Compressible Strips, Chiral Luttinger Liquids,
and All That Jazz
A.H. MacDonald

arXiv:cond-mat/9512043v1 6 Dec 1995

Department of Physics, Indiana University, Bloomington, IN 47405, USA.
(January 16, 2018)

Abstract
When the quantum Hall eﬀect occurs in a two-dimensional electron gas, all
low-energy elementary excitations are localized near the system edge. The
edge acts in many ways like a one-dimensional ring of electrons, except that a
ﬁnite current ﬂows around the ring in equilibrium. This article is a brief and
informal review of some of the physics of quantum Hall system edges. We
discuss the implications of macroscopic compressible and incompressible strip
models for microscopic chiral Luttinger liquid models and make an important distinction between the origin of non-Fermi-liquid behavior in fractional
quantum Hall edges and in usual one-dimensional electron gas systems.
PACS numbers: 73.40.Hm

Typeset using REVTEX
1

I. INTRODUCTION

The quantum Hall eﬀect1 is an anomaly which occurs in the transport properties of twodimensional electron system 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,
responsible for dissipation in the system, vanishes at low temperatures. Our understanding
of this transport anomaly is not absolutely complete. Nevertheless, there is fairly broad
agreement that edge state2,3 theories, in which the nonequilibrium transport currents are
carried near the edge of the system, capture the essence of the phenomenon. We present a
version of the edge state picture which accommodates both integer and fractional quantum
Hall eﬀects in Section II of this article. It follows from the edge state picture that when the
quantum Hall eﬀect occurs, the ground state of the two-dimensional electron system must
possess a set of low-energy excitations which are localized near the edge of the system. In
this article we discuss a sequence of pictures of edge excitations in the quantum Hall eﬀect.
The order in which we present the pictures is loosely one of increasing sophistication and
power, although there are important lessons at each step and the more powerful descriptions
are not in every sense the ones with greater generality. In Section III we discuss edge states
for a system of non-interacting electrons in which the particles are conﬁned to a ﬁnite
area by an external potential. In Section IV we grapple with the Coulomb interaction
between electrons. The long-range of this interaction can have overwhelming importance
in determining the edge electronic structure when the edge charge density proﬁle has a
width which is long compared to microscopic lengths, the so-called ‘smooth edge’ regime.
The discussion of Section IV uses both Thomas-Fermi and Hartree-Fock approximations to
treat electron-electron interactions. A many-particle description beyond the Hartree-Fock
approximation is necessary to describe edge states in the case of the fractional quantum Hall
eﬀect. In Section V we discuss the ground state and excited states of edges in the fractional
case using a language of many-body wavefunctions. This approach shows that the edges
are very similar in integer and fractional cases but fails to reveal some important diﬀerences
2

which appear in quantities like the tunneling density-of-states at the edge of the system. We
address these issues by using Wen’s4,5 chiral Luttinger liquid theory to discuss integer and
fractional edges in Section VI. In Section VII we present some concluding remarks.

II. INCOMPRESSIBILITY AND EDGE STATES

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 independent positive and negative
charges which 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 key to the quantum Hall eﬀect. This property
requires the existence of the gapless edge excitations which are the subject of this article.
To make this more concrete, consider a large but ﬁnite two-dimensional electron gas 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-reversalinvariance 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
3

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 =

∂M
∂N
|B δµ =
|µ δµ.
∂µ
∂B

(2)

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:
∂n
δI
=c
|µ .
δµ
∂B

(3)

The fact that δI/δµ = 0 implies that whenever the charge gap occurs at a density which
depends on magnetic ﬁeld, there must be gapless excitations at the edge of the system. From
a more microscopic point of view, Eq. (3) arises because the edge currents are related to the
way in which the spectrum evolves with changes in the the vector potential and hence in
the magnetic ﬁeld.2
This property of the edge states is expected to persist even if 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) was proposed as an explanation for the quantum Hall eﬀect by Pavel Stˇeda and is
r
commonly known as the Stˇeda formula.6
r
4

In using this picture to explain 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. In our view, however, there is no diﬃculty with the conclusion relevant to the present
paper; gapless edge excitations satisfying Eq. (3) must be present whenever the quantum
Hall eﬀect occurs.

III. NON-INTERACTING ELECTRON PICTURE

Throughout this article 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
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 edges and will not discuss the physics
of interaction or scattering between edges.7 In this geometry it is convenient to choose the
symmetric gauge, A = B(−y, x, 0)/2. For Vconf (r) ≡ 0, the single-electron kinetic energy
eigenfunctions and eigenenergies in this gauge have deﬁnite angular momentum and are
known analytically.3 The spectrum consists of a set of degenerate Landau levels. The kinetic
energy in the n-th Landau level is hωc (n+1/2) where ωc = eB/mc is the cyclotron frequency;
¯
states occur in the n-th Landau level with8 angular momentum m = −n, −n+ 1, · · ·. In each
Landau level states with larger angular momentum are localized further from the origin. For
example, the symmetric gauge wavefunctions in the lowest Landau level are:
φm (z) = (

1
2πℓ2 2m m!

)1/2 z m exp(−|z|2 /4ℓ2 )

(4)

where 2πℓ2 B = hc/e is the magnetic ﬂux quantum and ℓ is the magnetic length which serves
as the fundamental microscopic length scale in the strong magnetic ﬁeld regime. It is easy
5

to verify that, for large m , φm is localized 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, which we will adopt consistently in this article, the conﬁnement
potential does not mix diﬀerent Landau levels. Since there is only one state with each
angular momentum in each Landau level, that means that 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. [2]. 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
¯
h
the lowest energy unoccupied Landau level (E = 5¯ ωc /2). The states we are interested in
h
in this article are the ground state and the low energy excited states obtained by making
one or more particle-hole excitations at the edge. We will actually 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.9
We will also neglect the spin degree of freedom of the electrons throughout this article.
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

(5)

and all other states in the Hilbert space (truncated to the lowest Landau level) have larger
angular momentum.10 For large disks and total angular momentum near M0 the excitation
energy of a non-interacting electron state will be
∆E = γM

(6)

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:
6

γ = eEedge

dRm
= eEedge ℓ2 /R
dm

(7)

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 which 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
¯

(8)

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. [3]. 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 which occurs at M = 4 and is illustrated in Fig. [3].

IV. COMPRESSIBLE AND INCOMPRESSIBLE STRIPS:
THE THOMAS-FERMI PICTURE

It is very instructive to apply a Thomas-Fermi approximation to the edge of an electron
disk in the quantum Hall regime.11 A modern framework for discussing the Thomas-Fermi
approximation is provided by density-functional theory, which has been generalized in recent
years to accommodate magnetic ﬁelds.12 The Thomas-Fermi approximation is intended to
7

be applicable to the case where the charge density changes very slowly on atomic length
scales. In this regime we can hope that the local-density-approximation, where the energy
density at r is assumed to be equal to the energy density of a uniform system with density
n = n(r), is valid. The main lessons to be learned from the Thomas-Fermi approximation
for quantum Hall edges are connected with the long-range of the Coulomb interaction and
our discussion here is always for the case of these physically realistic interactions. When the
Thomas-Fermi approximation is applied for systems of charged particles, the electrostatic
interaction energy, for which a local density approximation is obviously inappropriate, must
be treated separately. The Thomas-Fermi energy functional is therefore
E[n] = EH [n] +

d2 rn(r)ǫ(n(r)) +

d2 rn(r)Vext (r),

(9)

where EH [n] is the electrostatic (Hartree) energy, Vext is the external potential, and ǫ(n)
is the energy per-particle of a uniform density system (in a magnetic ﬁeld in our case)
which is placed in an electrically neutralizing positively charged background to eliminate
the electrostatic energy. Minimizing this energy functional with respect to n(r) at ﬁxed
total particle number gives the Thomas-Fermi approximation expression for the density
proﬁle,
Vext (r) + VH (r) + µ2D (n(r)) = µ.

(10)

Here VH (r) is the (non-local) electrostatic potential, µ is the chemical potential, and
µ2D (n) =

d nǫ(n)
dn

(11)

is the density-dependent chemical potential of a uniform density electron system in a neutralizing background.
Eq. (11) can and has been used to discuss the density proﬁle at the edge in the case
of both integer11 and fractional13 quantum Hall eﬀects. We will discuss here the ThomasFermi version of the random-phase-approximation, where we include only the electrostatic
and kinetic energy contributions to the energy and fractional quantum Hall eﬀect features
are not captured. In this approximation (neglecting spin for simplicity)
8

µ2D (n) = hωc ([ν] + 1/2)
¯

(12)

where [ν] is the integer part of the Landau level ﬁlling factor ν = 2πℓ2 N/A. Let’s assume
for the moment that the non-interacting electron ground state remains the ground state in
the presence of interactions. Looking on the macroscopic length scale appropriate to the
Thomas-Fermi approximation, the charge density of this state is constant: n = (2πℓ2 )−1 for
R < RN and n = 0 for R > RN . The electrostatic potential produced by this charge density
is
VH (r) =

e2 √
r2
1 −1
, 1; 2 ).
2NF ( ,
ǫℓ
2 2
RN

(13)

where F (a, b, c; x) is the conﬂuent hypergeometric function.
The important point for us is that the scale of this potential is larger than the microscopic
√
interaction energy scale e2 /ℓ by a factor proportional to N. Since e2 /ℓ and hωc are of the
¯
same order it follows that except for the case of small N ‘quantum dot’ systems,14–17 the
maximum density droplet wavefunction cannot be the ground state unless the sum of an
external potential and the Hartree electrostatic potential is close to constant, i.e. unless
the external potential is similar to that from a neutralizing positive background which is
co-planar with the electrons. When this is not the case, the charge density proﬁle will
be close to that which would be obtained in a purely electrostatic theory by solving the
equation Vext (r) + VH (r) = µ. When µ2D (n) is included, the solution of the Thomas-Fermi
equation for the charge-density will consist of relatively broad regions where Vext (r)+VH (r) =
µ−¯ ωc (N +1/2) for some ﬁxed integer N and the charge density varies, and relatively narrow
h
regions separating adjacent values of N where the charge density is ﬁxed.
In this picture of quantum Hall system edges, regions where the electron density varies
are referred to as compressible strips and regions where the electron density is ﬁxed are
referred to as incompressible strips. Evidently, low-energy excitations can be created in
the compressible strips by varying the charge density along the edge or by altering slightly
the charge density proﬁle across the edge. To relate the Thomas-Fermi theory results to
microscopic theory, we must, at a minimum, use a Hartree or Hartree-Fock approximation.
9

The results of such a theory are illustrated schematically in Fig. [4]. In this illustration we
have in mind a ‘smooth edge’ where the external potential is created by charges on gates
or surface states some distance from the two-dimensional electron system. The extreme
limit of the ‘smooth edge’ model systems is one where the external potential is taken to
be parabolic. For this parabolic conﬁnement model, which is often applicable to quantum
dot systems, the electrostatic approximation to the Thomas-Fermi equations can be solved
analytically:18,11
n(r) = n0 1 − (r/R)2 ,

(14)

where n0 and R, the radius of the electron disk, are constants ﬁxed by the total electron
number and the curvature of the parabolic potential. When µ2D (n) is included in the
Thomas-Fermi equations, the density will distort slightly and develop the compressible and
incompressible strips discussed above and illustrated in Fig. [4]. For smooth edges, electrostatic considerations will be so dominant that nearly the same density proﬁle will be
produced by any microscopic theory. The simplest microscopic theory that includes interactions is a Hartree19 or Hartree-Fock theory in which the energy of each single particle state
is modiﬁed by electrostatic (or Hartree) and, in the Hartree-Fock case, exchange corrections.
In such a theory the regions in space where the local density does not correspond to an integral Landau level ﬁlling factor, i.e. the compressible strips must have fractional occupation
numbers for the Hartree-Fock single-particle states. It follows that the total variation of the
Hartree-Fock energy across the angular momenta in each compressible strip must be smaller
than the thermal energy kB T . One point of view on this result is to regard the system
as locally metallic in the compressible strips so that the conﬁnement potential is strongly
screened.20 Of course, the Hartree-Fock approximation is not justiﬁed in this regime but it
does provide one valid lesson. Throughout the compressible region, single-particle states
with nearby angular momenta will have a ﬁnite probability of being occupied or unoccupied
as the result of thermal or, in a more general theory, quantum ﬂuctuations. The sharp Fermi
edge between occupied and unoccupied states that we would have for non-interacting elec10

trons is lost. It seems clear that the spectrum of low-energy excitations can be exceedingly
complicated in the smooth edge regime.
In the following sections we will implicitly assume that the external conﬁning potential
permits the electron density at the edge to fall oﬀ on a microscopic length scale, i.e. that
we are in the ‘abrupt edge’ regime. We will also implicitly assume that the interactions
between the particles are short-ranged, appealing if pressed to the ubiquitous presence of
nearby gates which dress all electrons with image charges. You have been fairly warned
that these assumptions can be dangerous especially when electrostatic imperatives force an
electron-density that changes slowly on microscopic length scales. The task of determining
the excitation energy (temperature), if any, below which the ‘abrupt edge’ models we will
now discuss apply for ‘smooth edge’ systems remains an important challenge. In quantum
dot systems the transition between ‘abrupt edge’ and ‘smooth edge’ regimes is initiated by
edge reconstructions.16 Similar ‘reconstructions’ may occur at the edges of bulk systems and
when they occur they will complicate the excitation spectrum and all physical properties.

V. MANY-BODY WAVEFUNCTION PICTURE

In this section we 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 noninteracting 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
are can be comforting but are, in my view, misleading. Nevertheless, we will see that there
is a one-to-one 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
11

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),

(15)

where we have adopted ℓ as the unit of length and P (z1 , . . . , zN ) is a polynomial in the twodimensional complex coordinates. This property21 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 follows22 that
P (z1 , . . . , zN ) =
i<j

(zi − zj ) Q[z]

(16)

where Q[z] is any polynomial which 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
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 which, as in Eq. (16), 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. (16) 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 which favor states with larger total angular momentum. When this is the
12

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
polynomials23 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 here24 to the simplest fractional quantum Hall eﬀects which 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 which occurs at these ﬁlling factors was explained in the pioneering
paper of Laughlin.25,3 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 which avoid placing pairs in this state will have low energy. If this
condition26 is satisﬁed,
P (z1 , . . . , zN ) ≡

i<j

(zi − zj )3 Q[z]

(17)

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. 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
place17 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. As long as the external potential is such that
we are in the ‘abrupt edge’ regime, the edge states in the realistic case will be in one-to-one
correspondence with those of Eq. (17).
The wavefunction
Ψ[z] ≡ Q[z]

l

exp(−|zl |2 /4)
13

(18)

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=

(19)

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

(20)

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 which 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)

(21)

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 . . .
1 − xk
k=1
14

= (1 + x + x2 + x3 + . . .)(1 + x2 + x4 + . . .)(1 + x3 + x6 + . . .) . . .
= (1 + x + 2x2 + 3x3 + 5x4 + 7x5 + 11x6 + 15x7 + 22x8 + 30x9 + 42x10 + . . .

(22)

√2
1/2
For17 large M g(M) ∼ e 3 ·π·M . The function g(M) is well known to number theorists
from the theory of partitions27 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
picture of quantum Hall edges which 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.

VI. CHIRAL LUTTINGER LIQUID PICTURE

The chiral Luttinger liquid picture5 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.28 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 which
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

αRR 2
αLL 2
δnL (x) +
δnR (x) + αLR δnL (x) δnR (x) .
2
2
15

(23)

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. (23) is incorrect. 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
dx δn2 (x) =
R

1
n−qR .nqR
L q=0

(24)

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

αLL
2L

n−qR nqR .

(25)

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 =

(26)

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
16

(27)

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 which 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. (3), for M = 2 we have
nqR |Ψ0 = | . . . • • ◦ •| • ◦ ◦ . . .
+| . . . • • • ◦| ◦ • ◦ . . . .

(28)

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:28
[n−q′ R , nqR ] =

qL
δq,q′ .
2π

(29)

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 =
a† =
q

2π
n−qR
qL
2π
nqR
qL

(30)
(31)

With these deﬁnitions Eq. (29) yields
[aq′ , a† ] = δq,q′
q
so that the density waves satisfy bosonic commutation relations. Also note that
17

(32)

qL
ˆ
[M, aq ] = − aq
2π
qL †
†
ˆ
[M, aq ] =
a
2π q

(33)
(34)

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 =

(35)

q>0

where
v=

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

(36)

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
v=

h kF
¯
≡ vF
m∗

(37)

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 physics28 occurs only
when left and right-moving electrons are allowed to interact. Most notably, arbitrarily weak
interactions destroy the sharp Fermi edge which is the hallmark of Fermi liquids and which
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
two-dimensional electron density along a line perpendicular to the edge. Results discussed
in earlier sections can be discussed instead in the language of chiral Luttinger liquids. The
set of boson states are the states of the chiral phonon system which has modes with only
one sign of momentum and velocity.
18

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

(38)

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 ν = 1/m
and there are large quantum ﬂuctuations in the local conﬁguration of the system even in
the interior. However, we know24 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. (29) 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

(39)

which diﬀers from Eq. (29) 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 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. (39) 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
19

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 which 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.

(40)

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
E[n] = E0 + µδn + α

(δn)2
2

(41)

Minimizing with respect to δn we ﬁnd that
δµ
α

(42)

ev
δI
=
δµ
α

(43)

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

(44)

The extra factor of ν appearing in this equation compared to Eq. (36) requires the same
factor of ν to appear in Eq. (39). We discuss below the qualitative changes in the physics4,5
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,24,4,29 even though (nearly) all steps in the argument are superﬁcially
20

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 onedimensional 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 which 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 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 low-energy longwavelength eﬀective theory. The reader is reminded however, of the smooth edge regime,
where in our view both the many-particle wavefunction mapping and the chiral Luttinger
liquid theory of the edge are likely to fail.
An important aspect of Luttinger liquid theory is the expression for electron ﬁeld operators in terms of bosons.28 This relationship is established by requiring the exact identity
ˆ
ˆ
[ρ(x), ψ † (x′ )] = δ(x − x′ ) ψ † (x′ )

(45)

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.

(46)

In order to satisfy Eq. (45), the ﬁeld operator must be given by
−1
ˆ
ψ † (x) = ceiν φ(x)

(47)

where dφ(x)/dx = n(x) and c is a constant which cannot be determined by the theory.
The factor of ν −1 in the argument of the exponential of Eq. (47) is required because of the
21

factor of ν in the commutator of density Fourier components which 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 importance. It is worth remarking4 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 singlebranch chiral boson eﬀective Hamiltonian can be correct only when ν = 1/m for odd m.
Wen5 has surveyed, using this criterion, the multi-branch generalization of the simplest
eﬀective Hamiltonian theory which are possible at any given rational ﬁlling factor. His
conclusions are consistent with arguments24 based on the microscopic theory of the fractional
quantum Hall eﬀect.
Eq. (47) has been carefully checked numerically30 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 which 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 −

(48)

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

(49)

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 calculation4,5 yields a spectral function which
grows like (ǫ − µ)ν

−1 −1

. It is intuitively clear that the spectral function should be small at
22

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.

VII. ENDNOTE

There is, as always, a lot more which could be said. However other duties, including
even ones for which I am paid, are insisting that I must stop here. These notes have focused
on the microscopic origins, and also possibly (in the ‘smooth edge’ case) the limitations of
Luttinger liquid theories for quantum Hall edges. A diﬀerent and at least equally interesting
article could be written on applications of Luttinger liquid models in the fractional quantum Hall regime. Among these it appears that those which describe7 tunneling31 between
quantum Hall edge systems due to disorder are the most promising for experimental tests.
Indeed, initial experiments32 appear to conﬁrm theoretical predictions. It is likely that more
experimental work will be added to the literature soon. The important question of the role of
long-range interactions in Luttinger liquid theories, which we have for the most part dodged
here, is also beginning to be addressed.33
These informal notes are intended to be widely accessible. I hope that they will be of
some use to people who are expert on either theoretical or experimental aspects of the quantum Hall eﬀect but have not been following theories of quantum Hall edges. Likewise I hope
that they can be useful to experts on the one-dimensional electron gas who have not been
following the quantum Hall eﬀect. Comments, critical or complimentary, are welcome. The
ideas here have been shaped by discussions with members of the condensed matter theory
group at Indiana University, especially S.M. Girvin, R. Haussmann, S. Mitra, K. Moon, J.J.
Palacios, D. Pfannkuche, E. Sorensen, K. Tevosyan, K. Yang, and U. Z¨ licke. Discussions
u
with L. Brey, M. Fisher, M. Johnson, C. Kane, L. Martin, J. Oaknin, C. Tejedor, S.R.-E.
23

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.

24

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R.B. Laughlin, Phys. Rev. B 23, 5632 (1981); B.I. Halperin, Phys. Rev. B 25, 2185 (1982);
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3

See for example, A.H. MacDonald, in Les Houches, Session LXI, 1994, Physique
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4

X.G. Wen, Phys. Rev. B 41, 12838 (1990); D.H. Lee and X.G. Wen, Phys. Rev. Lett. 66,
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5

For reviews see X.G. Wen, Int. J. Mod. Phys. B6, 1711 (1992); X.G. Wen, to appear in
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6

P. Stˇeda, J. Phys. C, 15, L717 (1982); L. Smrˇka and P. Stˇeda, J. Phys. C, 10, 2153
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7

Lack of time and space limits the scope of this article to systems without internal boundaries which have a single outer edge. There is, appropriately, considerable interest in
annular and equivalent geometries which have inner and outer edges carrying currents in
opposite directions. These geometries allow the problem of scattering between edges by
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Glazman, Phys. Rev. B 49, 1966 (1994).

8

For negatively charged particles the angular momenta in the lowest Landau level are
25

negative for a magnetic ﬁeld in the z direction. We choose a magnetic ﬁeld in the −ˆ
ˆ
z
direction to have the convenience of positive angular momenta in the lowest Landau level.
9

For recent work on ‘multi-branch’ systems see C.L. Kane and Matthew P.A. Fisher, Phys.
Rev. B 51, 13449 (1995) and work cited therein.

10

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).

11

C.W.L. Beenakker, Phys. Rev. Lett. 64, 216 (1990); D.B. Chklovskii, B.I. Shklovskii, and
L.I. Glazman, Phys. Rev. B 46, 4026 (1992); ibid 46, 15606(E) (1992); P.L. McEuen, E.B.
Foxman, Jari Kinaret, U. Meirav, M.A. Kastner, Ned S. Wingreen, and S.J. Wind, Phys.
Rev. B. 45, 11419 (1992); D.B. Chklovskii, K.A. Matveev, and B.I. Shklovskii, Phys. Rev.
B 47, 12605 (1993); M.M. Fogler, E.I. Levin, and B.I. Shklovskii, Phys. Rev. B 49, 13767
(1994); Karheinz Lier and Rolf R. Gerhardts, Phys. Rev. B 50, 7757 (1994).

12

See for example M. Ferconi and G. Vignale, Phys. Rev. B 50, 14722 (1994).

13

M. Ferconi, M.R. Geller, and G. Vignale, preprint (1995).

14

For reviews on quantum dots see U. Merkt, Advances in Solid State Physics, 30, 77
(1990); Tapash Chakraborty, Comments on Condensed Matter Physics 16, 35 (1992);
M.A. Kastner, Rev. Mod. Phys. 64, 849 (1992); N.F. Johnson, J. Phys. C 7, 965 (1995).
Systems containing of order 100 or fewer electrons are generally referred to as quantum
dots.

15

For experimental studies of quantum dots in strong magnetic ﬁelds see P.L. McEuen, E.B.
Foxman, U. Meirav, M.A. Kastner, Y. Meir, Ned S. Wingreen, and S.J. Wind, Phys.
Rev. Lett. 66, 1926 (1991); ; E.B. Foxman, P. L. McEuen, U. Meirav, Ned S. Wingreen,
Yigal Meir, Paul A. Belk, N. R. Belk, and M.A. Kastner, and S.J. Wind, Phys. Rev. B
47, 10020 (1993); R.C. Ashoori, H.L. St¨rmer, J.S. Weiner, L.N. Pfeiﬀer, S.J. Pearton,
o
K.W. Baldwin, and K.W. West, Phys. Rev. Lett. 68, 3088 (1992); R.C. Ashoori, H.L.

26

St¨rmer, J.S. Weiner, L.N. Pfeiﬀer, K.W. Baldwin, and K.W. West, Phys. Rev. Lett. 71,
o
613 (1993); O. Klein, C. de Chamon, D. Tang, D.M. Abusch-Magder, X.-G. Wen, and
M.A. Kastner, Phys. Rev. Lett. 74, 785 (1995). T. Schmidt et al., Phys. Rev. B 51, 5570
(1995).
16

For theoretical studies of quantum dot systems in strong magnetic ﬁelds see J. J. Palacios
and P. Hawrylak, Phys. Rev. B 51, 1769 (1995); J. J. Palacios et al., Phys. Rev. B 50,
5760 (1994); J. H. Oaknin et al., Phys. Rev. Lett. 74, 5120 (1995); C. de Chamon and
X.-G. Wen, Phys. Rev. B 49, 8227 (1994); J. Dempsey, B.Y. Gelfand, and B.I. Halperin,
Phys. Rev. Lett. 70, 3639 (1993); S.-R. Eric Yang, A.H. MacDonald, and M.D. Johnson,
Phys. Rev. Lett. 71, 3194, (1993).

17

A.H. MacDonald and M.D. Johnson, Phys. Rev. Lett. 70, 3107.

18

S.S. Nazin and V.B. Shikin, Zh. Eksp. Fiz. 85, 530 (1983) [Sov. Phys. JETP 58, 210
(1983)]; V.B. Shikin, T. Demel’ and D. Heitmann, Zh. Eksp. Teor. Fiz. 96, 1406 (1989)
[Sov. Phys. JETP 69, 797 (1989)].

19

L. Brey, J.J. Palacios, and C. Tejedor, Phys. Rev. B 47, 13884 (1993).

20

U. Wulf, V. Gudmundsson, and R.R. Gerhardts, Phys. Rev. B 38, 4218 (1988); A.
Manolescu, Phys. Rev. B 45, 11829 (1992).

21

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).

22

A.H. MacDonald and D.B. Murray, Phys. Rev. B 32 2291 (1985).

23

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).

24

For a more general discussion see A.H. MacDonald, Phys. Rev. Lett. 64, 220 (1990).

25

R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).

27

26

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].

27

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.

28

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
and V. Meden, submitted to Am.J. Phys. (1995).

29

M.D. Johnson and A.H. MacDonald, Phys. Rev. Lett. 67, 2060 (1991).

30

J.J. Palacios and A.H. MacDonald, preprint (1995) and work cited therein.

31

K. Moon, H. Yi, C.L. Kane, S.M. Girvin, M.P.A. Fisher, Phys. Rev. Lett. 71, 4381 (1993);
P. Fendley, A.W.W. Ludwig, and H. Saleur, Phys. Rev. Lett. 74, 3005 (1995).

32

F.P. Milliken, C.P. Umbach, and R.A. Webb, Solid State Comm. (in press) (1995).

33

L.I. Glazman, I.M. Ruzin, and B.I. Shklovskii, Phys. Rev. B 45, 8454 (1992); H.J. Schulz,
Phys. Rev. Lett. 71, 1864 (1993); Yuval Oreg and Alexander M. Finkel’stein, Phys. Rev.
Lett. 74, 3668 (1995); K. Moon and S.M. Girvin, preprint (1995); U. Z¨ licke and A.H.
u
MacDonald, in preparation (1995).

28

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. 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.
FIG. 3. 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.
FIG. 4. Schematic illustration of the Thomas-Fermi theory and Hartree-Fock theory pictures
of the edge of a quantum Hall system with Landau level ﬁlling factor ν = 2 in the bulk. The
Thomas-Fermi theory gives an approximation for the charge density proﬁle across the edge. Regions where the charge density varies, indicated by dashed lines, correspond to constant values
for Vext + VH and are known as compressible strips. The edge excitations occur in the compressible strips. Hartree-Fock theory produces approximate values for the quasiparticle energies as a
function of angular momentum in the edge region. When the electron density at the edge varies
gradually on an atomic length scale the Hartree-Fock eigenvalues will have a weak dependence on
angular momentum where they cross the Fermi level.

29

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 which 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

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