Physics 563: Phase Transitions and the Renormalization Group
Term essays 2013
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 : Vatsal Dwivedi
Title : Renormalization group for Interacting Fermions
The renormalization group approach of integrating out degrees of freedom successively has been crucial in understanding the critical phenomena. For bosonic case, this integration is done over momentum shells in the Wilsonian RG, but things are more complicated for fermions as their ground state consists of a Fermi surface around which the integrals are to be performed, as opposed to around a point for the bosonic case. In this essay, we study the stability of a nonrelativistic fermionic system to interaction within a renormalization group framework, as discussed by R Shankar. The basic approach of RG here is analogous to that of integrating out on a momentum shells in a scalar field theory with a \phi^4 interaction. The application of RG in 2 or 3 dimensions leads to Landau-Fermi liquid theory, with only relevant operators being those of BCS type (Cooper pair instability).
Author: Ye Zhuang
Title: Critical Phenomena of Mott transition
Due to interaction between electrons, half-filled band materials can be insulators at finite temperature. The metal-insulator transition in such system is known as Mott transition. The order parameter for Mott transition is believed in the Ising universality class which has been verified both theoretically and experimentally. However, recent experiments find that Mott transitions in 2D system may belong to other universality class. In this essay, we discuss the Mott transition in Ising universality class and the possibility of it being in different universality classes.
In this paper I look at the BKT transition in the 2D XY model, first qualitatively, then using renormalization group techniques, and then finally comparing some recent experiments.
The renormalization group flow of most systems is characterized by attractive or repelling fixed points. Nevertheless, some systems can trace a different trajectory in coupling constant space corresponding to limit cycles or chaotic flow. I will focus on these types of non-conventional behaviours in the one-dimensional Ising model with complex coupling constants, and Efimov states. We map the regions of chaotic, normal, and point out the limit cycle flow spots for these types of systems.
In complex systems the degree of homogeneity (vs. heterogeneity) and connectivity (vs. modularity) determines whether or not there is a phase transition from one state to another. These are called critical transitions, and there are current efforts to understand both what factors are significant in causing these transitions and what factors are significant in predicting the fragility of these systems, or the susceptibility to the induction of a phase transition by some external shock. Complex phenomena in a wide range of fields can be studied using these ideas combined with the idea of critical slowing down. Approaches to complex systems in several examples will be discussed, with a focus on living systems.
Author: Juan Alberto Garcia
Title: Phases of the Bilinear-biquadratic Spin-1 Chains
Measurements done on LiVGe2O6 by Millet et al. [Phys. Rev. Lett. 83, 4176 (1999)] on phases of the Singlet ground state of the Spin-1 Chain can be described by the Bilinear-biquadratic spin-1 chain. I will discuss these results, among the results of others and talk about theoretical calculations done by A. Lauchli [Phys. Rev. B 74, 144426 (2006)] that describe the phase diagram of the bilinear-biquadratic model. At a particular point in the phase diagram the model becomes SU(3) globally symmetric and is known as the Lai-Sutherland model, it is Bethe ansatz solvable as shown in [A Schmitt et al., J. Phys. A: Math. Gen. 29 (1996) 3951-3962]. Finally I will describe the thermo- dynamics of the model as where calculated by Lou et al. [Phys. Rev. Lett. 85, 11 (2000)].
Author: Suraj Hegde
Title: Critical phenomena in stationary black hole spacetimes
This essay aims to be a study in the phase transitions and critical behaviour aspects of sta- tionary black hole spacetimes like Kerr-Newman Black Holes.Though there are divergences in the thermodynamic susceptibilities of the black holes,which might signal a phase transition,there are are certain intricacies involved in its interpretation.Using the thermodynamic fluctuation theory it is seen that divergences at a continuous turning point of the thermodynamic function need not be a critical point but only indicates a change in stability.This gives us insight into the understanding the relation between fluctuations and critical phenomena.It is found that the extremal phase of the Kerr-Newman black hole corresponds to a critical point of a continuous phase transition.Once can also obtain the critical exponents and scaling laws at that point. Also the formalism of fluctuation theory can be extended to give a geometric interpretation of some of the critical phenomena. In the end,just a mention is made of a recent development in considering the entanglement entropy of the black hole under the framework of Renormalisation Group.
Author: Isaac Brodsky
Title: Percolation theory
This essay describes percolation theory. Once percolation theory is defined, we explore applications to the renormalization group, computer simulations of potts models, and randomly punctured conducting sheets.
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