Physics 563: Phase Transitions and the Renormalization Group

Term essays 2017

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: Dmytro Bandak

Title: Quantum Phase Transitions as Exemplified by Heavy Fermionic Materials


In this term paper I discuss what is meant by a quantum phase transition, as well as its similarities and differences with a conventional thermal phase transition. In order to make the subject concrete the quantum critical point is considered for a model of metallic magnetism and its implications on the phase diagram discussed. To demonstrate the relevance of quantum phase transition to real materials, experimental data on scaling in heavy fermionic materials is presented and universality in this data discussed.

Author: Bikash Padhi

Title: Black Hole Thermodynamics and the Hawking-Page Phase Transition


We briefly review general relativity, present some basic ideas of AdS/CFT correspondence, thereby motivating the importance of anti de Sitter (AdS) spacetime. We show how to compute thermodynamic quantities such as temperate, free energy for a given background geometry. By computing free energy, we show in AdS space, if a black hole is smaller than a critical size then it becomes unstable and evaporates into thermal radiation. This first order phase transition is driven by breaking of conformal symmetry of AdS.

Author: Brian Busemeyer

Title: Critical Behavior in Cloud Formation


The connection between environmental changes and cloud formations remains an important, but poorly understood, component of climate research. Much of the difficulty in describing cloud formation lies in the multi-scale character of the underlying features of cloud formation. Recently, both observations of cloud formation via satellite and detailed simulations of cloud dynamics have suggested the transition between clear and cloudy skies carries signatures of a phase transition. These signatures include universal power law scaling, connections to predator-prey dynamics, and singularities in the response to external perturbations. I explore three investigations into these discoveries and discuss the implications of their assumptions and their results

Author: Purba Chatterjee

Title: The Flocking Transition: A Review of the Vicsek Model


This essay reviews important results from studies on the Vicsek model, which describes a flocking phase transition in self-propelled particles as a function of particle density or noise. The nature of the transition to collective motion is investigated, and the phase diagram in noise-density space is analyzed. It is shown that the flocking transition, which was originally considered to be continuous by Vicsek and collaborators, is actually discontinuous and fluctuation driven. The likeness of the flocking transition in the Vicsek model to a liquid-gas phase transition is discussed, and a qualitative study of the phase coexistence regime is performed. Lastly, the Vicsek model is compared to an active Ising model which shows a similar flocking transition, albeit with a different phase coexistence profile. The reason for the difference in the phase coexistence profiles of the two models is discussed and shown to be stochasticity driven.

Author: Eli Chertkov

Title: Phase transitions in random satisfiability problems


Boolean satisfiability is a difficult computational problem that is actively studied due to its many practical applications and its deep connections to computational complexity theory. In this essay, we demonstrate how techniques from the study of phase transitions can be used to better understand random instances of boolean satisfiability problems. We discuss the relation of satisfiability problems to Ising models and spin glasses and discuss how finite-size scaling can be applied to study satisfiable- unsatisfiable transitions in random instances of the k-satisfiability problem. We per- form numerical calculations using a modern satisfiability problem solving algorithm to demonstrate the finite-size scaling approach in detail. We numerically show that these transitions appear continuous and discontinuous in the order parameter for k = 2 and k >= 3 respectively. We also empirically demonstrate that problem instances near the critical region of the phase transition are the most difficult to solve, which is useful for benchmarking satisfiability algorithms.

Author: Chong Han

Title: Dewetting


A substrate that is covered by a film of liquid could dewet if system parameters change. Depending on the surface and liquid, dewetting could result in different patterns. The formation of dry patches starts with nucleation which could be caused by thermal noise, defects or spontaneous capillary waves. The conditions for each pattern has been examined and experimental results confirm theoretical predictions. The dynamics of hole formation is also studied and compared to hydrodynamics theory.

Author: Jahan Claes

Title: Phase transitions in social networks


In both evolution and economics, populations sometimes cooperate in ways that do not benefit the individual, and sometimes fail to cooperate in ways that would benefit the population. One method to explain the success or failure of cooperation is the social network model, in which simplified two-person interactions take place over a network of social connections. In this paper, I explore some of the experimental evidence of cooperation in microorganisms and humans. Then, I explain the theoretical models used to predict when cooperation will occur, with particular attention to the prediction of phase transitions.

Author: Souvik Datta

Title: The Exact Renormalization Group


Various aspects of the Exact Renormalization Group (ERG) are explored. In part I, we start with a review of the concepts underlying the framework, paying special attention to developing an intuitive picture of rescalings in an ERG algorithm (part II). We proceed to uncover Polchinski�s ERG equation (and its related cousins) in part III and in the process, obtain an interpretation of a continuum blocking function. In part IV, we attempt at solving the flow equation using non-perturbative truncations and encounter the ever-so-illuminating hurdles of handling such calculations. We conclude with hunting for fixed points (some really cool techniques!) for ERGEs which don't have analytic solutions.

Author: Oleg Dubinkin

Title: Renormalization group and sine-Gordon model


This essay describes the renormalization group approach to a 1D sine- Gordon model. This model is shown to be dual on the RG level to 2D X- Y spin model and exhibits the same Kosterlitz-Thouless transition near the critical point. We also discuss what role sine-Gordon model plays in studying boundaries of 2D topological materials, discuss its RG properties, and show how KT phase diagram analysis could possibly allow us to detect topological phases of matter experimentally.

Author: Davneet Kaur

Title: Collective behavior of many species ecosystems


Ecoevolutionary dynamics have long been of interest. However, our understanding of diverse systems has largely been based off of induction of our understanding of simpler systems with few species. This approach fails to reveal many important emergent phenomena. In this paper, I will first discuss two complementary ways of breaking down and understanding diverse complex systems. That is, analyzing the system in terms of regimes in phase space and in terms of the MacArthur Competition Equations. Following this discussion, I will briefly outline two recent works. In the first paper, the authors analytically solve the MacArthur Model in the high-diversity limit and reveals the presence of a phase where the immediate environment of individuals becomes fully decoupled from the outside world. In the second paper, the authors propose and experimentally test an assembly rule for predicting survival in multi-species competitions from the outcomes of pairwise competitions. I will conclude with a summary of the material and some closing thoughts.

Author: JiYoung Kim

Title: Holographic Wilsonian Renormalization Group


Strongly coupled systems are difficult to study because the perturbation of the systems does not work with strong couplings. However, the gauge/gravity duality and the ultraviolet-infrared connection enable to study the low energy regime of the gravity. The fundamentals of holographic Wilsonian renormalization is introduced with the example of free massive scalar fields. Finally, the crucial point of the holographic Wilsonian renormalization group and further questions will be discussed compared to the previous renormalization group methods.

Author: Nicholas Laracuente

Title: Phase transitions in the polynomial hierarchy


This essay describes a subset of the theories and computational experiments relating complexity theory to phase transitions surrounding the famous P vs. NP problem. Primary areas of focus are: 1) the meaning and physical significance of robust asymptotic complexity theory 2) observations and theories of phase transitions in complexity 3) past attempts to characterize the class of problems known collectively as boolean satisfiability.

Author: Zhiru Liu

Title: Machine learning and the renormalization group


In this essay, we reviewed the recent attempts on relating Machine Learning to Renormalization Group. Restricted Boltzmann Machines, a type of neural network, was shown to be connected to Variational Renormalization Group. A treatment of Principal Component Analysis analogous to momentum shell Renormalization Group uncovered a possible fixed point.

Author: Luo Di

Title: Machine learning, renormalization group and phase transitions


In this essay, we review recent research on the interaction between machine learning, renormalization group and phase transition. There are two important questions to be ad- dressed. The first one is how to understand and improve machine learning algorithms from the perspective of physics while the second one is how to apply machine learning to study physics problems. For the first direction, an attempt to construct a mapping between variational renormalization group and deep learning is presented. For the second direction, an unsupervised learning algorithm has been demonstrated to study the 2D Ising Model phase transition. Discussions on other possibilities between machine learning and physics have also been included.

Author: Alan M. Luu

Title: Phase transitions in artificial intelligence


Artificial intelligence often involves search and computation over large networks. This essay discusses a family of closely related papers on the application of ideas from the physics of phase transitions to artificial intelligence algorithms, specifically, heuristic search and activation spreading nets. The authors of these papers build theoretical models of these network algorithms with parameters associated with general properties of the network. Additionally, they run computational simulations to show that these algorithms display divergent behavior characteristic of phase transitions for certain values of the parameters.


Author: William Morong

Title: Renormalization flow of the Anderson transition

Abstract: I introduce the Anderson localization problem, and discuss the early scaling theory of localization as well as some subsequent refinements involving RG. I discuss the comparison to experiment and simulation and some of the challenges in getting a clear experimental measurement of the localization transition.

Author: Michael O'Boyle

Title: Non-equilibrium scaling in stellar phenomena


Many non-equilibrium systems show power law behavior in some way, whether it is noise, durations of avalanche events, or energy spectra. Inspired by the success of scaling laws in continuous phase transitions, self-organized criticality postulates that this power law behavior results from proximity to a kind of critical point. Initially appealing for its simplicity, experiments have brought its plausibility into question. Renormalization group methods inspired by those developed for continuous phase transitions have emerged as more successful alternative. This paper compares these two approaches to non-equilibrium scaling by considering their application to astrophysical phenomena, specifically solar flares and dimming events in the star KIC 8462852.

Author: Brendan Rhyno

Title: Two-impurity Kondo Physics


In this essay we review the two-impurity Kondo problem. Our goal is to proceed in some pedagogical manner; hence, we start with a discussion of the (single-impurity) Kondo problem. We will discuss the fixed points of the model and motivate the Anderson model as a natural generalization of this system. Upon introducing a second magnetic impurity, we will discuss the Ruderman-Kittel-Kasuya-Yosida (RKKY) in- teraction. Finally, we will discuss recent experimental efforts made to explore the phase diagram of the two-impurity Kondo model.

Author: Astha Sethi

Title: Role of incommensuration in the CDW and superconducting states of 1T-TiSe 2


A brief review of some of the most recent experiments on the charge density wave (CDW) transition in 1T-TiSe 2 is presented. With increasing pressure or intercalation, the CDW can be suppressed and a superconducting (SC) phase emerges. In the pristine 1T-TiSe 2 , a commensurate CDW develops around 202 K. These experiments show that by tuning these external parameters like pressure, or doping, the commensurate CDW melts into an incommensurate CDW phase through the formation of domain walls. The incipient incommensurate CDW phase coincides with the onset of superconductivity, thus providing insight into the importance of incommensurability in formation of SC phase.

Author: Li Shu

Title: Barkhausen noise in magnetic systems


Barkhausen noise has attracted growing attentions as an example of complex disordered systems. The renormalization group approach for Barkhausen noise has been developed for the past decades to understand the critical phenomena and applied to other similar systems such as deformation behaviors for solids under tensile stress. In this paper, experimental results and theoretical analysis in the field of Barkhausen noise within a renormalization group framework are reviewed. The discussion of results and future work regarding potential application to other systems are also studied.

Author: Archit Vasan

Title: Identifying critical behavior in viral capsid assembly


This essay reviews key aspects of physical virology as they apply to phase transitions in assembly mechanisms of icosahedral viral capsids. Viral architecture is first briefly reviewed with a description of the CK selection rules and some exceptions to them. The assembly mechanisms are then analyzed using empty capsids, loaded capsids and under varying ion and pH conditions.

Author: Benjamin Villalonga Correa

Title: Strong disorder RG for the many-body localized phase


The strong disorder renormalization group offers a theoretical approach to the study of the dynamical properties of local, disordered lattice models. It has been recently applied to the problem of many-body localization, where it provides compelling results that are in agreement with the exotic properties found in these systems through numerical studies.

Author: Xiangyu Song

Title: The spreading of epidemics in complex environments


The spreading of epidemics in complex networks has been extensively studied in the last few decades. Depending on the nature of the disease and the network it spreads on, there are different critical behaviors. In this essay I provide a brief introduction to the study of complex networks and the susceptible-infectious- susceptible(SIS) epidemic model. I discuss the different critical behaviors when SIS model is applied to exponential network and scale-free network. The exis- tence of non-zero epidemic threshold in exponential networks and the lack of such threshold in scale-freee networks can help understanding computer virus epidemics.

Author: Yubo (Paul) Yang

Title: Towards an understanding of living creatures


This paper investigates the surprising prevalence of near-criticality in nature. The motivation for this paper stems from experimental observations of apparent criticality in living systems. Namely, the statistical models designed to describe natural collective behavior are often tuned to the critical point of the underlying model. I will present and discuss the experimental evidence that supports the above claim.

Author: Jimmy Yuan

Title: Berezinskii-Kosterlitz-Thouless transition in a trapped Rubidium gas


The BKT transition is a topological phase transition that arises naturally from the XY spin model, a common toy model that describes a variety of 2D systems. We derive the behavior of the two-point correlation function via low-temperature expansions, and we explore the critical behavior using the renormalization group. It can be shown that that above the critical temperature for the BKT transition, but not too much higher, defects in the form of topologically-charged vortex pairs spontaneously are created, giving rise to quasi-long-range order. Experimental evidence has been found for this in an optically trapped rubidium gas.


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