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Unformatted text preview: Contents of Lecturenotes IV 1. The O (3) Model and the Heat Bath Algorithm 1 The O (3) Model and the Heat Bath Algorithm We give an example of a model with a continuous energy function. The 2 d version of the model is of interest to field theorists because of its analogies with the fourdimensional YangMills theory. In statistical physics the ddimensional model is known as the Heisenberg ferromagnet. Expectation values are calculated with respect to the partition function Z = Z Y i ds i e E ( { s i } ) . (1) The spins ~s i = s i, 1 s i, 2 s i, 3 are normalized to ( ~s i ) 2 = 1 (2) and the measure ds i is defined by Z ds i = 1 4 Z +1 1 d cos( i ) Z 2 d i , (3) where the polar ( i ) and azimuth ( i ) angles define the spin s i on the unit sphere. 2 The energy is E = X h ij i (1 ~s i ~s j ) , (4) where the sum goes over the nearest neighbor sites of the lattice. We would like to update a single spin ~s . The sum of its 2 d neighbors is ~ S = ~s 1 + ~s 2 + . . . + ~s 2 d 1 + ~s 2 d . Hence, the contribution of spin ~s to the action is 2 d ~s ~ S . We propose a new spin ~s with the measure (3) by drawing two uniformly distributed random numbers r [0 , ) for the azimuth angle and...
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 Fall '08
 Berg
 Energy, Heat

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