Date of Award
Doctor of Philosophy (PhD)
We study various nontrivial facets of Ҥegeneracyӭ a concept of paramount importance in numerous physical systems.
In the first part of this thesis, we challenge the folklore that if the ground state degeneracy of a physical system depends on topology then this system must necessarily realize an unconventional, so-called Ҵopological quantumӬ order. To this end, we introduce a classical rendition of the Toric Code model that displays such a topological degeneracy yet exhibits conventional Landau order. As the ground states of this classical system may be distinguished by local measurements, this example illustrates that, on its own, topological degeneracy is not a sufficient condition for topological quantum order. This conclusion is generic and applies to many other models.
In the second part of this thesis, we prove that under fairly modest conditions, all ҤualitiesӠare conformal. This general result has enormous practical consequences. For example, one can establish that weak- and strong-coupling series expansions of arbitrarily large finite size systems are trivially related. As we explain, this relation partially solves or, equivalently, localizes the computational complexity of evaluating the series expansions to only a subset of those coefficients. The coefficients in the strong-coupling series expansions are related to the degeneracy of the system. Thus, our results may facilitate the computation of the degeneracies of the various levels.
We end this thesis by establishing a unified framework for studying general disordered systems with either discrete or continuous coupling distributions. We introduce a ҢinomialӠspin glass wherein the couplings are the sum of ҭӠidentically distributed Bernoulli random variables. We demonstrate that for short-range Ising models on d-dimensional hypercubic lattices, the ground-state entropy density for N spins is bounded from above by ( sqrt(d/2m)+1/N )ln2. This confirms the long hand suspicion that the degeneracy of real (finite dimensional) spin glasses with Gaussian couplings is not extensive. Exact calculations reveal the presence of a crossover length scale L*(m) below which the binomial spin glass is indistinguishable from the Gaussian system. Our analytical and numerical results underscore the non-commutativity of the thermodynamic and continuous coupling limits.
Chair and Committee
Carl Bender, Gerardo Ortiz, Alexander Seidel, Li Yang,
Vaezi, Seyyed Mohammad Sadegh, "Dualities, Topological Properties, and Degeneracies of Classical and Quantum Lattice Models" (2018). Arts & Sciences Electronic Theses and Dissertations. 1659.