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.

The Land of the Savior: Óscar Romero and the Reform of Agriculture

This study approaches Óscar Romero by attending to his intimate involvement in and concern for the problematic surrounding the reform of Salvadoran agriculture and the conflict over property and possession underlying it. In this study, I situate Romero in relation to the concentration of landholding and the production of landlessness in El Salvador over the course of the twentieth century, and I examine his participation in the longstanding societal and ecclesial debate about agrarian reform provoked by these realities. I try to show how close attention to agrarian reform and what was at stake in it can illumine not only the conflict that occasioned Romero’s martyrdom but the meaning of the martyrdom itself.

Understanding Romero’s involvement in the debate about agrarian reform requires sustained attention to how it takes its bearings from the line of thinking about property and possession for which Pope Leo XIII’s 1891 encyclical Rerum novarum stands as a new beginning. The enclyclical tradition developing out of Leo’s pontificate is commonly referred to as Catholic social doctrine or Catholic social teaching. Romero’s and the Church’s participation in the debate about agrarian reform in El Salvador is unintelligible apart from it.

What Romero and the encyclical tradition share, I argue, is an understanding of creation as a common gift, from which follows a distinctive construal of property and the demands of justice with respect to possessing it. On this view, property does not name, as it is often taken to mean, the enclosure of what is common for the exclusive use of its possessors—something to be held by them over and against others. Rather, property and everything related to its holding derive from the claim that creation is a gift given to human creatures in common. The acknowledgement of creation as a common gift gives rise to what I describe in this study as a politics of common use, of which agrarian reform is one expression.

In Romero’s El Salvador, those who took the truth of creation as common gift seriously—those who spoke out against or opposed the ubiquity of the concentration of land and who clamored for agrarian reform so that the landless and land-poor could have access to land to cultivate for subsistence—suffered greatly as a consequence. I argue that, among other things, their suffering shows how, under the conditions of sin and violence, those who work to ensure that others have access to what is theirs in justice often risk laying down their lives in charity. In other words, they witness to the way that God’s work to restore creation has a cruciform shape. Therefore, while the advocacy for agrarian reform begins with the understanding of creation as common gift, the testimony to this truth in word and in deed points to the telos of the gift and the common life in the crucified and risen Lord in which it participates