My research at the Institute for Theoretical Physics: the quantum Hall effect

Between 1995 and 1999 I worked on the quantum Hall effect at the Institute for Theoretical Physics, University of Amsterdam. I had a much-visited webpage there, which I have now put online again in a slightly updated version.

Content: For people who can read Dutch, there is a short article by Kareljan Schoutens, published in the march 1997 issue of the faculty quasi-periodical "Afleiding".

The classical Hall effect

The Hall effect was discovered by Edwin Hall in 1879.

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.

The quantum Hall effect

The discovery of the quantised Hall effect in 1980 won von Klitzing the 1985 Nobel prize.

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

nu := electron density*h / magnetic field*e
shows extremely flat plateaux at integer multiples of e²/h around integer values of the ratio `nu'. (h is Planck's constant, e is the electron charge.) Furthermore, the `ordinary' conductance plotted as a function of nu is zero everywhere except where the Hall conductance has a transition from one plateau to another. In other words, there are whole intervals of nu where the voltage drop is completely at right angles to the current, with Hall conductance very accurately quantised in terms of the fundamental conductance quantum e²/h; in between these intervals, the longitudinal conductance has a peak, while the Hall conductance goes from one plateau value to another.

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]

Why do we care?

Doing research is hard work and condensed matter systems are particularly opaque. So the question naturally arises: why are we working on this, are we masochists or what? To which the answer is of course, Yes; we are theoretical physicists, remember? But apart from that, there are many reasons to be interested in the quantum Hall effect.
• A very obvious first reason is that a large and growing part of the world's information storage and manipulation depends on the movement of electrons through semiconductors. Everything that could possibly be known about the subject should therefore be known.
• For those who immediately want to know whether something has practical applications: The integer quantum Hall effect is now used as the international standard of resistance. The incredibly accurate quantisation of the Hall resistance to approximately one part in 108 makes this possible.
• The constant e²/h is proportional to the `fine structure constant' in electrodynamics, which basically tells how strongly light interacts with matter. The quantum Hall effect provides an independent way of accurately measuring this constant.
• To a theoretical physicist, the fractional effect is a mouth-watering feast of new theories, nice mathematics, exotic statistics and topology galore. I will try to explain this below.

Simple theory for the integer effect

The discovery of the quantum Hall effect showed that the theories of electron transport in disordered two-dimensional systems were inadequate. In the early eighties a simple explanation was found for the integer effect in terms of noninteracting electrons, i.e. electrons that do not repel each other. Such particles `feel' each other only through Pauli's exclusion principle, which says that no two fermions can occupy exactly the same quantum state. In this approximation you only have to figure out what the possible states are for one electron and then combine them in a simple way to form a many electron state; you just fill the available states with electrons, beginning with the lowest energy.

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]

Simple theory for the fractional effect

The simple explanation for the integer effect completely fails to predict the fractional effect. In the fractional effect the Coulomb repulsion between electrons plays an essential role. The similarity between the experimental results for the integer and fractional case, however, simply begs for a common description. And indeed one exists. It is called the "composite fermion" theory, formulated by Jain in 1989, and it states that mumblemumble somehow mumblemumble the Coulomb repulsion has the net effect of attaching an even number of magnetic flux quanta to every electron. Such composite objects obey the Pauli principle, which is where the name composite fermion comes from. Since a large part of the magnetic field has gone into defining the new composite particles, the field that these particles feel is much smaller than the original one; in fact it exactly mimics the integer effect's field strength. In this way the fractional quantum Hall effect is explained as the integer effect for composite fermions. This elegant picture is widely accepted, even though the equivalence between electrons and composite fermions is not of course 100% exact.

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]

An avalanche of interesting physics

The simplest theoretical explanations for the Hall effects already generated new ideas like magnetic flux attachment, incompressible quantum fluids and the importance of the `size' of wave functions (instead of only the question how many electron states exist in a certain energy interval). It didn't stop there. By taking these ideas a little step further, interesting predictions were made and links were discovered with other branches of theoretical physics.

• The incompressible quantum fluid and composite fermion picture immediately predicts the existence of particle-like disturbances of the electron gas ("quasiparticles") with very unusual properties. These quasiparticles can have a fractional charge like 1/3 or 1/5 and also fractional statistics. The term `statistics' refers to the behaviour of a wave function under exchange of identical particles. All known particles are either fermions (the wave function acquires a factor -1 under exchange) or bosons (factor +1). The quasiparticles in the fractional effect can have factors like (-1)1/3. The only other physical system that we know of today where fractional statistics may be found is liquid helium.
• It turns out that interesting things happen at the edge of a quantum Hall system. This can be roughly understood by noting that you can not cheaply change the density of an incompressible fluid, but you can change its form; and that happens at the surface. The low-energy disturbances at the edge are described by bosonic particles moving in one direction, "chiral edge bosons". Roughly speaking, these bosons can combine to form a quasiparticle on the edge that behaves like a composite fermion or like a particle with fractional charge and statistics.
• The theory of edge bosons (when neglecting Coulomb interactions) has so-called conformal symmetry. I will not elaborate on this, but symmetry always comes in handy when you want to solve a problem, and two-dimensional conformally invariant systems have an infinite number of symmetries. Conformal field theory, developed at lightning speed after Belavin, Polyakov and Zamolodchikov's famous 1984 paper, has been successfully applied to many problems in statistical mechanics and condensed matter.
• Particles with unusual statistics can be described by Chern-Simons field theory. This is a topological theory, which means that it cares only about the electrons' global motion, for instance how often they circle around one another. Flux attachment is incorporated in a natural way. By arguments of gauge invariance, Chern-Simons theory is directly related to chiral edge bosons.
• The chiral edge bosons are related to the Calogero-Sutherland model and Luttinger liquids.
• In everything mentioned above, the electron spin has been neglected. (In a strong enough magnetic field all spins point in the same direction.) If one does take these spins into account, one finds interesting spin structures called Skyrmions.

Recent experiments

[This was still under construction in 1999 and therefore not so "recent" any more.]

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 ~ V3 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 ~ V1/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.)

Quantum Field Theory

[An apology: In spite of all my good intentions, this part is completely unreadable for non-physicists and perhaps even for many physicists.]

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.

My own modest contributions

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.

— Terry Pratchett, Mort

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.

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.

My research in a nutshell

The starting point of my research can be roughly summarised in one sentence by saying that we have discovered a new symmetry in Finkelstein's theory for interacting electrons in a disordered medium and that we have extended it in such a way that the electrons can be coupled to gauge fields. The rest of my activities has basically consisted of capitalising on this to obtain new results for both the bulk and edge of quantum Hall systems.

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.

Understanding the tunneling experiment

One of the nice things about the work I've been doing is that, although it may look like a lot of arcane formalism, it can actually be directly related to experiments. In our approach to tunneling processes we found that there is an important difference between a tunneling experiment and a measurement of the Hall conductance. The Hall conductance is a non-equilibrium property (electrons are injected into an edge channel and don't get time to equilibrate with the localised bulk states), while by tunneling one probes the equilibrium energy eigenstates. Due to the presence of localised bulk states and the Coulomb interactions between all states, the many body eigenfunctions are not restricted to the edge. A tunneling experiment therefore feels the bulk as well as the edge.

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 = nu0 + delta, with nu0 the center of a plateau, then the noninteracting theory would contain a constant nu0 and the interactions would modify this to nu0 + 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 ~ V1/nu, in agreement with the experiments.

See my list of publications for more details.