Interplay between FQH Ground States, Regular Graphs, Binary Invariants, and [formula] -Algebras

Abstract

Fractional quantum Hall (FQH) phases are arguably the most outstanding example of topologically ordered matter. The ground state’s delicate structure, conjointly with anyonic statistics of local excitations, are two of the intriguing, if not defining, features of these phases. For the eventual goal of classifying topologically ordered matter, a better understanding of FQH ground states is critical. This thesis takes a closer look at FQH ground states, their properties, and the intersection with several other mathematical fields. We report on four novel findings in this document. (1) We present a graph-theoretic point of view toward clustering FQH ground state wavefunctions. In particular, we show that every ground state is essentially a superposition of regular graphs. In this paradigm, algebro-analytic properties of polynomial wavefunctions are synonymous with graph-theoretic properties. (2) We introduce a new possible local property of ground states called separability. Utilizing separability, a pseudopotential Hamiltonian, refining the existing projection Hamiltonians, is proposed. This new Hamiltonian has a much higher chance of realizing a unique densest zero-energy state when the traditional projection Hamiltonians fail to do so. (3) We also study model FQH ground states that are essentially chiral conformal blocks in [formula] parafermionic theories. We design an easy-to-work-with ansatz for multi-parafermion operator product expansions. Putting the ansatz to use, we unravel several polynomial structures within the [formula] -algebras. (4) We identify the second quantized version of FQH ground state wavefunctions with the so-called binary invariants. The wealth of knowledge on the theory of invariants of binary forms in the literature allows us to design an ansatz for the principal (i.e., smallest non-trivial) model FQH ground states which are conformal blocks in a [formula] algebra. In particular, we fully determine the principal [formula] wavefunctions. A partial characterization of [formula] wavefunctions with 𝑟 = 3, 4 is also provided.