Physics of Hexagonal Limit-Periodic Phases: Thermodynamics, Formation and Vibrational Modes

Limit-periodic (LP) structures exhibit a type of nonperiodic order yet to be found in a natural material. A recent result in tiling theory, however, has shown that LP order can spontaneously emerge in a two-dimensional (2D) lattice model with nearest-and next-nearest-neighbor interactions. In this dissertation, we explore the question of what types of interactions can lead to a LP state and address the issue of whether the formation of a LP structure in experiments is possible. We study emergence of LP order in three-dimensional (3D) tiling models and bring the subject into the physical realm by investigating systems with realistic Hamiltonians and low energy LP states. Finally, we present studies of the vibrational modes of a simple LP ball and spring model whose results indicate that LP materials would exhibit novel physical properties.

A 2D lattice model defined on a triangular lattice with nearest- and next-nearest-neighbor interactions based on the Taylor-Socolar (TS) monotile is known to have a LP ground state. The system reaches that state during a slow quench through an infinite sequence of phase transitions. Surprisingly, even when the strength of the next-nearest-neighbor interactions is zero, in which case there is a large degenerate class of both crystalline and LP ground states, a slow quench yields the LP state. The first study in this dissertation introduces 3D models closely related to the 2D models that exhibit LP phases. The particular 3D models were designed such that next-nearest-neighbor interactions of the TS type are implemented using only nearest-neighbor interactions. For one of the 3D models, we show that the phase transitions are first order, with equilibrium structures that can be more complex than in the 2D case.

In the second study, we investigate systems with physical Hamiltonians based on one of the 2D tiling models with the goal of stimulating attempts to create a LP structure in experiments. We explore physically realizable particle designs while being mindful of particular features that may make the assembly of a LP structure in an experimental system difficult. Through Monte Carlo (MC) simulations, we have found that one particle design in particular is a promising template for a physical particle; a 2D system of identical disks with embedded dipoles is observed to undergo the series of phase transitions which leads to the LP state.

LP structures are well ordered but nonperiodic, and hence have nontrivial vibrational modes. In the third section of this dissertation, we study a ball and spring model with a LP pattern of spring stiffnesses and identify a set of extended modes with arbitrarily low participation ratios, a situation that appears to be unique to LP systems. The balls that oscillate with large amplitude in these modes live on periodic nets with arbitrarily large lattice constants. By studying periodic approximants to the LP structure, we present numerical evidence for the existence of such modes, and we give a heuristic explanation of their structure.

State Political Parties in American Politics: Innovation and Integration in the Party System

What role do state party organizations play in twenty-first century American politics? What is the nature of the relationship between the state and national party organizations in contemporary elections? These questions frame the three studies presented in this dissertation. More specifically, I examine the organizational development of the state party organizations and the strategic interactions and connections between the state and national party organizations in contemporary elections.

In the first empirical chapter, I argue that the Internet Age represents a significant transitional period for state party organizations. Using data collected from surveys of state party leaders, this chapter reevaluates and updates existing theories of party organizational strength and demonstrates the importance of new indicators of party technological capacity to our understanding of party organizational development in the early twenty-first century. In the second chapter, I ask whether the national parties utilize different strategies in deciding how to allocate resources to state parties through fund transfers and through the 50-state-strategy party-building programs that both the Democratic and Republican National Committees advertised during the 2010 elections. Analyzing data collected from my 2011 state party survey and party-fund-transfer data collected from the Federal Election Commission, I find that the national parties considered a combination of state and national electoral concerns in directing assistance to the state parties through their 50-state strategies, as opposed to the strict battleground-state strategy that explains party fund transfers. In my last chapter, I examine the relationships between platforms issued by Democratic and Republican state and national parties and the strategic considerations that explain why state platforms vary in their degree of similarity to the national platform. I analyze an extensive platform dataset, using cluster analysis and document similarity measures to compare platform content across the 1952 to 2014 period. The analysis shows that, as a group, Democratic and Republican state platforms exhibit greater intra-party homogeneity and inter-party heterogeneity starting in the early 1990s, and state-national platform similarity is higher in states that are key players in presidential elections, among other factors. Together, these three studies demonstrate the significance of the state party organizations and the state-national party partnership in contemporary politics.