Theses and Dissertations

Xingjun Zhang

Issuing Body

Mississippi State University

Advisor

Novotny, Mark A.

Committee Member

Fowler James, E.

Committee Member

Winton, Raymond S.

Committee Member

Clay, Rudolf Torsten

Committee Member

Gwaltney, Steven

Date of Degree

12-10-2005

Document Type

Dissertation - Open Access

Major

Engineering Physics

Degree Name

Doctor of Philosophy

College

James Worth Bagley College of Engineering

Department

Department of Physics and Astronomy

Abstract

In this dissertation, the critical scaling behavior of magnetic Ising models with long range interactions is studied. These long range interactions, when imposed in addition to interactions on a regular lattice, lead to small-world graphs. By using large-scale Monte Carlo simulations, together with finite-size scaling, the critical behavior of a number of different models is obtained. The Ising models studied in this dissertation include the z-model introduced by Scalettar, standard small-world bonds superimposed on a square lattice, and physical small-world bonds superimposed on a square lattice. From the scaling results of the Binder 4th order cumulant, the order parameter, and the susceptibility, the long-range interaction is found to drive the systems behavior from Ising-like to mean field, and drive the critical point to a higher temperature. It is concluded that with a large amount of strong long-range connections (compared to the interactions on regular lattices), so the long-range connection density is non-vanishing, systems have mean field behavior. With a weak interaction that vanishes for an infinite system size or for vanishing density of long-range connections the systems have Ising-like critical behavior. The crossover from Ising-like to meanield behavior due to weak long-range interactions for systems with a large amount of long-range connections is also discussed. These results provide further evidence to support the existence of physical (quasi-) small-world nanomaterials.

URI

https://hdl.handle.net/11668/17390

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