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).
One way to investigate the ordering transition of a spin system
is by so-called
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.
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.
A simulation of the square-lattice Ising model with nearest-neighbor
interactions is shown below (adaptation of the applet of
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:
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
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,
- 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,
and J.R. Heringa,
Ising universality in three
dimensions: a Monte Carlo study, Journal of Physics A28, 6289
- 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
Bistability in an Ising model with non-Hamiltonian dynamics,
Physical Review B45, 5707 (1992)