Zee A. Quantum Field Theory in a Nutshell

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VI.2 Quantum Hall Fluids

Interplay between two pieces of physics

Over the last decade or so, the study of topological quantum fluids (of which the Hall fluid

is an example) has emerged as an interesting subject. The quantum Hall system consists

of a bunch of electrons moving in a plane in the presence of an external magnetic field

B perpendicular to the plane. The magnetic field is assumed to be sufficiently strong so

that the electrons all have spin up, say, so they may be treated as spinless fermions. As is

well known, this seemingly innocuous and simple physical situation contains a wealth of

physics, the elucidation of which has led to two Nobel prizes. This remarkable richness

follows from the interplay between two basic pieces of physics.

1. Even though the electron is pointlike, it takes up a finite amount of room.

Classically, a charged particle in a magnetic field moves in a Larmor circle of radius

r determined by evB = mv

/r. Classically, the radius is not fixed, with more energetic

particles moving in larger circles, but if we quantize the angular momentum mvr to be

h = 2π (in units in which  is equal to unity) we obtain eBr

∼ 2π. A quantum electron

takes up an area of order πr

∼ 2π

/eB .

2. Electrons are fermions and want to stay out of each other’s way.

Not only does each electron insist on taking up a finite amount of room, each has to

have its own room. Thus, the quantum Hall problem may be described as a sort of housing

crisis, or as the problem of assigning office space at an Institute for Theoretical Physics to

visitors who do not want to share offices.

Already at this stage, we would expect that when the number of electrons N

is just

right to fill out space completely, namely when N

πr

∼ N

(2π

/eB) ∼ A, the area of the

system, something special happens.

VI.2. Quantum Hall Fluids | 323

Landau levels and the integer Hall effect

These heuristic considerations could be made precise, of course. The textbook problem of

a single spinless electron in a magnetic field

−[(∂

− ieA

)

+ (∂

− ieA

)

]ψ = 2mEψ

was solved by Landau decades ago. The states occur in degenerate sets with energy



n +



, n = 0, 1, 2,

...

, known as the nth Landau level. Each Landau level has

degeneracy BA/2π, where A is the area of the system, reflecting the fact that the Larmor

circles may be placed anywhere. Note that one Landau level is separated from the next by

a finite amount of energy (eB/m).

Imagine putting in noninteracting electrons one by one. By the Pauli exclusion principle,

each succeeding electron we put in has to go into a different state in the Landau level. Since

each Landau level can hold BA/2π electrons it is natural (see exercise VI.2.1) to define a

filling factor ν ≡ N

/(BA/2π). When ν is equal to an integer, the first ν Landau levels are

filled. If we want to put in one more electron, it would have to go into the (ν +1)st Landau

level, costing us more energy than what we spent for the preceding electron.

Thus, for ν equal to an integer the Hall fluid is incompressible. Any attempt to compress

it lessens the degeneracy of the Landau levels (the effective area A decreases and so the

degeneracy BA/2π decreases) and forces some of the electrons to the next level, costing

us lots of energy.

An electric field E

imposed on the Hall fluid in the y direction produces a current

= σ

in the x direction with σ

= ν (in units of e

/h). This is easily understood in

terms of the Lorentz force law obeyed by electrons in the presence of a magnetic field. The

surprising experimental discovery was that the Hall conductance σ

when plotted against

B goes through a series of plateaus, which you might have heard about. To understand

these plateaus we would have to discuss the effect of impurities. I will touch upon the

fascinating subject of impurities and disorder in chapter VI.8.

So, the integer quantum Hall effect is relatively easy to understand.

Fractional Hall effect

After the integer Hall effect, the experimental discovery of the fractional Hall effect,

namely that the Hall fluid is also incompressible for filling factor ν equal to simple odd-

denominator fractions such as

and

, took theorists completely by surprise. For ν =

only one-third of the states in the first Landau level are filled. It would seem that throwing

in a few more electrons would not have that much effect on the system. Why should the

ν =

Hall fluid be incompressible?

Interaction between electrons turns out to be crucial. The point is that saying the first

Landau level is one-third filled with noninteracting spinless electrons does not define a

unique many-body state: there is an enormous degeneracy since each of the electrons can

324 | VI. Field Theory and Condensed Matter

go into any of the BA/2π states available subject only to Pauli exclusion. But as soon as we

turn on a repulsive interaction between the electrons, a presumably unique ground state

is picked out within the vast space of degenerate states. Wen has described the fractional

Hall state as an intricate dance of electrons: Not only does each electron occupy a finite

amount of room on the dance floor, but due to the mutual repulsion, it has to be careful

not to bump into another electron. The dance has to be carefully choreographed, possible

only for certain special values of ν.

Impurities also play an essential role, but we will postpone the discussion of impurities

to chapter VI.6.

In trying to understand the fractional Hall effect, we have an important clue. You will

remember from chapter V.7 that the fundamental unit of flux is given by 2π , and thus the

number of flux quanta penetrating the plane is equal to N

=BA/2π. Thus, the puzzle is

that something special happens when the number of flux quanta per electron N

=ν

−1

is an odd integer.

I arranged the chapters so that what you learned in the previous chapter is relevant to

solving the puzzle. Suppose that ν

−1

flux quanta are somehow bound to each electron.

When we interchange two such bound systems there is an additional Aharonov-Bohm

phase in addition to the (−1) from the Fermi statistics of the electrons. For ν

−1

odd these

bound systems effectively obey Bose statistics and can be described by a complex scalar

field ϕ. The condensation of ϕ turns out to be responsible for the physics of the quantum

Hall fluid.

Effective field theory of the Hall fluid

We would like to derive an effective field theory of the quantum Hall fluid, first obtained

by Kivelson, Hansson, and Zhang. There are two alternative derivations, a long way and a

short way.

In the long way, we start with the Lagrangian describing spinless electrons in a magnetic

field in the second quantized formalism (we will absorb the electric charge e into A

L = ψ

i(∂

− iA

)ψ+

†

(∂

− iA

)

ψ +V(ψ

†

ψ) (1)

and massage it into the form we want. In the previous chapter, we learned that by intro-

ducing a Chern-Simons gauge field we can transform ψ into a scalar field. We then invoke

duality, which we will learn about in the next chapter, to represent the phase degree of

freedom of the scalar field as a gauge field. After a number of steps, we will discover that

the effective theory of the Hall fluid turns out to be a Chern-Simons theory.

Instead, I will follow the short way. We will argue by the “what else can it be” method

or, to put it more elegantly, by invoking general principles.

Let us start by listing what we know about the Hall system.

1. We live in (2 + 1)-dimensional spacetime (because the electrons are restricted to a plane.)

2. The electromagnetic current J

is conserved: ∂

= 0.

VI.2. Quantum Hall Fluids | 325

These two statements are certainly indisputable; when combined they tell us that the

current can be written as the curl of a vector potential

2π



μνλ

∂

(2)

The factor of 1/(2π) defines the normalization of a

. We learned in school that in 3-

dimensional spacetime, if the divergence of something is zero, then that something is

the curl of something else. That is precisely what (2) says. The only sophistication here is

that what we learned in school works in Minkowskian space as well as Euclidean space—it

is just a matter of a few signs here and there.

The gauge potential comes looking for us

Observe that when we transform a

by a

→a

−∂

, the current is unchanged. In other

words, a

is a gauge potential.

We did not go looking for a gauge potential; the gauge potential came looking for us!

There is no place to hide. The existence of a gauge potential follows from completely

general considerations.

3. We want to describe the system field theoretically by an effective local Lagrangian.

4. We are only interested in the physics at long distance and large time, that is, at small

wave number and low frequency.

Indeed, a field theoretic description of a physical system may be regarded as a means of

organizing various aspects of the relevant physics in a systematic way according to their

relative importance at long distances and according to symmetries. We classify terms in a

field theoretic Lagrangian according to powers of derivatives, powers of the fields, and so

forth. A general scheme for classifying terms is according to their mass dimensions, as

explained in chapter III.2. The gauge potential a

has dimension 1, as is always the case for

any gauge potential coupled to matter fields according to the gauge principle, and thus (2)

is consistent with the fact that the current has mass dimension 2 in (2 + 1)-dimensional

spacetime.

5. Parity and time reversal are broken by the external magnetic field.

This last statement is just as indisputable as statements 1 and 2. The experimentalist

produces the magnetic field by driving a current through a coil with the current flowing

either clockwise or anticlockwise.

Given these five general statements we can deduce the form of the effective Lagrangian.

Since gauge invariance forbids the dimension-2 term a

in the Lagrangian, the

simplest possible term is in fact the dimension-3 Chern-Simons term 

μνλ

∂

. Thus,

the Lagrangian is simply

L =

4π

a∂a +

...

(3)

where k is a dimensionless parameter to be determined.

326 | VI. Field Theory and Condensed Matter

We have introduced and will use henceforth the compact notation a∂b ≡

μνλ

∂

b∂a for two vector fields a

and b

The terms indicated by (

...

) in (3) include the dimension-4 Maxwell term (1/g

)(f

−

βf

) and other terms with higher dimensions. (Here β is some constant; see exer-

cise VI.1.1.) The important observation is that these higher dimensional terms are less

important at long distances. The long distance physics is determined purely by the Chern-

Simons term. In general the coefficient k may well be zero, in which case the physics is

determined by the short distance terms represented by the (

...

) in (3). Put differently, a

Hall fluid may be defined as a 2-dimensional electron system for which the coefficient of

the Chern-Simons term does not vanish, and consequently is such that its long distance

physics is largely independent of the microscopic details that define the system. Indeed,

we can classify 2-dimensional electron systems according to whether k is zero or not.

Coupling the system to an “external” or “additional” electromagnetic gauge potential A

and using (2) we obtain (after integrating by parts and dropping a surface term)

L =

4π



μνλ

∂

−

2π



μνλ

∂

4π



μνλ

∂

−

2π



μνλ

∂

(4)

Note that the gauge potential of the magnetic field responsible for the Hall effect should

not be included in A

; it is implicitly contained already in the coefficient k.

The notion of quasiparticles or “elementary” excitations is basic to condensed matter

physics. The effects of a many-body interaction may be such that the quasiparticles in the

system are no longer electrons. Here we define the quasiparticles as the entities that couple

to the gauge potential and thus write

L =

4π

a∂a + a

−

2π



μνλ

∂

...

(5)

Defining

≡ j

− (1/2π)

μνλ

∂

and integrating out the gauge field we obtain (see

VI.1.4)

L =





μνλ

∂



(6)

Fractional charge and statistics

We can now simply read off the physics from (6). The Lagrangian contains three types

of terms: AA, Aj , and jj. The AA term has the schematic form A(∂∂∂/∂

)A. Using

∂∂ ∼ ∂

and canceling between numerator and denominator, we obtain

L =

4πk

A∂A (7)

Varying with respect to A we determine the electromagnetic current

4πk



μνλ

∂

(8)

VI.2. Quantum Hall Fluids | 327

We learn from the μ = 0 component of this equation that an excess density δn of electrons

is related to a local fluctuation of the magnetic field by δn =(1/2πk)δB; thus we can identify

the filling factor ν as 1/k, and from the μ = i components that an electric field produces

a current in the orthogonal direction with σ

= (1/k) =ν.

The Aj term has the schematic form A(∂∂/∂

)j. Canceling the differential operators,

we find

L =

(9)

Thus, the quasiparticle carries electric charge 1/k.

Finally, the quasiparticles interact with each other via

L =



μνλ

∂

(10)

We simply remove the twiddle sign in (6). Recalling chapter VI.1 we see that quasiparticles

obey fractional statistics with

(11)

By now, you may well be wondering that while all this is fine and good, what would

actually tell us that ν

−1

has to be an odd integer?

We now argue that the electron or hole should appear somewhere in the excitation

spectrum. After all, the theory is supposed to describe a system of electrons and thus

far our rather general Lagrangian does not contain any reference to the electron!

Let us look for the hole (or electron). We note from (9) that a bound object made up of

k quasiparticles would have charge equal to 1. This is perhaps the hole! For this to work,

we see that k has to be an integer. So far so good, but k doesn’t have to be odd yet.

What is the statistics of this bound object? Let us move one of these bound objects half-

way around another such bound object, thus effectively interchanging them. When one

quasiparticle moves around another we pick up a phase given by θ/π = 1/k according to

(11). But here we have k quasiparticles going around k quasiparticles and so we pick up a

phase

= k (12)

For the hole to be a fermion we must require θ/π to be an odd integer. This fixes k to be

an odd integer.

Since ν =1/k, we have here the classic Laughlin odd-denominator Hall fluids with filling

factor ν =

...

. The famous result that the quasiparticles carry fractional charge and

statistics just pops out [see (9) and (11)].

This is truly dramatic: a bunch of electrons moving around in a plane with a magnetic

field corresponding to ν =

, and lo and behold, each electron has fragmented into three

pieces, each piece with charge

and fractional statistics

328 | VI. Field Theory and Condensed Matter

A new kind of order

The goal of condensed matter physics is to understand the various states of matter. States

of matter are characterized by the presence (or absence) of order: a ferromagnet becomes

ordered below the transition temperature. In the Landau-Ginzburg theory, as we saw in

chapter V.3, order is associated with spontaneous symmetry breaking, described naturally

with group theory. Girvin and MacDonald first noted that the order in Hall fluids does not

really fit into the Landau-Ginzburg scheme: We have not broken any obvious symmetry.

The topological property of the Hall fluids provides a clue to what is going on. As explained

in the preceding chapter, the ground state degeneracy of a Hall fluid depends on the

topology of the manifold it lives on, a dependence group theory is incapable of accounting

for. Wen has forcefully emphasized that the study of topological order, or more generally

quantum order, may open up a vast new vista on the possible states of matter.

Comments and generalization

Let me conclude with several comments that might spur you to explore the wealth of

literature on the Hall fluid.

1. The appearance of integers implies that our result is robust. A slick argument can

be made based on the remark in the previous chapter that the Chern-Simons term does

not know about clocks and rulers and hence can’t possibly depend on microphysics such

as the scattering of electrons off impurities which cannot be defined without clocks and

rulers. In contrast, the physics that is not part of the topological field theory and described

by (

...

) in (3) would certainly depend on detailed microphysics.

2. If we had followed the long way to derive the effective field theory of the Hall fluid, we

would have seen that the quasiparticle is actually a vortex constructed (as in chapter V.7)

out of the scalar field representing the electron. Given that the Hall fluid is incompressible,

just about the only excitation you can think of is a vortex with electrons coherently whirling

around.

3. In the previous chapter we remarked that the Chern-Simons term is gauge invariant

only upon dropping a boundary term. But real Hall fluids in the laboratory live in samples

with boundaries. So how can (3) be correct? Remarkably, this apparent “defect” of the theory

actually represents one of its virtues! Suppose the theory (3) is defined on a bounded 2-

dimensional manifold, a disk for example. Then as first argued by Wen there must be

physical degrees of freedom living on the boundary and represented by an action whose

change under a gauge transformation cancels the change of



x(k/4π)a ∂a. Physically,

it is clear that an incompressible fluid would have edge excitations

corresponding to waves

on its boundary.

X. G. Wen, Quantum Field Theory of Many-Body Systems.

The existence of edge currents in the integer Hall fluid was first pointed out by Halperin.

VI.2. Quantum Hall Fluids | 329

4. What if we refuse to introduce gauge potentials? Since the current J

has dimension

2, the simplest term constructed out of the currents, J

, is already of dimension 4;

indeed, this is just the Maxwell term. There is no way of constructing a dimension 3 local

interaction out of the currents directly. To lower the dimension we are forced to introduce

the inverse of the derivative and write schematically J(1/∂)J , which is of course just the

non-local Hopf term. Thus, the question “why gauge field?” that people often ask can be

answered in part by saying that the introduction of gauge fields allows us to avoid dealing

with nonlocal interactions.

5. Experimentalists have constructed double-layered quantum Hall systems with an

infinitesimally small tunneling amplitude for electrons to go from one layer to the other.

Assuming that the current J

(I =1, 2) in each layer is separately conserved, we introduce

two gauge potentials by writing J

2π



μνλ

∂

Iλ

as in (2). We can repeat our general

argument and arrive at the effective Lagrangian

L =



I ,J

4π

∂a

...

(13)

The integer k has been promoted to a matrix K . As an exercise, you can derive the Hall

conductance, the fractional charge, and the statistics of the quasiparticles. You would not

be surprised that everywhere 1/k appears we now have the matrix inverse K

−1

instead.

An interesting question is what happens when K has a zero eigenvalue. For example, we

could have K =





. Then the low energy dynamics of the gauge potential a

−

≡a

−a

is not governed by the Chern-Simons term, but by the Maxwell term in the (

...

) in (13).

We have a linearly dispersing mode and thus a superfluid! This striking prediction

was

verified experimentally.

6. Finally, an amusing remark: In this formalism electron tunneling corresponds to the

nonconservation of the current J

−

≡J

−J

=(1/2π)

μνλ

∂

−λ

. The difference N

−N

of the number of electrons in the two layers is not conserved. But how can ∂

−

= 0 even

though J

−

is the curl of a

−λ

(as I have indicated explicitly)? Recalling chapter IV.4, you the

astute reader say, aha, magnetic monopoles! Tunneling in a double-layered Hall system in

Euclidean spacetime can be described as a gas of monopoles and antimonopoles.

(Think,

why monopoles and antimonopoles?) Note of course that these are not monopoles in the

usual electromagnetic gauge potential but in the gauge potential a

−λ

What we have given in this section is certainly a very slick derivation of the effective

long distance theory of the Hall fluid. Some would say too slick. Let us go back to our

five general statements or principles. Of these five, four are absolutely indisputable. In

fact, the most questionable is the statement that looks the most innocuous to the casual

reader, namely statement 3. In general, the effective Lagrangian for a condensed matter

system would be nonlocal. We are implicitly assuming that the system does not contain a

massless field, the exchange of which would lead to a nonlocal interaction.

Also implicit

X. G. Wen and A. Zee, Phys. Rev. Lett. 69: 1811, 1992.

X. G. Wen and A. Zee, Phys. Rev. B47: 2265, 1993.

A technical remark: Vortices (i.e., quasiparticles) pinned to impurities in the Hall fluid can generate an

interaction nonlocal in time.

330 | VI. Field Theory and Condensed Matter

in (3) is the assumption that the Lagrangian can be expressed completely in terms of the

gauge potential a. A priori, we certainly do not know that there might not be other relevant

degrees of freedom. The point is that as long as these degrees of freedom are not gapless

they can be safely integrated out.

Exercises

VI.2.1 To define filling factor precisely, we have to discuss the quantum Hall system on a sphere rather than on

a plane. Put a magnetic monopole of strength G (which according to Dirac can be only a half-integer or

an integer) at the center of a unit sphere. The flux through the sphere is equal to N

=2G. Show that the

single electron energy is given by E

ω

)



l(l +1) − G



/G with the Landau levels corresponding

to l = G, G + 1, G +2, . . . , and that the degeneracy of the lth level is 2l + 1. With L Landau levels filled

with noninteracting electrons (ν =L) show that N

= ν

−1

− S, where the topological quantity S is

known as the shift.

VI.2.2 For a challenge, derive the effective field theory for Hall fluids with filling factor ν =m/k with k an

odd integer, such as ν =

. [Hint: You have to introduce m gauge potentials a

Iλ

and generalize (2) to

= (1/2π)

μνλ

∂



I =1

Iλ

. The effective theory turns out to be

L =

4π



I ,J =1

∂a



I =1

Iμ

...

with the integer k replaced by a matrix K . Compare with (13).]

VI.2.3 For the Lagrangian in (13), derive the analogs of (8), (9), and (11).

VI.3 Duality

A far reaching concept

Duality is a profound and far reaching concept

in theoretical physics, with origins in

electromagnetism and statistical mechanics. The emergence of duality in recent years in

several areas of modern physics, ranging from the quantum Hall fluids to string theory,

represents a major development in our understanding of quantum field theory. Here I

touch upon one particular example just to give you a flavor of this vast subject.

My plan is to treat a relativistic theory first, and after you get the hang of the subject, I

will go on to discuss the nonrelativistic theory. It makes sense that some of the interesting

physics of the nonrelativistic theory is absent in the relativistic formulation: A larger

symmetry is more constraining. By the same token, the relativistic theory is actually much

easier to understand if only because of notational simplicity.

Vortices

Couple a scalar field in (2 + 1)-dimensions to an external electromagnetic gauge potential,

with the electric charge q indicated explicitly for later convenience:

L =

|(∂

− iqA

)ϕ|

− V(ϕ

†

ϕ) (1)

We have already studied this theory many times, most recently in chapter V.7 in connection

with vortices. As usual, write ϕ =|ϕ|e

iθ

. Minimizing the potential V at |ϕ|=v gives the

ground state field configuration. Setting ϕ = ve

iθ

in (1) we obtain

L =

(∂

θ −qA

)

(2)

For a first introduction to duality, I highly recommend J. M. Figueroa-O’Farrill, Electromagnetic Duality for

Children, http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/EDC.html.