Dualities, Topological Properties, and Degeneracies of Classical and Quantum Lattice Models

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

Why are all dualities conformal? Theory and practical consequences

We relate duality mappings to the "Babbage equation" F(F(z)) = z, with F a map linking weak- to strong-coupling theories. Under fairly general conditions F may only be a specific conformal transformation of the fractional linear type. This deep 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, i.e., after generating one of those series the other is automatically determined through a set of linear constraints between the series coefficients. This latter relation partially solve or, equivalently, localize the computational complexity of evaluating the series expansion to a simple fraction of those coefficients. As a bonus, those relations also encode non-trivial equalities between different geometric constructions in general dimensions, and connect derived coefficients to polytope volumes. We illustrate our findings by examining various models including, but not limited to, ferromagnetic and spin-glass Ising, and Ising gauge type theories on hypercubic lattices in 1< D <9 dimensions.Comment: 41 pages (18 (main text)+23 (suppl. information)), 2 figures, 8 tables; to appear in Nuclear Physics

Binomial Spin Glass

To establish a unified framework for studying both discrete and continuous coupling distributions, we introduce the {\it binomial} spin glass, a class of models where the couplings are sums of $m$ identically distributed Bernoulli random variables. In the continuum limit $m \to \infty$, the class reduces to one with Gaussian couplings, while $m=1$ corresponds to the $\pm J$ spin glass. 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 $(\sqrt{d/2m} + 1/N)\ln2$, and further show that the actual entropies follow the scaling behavior implied by this bound. We thus uncover a fundamental non-commutativity of the thermodynamic and continuous coupling limits that leads to the presence or absence of degeneracies depending on the precise way the limits are taken. Exact calculations of defect energies reveal a crossover length scale $L^\ast(m) \sim L^\kappa$ below which the binomial spin glass is indistinguishable from the Gaussian system. Since $\kappa = -1/(2\theta)$, where $\theta$ is the spin-stiffness exponent, discrete couplings become irrelevant at large scales for systems with a finite-temperature spin-glass phase