations within this pic-

ture have consistently proven to be remarkably accurate when compared with

other, more direct, schemes as well as with experiments, and have provided

strong support for this approach. The basic principle is to replace the

strongly interacting electrons by some other particles, which are chosen to

be weakly interacting. That is, we try to construct some composite particle

such that the fractional quantum Hall ground states are well described by a

gas of such noninteracting particles. This is reminiscent of Fermi liquid

theory, in which a system of rather strongly interacting electrons can be

described as a weakly interacting system of quasiparticles. In Fermi liquid

theory the interaction between electrons leads to quasiparticles of di¬erent

e¬ective mass and with a modi¬ed energy dispersion relation. Similarly, we

would here like to ¬nd new particles such that the strong interactions of the

true electrons have been transformed into kinetic energy of these new parti-

cles. The question then is to identify the kind of particle that would be a good

candidate for a similar description of fractional quantum Hall states. Jain

observed that we can write the Laughlin wavefunction for, say, n ¼ 1=3, in

the following way:

Y Y jzk j2

É3 ¼ °zi À zj Þ3 exp À 2

4˜B

i<j k

Y Y Y jzk j2

¼ °zi À zj Þ2 °zi À zj Þ exp À 2 :

4˜B

i<j i<j k

That is, we can think of the Laughlin wavefunction for the ¼ 1=3 state as

obtained by starting with the ¬lled Landau level of ¼ 1 and multiplying

Q

that unique wavefunction by the factors i<j °zi À zj Þ2 . But, according to our

earlier discussion, this factor has the same e¬ect as attaching two ¬‚ux quanta

to each electron (as seen by the other electrons). So the ¼ 1=3 wavefunction

can be thought of as a ¬lled ¼ 1 Landau level for particles that consist of

379