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 J_ij, 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({s_l})= -

_ij J_ij s_is_j - i

_i h_is_i. (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 s_i 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)/k_BT (2)

Probability Distribution (Ensemble Dependent)

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

_G e^-H(G)/k_BT (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:

The outcome of each trial belongs to a finite set of outcomes, {G₁, ....,G_m, G_n....}.
The outcome of each trial depends only on the outcome of the trial immediately proceeding it.

Thus, G_m and G_n are linked by some transition probability, PI_mn. Here

_n PI_mn=1, so that there is SOME OUTCOME.

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

_m p_mPI_mn=p_n,

_n p_nPI_nq=p_q, and so on.

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

PI_mn = T_mn ( 1 ) p_n > p_m m.ne.n
PI_mn = T_mn (p_n/p_m) p_n < p_m m.ne.n Here, T_mn (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 PI_mm, 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 p_n/p_m= e^-
E_mn/k_BT. In other words p= min( 1, e^-
E_mn/k_BT ). This allows the system to explore other minima rather than just local ones.

Notice this ratio p_n/p_m does not depend on the partition function, which is unknown. Furthermore, the ratio requires only determination of E_mn 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 J_ij = 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, M_i, is < s_i >. Hence, using Eqs. 1-4, you can show that

M_i = < s_i > = < tanh B(

_j J_ij s_j + h_j) > 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 s_i s_j we suppose that site i "feels" the average effect of all other sites, so s_i < s_j >. In this approximation the statistical averages can be performed because correlations between specific sites have been ignored. In any case, M_i = < s_i > = tanh B(

_j J_ij < s_j > + h_j) . 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 T_Critical, 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 T_Critical where above M=0 and below M is finite.

In 1-D, however, there is no phase transition. The fully-correlated state with all spins up can be easily destroyed by just moving one spin down. Hence only at T=0 K, are all spins aligned.
In 2-D, there is finite phase transition, along with usual latent heat, etc. This has been solved exactly by Onsager, but only for h=0. The critical behavior also depends on type of lattice. On a square lattice with n.n. only, the spins align when (k_BT)^-1 > 0.4407/J, in zero field.
In 3-D there is no analytic solution, and it may be impossible to do so. Hence the importance of simulation.
Above T_Critical, there is short-range order arising from the correlations that form, and indicated by the pair correlations. Below T_Critical, there is long-range order that forms due to these correlations becoming infinite ranged, indicated by the appropriate order parameter, which is M for the Ising model.

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:

Liquid-Vapor transformation, where clustering of s_i= 1 represents condensation of the gas molecules due to their attractive intereaction. As gas molecules interact and no longer behave as non-interacting gas (i.e., PV=nRT), then there can be transition to liquid phase.
Liquid Mixtures. "Critical opalescence" arises from the short-range correlation of disordered (mixed) phase and there is scattering of light (sample is opaque) and below transition there is long-range correlation in the single phase mixture and sample is transparent because light does not scatter.
An Alloy, if one considers up=A atoms and down=B atoms. In this case, we have a binary alloy and the magnetic field, h, is the chemical potential (for h=0, we have 50-50 alloy). Nonetheless, non-zero field (finite chemical potential difference) is important generally. The solution of this finite-field problem is not even known in 2-D. Hence, numerical simulations are required.
Liquid Crystal and Polymers in some instances can also be modeled via lattices. For example, molecules that rotate (as with polarization of liquid crystals) but more-or-less form on a lattice are treated this way.

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October 9 1998 by D.D. Johnson
(updated Oct 28, 1999)
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