University of Illinois Urbana-Champaign

Numerical studies of phase transitions and critical phenomena in fermionic and polymeric systems

Cannon, Joel W.

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https://hdl.handle.net/2142/19638

Description

Title
Numerical studies of phase transitions and critical phenomena in fermionic and polymeric systems
Author(s)
Cannon, Joel W.
Issue Date
1989
Doctoral Committee Chair(s)
Fradkin, Eduardo H.
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Physics, Condensed Matter
Language
eng
Abstract
  • In this thesis we study three problems in phase transitions and critical phenomena.
  • We study the phase diagram of the Hubbard model using monte-carlo simulation, and a novel application of the Lanczos algorithm. We demonstrate the presence of a tri-critical point, and estimate the phase boundary to occur at roughly V = 2.92 $\pm$.01 at U = 5.5, and V = 1.65 $\pm$.05 at U = 3.0. We estimate the tri-critical point to occur above U = 3.0.
  • "We study the finite-sized dependence of the energy of the ground and first excited states for various values of U to observe the crossover of the conformal anomaly, c, from free fermion to Heisenberg behavior. We have inferred c along this line using the finite-sized dependence of the ground state energy. We obtain the correct asymptotic values, but find a maximum in the value of c at approximately U = 2 in apparent contradiction to Zamolodchikov's ""c theorem"". We show that this behavior is explained by postulating that our trajectory slices the RG trajectory in such a way that we retrace the irrelevant operators back on their trajectories for a short distance, and the presence of these operators invalidates the equation$$E\sb0(N) = E\sb{0}(\infty) - {\pi cv\sb{f}\over 6}{1\over N\sp2}\eqno(0.1)$$"
  • We calculate the probability distribution functions for linear, 3-, and 4-star polymers and find that the configurations become less prolate and more anisotropic as the number of rays increase. We find that the parameterization developed by Aronovitz and Nelson is a useful and consistent way of characterizing polymer shape.
  • I have for the first time used ground state eigenvectors, calculated with the Lanczos algorithm, to define probability distribution functions for the order parameter of a system.
  • Algorithmic improvements include: (i) development of an efficient data structure for performing monte-carlo simulations on discrete systems; (ii) A method to obtain the ground state eigenvector via the Lanczos method with a small increase in required memory; (iii) A data structure to allow efficient monte-carlo simulation of a lattice polymer; and (iv) An algorithm for simulation of cross-linked polymers. In addition, we obtain numerical solutions for the ground state of the finite size Bethe ansatz equations for the Hubbard model. (Abstract shortened with permission of author.)
Type of Resource
text
Permalink
http://hdl.handle.net/2142/19638
Copyright and License Information
Copyright 1989 Cannon, Joel W.

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