Continuous Systems: Heisenberg Spin Model
We give an example of a model with a continuous energy function. The 2d version of the model is known as σ-model and of interest in field theory because of its analogies with 4d Yang-Mills theory. In statistical physics the d-dimensional model is known as Heisenberg ferromagnet. Expectation values are calculated with respect to the partition function Z Y −βE({si}) Z = dsi e . (1) i si,1 2 The spins ~si = si,2 are normalized to (~si) = 1 (2) si,3 Z 1 Z +1 Z 2π and the measure dsi is defined by dsi = d cos(θi) dφi , (3) 4π −1 0 where the polar (θi) and azimuth (φi) angles define the spin si on the unit sphere.
1 The energy is X E = (1 − ~si~sj) , (4)
Metropolis Updating
We would like to update a single spin ~s. The sum of its 2d neighbors is
S~ = ~s1 + ~s2 + ... + ~s2d−1 + ~s2d .
Hence, the contribution of spin ~s to the action is 2d − ~sS.~ We propose a new spin 0 ~s with the measure (3) by drawing two uniformly distributed random numbers
φr ∈ [0, 2π) for the azimuth angle and cos(θr) = xr ∈ [−1, +1) for the cosine of the polar angle .
2 0 This defines the probability function f(~s ,~s) of the Metropolis process, which 0 accepts the proposed spin ~s with probability
0 0 1 for S~s~ > S~s,~ w(~s → ~s ) = 0 0 e−β(S~s~ −S~s~ ) for S~s~ < S~s.~
If sites are chosen with the uniform probability distribution 1/N per site, where N is the total number of spins, it is obvious that the procedure fulfills detailed balance.
It is noteworthy that the procedure remains valid when the spins are chosen in the systematic order 1,...,N, then 1,...,N again, and so on. Balance still holds, whereas detailed balance is violated (exercise).
3 Heathbath Updating Repeating the Metropolis algorithm again and again for the same spin ~s leads to the equilibrium distribution of this spin, which reads
0 0 Z 0 P (~s ; S~) = const eβS~s~ with P (~s ; S~) ds0 = 1 .
0 One would prefer to choose ~s directly with the probability
0 0 0 W (~s → ~s ) = P (~s ; S~) = const eβ ~s S~ ,
0 as ~s is then Boltzmann distributed with respect to its neighbor spins. The algorithm, which creates this distribution, is called the heatbath algorithm (in Math Gibbs sampler). Implementation of this algorithm becomes feasible when the energy function is 0 sufficiently simple to allow for an explicit calculation of the probability P (~s ; S~).
4 This is an easy task for the Heisenberg model. Let
0 α = angle(~s , S~), x = cos(α) and S = β|S~| .
0 For S = 0 a new spin ~s is simply obtained by random sampling. We assume in the following S > 0. The Boltzmann weight becomes exp(xS) and the normalization constant follows from Z +1 2 dx exS = sinh(S) . −1 S
Therefore, the desired probability is
0 S P (~s ; S~) = exS =: f(x) 2 sinh(S)
and the method of the first lecture can be used to generate events with the probability density f(x).
5 With x x Z Z S 0 exp(+xS) − exp(−S) y = F (x) = dx0 f(x0) = dx0 ex S = −1 −1 2 sinh(S) 2 sinh(S)
a uniformly distributed random number yr ∈ [0, 1) translates into
1 xr = cos αr = ln [ exp(+S) − yr exp(+S) + yr exp(−S)] . (5) S
0 Finally, one has to give ~s a direction in the plane orthogonal to S. This is done by choosing a random angle βr uniformly distributed in the range 0 ≤ βr < 2π. Then, 0 0 xr = cos αr and βr completely determine ~s with respect to S~. Before storing ~s 0 in the computer memory, we have to calculate coordinates of ~s with respect to a Cartesian coordinate system, which is globally used for all spins of the lattice. This is achieved by a linear transformation. We define
S3 p S1 S2 cos θ = , sin θ = 1 − cos2 θ, cos φ = and sin φ = . S S sin θ S sin θ
6 Unit vectors of a coordinate frame K0, with zˆ in the direction of Sˆ and yˆ in the x − y plane, are then defined by
sin θ cos φ cos θ cos φ − sin φ 0 0 0 zˆ = sin θ sin φ , xˆ = cos θ sin φ andy ˆ = cos φ . cos θ − sin θ 0
0 Expanding ~s in these units vectors, we have
0 0 0 ~s = sin αr cos βr xˆ + sin αr sin βr yˆ + cos αr zˆ
sin αr cos βr cos θ cos φ − sin αr sin βr sin φ + cos αr sin θ cos φ = sin αr cos βr cos θ sin φ + sin αr cos βr cos φ + cos αr sin θ sin φ (6) − sin αr cos βr sin θ + cos αr cosθ and the three numbers of the column vector (6) are stored in the computer.
7 Self-Averaging Illustration for the 2d O(3) model
We compare the peaked distribution function of the mean action per link for different lattice sizes. The property of self-averaging is observed: The larger the lattice, the smaller the confidence range. The other way round, the peaked distribution function is very well suited to exhibit observables for which self-averaging does not work, as for instance encountered in spin glass simulations.
0.5 L=20 L=34 L=60 0.4 L=100
0.3 q F 0.2
0.1
0 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 actm
Figure 1: O(3) σ-model at β = 1.1 (assignments a0304 06 and a0304 08).
8 3D 3-state Potts model
60000 h(actm) 50000
40000
30000 Histogram 20000
10000
0 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 actm
To illustrate features of a first order phase transition for the 3d 3-state Potts model on a 243 lattice. We use the 1-hit Metropolis algorithm to simulate at β = 0.275229525, perform 20 000 sweeps for reaching equilibrium, then 64×10 000 sweeps with measurements and plot an “action variable” histogram with its error bars (assignment a0303 10 of the book).
9 The histogram exhibits a double peak structure which is typically obtained when systems with first order transitions are simulated on finite lattices in the neighborhood of so called pseudo-transition temperatures. These are finite lattice temperature definitions, which converge with increasing system size towards the infinite volume transition temperature. Equal heights of the maxima of the two peaks is a popular definition of a pseudo-transition temperature for first order phase transitions.
Homework: A finite size scaling (FSS) determination of pseudocritical beta values defined by equal heights plots.
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