**Physics 563: Phase Transitions and the Renormalization Group**

**Term essays 2012**

*These essays were written by students taking Physics 563 Phase Transitions and the Renormalization Group at the University of Illinois at Urbana-Champaign. The copyright of each essay is due to the author. *

*Please acknowledge the essay title, author, and this course in any citation to these articles. *

*The information, opinions and interpretations expressed are those of the authors, not necessarily those of the instructor. *

Author: Andrew Blanchard

Title: Simple Models of Protein Folding

Abstract: Lattice models with simplified interaction potentials have been used to analyze the ability of certain amino acid sequences to adopt a unique configuration in space [1, 2, 3]. Furthermore, phenomenological models have been used to predict protein folding kinetics amongst a subset of energetically favorable states [4, 5]. In the following, I will specifically discuss both the use of two dimensional lattice models and simple rate matrices to describe the transition of disordered proteins to a unique native state (or subset of states). Furthermore, I will discuss the use of both molecular dynamics simulations and experimental techniques to observe specific pathways for protein folding and provide direct connections to theory.

Name: Wenchao Xu

Title: Phase Transition in Two-dimensional Bose gas

Abstract: Dimensionality of system plays a vital role in physics. Recent developments in ultracold gases provide the possibilities to explore low-dimensional region. In this paper, I will focus on two kinds of phase transitions in two-dimensional Bose gases: one is the Berezinskii-Kosterlitz-Thouless (BKT) phase transition; whose disordered state is characterized by the proliferation of topological vortices pairs; the other is the normal- to-super?fluid transition occurred in optical lattice at the zero-temperature limit, the physics of which can be described by Bose-Hubbard Model. I will present theoretical approach to these two problems, and then discuss recent experimental results which provide a clear demonstration for the critical phenomena.

Author: Jiawen Liu

Title: Dynamic scaling analysis of Binary fluid model

Abstract

This paper examines the dynamic scaling behavior of the binary fluid model, and hence show that the static scaling hypothesis breaks down. Mode-coupling method is examined in detail. By following Kandanoff and Swift's method, we show that the transport coefficients diverge near the vicinity of the critical region. We also find that the scaling law of the transport coefficients, $\eta\lambda^*\propto\xi^{4-d-\eta}$.

Author: Kiaran Dave

Title: Critical Behavior of Viscosity at the Liquid-Gas Critical Point of Xe

Abstract

A brief survey is given of the theoretical calculation of the critical behavior of the shear viscosity at the liquid-gas critical point, as well as experimental attempts to measure the critical exponent.

Author: Wladimir Benalcazar

Title: Phase transitions in lipid bilayers

Abstract

Despite the simplicity in their structure, lipid bilayers can arrange in a rich variety of phases, including rippled phases that exhibit supramolecular periodic structures. A number of theoretical models have been produced to characterize these phases over the last three decades, some of which show consistency with x-ray diffraction data. Here I describe a phenomenological Landau free energy theory of lipid bilayers, whose phases show reasonable agreement with experiments, and which predicts the existence of some novel phases, unobserved so far (to the best of my knowledge).

Author: Vikyath Deviprasad Rao

Title: Critical Phenomena in Complex Networks

Abstract:

Networks arise in wide variety of contexts across disciplines, from social networks in sociology and transport networks in engineering, to networks for disease spreading in epidemiology and metabolic networks in biology. In this essay we provide an introduction to the systematic study of networks. We discuss a few important network models, focusing particularly on features that are similar to those that arise in statistical physics. We also discuss the application of renormalization group methods to examine the critical properties of a simple network model.

Author: Xiaoxiao Wang

Title: Physical Examples of Phase Transition in One-Dimensional Systems with Short Range interaction

Abstract

In this paper, we study the thermodynamic phase transition in one dimensional systems with short range interaction. We begin by reviewing some famous non-existence result, such as Landau's argument and van Hove's theorem. And we point out that the Perron-Frobenius theorem is generally used to determine whether there is phase transition in such kind of system. Then several examples that Perron-Frobenius theorem doesn't apply and exhibit phase transitions will be present.

Name: Thomas Tuegel

Title: Phase Transitions in Complex Networks

Abstract:

Recent technological developments have enabled the creation and study of complex networks in information technology, biology, and other areas, renewing interest in the study of this field. The basic theory of networks is outlined, with the goal of describing the structure of various networks. Phase transitions in network structures themselves, such as the emergence of the giant connected component and the condensation of edges, are described. Methods used to study simple models on networks are described. The resulting descriptions of phase transitions and critical behavior in the Ising model and simple epidemiological models are reported.

Author: L. Ross

Title: Topological Aspects of Phase Transitions

Abstract:

In the past decade and a half, an alternative analysis of phase transitions, based in the topological behavior of system configuration spaces, has been introduced. This method offers a new understanding of many phase transitions, particularly those in finite systems, using topology and manifold behavior. While the broadest hypotheses offered by researchers have proven overly optimistic, the methods proposed have proven useful in many cases and offer some intriguing insights into the origins of phase transitions.

Author: Brandon Langley

Title: Phase Transitions Methodology Applied to Localization

Abstract

This paper is intended to explore localization transitions, phenomena where system states become localized due to the presence of disorder. The paper will divulge various results and ideas from the application of traditional phase transitions methods, including the dimensionality dependence from the renormalization group in the metal-insulator transition and the application and subsequent modification to Goldstone's theorem.

Author: Arka Banerjee

Title: Self-organized criticality in sandpile models

Abstract: The sandpile model, introduced in 1987, was the first model to exhibit self-organized critical behavior, that is, the system moved towards its critical point without the need to tune any adjustable external parameter. In this paper, we look at the why these models exhibit such non-intuitive behavior. We also look at some of the phenomenology near the critical point, such as scaling laws and critical exponents. Finally, we look at some experimental realizations of the sandpile model.

Author: Seyyed Mohammad Hassan Halataei

Title : Quantum Phase Transitions

Abstract:

This essay describes what a quantum phase transition is and one way of proving its very existence for an Ising Model. Quantum-classical mapping is discussed and showed that quantum problem in d spatial dimension can be reduced to classical problem in (d+z) effective dimension. The existence of an exact solution for one dimensional quantum Ising model is cited and it is compared to the previous estimates for density of kinks.

Author: K. Michael Martini

Title: Limit Cycles in the Renormalization Group

Abstract:

This paper will review the consequences of having limit cycles in the renormalization group. Characteristics such as discrete symmetry invariance and log periodic behavior of observables are expected in systems that exhibit limit cycles in their renormalization group. Some realizations of limit cycles will be explained. Specifically limit cycles in RG flows will be used to explain the Efimov effect.

Author: Anirbit
Mukherjee

Title: The Landau-Ginzburg/Calabi-Yau Phase Transition

Abstract

In this article we shall explain how the Abelian ${\cal N}=2$ twisted supergauge theory in $1+1$ dimensions shows phase-transition in the very precise sense of the moduli space of the theory changing discontinuously as a function of the Fayet-Illiopoulous parameter ($r$) and the topological $\theta$-term. This model allows for exact evaluation of renormalization effect on the critical point. It also gives a very analytically controlled scenario of seeing the two most important features of the liquid-vapour transition that the two phases have the ``same" symmetries (here in a very precise sense) and there being the possibility of going around the critical point and thus transiting between the phases without encountering any singularity. The analogy is so compelling that the discoverer of this profound effect, Edward Witten, on page 29 of his revolutionary paper \cite{EW1} comments, \emph{``..like liquid and gas, Calabi-Yau and Landau-Ginzburg look like different phases.."

Author: Huihuo Zheng

Title: Entanglement in quantum phase transition

Abstract:

A quantum phase transition (QPT) happens when the zero-temperature quantum fluctuations in a quantum many-particle system cause a transition from one type of ground state to another. Such transitions are induced by the change of a physical parameter. Long-range correlations in the ground state also develop at quantum critical regime. It turns out the property responsible for the long-range correlations is entanglement. The system state is strongly entangled at the critical point. Therefore we expect that systems near quantum critical points can be characterized in terms of entanglement. In this paper, we studied the use of entanglement measures for identifying and characterizing quantum phase transitions, and examined the scaling behavior of entanglement near quantum critical point, and finally studied the renormalization of entanglement under real space renormalization.

Author: Hong-Yan Shih

Title: Robustness of interacting networks

Abstract:

In the real world the systems are composed of networks which are coupled together. The analytic works studying the robustness of a system of interdependent and interconnected networks under failures or attacks are reviewed. Through the generating formalism for percolation process, in specific analytic models, the final fraction of functional nodes in the networks is found to have unusual transition between first and second order phase transitions as a function of the number of interdependent networks, the initial fraction of the remaining nodes and the dependency of couplings between interacting networks.