The Ising Model of a Ferromagnet from 1920 to the Present Day

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The Ising Model of a Ferromagnet from 1920 to the Present Day

Distinguished Colloquium Series in Interdisciplinary & Applied Mathematics Stanislav Smirnov Professor of Mathematics, University of Geneva The Ising Model of a Ferromagnet from 1920 to the Present Day The Ising model is an archetypical model of the order-disorder phase transition: though simple to formulate, it exhibits a complex behavior, much like the real-world phenomena in solid-state physics, ferromagnetism, chemistry, biology, computer science. In 1920 the physicist Wilhelm Lenz proposed to model ferromagnetic materials by a lattice of plus-minus spins with interacting neighbors. His student Ernst Ising solved the model in dimension one four years later. The one-dimensional behavior turned out to be trivial, with no phase transition when the interaction strength changes, and for a decade peo- ple searched for other possible models. However, a ferromagnetic phase transition was established by Rudolf Peierls in higher dimensions, and in 1944 Lars Onsager famously calculated the free energy for the Ising model in two dimensions. Since then the Ising model became widely studied, as it exhibits compli- cated phase transition behavior, yet allows a very detailed analysis in dimension two. We will give a historical introduction to the model, and then describe some recent results. Stanislav Smirnov is a professor at the University of Geneva, Switzerland, and holds a part-time position at the St. Petersburg State University. He received his PhD in 1996 at the California Institute of Technology under the direction of Nikolai Makarov, and worked at Yale and KTH Stockholm before moving to Geneva in 2003. Smirnov’s research interests include analysis, dynamical sys- tems, probability and mathematical physics. In 2010 he was awarded the Fields Medal for his work on conformal invariance of the percolation and Ising models at criticality. Organizing Committee: Don Goldfarb (IEOR) Thursday, October 23, 3:30pm Eitan Grinspin (Computer Science / APAM) Ioannis Karatzas (Mathematics) Davis Auditorium, CEPSR (Shapiro Center) Michael I. Weinstein (APAM / Mathematics) http://engineering.columbia.edu/IAM-Colloquium .

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An Introduction to the Ising Model Barry A. Cipra the American
An Introduction to the Ising Model Barry A. Cipra The American Mathematical Monthly, Vol. 94, No. 10. (Dec., 1987), pp. 937-959. Stable URL: http://links.jstor.org/sici?sici=0002-9890%28198712%2994%3A10%3C937%3AAITTIM%3E2.0.CO%3B2-V The American Mathematical Monthly is currently published by Mathematical Association of America. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/maa.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is an independent not-for-profit organization dedicated to and preserving a digital archive of scholarly journals. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Mon Apr 30 12:49:36 2007 An Introduction to the Ising Model BARRYA. CIPRA,St. Olu/College, Northfield, Minnesotu 55057 Burry Cipru received a Ph.D, in mathematics from the University of Maryland in 1980. He has been an instructor at M.I.T. and at the Ohio State University, and is currently an assistant professor of mathematics at St.
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