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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Contents of Lecturenotes IV 1. The O (3) σ Model and the Heat Bath Algorithm 1 The O (3) σ 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- βE ( { 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 π Z +1- 1 d cos( θ i ) Z 2 π dφ i , (3) where the polar ( θ i ) and azimuth ( φ 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 φ r ∈ [0 , π ) for the azimuth angle and...
View Full Document

{[ snackBarMessage ]}

### Page1 / 9

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online