IUCM96-002

Non Possum Comprimi Ergo Sum: Skyrmions and Edge States in the
Quantum Hall Effect

arXiv:cond-mat/9601144v1 30 Jan 1996

A.H. MacDonald
Indiana University, Department of Physics, Bloomington, IN 47405, USA.
(October 5, 2017)

Abstract
When the chemical potential of an electron system has a discontinuity at
a density n∗ , the system is said to be incompressible and a finite energy is
required to create mobile charges in the bulk of the system. The quantum
Hall effect is associated with incompressibilities in a two-dimensional electron
system that occur at magnetic-field dependent densities, n∗ (B). In these
notes we discuss two aspects of the physics of quantum Hall systems that
follow directly from this association.

Typeset using REVTEX
1

I. INTRODUCTION

The quantum Hall effect [1,2] is an anomaly that occurs in the transport properties of
two-dimensional electron system (2DES) in the regime of strong perpendicular magnetic
fields. At certain magnetic fields it is found that the voltage drop in the system in the
direction of current flow, which is responsible for dissipation, vanishes at low temperatures.
Our understanding of this transport anomaly is not absolutely complete, however there is
fairly broad agreement that the effect can occur only when the electronic system has, in the
absence of disorder, jumps in its chemical potential at certain densities (n∗ ), that depend
on magnetic field. In Section II of these notes we specifically discuss the relationship between incompressibility and transport properties in a 2DES. The remaining sections discuss
two aspects of the physics of quantum Hall systems that are direct consequences of this
relationship.
Quantum Hall systems can be ferromagnetic. Under appropriate circumstances the spinmoments of the electrons align spontaneously, i.e. in the absence of Zeeman coupling to a
magnetic field. We will refer to the 2DES in this case as a quantum Hall ferromagnet.
Quantum Hall ferromagnets provide a particularly simple example of a two-dimensional
itinerant electron ferromagnet, and therefore represent an attractive target for theories of the
quantum statistical mechanics of such systems [3–5]. They also have a number of interesting
unique properties, the most striking of which is that their instantons carry an electronic
charge. This property is an immediate consequence of incompressibility at a magnetic-field
dependent density. In Section III of these notes we discuss the electrically charged instantons
of quantum Hall ferromagnets and some of the observable consequences of their presence.
Another consequence of incompressibility at a magnetic field dependent density is that
quantum Hall systems necessarily have gapless excitations localized at their edges. The lowenergy physics of quantum Hall systems is therefore like that of a one-dimensional electron
system which, because of time-reversal symmetry breaking by the magnetic field, can carry
a current even in equilibrium. In Section IV we will discuss the physics of quantum Hall
2

edges, including their description in terms of the Luttinger liquid models appropriate to
one-dimensional fermion systems. Section V contains some concluding remarks.
These notes do not attempt a systematic introduction to the physics of the quantum Hall
effect. Interested readers can consult a previous effort of mine [6] that is cited frequently
below. If greater depth is desired, readers can consult one of the excellent books covering
different aspects of the subject [2]. I have written previously at this level but at greater
length on quantum Hall edge states [7]; parts of the present notes borrow from that text.

II. INCOMPRESSIBILITY AND THE QUANTUM HALL EFFECT

The thermodynamic compressibility of a system of interacting particles is proportional
to the derivative of the chemical potential with respect to density. It can happen that at zero
temperature the chemical potential has a discontinuity at a density n∗ : the energy to add a
particle to the system (µ+ ) differs, at this density, from the energy to remove a particle from
the system (µ− ). The system is then said to be incompressible. In an incompressible system
a finite energy is required to create unbound positive and negative charges that are capable
of carrying current through the bulk. The number of these free charges present in the system
will have an activated temperature dependence and will vanish for T → 0. Incompressible
systems are usually insulating. Paradoxically, as we explain below, incompressibility is
precisely the condition required for the quantum Hall effect to occur. The twist is that in
the case of the quantum Hall effect, the density n∗ at which the incompressibility occurs
must depend on magnetic field. In my view, incompressibility at a magnetic-field-dependent
density is the sine qua non of the quantum Hall effect.
For non-interacting electrons, the single-particle energy spectrum of a 2DES in a magnetic field consists of Landau levels, separated by hωc ≡ heB/m∗ c energy gaps and with a
¯
¯
macroscopic degeneracy Nφ = AB/Φ0 = A/(2πℓ2). (Here B is the magnetic field strength,
Φ0 = hc/e is the magnetic flux quantum, A is the area of the system, and m∗ is the electron
mass.) Both the energy gaps between Landau levels and the degeneracy of the Landau levels
3

are proportional to B. Chemical potential discontinuities occur whenever the density is an
integral multiple of n∗ = B/Φ0 . We show below that this property requires the existence
of the gapless edge excitations discussed in Section IV. It is conventional in discussing the
quantum Hall effect to use a magnetic field dependent density unit by defining the Landau
level filling factor ν ≡ n/n∗ = 2πℓ2 n. For non-interacting electrons, incompressibilities occur
at integer filling factors. When interactions are included, incompressibilities can also occur
at fractional filling factors and, as we will discuss, physical properties near integer filling
factors can be qualitatively altered.
The relationship between incompressibility and the transport anomalies that give the
quantum Hall effect its name can be understood by the following argument [8]. Consider
a 2DES at zero temperature, as illustrated in Fig. [1]. We consider the case in which
the chemical potential lies in the ‘charge gap’; µ ∈ (µ− , µ+ ). We want to consider the
change in the equilibrium local currents, present in the system because of the breaking of
time-reversal-invariance by the magnetic field, when we make an infinitesimal change in the
chemical potential, δµ. Because µ lies in the charge gap the change in the local current
density anywhere in the bulk of the system must be zero. The current density can change, if
it does anywhere, only at the edge of the system. It follows from charge conservation that, if
there is a change in the current flowing along the edge of the system, it must be the same at
any point along the edge. We can relate this change in current to the change in the orbital
magnetization:
δI =

c
δM.
A

(1)

Eq. (1) is just the equation for the magnetic moment of a current loop. However,
δM =

∂N
∂M
|B δµ =
|µ δµ.
∂µ
∂B

(2)

(N is the number of electrons.) The second equality in Eq. (2) follows from a Maxwell
relation. Combining Eq. (1) and Eq. (2) we obtain the following result for the rate at which
the equilibrium edge current changes with chemical potential when the chemical potential
lies in a charge gap:
4

δI
∂n∗
=c
|µ .
δµ
∂B

(3)

The fact that δI/δµ = 0 implies that whenever the charge gap occurs at a density that
depends on magnetic field, there must be gapless excitations at the edge of the system.
Properties of these low-energy edge excitations are discussed in Section IV.
Eq.( 3) is expected to apply to the edge states even when the chemical potential lies
only in a mobility gap and not in a true gap, as illustrated schematically in Fig. [1]. A net
current can be carried from source to drain across the system by changing the local chemical
potentials only at the edges and having different chemical potentials along the two edges
connecting source and drain. When bulk states are localized, the two edges and the bulk
are effectively decoupled from each other. Eq. (3) then also applies to transport currents,
relating the current carried from source to drain to the chemical potential difference between
the two edges, equal to eVH where VH is the Hall voltage. There is no voltage drop along
an edge since each edge is in local equilibrium and hence no dissipation inside the sample.
Eq. (3) is commonly known as the Stˇeda-Widom formula [9]. In using this picture to explain
r
transport experiments in bulk systems it is necessary to claim that the transport current
will be carried entirely at the edge of the system even when bulk states occur at the Fermi
level, as long as these states are localized. There are difficulties with this argument as a
complete explanation for all transport phenomena associated with the quantum Hall effect,
but that is another story and we will not pursue it here.

III. QUANTUM HALL FERROMAGNETS

A. Energy Scales

The quantum Hall regime is usually understood as the regime in which no qualitative
change in physical properties results from mixing of Landau levels by either interactions or
disorder. It is common in theoretical studies to truncate the Hilbert space to a single orbital
Landau level and include mixing , if at all, only when making quantitative estimates for
5

comparison with experiment. The quantum Hall regime, then, assumes that the Landau
level separation, hωc is larger than other energy scales of interest. On the other hand the
¯
ferromagnetic state is generally defined in terms of the properties of an electronic system
in the absence of a magnetic field. The term Quantum Hall Ferromagnet appears to be an
oxymoron. To understand why it is not only sensible but also of more than academic interest
to add this category to our taxonomy of electronic states, it is necessary to consider the
relevant energy scales for the case of the semiconductor systems in which 2DES’s are realized.
For a free-electron system in a magnetic field, the Zeeman splitting of spin-levels gµB B
and the Landau level separation hωc are identical, apart from small relativistic corrections.
¯
Electrons in states near the conduction band minimum of a semiconductor behave like free
electrons [10] except that band effects renormalize the electron mass m∗ and the g-factor.
In the case of the GaAs systems, where the quantum Hall effect is most often studied, band
effects increase the Landau level separation by a factor of ∼ 20 and reduce the Zeeman
splitting by a factor of ∼ 4. As a result for typical experimental situations, the Landau
level separation (in temperature units) is ≈ 200K, and the characteristic scale for electronelectron interactions is ≈ 100K while the Zeeman splitting is only ≈ 2K. We call a system
a quantum Hall ferromagnet if the electronic spins in the incompressible ground state with
density n∗ align in the absence of Zeeman coupling. In many cases, the properties of a
quantum Hall ferromagnet with such a small Zeeman coupling do not differ noticeably from
the properties when the Zeeman coupling is set to zero. In other cases, the small Zeeman
coupling plays an important role but it is still useful to treat the system as a ferromagnet
in the presence of a small symmetry breaking field.

B. Ferromagnetic Ground States

The Hartree-Fock approximation, in which many-electron states are approximated by
single Slater determinants, provides a simple explanation for itinerant electron ferromagnetism that is often qualitatively correct. For non-interacting electrons, energy is minimized
6

by occupying both spin-states of each single particle energy level. (If the number of majorityspin electrons exceeds the number of minority-spin electrons it is necessary to occupy higher
energy single-particle levels.) The ground state thus has equal numbers of majority-spin
and minority-spin electrons if the total number of electrons is even and the difference is
one if the total number of electrons is odd. This statement applies for an arbitrary spinquantization axis. As long as there is no spin-orbit coupling, the Hamiltonian is invariant
under global spin rotations and the total spin of all electrons is a good quantum number.
The above statement is equivalent to the observation that for non-interacting electrons the
ground state always has total spin S = 0 if the number of electrons is even and total spin
S = 1/2 if the number of electrons is odd. However, interaction energies are lower in singleSlater-determinant states with higher values of the total spin and, generally, is minimized
in fully spin-polarized states with S = N/2. As in the familiar Hund’s rules from atomic
physics, higher spin states tend to have lower interaction energies because like-spin electrons
are prevented from being at the same position by the Pauli exclusion principle and therefore
have more energetically favorable spatial correlations. In the Hartree-Fock approximation,
or in closely related spin-density-functional approximations, an itinerant electron system is
expected to be ferromagnetic if the reduction in interaction energy due to creating a finite
spin-polarization state exceeds the cost in single-particle kinetic energy, or more generally
‘band’ energy. Because of Landau level degeneracy, the cost in kinetic energy of creating a
finite spin polarization for electrons in a magnetic field is precisely zero unless ν is an even
integer. Hartree-Fock or similar approximations would predict a ferromagnetic ground state
for electrons at nearly any value of ν. In fact, this conclusion is incorrect. For example, at
certain filling factors it is known [11] that the interaction energy is minimized in a S = 0
state. We do believe, however, that there exist finite ranges of filling factor over which the
ground state has S/N = 0. For a deeper understanding of this behavior, we need a more
rigorous argument.
The approach we now describe is in the same spirit as the illuminating outlook on the
spin-polarized fractional quantum Hall effect that arises from appropriate hard-core model
7

Hamiltonians [12,13]. As discussed for the case of interest below, these models have zero
energy many-particle eigenstates that are often known analytically, are separated from other
many-particle states by a finite gap, and have a degeneracy that increases with decreasing
N. The incompressible state responsible [6] for a quantum Hall effect transport anomaly
in such a model is the nondegenerate maximum N zero energy eigenstate. The zero energy
eigenstates at lower densities constitute the portion of the spectrum that involves only the
degrees of freedom of the, in general fractionally charged [14], quasiholes of the incompressible state. It is assumed that the difference between the model Hamiltonian and the true
Hamiltonian is a sufficiently weak perturbation that the quasihole states are still well separated from other states in the Hilbert space, although accidental degeneracies will be lifted
in the spectrum of the true Hamiltonian. Here we apply this approach to argue that the
ground state at ν = 1 is a quantum Hall ferromagnet with S = N/2.
For our analysis we use the symmetric gauge in which the single-particle orbitals [6] in
the lowest Landau level are
φm (z) =

zm
exp(−|z|2 /4),
m+1 πm!)1/2
(2

(4)

where [15] m = 0, 1, · · · , Nφ − 1, z = x + iy, and x and y are the Cartesian components of
the two-dimensional coordinate. We study here a hard-core model for which the interaction
is:
V = 4πV0
i<j

δ (2) (ri − rj )

(5)

At strong magnetic fields the low-energy Hamiltonian is simply the projection of this interaction onto the lowest Landau level [12]. Many-particle wavefunctions that are zero energy
eigenstates of this Hamiltonian must vanish when any two-particles are at the same position
and must therefore have the difference coordinate for each pair of particles as a factor:
Ψ[z, χ] = [
i<j

(zi − zj )] ΨB [z, χ].

(6)

We note that the each complex coordinate appears to the power N − 1 in the factor in
square brackets in Eq. (6) and that this factor is completely antisymmetric. It follows that
8

ΨB [z] must be a wavefunction for N bosons and that these bosons can be in states with
angular momenta from 0 to Nφ − N. This simple observation leads to the conclusions we
reach below.
In these notes we discuss only the case where N = Nφ ; the same approach can [16]
be extended to N = Nφ to elucidate the physics of charged excitations in quantum Hall
ferromagnets but here we will follow another line for this part of the discourse. For N = Nφ
all bosons must be in orbitals with m = 0. ΨB [z, χ] must then be proportional to a symmetric
many-particle spinor and therefore have total spin quantum number S = N/2. The orbital
part of the fermion wavefunction can be recognized as the Slater determinant with all orbitals
from m = 0 to m = Nφ −1 occupied. We are able to conclude that the ground state is a strong
ferromagnet with no orbital degeneracy. The ease with which this conclusion can be reached
contrasts markedly with the case of the Hubbard model where enormous effort has yielded
relatively few firm results [17]. When Zeeman coupling is included in the Hamiltonian, the
ground state will be the member of this N + 1 fold degenerate multiplet for which all spins
are aligned with the magnetic field, i.e. the state with Sz = S = N/2.

C. Charged Instantons

In this section we follow a line of argument that emphasizes the role of incompressibility at
a magnetic field dependent density in the unusual properties of quantum Hall ferromagnets.
For the most part we follow Sondhi et al., in applying the non-linear σ model (NLσ) fieldtheoretical description of a ferromagnet to the present case [18]. The NLσ model is intended
to capture the long-wavelength low-energy physics of isotropic ferromagnets. In the NLσ
model, the energy is expressed as a functional of a unit vector m(r), which specifies the
ˆ
direction of the ordered spin-moment as a function of position:
E = E0 +

ρs
2

d2 r |∇m(r)|2 .
ˆ

(7)

We will refer to a particular configuration of the ferromagnet specified by a function m(r)
ˆ
as a spin-texture. Here E0 is the ground state energy, which is independent of the direction
9

of the ordered moment as long as it is constant in space, and ρs , the spin-stiffness, is a
phenomenological constant that must be determined from experiment or calculated from a
microscopic model. The low-energy long-wavelength physics of ferromagnets is dominated
by variations in the direction of the ordered moment that are slow on microscopic length
scales and that can cost vanishingly small energies.
An important aspect of any field theory is the enumeration of its instantons [19]. Instantons are finite excitation energy extrema of an energy (or action) functional, in which the
variation of the field is localized in space. Typically, the stability of an instanton is associated with a topological classification of field configurations. For NLσ models in two space
dimensions and with three-dimensional unit vector fields, all finite energy spin configuration can be classified by an integer valued topological index, sometimes called a topological
charge, which specifies the number of times the order parameter field is wrapped around the
unit sphere when the position r is varied over two-dimensional space. Field configurations
with different topological indices cannot be continuously deformed into one another. The
sign of the topological index depends on the sense of the closed paths traced out on the surface of the unit sphere when r traces a closed path in space. An explicit expression for the
topological index, Q[m], associated with a spin-texture can be derived by first calculating
the solid angle enclosed on the sphere by m(r) when r encloses an infinitesimal area element
ˆ
in space, and then integrating over space:
Q[m] =

1
4π

d2 r m(r) · [∂x m(r) × ∂y m(r)].
ˆ
ˆ
ˆ

(8)

Eq.[ 8] follows from the observation that for a unit vector field, ∂i m(r) is orthogonal to m(r).
ˆ
ˆ
For the 2D NLσ model it is possible to derive analytic expressions for the lowest energy spin-textures of a given topological charge [19,20]. The lowest energy textures with
unit magnitude topological charge, commonly called Skyrmions, have a spin-texture of the
following form:
mx = 2xλ/(r 2 + λ2 )
my = ±2yλ/(r 2 + λ2 )
10

mz = (r 2 − λ2 )/(r 2 + λ2 )

(9)

Here λ is an arbitrary length scale, m has an arbitrary global orientation fixed here by setting
ˆ
m = z for r → ∞, and the skyrmion is centered at an arbitrary point chosen as the origin of
ˆ
ˆ
the coordinate system. It is easy to verify that these spin textures have topological charge
Q = ±1 and excitation energy E − E0 = 4πρs . The form of the Skyrmion spin texture is
illustrated in Fig.[ 2].
The NLσ model considerations in the above paragraph are appropriate for any 2D ferromagnet. Skyrmions lead to contributions to the physical properties of 2D ferromagnets
that have a non-analytic temperature dependence of the form exp(−4πρs /kB T ) at low temperatures. These non-analytic terms are interesting, since they cannot be captured by
perturbative theories, but they typically produce only subtle corrections to a temperature
dependence that is dominantly controlled by thermally excited spin-wave excitations. In
the case of quantum Hall ferromagnets, however, Skyrmions carry an electrical charge and
as a consequence play a more prominent role in determining the physical properties of the
system. To understand why Skyrmions carry a charge it is useful to consider first the case
of a non-interacting electron in the presence of a magnetic field that couples to its orbital
degrees of freedom and an independent strong Zeeman field that couples to its spin and
whose orientation, specified by m(r), varies slowly in space. The direction of the electron
ˆ
spin will vary in space to maintain alignment with the external magnetic field. As is familiar
from the calculation of spin Berry phases, the changing spin direction changes the adiabatic
orbital Hamiltonian that now [21] takes the form:
H=

e
1
(p + (A + AB ))2 .
2m
c

(10)

Here the effective magnetic field due to the varying spin orientation,
∇ × AB =

Φ0
m · [∂x m(r) × ∂y m(r)],
ˆ
ˆ
ˆ
4π

(11)

is proportional to the topological index density of the unit vector field m(r). In a Hartreeˆ
Fock approximation, the exchange interaction will produce a strong effective magnetic field
11

that points in the same direction as the local order parameter, so we can associate m(r)
ˆ
above with the order parameter field. It follows that the effective value of Nφ is changed
by one when the order parameter has a texture with unit topological index. Moreover, as
we have emphasized in these notes, the charge gap of the quantum Hall effect occurs at
an electron density that depends on magnetic field. For a quantum Hall effect at Landau
level filling factor ν we conclude that the electron number at which the charge gap occurs
changes by νQ[m] when the order parameter field has topological index Q[m]. The same
ˆ
ˆ
conclusion can be reached [18,22] using the Chern-Simons Landau-Ginzburg theory of the
quantum Hall effect [23,24] or by explicit calculation [25].
The Skyrmion excitations of quantum Hall ferromagnets are responsible for striking
physical effects because they carry a physical charge and are present in the ground state
for filling factors near those at which the incompressible state occurs. To be concrete, we
consider the quantum Hall ferromagnet that occurs at ν = 1. For N = Nφ the ground state
has S = N/2 as discussed above. For N = Nφ ±1, the ground state contains a single charged
Skyrmion. The Skyrmion can be introduced by changing the total electron number or, in
what is the typical experimental situation, by changing the magnetic field strength and hence
Nφ . In the NLσ model the energy of a Skyrmion is independent of its size. To represent
Skyrmions in quantum Hall ferromagnets, however, it is necessary to add additional terms
to the model to account for Zeeman coupling and for the Coulomb self-interaction energy
of a Skyrmion. Zeeman coupling favors small Skyrmions, since the spin near the center of
the Skyrmion is oriented in opposition to the Zeeman field. On the other hand the repulsive
Coulomb energy favors large Skyrmions. For typical experimental situations the optimal
Skyrmion size is not so much larger than microscopic lengths, invalidating the use of the
field-theory description for quantitative estimates. In a quantum description, the number
of reversed spins per skyrmion is quantized so that, when Skyrmion-Skyrmion interactions
can be neglected, we expect that the component of the total spin along the direction of the
Zeeman field is

12

Sz = N/2 − (K + θ)|N − Nφ |

(12)

Here θ = 1 for N > Nφ and θ = 0 for N < Nφ . |N − Nφ | is the number of Skyrmions
or antiskyrmions present in the system. The integer quantum number K will depend in
general on the relative size of Zeeman and Coulomb interaction terms and is the relevant
quantum measure of the Skyrmion size. For non-interacting electrons, or with interactions
treated in the Hartree-Fock approximation, K = 0 so that Sz always has the maximum value
allowed by the Pauli exclusion principle. (K is guaranteed by particle-hole symmetry [26] to
have the same value for N > Nφ and N < Nφ .) The NLσ model considerations of Sondhi
et al. [18] described above, and also earlier numerical exact diagonalization calculations
[27], suggest that K should be non-zero for quantum Hall ferromagnets and quite large if
the Zeeman energy is small. These predictions were dramatically confirmed when Barrett
et al. unexpectedly succeeded [28] in using optical pumping techniques to perform NMR
Knight shift measurements of the spin-polarization of two-dimensional electron systems in
the quantum Hall regime. The results of this experiment are illustrated in Fig. 3 and
correspond to K = 3, in quantitative agreement with microscopic predictions based on a
generalized Hartree-Fock approximation for single-Skyrmion states [29]. There seems to
be little doubt that the elementary charged excitations of quantum Hall ferromagnets are
Skyrmion-like objects that carry large spin quantum numbers. Recent transport [30] and
optical [31] experiments add additional support to this conclusion.
For large enough |N − Nφ | the Skyrmion-like objects will eventually interact strongly.
When the density of Skyrmions is low and the temperature is low, Skyrmions are expected
to form a lattice similar to the Wigner crystal state formed by electrons in the limit of very
strong magnetic fields. In Fig. 4 we compare theoretical calculations of the spin-polarizations
as a function of filling factor for several candidate Skyrmion lattice states with experimental
data. The theoretical results were obtained by Brey et al. using a generalized Hartree-Fock
approximation [32] and illustrate several important aspects of the physics of Skyrme crystals.
These authors find that the ground state of the Skyrme crystal is a square lattice rather than
13

a triangular lattice as found for the electron Wigner crystal. Furthermore, as illustrated in
Fig. 4, the spin-polarization of the square lattice Skyrme crystal is much smaller, for a given
Zeeman coupling strength, than for the lowest energy triangular lattice state. The preference
for a square lattice can be understood qualitatively in terms of the NLσ model description
of Skyrmion states. For that model Skyrmions are centered at an arbitrary point, have an
arbitrary size, and are invariant under arbitrary global spin rotations. When Coulomb and
Zeeman energies are included the optimal Skyrmion size is fixed and the spin moment must
be aligned with the Zeeman field far from the Skyrmion center. However, the energy of each
Skyrmion is still invariant under global rotations of the moment about an axis aligned with
the Zeeman field. For a Skyrme lattice the relative values of these rotation angles must be
adjusted to minimize the total energy. It turns out that the interaction energy between a
pair of Skyrmions is reduced when they have opposing orientations for the component of the
ordered moment perpendicular to the Zeeman field. This arrangement allows the ordered
moment orientation to vary more smoothly along the line connecting Skyrmion centers.
The tendency toward opposing orientations is frustrated on triangular lattice, hence the
energetic preference for a square lattice. The stronger short-range repulsive interaction in
the aligned orientation ferromagnetic lattice case, results in smaller Skyrmions and therefore
more spin-polarized states. The spin-polarizations calculated for the opposing orientation,
square lattice case shown in Fig. 4 appear to be in excellent agreement with experiment over
a wide range of filling factors near ν = 1.

IV. EDGE EXCITATIONS OF AN INCOMPRESSIBLE QUANTUM HALL FLUID

A. Non-Interacting Electron Picture

Throughout this section we will consider a disk geometry where electrons are confined to
a finite area centered on the origin by a circularly symmetric confining potential, Vconf (r). We
have in mind the situation where Vconf (r) rises from zero to a large value near r = R, where R

14

is loosely speaking the radius of the disk in which the electron system is confined. We choose
this geometry, for which the electron system has a single edge, since we limit our attention
here to the properties of an isolated quantum Hall edge and will not discuss the physics
of interaction or scattering between edges [33]. In this geometry it is convenient to choose
the symmetric gauge for which angular momentum is a good quantum number. For Vconf (r)
the single-electron kinetic energy operator has the macroscopically degenerate Landau levels
separated by hωc and in each Landau level states with larger angular momentum are localized
¯
further from the origin. We recall from Eq.( 4) that wavefunctions with angular momentum
m are localized [6] near a circle with radius Rm =

2(m + 1)ℓ. (Note that for large m the

separation between adjacent values of Rm is ℓ2 /Rm << ℓ.) In the strong magnetic field
limit the confinement potential does not mix different Landau levels. Since there is only one
state with each angular momentum in each Landau level the only effect of the confinement
potential is to increase the energy of the symmetric gauge eigenstates when Rm becomes
larger than ∼ R. The typical situation is illustrated schematically in Fig. [5]. Here the n = 0
and n = 1 Landau levels are occupied in the bulk and the chemical potential µ lies in the gap
∆ = hωc between the highest energy occupied Landau level (E = 3¯ ωc /2) and the lowest
¯
h
energy unoccupied Landau level (E = 5¯ ωc /2). In this section we are interested only in the
h
ground state and the low energy excited states obtained by making one or more particle-hole
excitations at the edge. We will discuss only the simplest situation where a single Landau
level crosses the chemical potential at the edge of the system and the analogous single branch
situations in the case of the fractional quantum Hall effect [34]. We will also neglect the spin
degree of freedom of the electrons, which figured so prominently in the previous section.
An important property of the ground state of the non-interacting electron system in the
case of interest, is that it remains an exact eigenstate of the system (but not necessarily the
ground state!) when interactions are present. That is because the total angular momentum
K for this state is
N −1

M0 =
m=0

m = N(N − 1)/2
15

(13)

and all other states in the Hilbert space (truncated to the lowest Landau level) have larger
angular momentum [35]. For large disks and total angular momentum near M0 the excitation
energy of a non-interacting electron state will be
∆E = γM

(14)

where M ≡ K −M0 is the excess angular momentum and γ is the energy separation between
single-particle states with adjacent angular momenta and energies near the Fermi energy. γ
is related to the electric field, Eedge from the confining potential at the edge of the disk:
γ = eEedge

dRm
= eEedge ℓ2 /R
dm

(15)

This expression for γ can be understood in a more appealing way. In a strong magnetic
field charged particles execute rapid cyclotron orbits centered on a point that slowly drifts
in the direction perpendicular to both the magnetic field and the local electric field. For an
electron at the edge of the disk the velocity of this ‘E cross B’ drift is vedge = cEedge /B. The
energy level separation can therefore be written in the form
γ = hvedge /Redge = h/T
¯

(16)

where T is the period of the slow drift motion of edge electrons around the disk, in agreement
with expectations based on semiclassical quantization.
Since the excitation energy depends only on the angular momentum increase compared
to the ground state it is useful to classify states by M. It is easy to count the number of
distinct many-body states with a given value of M as illustrated in Fig. [6]. For M = 1
only one many-particle state is permitted by the Pauli exclusion principle; it is obtained by
promoting the ground state electron with m = N − 1 to m = N. For M = 2, particle hole
excitations are possible from m = N − 1 to m = N + 1 and from m = N − 2 to m = N.
In general M many-particle states with excess angular momentum M can be created by
making a single-particle hole excitation of the ground state. For M ≥ 4 additional states
can be created by making multiple particle-hole excitations. The first of these is a state
with two particle-hole excitations that occurs at M = 4 and is illustrated in Fig. [6].
16

B. Many-Body Wavefunction Picture

We now discuss the edge excitation spectrum of interacting electrons using a language of
many-particle wavefunctions. For the case of the integer quantum Hall effect we will essentially recover the picture of the excitation spectrum obtained previously for non-interacting
electrons by counting occupation numbers. We could have used the Hartree-Fock approximation and occupation number counting to generalize these results to interacting electrons.
However, the Hartree-Fock approximation is completely at sea when it comes to the fractional case. Discussions of the fractional edge using an independent electron language can
be comforting but can also be misleading. Nevertheless, we will see that there is a one-toone correspondence between the edge excitation spectrum for non-interacting electrons at
integer filling factors and the fractional edge excitation spectrum.
Many-electron wavefunctions where all electrons are confined to the lowest Landau level
must be sums of products of one-particle wavefunctions from the lowest Landau level. From
Eq. (4) it follows that any N electron wavefunction has the form
Ψ[z] = P (z1 , . . . , zN )
ℓ

exp (−|zℓ |2 /4),

(17)

where we have adopted ℓ as the unit of length and P (z1 , . . . , zN ) is a polynomial in the twodimensional complex coordinates. This property [36] of the wavefunctions will be exploited
in this section. The first important observation is that since Ψ[z] is a wavefunction for many
identical fermions it must change sign when any two particles are interchanged, and therefore
must vanish as any two particles positions approach each other. Since P (z1 , . . . , zN ) is a
polynomial in each complex coordinate it follows [37] that
P (z1 , . . . , zN ) =
i<j

(zi − zj ) Q[z]

(18)

where Q[z] is any polynomial that is symmetric under particle interchange. It is important
to note that the total angular momentum of all the particles (K) is just the degree of
the polynomial P [z], i.e. the sum of the powers to which the individual particle complex
17

coordinates are raised. Since the total angular momentum is a good quantum number the
polynomial part of any many-electron eigenstate will be a homogeneous degree polynomial,
i.e. all terms in the many-particle polynomial must have the same degree. Additionally, the
total angular momenta corresponding to a polynomial that, as in Eq. (18), is the product
of two polynomials is the sum of the angular momenta associated with those polynomials.
From the discussion of Section III it is clear that for non-interacting electrons in any
monotonically increasing confinement potential, the lowest energy state will be the state
with the minimum total angular momentum. In Eq. (18) that corresponds to choosing Q[z]
to have degree zero, i.e. to Q[z] ∝ 1. It is easy to verify that the wavefunction when Q[z] is a
constant is in fact the Slater determinant formed by occupying the single-particle states with
m = 0, . . . , N − 1. For interacting electrons this state will remain the ground state provided
the confinement potential is strong enough to overcome the repulsive interactions between
electrons that favor states with larger total angular momentum. When this is the ground
state, low-energy excited states with excess angular momentum M = K − N(N − 1)/2 are
linear combinations of the states constructed by choosing all possible symmetric polynomials
[38] of degree M for Q[z].
We’ll discuss the enumeration of these polynomials in a moment but pause now to explain
how this analysis many be generalized to the case of the fractional quantum Hall effect. We
limit our attention here [39] to the simplest fractional quantum Hall effects that occur at
Landau level filling factors ν = 1/m for any odd integer m; in some senses the m = 1 case
can be regarded as a special case of the fractional quantum Hall effect. The physics of the
chemical potential jump that occurs at these filling factors was explained in the pioneering
paper of Laughlin [14,6]. For ν < 1/3, for example, it is possible to find states in the Hilbert
space in which pairs of electrons are never found in a state with relative angular momentum
equal to one. This is the two-body state in which two electrons are closest together. All
many-particle states that avoid placing pairs in this state will have low energy. If this
condition [40] is satisfied,

18

P (z1 , . . . , zN ) ≡

i<j

(zi − zj )3 Q[z]

(19)

for any symmetric polynomial Q[z]. If the system has an abrupt edge the ground state will
have Q[z] ≡ 1 just as in the non-interacting case. (Note that this approach to identifying
the ground state and low-energy excitations is very much the same as that used in Section
III to identify the ground state for ν near one when the electrons are not spin-polarized,
except possibly spontaneously!) The edge excitations correspond to the same set of symmetric polynomials as in the ν = 1 case. In the case of a model system with a short-range
interaction and a parabolic confinement potential, it is easy to place [41] the argument we
have sketched above on firm ground. It is known from numerical studies that the bulk chemical potential discontinuity survives when the model interaction is replaced by the realistic
Coulomb interaction. However, special care is required in considering edge excitations in
physically realistic systems with long range interactions and the reader is warned that the
simple models discussed below may not always apply. These issues are discussed at greater
length elsewhere [7].
The wavefunction
Ψ[z] ≡ Q[z]

l

exp(−|zl |2 /4)

(20)

is a wavefunction for N bosons in a strong magnetic field. Thus the enumeration of the edge
excitations in terms of symmetric polynomials discussed above is equivalent to enumerating
all many boson wavefunctions with a given value of the total angular momentum. The boson
angular momentum
∞

m nm

M=

(21)

m=0

where nm are the boson occupation numbers, is equivalent (for ν = 1/m) to the excess
angular momentum M = K − mN(N − 1)/2 of the fermion wavefunctions. In the state with
M = 0, n0 = N, all other boson occupation numbers are zero, and Q[z] is a constant. In
the boson language the ground state is a Bose condensate. The lone state with M = 1 has
n1 = 1,n0 = N − 1; the symmetric polynomial for this boson wavefunction is
19

Q[z] = z1 + z2 + . . . zN .

(22)

For the integer ν = 1 case, it can be shown explicitly that the corresponding many-fermion
state is the M = 1 state with a single particle-hole excitation at the edge, discussed in
Section III. The set of excitations at general values of M can be described equally well in
either fermion or boson languages. Some of the states that occur at small values of M are
listed in Table I.
In the parabolic confinement case the total energy depends only on the excess total
angular momentum: δE = γM. The number of many-boson states with total angular
momentum M, g(M) can be calculated by considering a system of non-interacting bosons
with single-particle energy γm so that E =

m (γm)

g(M) e−γM/kB T =

Z=
M

· nm = γM. The partition function is
xM g(M)

(23)

M

where x = e−γ/kB T . For N → ∞, the m = 0 state acts like a reservoir with chemical potential
µ = 0 so that the partition function calculation can be done in the grand canonical ensemble.
The degeneracies g(M) can be read off the power series expansion of the partition function:
∞

Z=

1
= (1 − x)−1 (1 − x2 )−1 (1 − x3 )−1 . . .
k
k=1 1 − x

= (1 + x + x2 + x3 + . . .)(1 + x2 + x4 + . . .)(1 + x3 + x6 + . . .) . . .
= (1 + x + 2x2 + 3x3 + 5x4 + 7x5 + 11x6 + 15x7 + 22x8 + . . . .

(24)

√2
1/2
For [41] large M g(M) ∼ e 3 ·π·M . The function g(M) is well known to number theorists
from the theory of partitions [42] in which it is known as the partition function, not to be
confused with the physics partition function above! For parabolic confinement potentials
and short-ranged repulsive interactions, the degeneracy of the edge excitations at a given
excess angular momentum is exact in both integer and fractional ν = 1/m cases. For general
confinement potentials and general electron-electron interactions these degeneracies will be
lifted. However, there is reason to expect that in the thermodynamic limit excitations with
M ≪ N 1/2 will be nearly degenerate. One way to see this is to use the chiral Luttinger liquid
20

picture of quantum Hall edges that we discuss in the following section. This approach will
allow us to do more than enumerate excitations of the system and, in particular will enable
us to discuss the density-of-states for tunneling into the edge of a quantum Hall system.

C. Chiral Luttinger Liquid Picture

The chiral Luttinger liquid picture [43] of quantum Hall systems is an adaptation of the
Luttinger liquid theory of one-dimensional electron systems. We start this section with a
brief outline of the portion of that theory that we require. Readers in search of greater
depth should look elsewhere [44]. As in higher dimensions, low excitation energies states
in a one-dimensional fermion system will involve only single-particle states near the Fermi
wavevector. Since the differences in wavevector among the relevant states at a given Fermi
edge are small, the excitations produced by rearranging them occur on length scales that are
long compared to microscopic lengths. It is therefore reasonable to argue that the energy
density in the system at any point in space should depend only on the local density of
left-moving (k < 0) and right-moving (k > 0) electrons, nL (x) and nR (x):
E[nL , nR ] = E0 +

dx

αLL 2
αRR 2
δnL (x) +
δnR (x) + αLR δnL (x) δnR (x) .
2
2

(25)

It is, perhaps, not completely obvious that the density provides a complete parameterization
of the low-energy excitations, and indeed in the fractional Hall case there are situations
where the analog of Eq. (25) is incorrect. [45] Here αLL , αLR and αRR are determined
by the second derivatives of the energy per unit length with respect to nL and nR for a
uniform system and can be determined in principle by a microscopic calculation. δnL (x)
and δnR (x) are differences of the density from the ground state density. Note that we have
as a convenience chosen the chemical potential to be zero in dropping a term proportional
to

dx(δnL (x) + δnR (x)). We start by considering the case where αLR = 0 so that the

left-moving electrons and right moving electrons are decoupled. Focus for this case on the
energy of the right moving electrons. We Fourier expand the density and note that
21

dx δn2 (x) =
R

1
n−qR .nqR
L q=0

(26)

so that the energy can be written in the form
ER = E0 +

αLL
2L

n−qR nqR .

(27)

q=0

The energy above can be used as an effective Hamiltonian for low-energy long-wavelength
excitations. The simplification at the heart of the Luttinger liquid theory is the observation
that when the Hilbert space is truncated to include only low-energy, long-wavelength excitations (in particular when the number of left-moving and right-moving electrons is fixed)
Fourier components of the charge density do not commute. For example consider the second
quantization expression for nqR in terms of creation and annihilation operators with k > 0:
c† ck .
k+q

nqR =

(28)

k>0

An example of the dependence of the effect of products of these operators on the order in
which they act is more instructive than the actual algebraic calculation of the commutators.
Note for example that
n−qR |Ψ0 = 0

(29)

where q > 0 and |Ψ0 is the state with all right-going electron states with k < kF occupied
and all right-going states with k > kF empty. (The alert reader will have noticed that
this state of ‘right-going’ electrons corresponds precisely to the ‘maximum density droplet’
states that occur in the quantum Hall effect.) n−qR annihilates this state because there
are no right-electron states with a smaller total momentum than |Ψ0 . On the other hand
for q = M2π/L, nqR |Ψ0 yields a sum of M terms in which single-particle hole excitations
have been formed in |Ψ0 . For example, if we represent occupied states by solid circles and
unoccupied states by open circles, as in Fig. (6), for M = 2 we have
nqR |Ψ0 = | . . . • • ◦ •| • ◦ ◦ . . .
+| . . . • • • ◦| ◦ • ◦ . . . .
22

(30)

Each of the M terms produced by nqR |Ψ0 is mapped back to |Ψ0 by n−qR . Therefore
nqR n−qR |Ψ0 = 0 whereas n−qR nqR |Ψ0 = M|Ψ0 . The general form of the commutation
relation is readily established by a little careful algebra [44]:
[n−q′ R , nqR ] =

qL
δq,q′ .
2π

(31)

This holds as long as we truncate the Hilbert space to states with a fixed number of rightgoing electrons and assume that states far from the Fermi edge are always occupied.
We can define creation and annihilation operators for density wave excitations of rightgoing electrons. For q > 0
aq =

2π
n−qR
qL

(32)

a† =
q

2π
nqR
qL

(33)

With these definitions Eq. (31) yields
[aq′ , a† ] = δq,q′
q

(34)

so that the density waves satisfy bosonic commutation relations. Also note that
qL
ˆ
[M, aq ] = − aq
2π
qL †
ˆ
a
[M, a† ] =
q
2π q

(35)
(36)

ˆ
where M is the total angular momentum operator. The contribution to the Hamiltonian
from right-going electrons is therefore
h vq a† aq
¯
q

HR =

(37)

q>0

where
v=

αRR
1 d2 E0
1 dµR
=
=
2
2π¯
h
2πL¯ dnR
h
2π¯ dnR
h

(38)

At low-energies the system is equivalent to a system of one-dimensional phonons traveling
to the right with velocity v. In the limit of non-interacting electrons
23

v=

h kF
¯
≡ vF
m∗

(39)

as expected.
Without interactions between left and right-moving electrons a Luttinger liquid is quite
trivial. In particular the ground state (|Ψ0 ) is a single-Slater determinant with a sharp
Fermi edge. For one-dimensional electron gas systems the interesting physics [44] occurs
only when left and right-moving electrons are allowed to interact. Most notably, arbitrarily
weak interactions destroy the sharp Fermi edge that is the hallmark of Fermi liquids and that
survives interactions in higher dimensions. In the case of quantum Hall edges, however, the
above restriction to electrons moving in only one direction is not a temporary pedagogical
device. The model with only right moving electrons discussed above can be taken over
mutatis mutandis as a model of the edge excitations for an electron system with ν = 1. The
role played by the one-dimensional electron density is taken over by the integral of the twodimensional electron density along a line perpendicular to the edge. In this way we arrive
at the same bosonized picture of the ground state and low-lying excitations at the edge
of a quantum Hall system as we reached previously by arguing in terms of many-particle
wavefunctions. The single boson states which appeared there are replaced by the states of
the chiral phonon system which has modes with only one sign of momentum and velocity.
For ν = 1 the analysis applies whether or not the electrons interact. We now turn
our attention to a discussion of the fractional case. Do all steps of the above discussion
generalize? We can argue that if we are interested only in low-energy long-wavelength
excitations, the energy can be expressed in the form
E = E0 +

α
n−q nq .
2L q=0

(40)

As we comment later, this expression can fail at the edge of fractional quantum Hall systems
although it is appropriate for ν = 1/m. What about the commutator? There is an important
difference in the line of argument in this case, since single-particle states far from the edge
of the system are not certain to be occupied. Instead the average occupation number is
24

ν = 1/m and there are large quantum fluctuations in the local configuration of the system
even in the interior. However, we know [39] from the discussion in terms of many-body
wavefunctions in the previous section that the low-energy excitations at ν = 1/m can be
described as the excitations of a boson system, exactly like those at ν = 1, which suggests
that something like Eq. (31) must still be satisfied when the Hilbert space is projected to
low energies. If we replace the commutator by its expectation value in the ground state we
obtain
[n−q′ , nq ] = ν ·

qL
δq,q′
2π

(41)

which differs from Eq. (31) only through the factor ν. It seems clear for the case of ν = 1/m
this replacement can be justified on the grounds that the interior is essentially frozen (but
in this case not simply by the Pauli exclusion principle) at excitation energies smaller than
the gap for bulk excitations. What we need to show is that Eq. (41) applies as an operator
identity in the entire low-energy portion of the Hilbert space. Below, however, we follow a
different line of argument.
Appealing to the microscopic analysis in terms of many-body wavefunctions we know
that the excitation spectrum for ν = 1/m is equivalent to that of a system of bosons.
We conjecture that the commutator [n−q′ , nq ] =∝ qδq,q′ . To determine the constant of
proportionality we will require that the rate of change of the equilibrium edge current with
chemical potential be eν/h. From the edge state picture of the quantum Hall effect discussed
in Section II, it is clear that this is equivalent to requiring the Hall conductivity to be
quantized at νe2 /h. Since our theory will yield a set of phonon modes that travel with a
common velocity v it is clear that the change in equilibrium edge current is related to the
change in equilibrium density by
δI = evδn.

(42)

When the chemical potential for the single edge system is shifted slightly from its reference
value (which we chose to be zero) the grand potential is given by
25

E[n] = E0 + µδn + α

(δn)2
2

(43)

Minimizing with respect to δn we find that
δµ
α

(44)

ev
δI
=
δµ
α

(45)

δn =
so that

In order for this to be consistent with the quantum Hall effect (δI = (eν/h)δµ) our theory
must yield a edge phonon velocity given by
v=

α
· ν.
h

(46)

The extra factor of ν appearing in this equation compared to Eq. (38) requires the same
factor of ν to appear in Eq. (41). We discuss below the qualitative changes in the physics
[46,43] of fractional edge states which are implied by this outwardly innocent numerical
factor.
It is worth remarking that the line of argument leading to this specific chiral Luttinger
liquid theory of the fractional quantum Hall effect is not completely rigorous. In fact we
know that this simplest possible theory with a single branch of chiral bosons does not
apply for all filling factors [39,46,47], even though (nearly) all steps in the argument are
superficially completely general. The reader is encouraged to think seriously about what
could go wrong with our arguments. Certainly the possibility of adiabatically connecting all
low-energy states with corresponding states of the non-interacting electron system, available
for one-dimensional electron gases and for quantum Hall systems at integer filling factors
but not at fractional filling factors, adds confidence when it is available. In our view, the
microscopic many-particle wavefunction approach that establishes a one-to-one mapping
between integer and fractional edge excitations (for ν = 1/m!) is an important part of the
theoretical underpinning of the Luttinger liquid model of fractional Hall edges. Once we
26

know that the edge excitations map to those of a chiral boson gas and that the fractional
quantum Hall effect occurs, it appears that no freedom is left in the construction of a lowenergy long-wavelength effective theory.
An important aspect of Luttinger liquid theory is the expression for electron field operators in terms of bosons [44]. This relationship is established by requiring the exact identity
ˆ
ˆ
[ρ(x), ψ † (x′ )] = δ(x − x′ ) ψ † (x′ )

(47)

to be reproduced by the effective low-energy theory. This equation simply requires the
electron charge density to increase by the required amount when an electron is added to the
system. The electron creation operator should also be consistent with Fermi statistics for
the electrons:
ψ † (x), ψ † (x′ ) = 0.

(48)

In order to satisfy Eq. (47), the field operator must be given by
−1
ˆ
ψ † (x) = ceiν φ(x)

(49)

where dφ(x)/dx = n(x) and c is a constant that cannot be determined by the theory. The
factor of ν −1 in the argument of the exponential of Eq. (49) is required because of the factor
of ν in the commutator of density Fourier components that in turn was required to make
the theory consistent with the fractional quantum Hall effect. When the exponential is
expanded the k − th order terms generate states with total boson occupation number k and
are multiplied in the fractional case by the factor ν −k ; multi-phonon terms are increased in
relative importance. It is worth remarking [46] that the anticommutation relation between
fermion creation operators in the effective theory is satisfied only when ν −1 is an odd integer.
This provides an indication, independent of microscopic considerations, that the simplest
single-branch chiral boson effective Hamiltonian can be correct only when ν = 1/m for
odd m. Wen [43] has surveyed, using this criterion, the multi-branch generalizations of the
simplest effective Hamiltonian theory which are possible at any given rational filling factor.
27

His conclusions are consistent with arguments [39] based on the microscopic theory of the
fractional quantum Hall effect.
Eq. (49) has been carefully checked numerically [48] and appears to be correct. The ν −1
factor leads to predictions of qualitative changes in a number of properties of fractional edges.
The quantity that is most directly altered is the tunneling density-of-states. Consider, for
example, the state created when an electron, localized on a magnetic length scale, is added
to the ground state at the edge of a N− electron system with ν = 1/m:
a†
√ n |ψ0
nν
n>0
1 phonon term 2 phonon terms
=1+
+
+ ....
ν 1/2
ν

ˆ
ψ † (0)|Ψ0 ∼ exp −

(50)

The tunneling density states is given by a sum over the ground and excited states of the
N + 1 particle system:
A(ǫ) =
n

δ(En − E0 − ǫ)| Ψn |ψ † (0)|Ψ0 |2

(51)

Because of the increased weighting of multiphonon states, which become more numerous
at energies farther from the chemical potential, the spectral function is larger at larger
ǫ − µ in the fractional case. An explicit calculation [46,43] yields a spectral function that
grows like (ǫ − µ)ν

−1 −1

. It is intuitively clear that the spectral function should be small at

low-energies in the fractional case since the added electron will not share the very specific
correlations common to all the low-energy states. It is amazing that by simply requiring
the low-energy theory to be consistent with the fractional quantum Hall effect we get a very
specific prediction for the way in which this qualitative notion is manifested in the tunneling
density of states.

V. ACKNOWLEDGMENTS

The ideas discussed here have been shaped by discussions with members of the condensed
matter theory group at Indiana University, especially M. Abolfath, S.M. Girvin, C. Hanna,
28

R. Haussmann, S. Mitra, K. Moon, J.J. Palacios, D. Pfannkuche, E. Sorensen, K. Tevosyan,
K. Yang, and U. Z¨ licke. Discussions with L. Brey, R. Cote, H. Fertig, M. Fisher, M.
u
Johnson, C. Kane, L. Martin, J. Oaknin, C. Tejedor, S.R.-E. Yang and X.-G. Wen are also
gratefully acknowledged. The responsibility for surviving misapprehensions rests with me.
This work was supported by the National Science Foundation under grant DMR-9416906.

29

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31

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[29] H.A. Fertig, L. Brey, R. Cˆt´, and A.H. MacDonald, Phys. Rev. B 50, 11018 (1994);
oe
ibid, submitted to Phys. Rev. B (1996).
[30] A. Schmeller and J.P. Eisenstein, Phys. Rev. Lett. 75, 4290 (1995).
[31] E.H. Aifer, B.B. Goldberg, and D.A. Broido, preprint (1995).
[32] L. Brey, H.A. Fertig, R. Cote, and A.H. MacDonald, Phys. Rev. Lett. 75, 2562 (1995).
[33] Lack of time and space limits the scope of our discussion to systems without internal
boundaries that have a single outer edge. There is, appropriately, considerable interest in
annular and equivalent geometries that have inner and outer edges carrying currents in
opposite directions. These geometries allow the problem of scattering between edges by
disorder, related to transport experiments, to be considered. For work on this problem
see C.L. Kane and M.P.A. Fisher, Phys. Rev. Lett. 68, 1220 (1992); Phys. Rev. B 46,
15233 (1992); M. Ogata and H. Fukuyama, Phys. Rev. Lett. 47, 4631 (1993); K.A.
Matveev, D. Yue, and L.I. Glazman, Phys. Rev. Lett. 49, 1966 (1994); K.A. Matveev,
D. Yue, and L.I. Glazman, Phys. Rev. B 49, 1966 (1994).
[34] For recent work on ‘multi-branch’ systems see C.L. Kane and Matthew P.A. Fisher,
32

Phys. Rev. B 51, 13449 (1995) and work cited therein.
[35] For an elementary discussion of some properties of this maximum density droplet state
see A.H. MacDonald, S.R. Eric Yang, and M.D. Johnson, Aust. J. Phys. 46, 345 (1993).
[36] The Hilbert space of the lowest Landau level is the Hilbert space of analytic functions:
S.M. Girvin and T. Jach, Phys. Rev. B 29, 5617 (1984).
[37] A.H. MacDonald and D.B. Murray, Phys. Rev. B 32 2291 (1985).
[38] Michael Stone, Phys. Rev. B 42, 8399 (1990); Michael Stone, Ann. Phys. (NY) 207, 38
(1991); Michael Stone H.W. Wyld, and R.L. Schult, Phys. Rev. B 45, 14156 (1992).
[39] For a more general discussion see A.H. MacDonald, Phys. Rev. Lett. 64, 220 (1990).
[40] The antisymmetry requirement generally requires P [z] must be an odd function of zi −zj
for any i and j so that P [z] contains only odd terms in a Taylor series expansion of
its dependence on this difference coordinate. The coefficient of the (zi − zj )1 term must
vanish if the pair never has relative angular momentum equal to one so that the leading
term is (zi − zj )3 . It then follows from analyticity that each difference coordinate to the
third power is a factor of P [z].
[41] A.H. MacDonald and M.D. Johnson, Phys. Rev. Lett. 70, 3107.
[42] See for example, George E. Andrews, The Theory of Partitions (Addison-Wesley, Reading, 1976). A famous theorem by Hardy and Ramanujan in the theory of partitions gives
an exact formula expressing g(M) in terms of a finite sum.
[43] For reviews see X.G. Wen, Int. J. Mod. Phys. B6, 1711 (1992); X.G. Wen, to appear in
Advances in Physics (1995).
[44] See for example, J. S´lym, Adv. Phys. 28, 201 (1979); F.D.M. Haldane, J. Phys. C 14,
o
2585 (1981); G.D. Mahan, Many-Particle Physics, (Plenum, New York, 1990) Chapter 4;
J. Voit, preprint (1995). [To appear in Reports on Progress in Physics.] K. Sch¨nhammer
o

33

and V. Meden, submitted to Am.J. Phys. (1995).
[45] The microscopic analysis of fractional quantum Hall edges in Ref. [39] shows that the
parameterization of the energy density at the edge in terms of the linear charge density
is successful only for the simple case where ν = 1/m that we have discussed in these
notes.
[46] X.G. Wen, Phys. Rev. B 41, 12838 (1990); D.H. Lee and X.G. Wen, Phys. Rev. Lett.
66, 1765 (1991); X.G. Wen, Phys. Rev. B 44, 5708 (1991).
[47] M.D. Johnson and A.H. MacDonald, Phys. Rev. Lett. 67, 2060 (1991).
[48] J.J. Palacios and A.H. MacDonald, Phys. Rev. Lett. 76, 118 (1996) and work cited
therein.

34

FIGURES
FIG. 1. A large but finite two-dimensional electron gas. In panel (a) the chemical potential lies
in a gap and the only low-energy excitations are localized at the edge of the system. In panel (b)
the chemical potential lies in a mobility gap so that there are low-energy excitations in the bulk
but they are localized away from the edge. In panel (c) a net current is carried from source to
drain by having local equilibria at different chemical potentials on upper and lower edges.
FIG. 2. Illustration of a Skyrmion spin texture. At the center of the Skyrmion m points in the
ˆ
down (−ˆ) direction. Far from the center of a Skyrmion m points in the up (ˆ) direction. Along
z
ˆ
z
a ray at angle θ in a circular coordinate system defined with respect to the Skyrmion center, m
ˆ
rotates about an axis in the (sin(θ) − cos(θ) direction from −ˆ to z . At fixed r the x − y projection
z
ˆ
ˆ ˆ
of m has fixed magnitude and rotates by ±2π when the angular coordinate winds by ±2π. At
ˆ
r = λ, m lies entirely in the x − y plane.
ˆ
ˆ ˆ
FIG. 3. Knight shift measurements by Barrett et al.

of the spin polarization of a

two-dimensional electron gas near filling factor ν = 1. Here S = A = K + 1/2 so that the
experiment is consistent with K = 3 for this sample. The dashed line in this figure shows the
dependence of spin-polarization on filling factor expected for non-interacting electrons and, in the
Hartree-Fock approximation, also for interacting electrons. The spin-polarization is assumed to be
proportional to the Knight shift of the 71 Ga nuclear resonance and to be complete at ν = 1. (After
Ref. [28]
FIG. 4. Dependence of spin-polarization P on filling factor for Skyrme lattice states. Here g is
the ratio of the Zeeman energy to the characteristic interaction energy e2 /ℓ and the values chosen
are typical of experimental systems. The open and closed circles are experimental results of Barrett
et al.. The legends indicate the nature of the Skyrme lattice state: the SLA state is a square lattice
state with opposing Skyrmion orientations; the TLF state is a triangular lattice state with aligned
Skyrmion orientations. (After Brey et al. in Ref. [32])

35

FIG. 5. Schematic spectrum for non-interacting electrons confined to a circular disk in a strong
magnetic field. In the limit of large disks the dependence of the energy on m can usually be
considered to be continuous. The situation depicted has Landau level filling factor ν = 2 in the
bulk of the system. The low-energy excitations are particle-hole excitations at the edge of the
system.
FIG. 6. Non-interacting many electron eigenstates for small excess angular momentum M
specified by occupation numbers for the single-particle states with energies near the chemical
potential µ. The vertical bars separate single-particle states with ǫm < µ from those with ǫm > µ.
A solid circle indicates that nm = 1 in both the ground state and in the particular excited state;
a shaded circle indicates that nm = 1 in the particular excited state but not in the ground state;
an empty circle indicates that nm = 0.

36

TABLES
TABLE I. Quantum occupation numbers in boson and fermion descriptions for edge excitations with small excess angular momentum M . gM is the number of states with excess angular
momentum M . The fermion occupation numbers are relative to the maximum density droplet
state. Only non-zero values are listed for both fermion and boson descriptions. L = N − 1 is the
highest angular momentum that is occupied in the maximum density droplet state.
M

gM

Fermion Description

Boson Description

1

1

nL+1 = 1, nL = −1

n1 = 1

2

2

nL+2 = 1, nL = −1; nL+1 = 1, nL−1 = −1

n2 = 1; n1 = 2

3

3

nL+3 = 1, nL = −1; nL+2 = 1, nL−1 = −1

n3 = 1; n2 = 1, n1 = 1;

nL+1 = 1nL−2 = −1

n1 = 3

nL+4 = 1, nL = −1; nL+3 = 1, nL−1 = −1

n4 = 1; n3 = 1, n1 = 1; n2 = 2

nL+2 = 1, nL−2 = −1; nL+1 = 1, nL−3 = −1

n2 = 1, n1 = 2; n1 = 4

4

5

nL+2 = 1, nL+1 = 1, nL = −1, nL−1 = −1

37

20

15

(S=A=3.4+/-0.3)

10

71Ga

Well Shift (kHz)

Interacting Electrons

5

T=1.55 K,
H=7.05 Tesla
Independent Electrons
(S=A=1)

0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

ν

...

. . . gM=1 = 1

• M=2

...

...

...

. . . gM=2 = 2
...

...

• M=1

• M=4

. . . gM=4 = 5
...

...

EDGE REGION
R

(a)

dA

dl

EQUILIBRIUM
µ IN GAP

EDGE REGION

(b)

EQUILIBRIUM
µ IN MOBILITY GAP
LOCALIZED STATES
µ1

µ2

(c)

LOCAL EQUILIBRIUM
ON ISOLATED EDGES

I
I

µ3

µ4

n=2
n=1
n=0
empty

∆

occupied
occupied

µ

1.0

SLA, g*= 0.015
TLF, g*= 0.015
SLA, g*= 0.020
Indep. electrons
Experiment
Experiment

Spin Polarization, P

0.8

0.6

0.4

0.2

0.0
0.6

0.8

1.0

1.2

ν

1.4

1.6

1.8