Information Theory, Inference, and Learning ... - Inference Group

Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links.31Ising ModelsAn Ising model is an array of spins (e.g., atoms that can take states ±1) thatare magnetically coupled to each other. If one spin is, say, in the +1 statethen it is energetically favourable for its immediate neighbours to be in thesame state, in the case of a ferromagnetic model, and in the opposite state, inthe case of an antiferromagnet. In this chapter we discuss two computationaltechniques for studying Ising models.Let the state x of an Ising model with N spins be a vector in which eachcomponent x n takes values −1 or +1. If two spins m and n are neighbours wewrite (m, n) ∈ N . The coupling between neighbouring spins is J. We defineJ mn = J if m and n are neighbours and J mn = 0 otherwise. The energy of astate x is[1 ∑E(x; J, H) = − J mn x m x n + ∑ ]Hx n , (31.1)2m,n nwhere H is the applied field. If J > 0 then the model is ferromagnetic, andif J < 0 it is antiferromagnetic. We’ve included the factor of 1/ 2 because eachpair is counted twice in the first sum, once as (m, n) and once as (n, m). Atequilibrium at temperature T , the probability that the state is x isP (x | β, J, H) =where β = 1/k B T , k B is Boltzmann’s constant, and1exp[−βE(x; J, H)] , (31.2)Z(β, J, H)Z(β, J, H) ≡ ∑ xexp[−βE(x; J, H)] . (31.3)Relevance of Ising modelsIsing models are relevant for three reasons.Ising models are important first as models of magnetic systems that havea phase transition. The theory of universality in statistical physics shows thatall systems with the same dimension (here, two), and the same symmetries,have equivalent critical properties, i.e., the scaling laws shown by their phasetransitions are identical. So by studying Ising models we can find out not onlyabout magnetic phase transitions but also about phase transitions in manyother systems.Second, if we generalize the energy function to[1 ∑E(x; J, h) = − J mn x m x n + ∑ ]h n x n , (31.4)2m,n nwhere the couplings J mn and applied fields h n are not constant, we obtaina family of models known as ‘spin glasses’ to physicists, and as ‘Hopfield400