Ph D Thesis: Quantum Hall Wave Functions on the Torus
Wednesday 06 May 2015
to 15:00 at
Mikael Fremling (Stockholm University, Department of Physics)
The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed
matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles
with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer.
The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological
order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early
stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of
these wave functions have given further insight to the nature of the FQHE.
An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on
planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also
construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend
this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions.
One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on
surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where
a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases,
such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions
with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class
of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states,
which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having
boundary modes with only one chirality.
Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in
terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is
taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our
construction solves the long standing problem of engineering torus wave functions for multi-component many-body states.
Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5,
with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular
parameter τ, that describes the geometry of the torus.
Having explicit torus wave functions allows us to use the methods developed by Read and Read \& Rezayi to numerically
compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport
coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere.
The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all
relevant topologically information.
We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as
we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in
viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative
and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results
in the thin torus limit.