The Fractional Quantum Hall Effect: from Electrons to Topological Field Theory
Project IV with Dr. Nabil Iqbal (2021-2022)
One of the most interesting ideas in physics is the notion that more is different: in other words, you might understand what a single electron does. But if you put 10^23 interacting electrons in a box, the resulting dynamics of the many-body system may be completely unexpected, seeming to have nothing to do with the motion of a single electron.
In this project, we will study one particularly dramatic example of this phenomenon, called the Fractional Quantum Hall Effect: in particular, if you put enough electrons in a box and apply a magnetic field, it is possible to obtain a qualitatively new state of matter, that behaves as though each electron -- conventionally thought of as an elementary and thus unsplittable particle -- has nevertheless somehow been split into fractional pieces, each of them carrying a fraction of the original charge.
This is a picture of a knot; it can also be understood as the trajectory in space-time of a particle/anti-particle pair that appear, braid around each other, and annihilate again -- these processes play an important role in the FQHE and are described by topological field theory.
The experimental discovery and subsequent theoretical understanding of the Fractional Quantum Hall Effect led to the 1998 Nobel Prize in physics. In this project we will study the (surprisingly) sophisticated theoretical physics behind this, starting from the quantum mechanical problem of electrons in magnetic fields (and the simpler Integer Quantum Hall Effect), then moving to its fractional cousin. We will understand how the concept of topological order allows electrons to "split into pieces" and study Topological Field Theories, special kinds of quantum field theories that are the correct way to describe the long-distance dynamics of the Quantum Hall System. We will learn that much current research in condensed matter physics (e.g. the 2016 Nobel Prize) deals with the conceptually new ideas of which the FQHE is just one manifestation.
Possible directions for development in the second half of the project include:
Topological field theories and their applications to quantum computing
Other kinds of topological order, such as topological insulators
The Landau paradigm of phase transitions and its interplay with topological phases of matter
David Tong's lecture notes on the Fractional Quantum Hall effect are a wonderful resource that we will follow for part of the course.
Here is a colloquium by Tong on the topic (and its relation to the 2016 Nobel Prize)
MATH3111 Quantum Mechanics III (or Physics equivalent).
MATH4061 Advanced Quantum Theory IV.
Some exposure to statistical mechanical ideas (e.g. through concurrently taking MATH4231) also wouldn’t hurt, but is definitely not necessary.