Wednesday, 18 July 2018

Why is it hard to solve the Ising-model in 3D?


The Ising model is a well-known and well-studied model of magnetism. Ising solved the model in one dimension in 1925. In 1944, Onsager obtained the exact free energy of the two-dimensional (2D) model in zero field and, in 1952, Yang presented a computation of the spontaneous magnetization. But, the three-dimensional (3D) model has withstood challenges and remains, to this date, an outstanding unsolved problem.



Answer



There is a result I only heard about recently: it has been proven that computing partition functions for the Ising-model in dimensions > 2 is NP-complete. (The paper can be found at http://www.cs.brown.edu/people/sorin/pdfs/Ising-paper.pdf; a more readable one is here http://www.siam.org/pdf/news/654.pdf - both can be found on the Wikipedia on the Ising model). I'm far from an expert on this, but the main idea is that a certain NP-complete graph theory problem on finding maximal sets of edges can be mapped to ground states of Ising-3D. Roughly, this means that you can't find ground states in polynomial time, and as most physicists know, if the difficulty of your problem scales exponentially, solving something exactly for large systems quickly becomes impossible.


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