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

**Term essays 2008**

*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: Guang Bian

Title: Realization of Bose-Einstein Condensation in dilute gases

Abstract: This essay discribes theoretical aspects of Bose-Einstein Condensation and the first experiment realization of BEC in dilute alkali gases. In addition, some recent experimental progress related to BEC are reported.

Author: Derek Vigil

Title: The Jamming Transition in Granular Materials

Abstract: We review the jamming transition in granular materials, especially as studied via simulation by Makse and O'Hern, and via experiment by Majmudar. In all three cases scaling behavior for pressure and mean contact excess with distance from the critical volume fraction is found. Makse finds a dependence of the critical volume fraction on the rate of approach to criticality, i.e compression(decompression) rate. O'Hern finds a dependence of the pressure critical exponent on the type of potential, but not on dimensionality, while all three cases find the mean contact excess exponent to be independent of both potential and dimension. Agreement is found between the experiments by Majmudar on granular systems composed of discs and the results of O'Hern for frictionless particles in a harmonic potential, indicating that the jamming transition is relatively insensitive to frictional effects. A comparison of the characteristics of the jamming transition and continuous phase transitions is made.

Name: Sarang Gopalakrishnan

Title: Dissipative Phase Transitions

Abstract: The transport properties of a quantum mechanical system coupled to an environment deviate sharply from those of the isolated system when the coupling exceeds a critical value. This effect is governed by a quantum phase transition, which takes place because the environment suppresses quantum fluctuations. This paper explains why dissipation leads to spontaneous symmetry breaking at zero temperature, and how dissipative couplings stabilize superconductivity in Josephson junction arrays and nanowires.

Author: Hefei Hu

Title: Phase transition in collective motion

Abstract: There has been a high interest in studying the collective behavior of organisms in recent years. When the density of living systems is increased, a phase transition from a disordered state into an ordered state occurs: i.e. units, which move in random directions below the transition, move together in the approximately same manner or direction. Several models and experiments are reviewed in this essay.

Author: Mehmet Oz

Title: Sznajd model and its application to politics

Abstract: Sznajd model of opinion formation in socities has found many applications, primarily in politics. Here, we review the basic model, show that its natural extension to two dimensions exhibits a phase transition, and then focus on its application to proportional elections where our model predicts the exponent for the power law distribution of number of votes obtained by different candidates in Brazilian proportional elections.

Author: ShengQuan Zhou

Title: Coulomb gas formulation of two dimensional Kosterlitz-Thouless phase transition

Abstract: The absence of conventional long range order in the classical two dimensional xy model is introduced. Two main types of excitations of this model, spin wave and vortices, are analyzed in the framework of generalized theory of elasticity. The Hamiltonian is then mapping onto the two dimensional Coulomb gas model and the critical temperature of the Kosterlitz-Thouless transition is located by a renormalization group technique.

Author: Seungmin Hong

Title: Aspects of Universality in Quantum Hall Effect

Abstract

In this essay, we intend to review aspects of universality and phase transition in the quantum Hall effect. In the perspective of scaling, we first notice the nature of the transition between two quantum Hall liquids, and briefly discuss theoretical understanding on it. In the fractional quantum Hall state, we review the theory of Laughlin's wavefunction, followed by the comment on universality class. In order to formulate this notion, an effective low energy field theory - the form of Landau-Ginzburg theory will be developed in terms of Chern-Simons gauge. It will then be shown that the most of known phenomenological features can be obtained from the mean field theory solution and the low energy fluctuation.

Author: Zhenyu Wang

Title: Renormalization in Complex Networks

Abstract

Complex networks are ubiquitous both in biology and sociology, and also widely constructed and exploited in science and technology. Common topological features are shared among various complex networks across disciplines. Renormalization of complex networks, yielding scaling laws and critical exponents, offers a new approach to category distinctive networks into universality classes, revealing similarities hard to capture in usually ways, whose analysis will help the understanding and facilitate improvement in different fields.

Author: Benjamin Hsu

Title: From Random Walks to Critical Phenomena and Conformal Field Theory

Abstract: In class, we approached critical phenomena from the view point of correlation functions and discussed renormalization group methods for obtaining values of critical exponents. Here I discuss a parallel idea that studies critical phenomena from properties of random curves which form the domain walls of the critical system. It will be shown that such random curves obey a stochastic differential equation which can be regarded as a conformal mapping. By studying the behavior of these conformal maps, such as the likelihood that a random walk connects two points or encompasses a certain point, one can make connections with quantities in conformal field theory. In addition, it will be shown that the critical exponent can be regarded as the diffusion constant for such random walks. This is a new field which has recently attracted the attention of theoretical physicists and many questions still remain. A general overview is presented here.

Author: Carlo C. Zuniga Zamalloa

Title: Optimal Channel Networks in Biology and Hydrology

Abstract:

A review on phase transitions in optimal channel networks -OCNs- is presented. The theory of OCNs is used to obtain scaling laws in biology (mass and metabolic rate). An analogy is then made between biological networks and rivers to find an equivalent scaling law to describe river morphology. Collapsed data from four different rivers is presented to test the accuracy of the scaling relation.

Author: David Lyon

Title: The Physics of the Riemann Zeta function

Abstract: One of the Clay Institute Millennium Prize Problems is the Riemann Hypothesis. Interest in this problem has led to collaboration between mathematicians and physicists to study the Riemann Zeta function and related classes of functions called Zeta functions and L-functions. A unified picture of these functions is emerging which combines insights from mathematics with those from many areas of physics such as thermodynamics, quantum mechanics, chaos and random matrices. In this paper, I will give an overview of the connections between the Riemann Hypothesis and Physics.

Name: Philip Powell

Title: Phase Transitions in Strongly-Interacting Matter and Cold Atomic Systems

Abstract: In this paper I discuss a number of qualitative similarities between strongly-interacting nuclear matter and cold atomic systems including a strong-to-weak coupling phase transition and nearly ideal hydrodynamics in the unitarity limit. I also address ways in which the fields may inform one another.

Author: Bing Lu

Title: Phase Transition in Nematic Elastomers

Abstract: Two kinds of phase transitions arising in nematic elastomers are discussed. It is also explained how the two different kinds of physics, one belonging to nematic liquid crystals, the other to nonlinear elasticity, mutually influence each other, resulting in the unexpected smoothening of the isotropic-nematic transition from first to second order, and the appearance of soft elasticity.

Author: Yiruo Lin

Title: Critical phenomena in gravitational collapse

Abstract:
I briefly review the critical phenomena in gravitational collapse with emphases on connections to critical phase transitions.

Author: Nir Friedman

Title: The Random Field Ising Model at Zero Temperature.

Abstract: Abstract Barkhausen noise and hysteresis are explained in detail, and requires that must be met to model it successfully. The random field Ising model is constructed using physical ideas. The dynamics of the model are discussed, as well as qualitative features and the reasoning behind why a critical point is expected. Theoretical methods, such as the epsilon expansion as well as scaling techniques are briefly discussed, as is the computation treatment of the random field Ising model. The results of theory, numerical simulations, and experiments are all compared. The usage of random field Ising model for systems that display power law scaling beyond mere magnetic materials is briefly demonstrated with an example from earthquake dynamics. Future directions for study are suggested at the end.

Author: Jitong Yu

Title: Phase Diagram of a Polarized Fermi Gas

Abstract: The polarized Fermi gas, which has recently been realized in ultra-cold atomic gases where the number of atoms in the two spin states is different, has attracted many experimental and theoretical works. This paper introduces the phase diagram of the polarized Fermi gas at T = 0 and at finite temperature as well. Open questions that haven't been solved are pointed out and available experiments are presented.

Author: Lyudmila Kushnir

Title: Self-Organization.

Abstract:
This essay addresses the question of self-organization in open systems, i.e. a process in which the internal organization of a system, increases in complexity without being guided or managed by an outside source. First some general issues about possibility of self-organization are discussed, then two examples are considered. The first example, reaction-diffusion model, is related to bio-chemistry, and the second one - to population dynamics.

Author: Stanimir Kondov

Title: Bose-Hubbard Model

Abstract:
The Bose-Hubbard hamiltonian will be introduced and justified as an effective description of some relevant physical systems (dilute alkali gases in optical lattices, arrays of Josephson junctions). I will also show a mean field theory solution and the resulting phase diagram. Finally, I will describe in detail one of the more important experiments with an optical lattice of 87Rb atoms.

Author: Young Il Joe

Title: CDW phase of TiSe_2

Abstract:
TiSe_2 undergoes a commensurate structural phase transition. This essay has some of experimental works supporting each of several theories have been proposed to explain the CDW phase formation.

Author: Wade Gottardi

Title:
One-dimensional quantum wires

Abstract: In this paper we investigate the properties of a one-dimensional quantum wire of interacting electrons in the Wigner crystal limit. Recent theoretical work has explored some of the classical and quantum phase transitions associated with this system. For example, there exists a critical density above which the system can lower its energy by forming a quasi-one-dimensional zig-zag chain. We also discuss the potential relevance of such behavior to the so-called $0.7$-structure.

Author: Weicheng Lv

Title: Deconfinement Phase Transition in QCD

Abstract: At low energy, the interactions between quarks are so strong that they have to form a bound state because of color confinement. However, at high energy, due to asymptotic freedom, the effective coupling becomes small. Quarks will be deconfined, leading to a state of quark-gluon plasma. This is the so-called deconfinement phase transition. Starting with some background, we will describe the basic physics behind this phase transition. Then lattice QCD is introduced to give us more quantitative results. Possible connections with condensed matter physics will also be considered.

Author: Mingwu Lu

Title: Quantum phase transition from a Mott insulator to a superfluid in bosons

Abstract: Bose Hubbard model is presented and basic natures of Mott insulating phase and superfluid phase are studied in this essay. Also how and when this quantum phase transition occurs is discussed. Experimental supports from ultracold atoms physics are explained, while some miscellaneous topics are touched in the end.

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