mcmc4 - Contents of Lecturenotes IV 1. The O (3) Model and...

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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 four-dimensional Yang-Mills theory. In statistical physics the d-dimensional 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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mcmc4 - Contents of Lecturenotes IV 1. The O (3) Model and...

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