Thermal activation, long-range ordering, and topological frustration of artificial spin ice

Frustrated systems, typically characterized by competing interactions that cannot all be simultaneously satisfied, are ubiquitous in nature and display many rich phenomena and novel physics. Artificial spin ices (ASIs), arrays of lithographically patterned Ising-like single-domain magnetic nanostructures, are highly tunable systems that have proven to be a novel method for studying the effects of frustration and associated properties. The strength and nature of the frustrated interactions between individual magnets are readily tuned by design and the exact microstate of the system can be determined by a variety of characterization techniques. Recently, thermal activation of ASI systems has been demonstrated, introducing the spontaneous reversal of individual magnets and allowing for new explorations of novel phase transitions and phenomena using these systems. In this work, we introduce a new, robust material with favorable magnetic properties for studying thermally active ASI and use it to investigate a variety of ASI geometries. We reproduce previously reported perfect ground-state ordering in the square geometry and present studies of the kagome lattice showing the highest yet degree of ordering observed in this fully frustrated system. We consider theoretical predictions of long-range order in ASI and use both our experimental studies and kinetic Monte Carlo simulations to evaluate these predictions. Next, we introduce controlled topological defects into our square ASI samples and observe a new, extended frustration effect of the system. When we introduce a dislocation into the lattice, we still see large domains of ground-state order, but, in every sample, a domain wall containing higher energy spin arrangements originates from the dislocation, resolving a discontinuity in the ground-state order parameter. Locally, the magnets are unfrustrated, but frustration of the lattice persists due to its topology. We demonstrate the first direct imaging of spin configurations resulting from topological frustration in any system and make predictions on how dislocations could affect properties in numerous materials systems.

Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

Essays in Personal Transportation Demand and Consumer Finance

This dissertation is composed of three essays covering two areas of interest. The first topic is personal transportation demand with a focus on price and fuel efficiency elasticities of mileage demand, challenging assumptions common in the rebound effect literature. The second topic is consumer finance with a focus on small loans. The first chapter creates separate variables for fuel prices during periods of increasing and decreasing prices as well as an observed fuel economy measure to empirically test the equivalence of these elasticities. Using a panel from Germany from 1997 to 2009 I find a fuel economy elasticity of mileage of 53.3%, which is significantly different from the gas price elasticity of mileage during periods of decreasing gas prices, 4.8%. I reject the null hypothesis or price symmetry, with the elasticity of mileage during period of increasing gas prices ranging from 26.2% and 28.9%. The second chapter explores the potential for the rebound effect to vary with income. Panel data from U.S. households from 1997 to 2003 is used to estimate the rebound effect in a median regression. The estimated rebound effect independent of income ranges from 17.8% to 23.6%. An interaction of income and fuel economy is negative and significant, indicating that the rebound effect may be much higher for low income individuals and decreases with income; the rebound effect for low income households ranged from 80.3% to 105.0%, indicating that such households may increase gasoline consumption given an improvement in fuel economy. The final chapter documents the costs of credit instruments found in major mail order catalogs throughout the 20th century. This study constructs a new dataset and finds that the cost of credit increased and became stickier as mail order retailers switched from an installment-style closed-end loan to a revolving-style credit card. This study argues that revolving credit's ability to decrease salience of credit costs in the price of goods is the best explanation for rate stickiness in the mail order industry as well as for the preference of revolving credit among retailers.