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Quantum Hall Fluids
Andrew York
11/10/08
Phys 611, Siopsis
The study of topological quantum fluids has emerged over the last decade or so as
an interesting application of Quantum Field Theory. The Hall fluid is an example of one
such fluid, and describes a system in which a collection of electrons move in a plane in
the presence of a magnetic field B directed normal to the plane. The magnetic field is
strong enough to align all the electron spins so they may be treated as spinless fermions.
In such a system, electrons take up a finite amount of space. Classically, we would say
they rotate in Larmor circles (where evB  mv 2 / r ) and a simple description using
quantization of angular momentum mvr  eBr 2  2 (after setting   1, h  2 ) implies
each electron would take up a minimum area of r 2 ~ 2 2 / eB . Further, one electron’s
space cannot overlap with another’s (they are fermions). Thus, a quantum Hall fluid
must have enough space to allocate to each electron in the system. This is a simple
description, but even from this it is obvious that the system becomes interesting when the
total available area A ~ N er 2 ~ N e (2 2 / eB) , where N e is the number of electrons in
the system. This paper will examine in some detail the properties of Hall fluids and the
physical justification for these properties. First I will explore the integer Hall effect,
which is the already obvious special circumstance that arises when the total area available
is equal to the area required by the individual electrons in the fluid. Then the bulk of the
paper will be devoted to understanding the fractional Hall effect, which occurs when the
surface area contains an odd fraction of the number of electrons needed to fill it
completely.
The textbook case of a spinless electron in a magnetic field is defined by the wave
equation:
(1)
 [( x  ieAx ) 2  ( y  ieAy ) 2 ]  2mE
1 eB
With the energy states (solved by Landau years ago) E n  (n  ) , n  0,1,2... known
2 m
as the nth Landau level. Because the Larmor circles may be placed anywhere, each
Landau level has a degeneracy of BA / 2 (where A is the area of the system). Thus we
can define a filling factor   N e /( BA / 2 ) that represents the number of completely
filled degenerate levels. When  is an integer,  energy levels are completely filled and
were we to add any additional electrons they would have to go into the  +1 level.
Clearly then, the Hall fluid is incompressible for integer values of  . Any attempt to
compress it would reduce the area (A), reduce the degeneracy ( BA / 2 ), and force some
of the electrons into the next energy level (costing lots of energy, since the energy
eB
spacing is
). This incompressibility at integer filling factors is known as the integer
m
quantum Hall effect.
Scientists were taken by surprise when experiments yielded yet another
incompressible quantum Hall fluid that occurred when the filling factor  was an odd-
1 1
denominator fraction (  , , etc.). This is known as the fractional quantum Hall
3 5
effect, and requires a more sophisticated field theory of the Hall fluid. This paper will
attempt to explain the fractional Hall effect as the natural consequence of a few general
principles of the Hall system.
This treatment of the Hall system starts with the following premises, which are
either obviously true or assumed to be true:
1) The system is a 2+1 dimensional spacetime (confined to a plane).
2) E-M current is conserved (   J   0 ).
3) A descriptive field theory should follow from the development of a good local
Lagrangian.
4) The important physics occurs at large distances and large time (small wave
number and low frequency). In other words, the lowest dimension terms will dominate
 
the Lagrangian.   a
5) Parity and time reversal are broken by the external magnetic field.
1) and 2) above are enough to conclude that the current can be written as the curl
of a vector potential:
1 
J 
  a 
(2)
2
This follows from the fact that whenever the divergence of something is 0 in 31
dimensions, it must be the curl of something else (the factor
is for normalization). It
2
is worth noting at this point that transforming a  by a   a      doesn’t change the
current, which means a  is a gauge potential.
Following 3) above, we should begin guessing the Lagrangian. It must be gauge
invariant, so the 2D term a a  won’t work. The simplest gauge-invariant term is the 3D
term  a  a , so the first term in the Lagrangian is:
L
k 
 a   a 
4
(3)
1
is normalization
4
over 3 dimensions. According to principle 4) above, we can ignore the higher
dimensional terms that may be in the Lagrangian. We also know this system’s current
couples to an external gauge potential A , so using equation (2) above we add a term to
Where k is an as-yet-undetermined dimensionless parameter and
1 
 A  a  to get (after integrating by parts and
2
dropping a surface term A a  ):
the Lagrangian  A J   
L
k 
1 
k 
1 
 a   a  
 A  a 
 a   a  
 a   A
4
2
4
2
(4)
Last, it is important to note that “quasiparticles” are a fundamental concept in
condensed matter physics. In this context, quasiparticle effects are the effects due to the
many-body interactions occurring in the medium. This is represented by a term a j  in
the Lagrangian. The complete Lagrangian (given assumption 4) above) is now:
k 
1 
L
 a   a   a  j  
 a   A
(5)
4
2
~
This Lagrangian can be simplified somewhat. Defining j  j   (1 / 2 )   A
(5) becomes:
k 
~
L
 a   a   a  j 
(6)
4
Then using the identity (central to QFT)
K
 De
1
   K   J 
2
k 
~
  , J  j , and integrating (6) over
2
 ~   
L  j  
k  2
e
1
J  K 1  J
2
and in this case
a , we get:
~
 j


2
Which allows us to use     ~  and re-expand to:
1 
1
   
L
 A  A  A j   j 
j
4k
k
k
2
(7)
(8)
In this Lagrangian, there is an AA term, an Aj term, and a jj term. From the AA
term, we can deduce by varying A that the electromagnetic current is:
1 

J em

  A
(9)
4k
The   0 part of this current tells us that an excess density of electrons is related to a
local fluctuation of the magnetic field by N e  (1 / 2k )B and comparing with the old
definition of filling factor   N e /( BA / 2 ) it becomes clear that the filling factor is
associated with 1/k, and from the   i component that an electric field produces current
in the orthogonal direction, so   1 / k   xy . The Aj term tells us the electric charge of
the quasiparticle is 1/k.
At this point it is instructive to step back and try to understand exactly what these
quasiparticles are. Essentially, they must be collections of particles (in this case either
electrons or holes of electrons) that move around in the system and interact like particles
with a charge of 1/k. Imagining the system, we have a plane partially filled with
electrons orbiting in Larmor circles. These electrons can in principle be distributed any
way possible so long as they are not overlapping, but in practice it makes sense that
electron repulsion will cause them to settle into some sort of unique ground state in which
they are uniformly distributed across the plane. Such ground states are only possible for
certain filling factors  (as was discovered experimentally). These quasiparticles
represent the hole or electron structures necessary to describe the system ground state.
Since they behave like particles and cannot overlap, they should obey the rules for
fermions. Their states can be interchanged with a phase factor determined by the Hopf
term common to all particles. The Hopf term looks like:

1   
 j
(10)
LHopf 
j 
2

4 


1 
And comparing (10) with (7), it is clear that
 , and thus the statistics parameter 
4 k
1 
that describes the interchange of particle states is related by 1 
for a single

4 k
quasiparticle. When exchanging k quasiparticles with k quasiparticles, we would expect



1
to pick up a phase k  1  k 2  k 2  k . k must be an odd integer for fermions, so k

 
k
must be an odd integer as well, and thus   1/ k will be an odd-denominator fraction.
This explains the fractional quantum Hall effect.