It is well known that a charged particle moving in a magnetic field feels a `Lorentz' force perpendicular to its direction of motion and the magnetic field. As a direct consequence of this Lorentz force, charged particles will accumulate to one side of a wire if you send current through it and hold it still in a (perpendicular) magnetic field. This is called the Hall effect. The voltage drop at right angles to the current is called the Hall voltage; The current divided by the Hall voltage is called the Hall conductance.
The Hall effect can be put to use in several ways.
One application is magnetic field strength measurement.
Since the Hall voltage is proportional to the current and the field
strength, sending a known current through a medium and measuring the
Hall voltage tells you the field strength.
Another nice thing is that you can reveal the nature of the mobile charges
in a current-carrying medium. The Lorentz force will push a moving hole
(positive charge)
and a moving electron (negative charge) in
exactly the same direction, since they travel in opposite ways;
from the sign of the Hall voltage you can tell if there are more mobile
holes than electrons or vice versa.
Investigating the conductance properties of two-dimensional electron
gases at very low temperature and high magnetic fields, his group
obtained curious results: The Hall conductance of such a system
plotted as a function of the ratio
In 1982, Tsui, Gossard and Störmer, working with samples that contained less impurities, discovered the so-called "fractional" quantum Hall effect. Here the conductance is quantised in fractional multiples of e²/h, like 1/3, 1/5, 2/3, 2/5 etc etc, always with odd denominator. Whereas the integer quantisation could perhaps have been expected, the fractional effect came as a total surprise. Tsui and Störmer were awarded the 1998 physics Nobel prize for the discovery, sharing it with Laughlin, who was the first to come with a theoretical description.
[Picture of the
experimental setup; 3.6Kb]
[Graph of experimental
data; 11Kb]
It turns out that the quantum states for an electron in a magnetic field, moving in a two-dimensional random potential energy landscape, fall into two classes: so-called "localised" and "extended" states. Roughly speaking, the localised states are bound to one or more `peaks' or `chasms' in the energy landscape and have an energy corresponding to the `height' were they are sitting. In contrast, an extended state spreads through the whole sample and its energy is that of a particle that does not feel the random potential. (The energy levels of an undisturbed electron are called Landau levels.) The localised states, being bound to one small region, cannot contribute to electron transport. By putting extra electrons into localised states it is therefore possible to change the parameter `nu' without changing the conductance! This explains the occurrence of plateaux. Only when the new electrons reach a Landau level does the conductance change, because now the extended states come into play. The fact that the plateaux of the Hall conductance lie exactly at integer multiples of e²/h can be explained by relating the sample with its random potential to a hypothetical situation without impurities, but I am not going to elaborate on this.
[Picture of the density of states; 3Kb]
Before the composite fermion theory was formulated, a many-electron wave function was written down by Laughlin in 1983 for the fractional plateaux around nu=1/(odd integer), in the idealised case where there are no impurities in the sample. Even though it neglects the disorder, this wave function gives a lot of insight. It clarifies how the Coulomb interaction makes it possible for the Hall quantisation to be non-integer. It also gives a hint how flux attachment works. And it shows that when a plateau occurs, the electrons form a so-called "incompressible quantum liquid" whose density, as the name implies, is not easily changed.
[Picture of the Laughlin 1/3 wave function; 56Kb]
Particles with fractional charge and statistics can at present only be probed at the edge. Chang et al. have done an experiment where electrons tunnel from a normal metal into the edge of a nu=1/3 quantum Hall sample. They found a current-voltage relation for the tunneling that goes like I ~ V for low voltage and like I ~ V^{3} for higher voltage, in accordance with theory. The temperature dependence was also in good agreement with theory.
In an experiment by Milliken et al., the tunneling current was measured between two edges of a nu=1/3 quantum Hall sample as a function of temperature and gate voltage.
In an experiment by Chang et al., electrons were tunneled from an
ordinary metal into quantum Hall samples with a very sharp edge, for
filling fractions between 1/4 and 1. They obtained the surprising result
I ~ V^{1/nu}. Surprising for two reasons:
First, one would naively expect the power of V (the so-called
tunneling exponent) to be quantised when the conductances are quantised.
Instead, the tunneling exponent varies continously with the (non-quantised)
filling fraction! Second, even right in the middle of conductance
plateaux the 1/nu result contradicts calculations made purely on the basis of
chiral edge boson theories, except for the simplest cases
nu=1 and nu=1/3.
We have proposed an
explanation
for this experiment, based on Coulomb
interactions between edge bosons and localised states in the bulk.
(Here I wanted to add a few words on
shot noise, nuclear magnetic resonance, Knight shift, Skyrmions etc, but never managed to find the time.)
The attentive reader will have noticed that in the `simple theories' the combination of disorder and Coulomb interactions has been carefully avoided. The reason is that there is nothing simple about this combination. It is, in fact, a notoriously difficult problem. The Coulomb interactions prevent you from using the single-particle wave functions with which you can attack the disorder, while the disorder breaks the symmetry that would help you tackle the interaction problem.
The only hope left is quantum field theory. Write down an action that contains all the ingredients: 2D electrons in a magnetic field, a random potential, Coulomb interaction and a Chern-Simons gauge field that will generate flux attachment. Put the action in an imaginary time path integral. Then perform the "shake and stir" of field theory: Take the disorder average by integrating over the random potential. Identify the massive modes and integrate them out in order to obtain an effective action for the physically interesting massless modes. Finally, do a renormalisation group analysis that tells you how observables will depend on length scale.
Each of these steps introduces its own problems, which are, in principle,
solvable. The solutions are sometimes quite peculiar.
The presence of disorder, for example, requires you to average
the logarithm of the partition sum, not the partition sum
itself. This forces you to perform, on top of everything else, the
so-called replica trick, taking N identical copies of the
system and then sending N to zero.
Somewhere in the derivation of the effective action,
you are forced to put a cutoff
on frequency space, destroying the gauge invariance of the theory.
Only by sending this cutoff to infinity at the end of all calculations
is the gauge invariance restored.
AND WHY DO YOU THINK I DIRECTED YOU TO THE STABLES? THINK CAREFULLY, NOW. Mort hesitated. He had been thinking carefully, in between counting wheel barrows. He'd wondered if it had been to coordinate his hand and eye, or teach him the importance, on the human scale, of small tasks, or make him realise that even great men must start on the bottom. None of these explanations sounded exactly right. "I think", he began. YES? "Well, I think it was because you were up to your knees in horseshit, to tell you the truth." Death looked at him for a long time. Mort shifted uneasily from one foot to the other. ABSOLUTELY CORRECT, snapped Death. CLARITY OF THOUGHT. REALISTIC APPROACH. VERY IMPORTANT IN A JOB LIKE OURS.I wrote my master's thesis in 1995 under the supervision of prof.dr.ir. F.A. Bais at the University of Amsterdam. It is called "Infinite symmetries in the quantum Hall effect". Although this work did not lead to a publication, it is a nice review of the quantum Hall effect (including the Chern-Simons theory, conformal symmetry, Kac-Moody algebra and W-algebra involved in its effective description) that has helped to lure several students to this great subject.— Terry Pratchett, Mort
Between 1995 and 1999 I did my PhD research on the quantum Hall effect. My PhD supervisor was Aad Pruisken. Other members of the condensed matter theory club at that time were prof. Kareljan Schoutens, Mischa Baranov, Sathya Guruswamy, Ronald van Elburg and Eddy Ardonne.
The coupling is by no means a simple procedure. Simple attempts give rise to infinities and problems with the U(1) gauge invariance. The way we did it was by first noting that the Finkelstein theory has a hidden symmetry (which we dubbed "F-invariance"). In the presence of long-range interactions, the theory is invariant under a spatially constant shift of the plasmon field. (The plasmon field is roughly speaking defined as the Coulomb potential at a certain point in the sample due to all the other charges elsewhere in the sample). What is required for the invariance to hold is a very special way of treating frequency cutoffs. The shift of the plasmon field is reminiscent of a gauge transformation of the electromagnetic scalar potential; this fact, together with the cutoff prescription, made it possible to include U(1) gauge fields in the theory.
Having gauge fields at your disposal is obviously a great advantage. It allows you to do linear response calculations and to perform the Chern-Simons flux attachment trick, which is exactly what we have done. Apart from that, the F-invariance enabled us to do renormalisation group calculations to two-loop order. My last QHE work was on edge states. In the limit of zero bulk density of states, our 2+1 dimensional theory becomes a 1+1 dimensional theory of chiral "relativistic" edge bosons that has the same structure as phenomenological edge models, but yields new insights.
For more information I refer to the publications listed below and references therein. I decided to highlight one of our results here, because we believe it settles a controversy alive among people working on quantum Hall edges.
We have done a calculation that shows that the Coulomb interactions can be effectively taken care of by writing down a theory that lives only on the edge but has modified constants. To be more precise, if we are sitting at filling fraction nu = nu_{0} + delta, with nu_{0} the center of a plateau, then the noninteracting theory would contain a constant nu_{0} and the interactions would modify this to nu_{0} + delta = nu. Another effect of the interactions turns out to be that the so-called "neutral modes", which are degrees of freedom that are not related to the charge of the electrons, get strongly suppressed. As a result of all this we find a current-voltage relation of the form I ~ V^{1/nu}, in agreement with the experiments.
See my list of
publications for more details.