Lattice Monte Carlo:     Ising Model


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The Lenz-Ising Hamiltonian

The Ising Model is the "fruit fly" of the study of phase transitions. The notes on potentials , such as Jij, discussed the generic points associated with performing lattice type Monte Carlo, in particular how potentials might be obtained for a variety of different problems that can be simulated via a lattice model. The basic Hamiltonian, or Energy functional, required is
H({sl})= -  ij Jij sisj     -  i i hisi.     (1)
where we assume that the term i=j is ignored. Here, we shall only talk about the Lenz (1920) or Ising (1925) type model (rather than a vector-type, or Heisenburg, model) in which si are the thermodynamic "spin variables" which can take the values ±1. Here, J can be (Ferromagentic) FM-like interactions tending to cluster like spins together, or (Anti-ferromagnetic) AFM tending to order spins periodically (e.g., ...up...dn...up...dn). (Ising only studied this model in 1-D, where there is no phase transition.) For generality, I have included an external magnetic field (in fact, I have made it site dependent which for mathematical convienience allows one to take derivatives with respect to field, useful for getting susceptibilities).
 

The Ensemble Average

Within any simulation the goal is to calculate some average of interest (which varies with model). Thus, for quantity A, the
     Ensemble Average over G states
< A > NVT G A(G) e-H(G)/kBT         (2)

     Probability Distribution (Ensemble Dependent)

p(G) = e-H(G)/kBT/Z(N,V,T)        (3)
     Partition Function
Z=  G e-H(G)/kBT        (4)


The Metropolis Algorithm

The Metropolis et al. (1954) came up with an efficient means of sampling the required distribution function. The method works because of its relation to a Markov Chain  (random walk). A Markov Chain requires: Thus, Gm and Gn are linked by some transition probability, PImn. Here  n PImn=1, so that there is SOME OUTCOME.

Repeated application of PI with a starting probability pm can generate all sequences in the Markov chain, i.e.,

m pmPImn=pn,    n pnPInq=pq,        and so on.

Thus, if you know pm and PImn you have everything to solve problem. Unfortunately, you usually do not know pm and rarely known anything about PImn. Within the Metropolis approach,

PImn = Tmn (   1    )      pn > pm      m.ne.n
PImn = Tmn (pn/pm)      pn < pm      m.ne.n
Here, Tmn (the 'a priori' probability) is a symmetric stochastic matrix which determines entirely the properties of the Markov Process. The symmetric property is not required, but using it establishes microscopic reversibility of the probabilities in the simulation. Furthermore, it is straightforward to show that PImm, the probability to stay in same state, is always finite. The possibility to stay in same state is required to model Markov process.

Metropolis et al. chose to evaluate < A >NVT = < A >MC run using an criteria which accepts (or rejects) a "Monte Carlo move" according to the required statistical weight. More physically, it accepts an energy lowering move with probability 1, so you always move "downhill" towards a minima. However, it accepts an "uphill" move randomly if a random number between (0,1) is less than pn/pm= e- Emn/kBT. In other words p= min( 1, e- Emn/kBT ). This allows the system to explore other minima rather than just local ones.

Notice this ratio pn/pm does not depend on the partition function, which is unknown. Furthermore, the ratio requires only determination of Emn rather than the entire configuration. When considering long-range interactions, these usually cancel out and never have to be considered, and, hence, the Metropolis method is computationally efficient because it is the calculation of potential that is the time consuming part.
 
 

Nearest-Neighbor Model and Simulation of Averages

The so-called Nearest-Neighbor Ising model is one in which Jij = J if i and j are nearest neighbors, and 0 otherwise.  Even for this very simple interaction the statistical mechanics and phase transformation can be very interesting depending on the type of Bravais lattice and dimensionality of lattice. In other words, you do not need complexity of potential to get complex phase behavior.

For the Nearest-Neighbor Ising Model, the interesting thermal averages are the Energy, E, and the Magnetization, M, and perhaps also the pair correlations (or susceptibility). The site Magnetization, Mi, is < si >. Hence, using Eqs. 1-4, you can show that

Mi = < si > = < tanh B( j Jij sj + hj) >
where the thermal average <...> is over the remaining spin variables not on site i. Let us explore a Mean-Field Theory approximation to this where where the average has been moved through the TANH function, or equivalently, we have ignored correlations in the variables, or, in other words, rather than handle si sj we suppose that site i "feels" the average effect of all other sites, so si < sj >. In this approximation the statistical averages can be performed because correlations between specific sites have been ignored. In any case,
Mi = < si > = tanh B( j Jij < sj > + hj) .
This is a transcendental equation whose solution may be found graphically. The R.H.S. TANH function is a slanted s-like curve and the L.H.S. is a straight line. Their intersection is a solution. We find two types. For temperatures above a certain TCritical, the solution is at M=0, whereas at some lower T there are two solution ±M. These two solutions are the two possible ground states of the Ising model: all +1 spins, or all -1 spins (both have same E). We find that there is then a phase transition that can occur, i.e. a TCritical where above M=0 and below M is finite. So based on intuition from MFT, we at least can expect some interesting critical behavior associated with the numerical investigation of the Ising Model. 

Explore the 2-D Lattice Monte Carlo with N.N. Ising Model using a JAVA Applet.
 
 

Relation to Physical Systems

The Ising model appears to be a simple model of little relevance to most physical problems. The closest physical problem is a magnetic phase transformation for a simple ferromagnet. However, it is a good model also for:   TOP
October 9 1998 by D.D. Johnson

(updated Oct 28, 1999)
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