A Generic Theory of the Integer Quantum Hall Effect

Abstract

The integer quantum Hall effect (IQHE) is usually modeled by a Galilean or rotationally invariant Hamiltonian. These are not generic symmetries for electrons moving in a crystal background and can potentially confuse non-topological quantities with topological ones and identify otherwise distinct geometrical properties. In this thesis we present a generic theory for the IQHE. First we show that a generic guiding-center coherent state, defined by a natural metric in each Landau level, has the form of an antiholomorphic function times a Gaussian factor. Then by numerically solving the eigenproblem for a quartic Hamiltonian and finding the roots of the antiholomorphic part we are able to define a topological spin $s_n = n+\tfrac{1}{2}$ where $n$ is the number of central roots that are enclosed by the semiclassical orbit. We derive a generic formula for the Hall viscosity in the absence of rotational symmetry and show that the previous interpretation of the scalar Hall viscosity as the ``intrinsic orbital angular momentum'' breaks down since the concept of angular momentum requires the presence of rotational symmetry. We also calculate generic electromagnetic responses and differentiate between universal terms that are diagonal with respect to Landau level index and non-universal terms that depend on inter-Landau-level mixing. We conclude that the generic theory offers a fundamental definition for the topological spin and reveals finer structure in the geometrical properties of the IQHE.