Simulations of Ising models

The Ising model

The Ising model is a simple model to study phase transitions. So-called spins sit on the sites of a lattice; a spin S can take the value +1 or -1. These values could stand for the presence or absence of an atom, or the orientation of a magnetic atom (up or down). The energy of the model derives from the interaction between the spins. We take the energy per pair of neighbors S and S' as J S S', where J is the spin-spin interaction. When the temperature T is high in relation with J, the spins are disordered: they take more or less random values. However, when the temperature T drops below the critical point, the spin system `orders' into a state of `broken symmetry': most of the spins have the same sign (when J>0).

Metropolis Algorithm

One way to investigate the ordering transition of a spin system is by so-called Monte Carlo procedures. The `classical' Monte Carlo method is called `importance sampling' and is due to Metropolis et al. In each step of this method, one proposes to flip (change the sign of) a single spin; the probability of acceptance is chosen such (still depending on J, T and the spins) that each state occurs with the right probability. A different Monte Carlo method due to Kawasaki does not involve the flipping of a spin, but instead the exchange of two neighboring spin variables. In critical systems, simulations by the Metropolis and Kawasaki methods tend to become very slow. This is so because large correlated region exist, which are hardly affected by single spin flips.

Cluster Algorithm

More recently, `cluster' algorithms were invented. These suppress critical slowing down because they typically do not flip single spins, but instead large regions of spins. In the method due to Swendsen and Wang, the system is split up into a number of clusters whose orientations are randomly chosen. Wolff flipped just one cluster starting from a randomly chosen site. These cluster methods are extremely efficient at criticality.

Even more recently a cluster method has been introduced in which two regions of spins are swapped. It can be seen as a cluster version of the Kawasaki method. It exists in both the many-cluster and one-cluster kind.

Simulations

A simulation of the square-lattice Ising model with nearest-neighbor interactions is shown below (adaptation of the applet of Kenji Harada by Jouke Heringa). To run this application, it may, depending on your browser settings, be necessary to change your Java security settings. The two possiblities S=+1 or -1 appear as blue or white. At low temperatures, controlled by the red bar, the spins prefer to be parallel.

In the Delft computational physics group, Ising simulations are performed for scientific purposes. Other systems under investigation are models with multispin interactions and non-equilibrium models.

Further details can be found in the following publications:

H.W.J. Blöte, J.R. Heringa and E. Luijten, Cluster Monte Carlo: Extending the range, Computer Physics Communications 147, 58 (2002)
H.W.J. Blöte, J.R. Heringa and M.M. Tsypin, Three-dimensional Ising model in the fixed-magnetization ensemble: a Monte Carlo study, Physical Review E62, 77 (2000)
J.R. Heringa and H.W.J. Blöte, Geometric Cluster Monte Carlo Simulation, Physical Review E57, 4976 (1998)
H.W.J. Blöte, J.R. Heringa, A. Hoogland, E.W. Meyer and T.S. Smit, Monte Carlo renormalization of the 3-D Ising Model: Analyticity and convergence, Physical Review Letters 76 , 2613 (1996)
H.W.J. Blöte, E. Luijten and J.R. Heringa, Ising universality in three dimensions: a Monte Carlo study, Journal of Physics A28, 6289 (1995)
J.R. Heringa and H.W.J. Blöte, Demonen in Monte Carlo, Nederlands Tijdschrift voor Natuurkunde 61, 163 (1995)
J.R. Heringa, H.W.J. Blöte and A. Hoogland, Critical properties of 3D Ising systems with non-Hamiltonian dynamics, International Journal of Modern Physics C5, 589 (1994)
J.R. Heringa and H.W.J. Blöte, Bond-updating mechanism in cluster Monte Carlo calculations, Physical Review E49, 1827 (1994)
F. Iglói, J.R. Heringa, M.M.F. Philippens, A. Hoogland and H.W.J. Blöte, Critical behavior of two Ising models with near neighbor exclusion, Journal of Physics A23, 6231 (1992)
J.R. Heringa, H. Shinkai, H.W.J. Blöte, A. Hoogland and R.K.P. Zia, Bistability in an Ising model with non-Hamiltonian dynamics, Physical Review B45, 5707 (1992)