Modulated phases in statistical models with chiral interactions - Castilho, William de (2024)
In this study, we introduce a series of models aimed at elucidating the phenomena of modulated structures and the transitions that give rise to these phases. Modulated phases represent a complex behavior observed in a diverse array of materials, spanning from rigid metals to soft matter like liquid crystals. The methodologies employed in this work are firmly rooted in the foundational principles of statistical physics. Employing a probabilistic framework, we explore various types of graph structures and strategies for representing the states of the particles within the proposed systems. Throughout the text, our primary focus lies on discrete state models, which offer a simplified yet faithful representation of the underlying phenomenology. With these considerations in mind, we investigate phase transitions and characterize modulated phases across a range of problems, shedding light on their intricate dynamics and properties. The primary analysis method used is the Cayley tree, a self-generated graph. We also explored renormalization group techniques and, for some models, more traditional mean-field methods. The work primarily focuses on liquid crystals but presents two magnetic models studied during the period: a spherical model with competitive interactions and a variation of the Ising model with two modes. In liquid crystals, we concentrated on describing cholesterics.