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Fradkin, Eduardo H. Cannon, Joel W. 2011-05-07T12:13:47Z 2011-05-07T12:13:47Z 10000-01-01 1989 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.) Made available in DSpace on 2011-05-07T12:13:47Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9010815.pdf: 4236012 bytes, checksum: b927b7b1d616c5f6aaf1bbde11b62ffe (MD5) Previous issue date: 1989 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:38:25Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:16:01-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9010815 (UMI)AAI9010815 http://hdl.handle.net/2142/19638 eng Copyright 1989 Cannon, Joel W. Physics, Condensed Matter Numerical studies of phase transitions and critical phenomena in fermionic and polymeric systems text Physics Physics University of Illinois at Urbana-Champaign Dissertation Ph.D.