Introduction To The
Hrothgar
Ising Model Unit

Contents

Motivation: A Model For Magnets

Just as electrical current flowing in a loop produces a magnetic field, the motion of electrons around a nucleus can produce a (tiny) magnetic field associated with an individual atom. In many respects, these atomic magnets are like ordinary magnets and can be thought of in terms of little magnet vectors (e.g., pointing from south to north poles).

In most ordinary matter, these little atomic magnets are pointing in random directions. The magnetic fields from the individual atoms cancel, and there is no overall magnetic field in any (macroscopic) clump of matter. However, in some materials, such as iron, it is possible for very large numbers of the little atomic magnets to line up, giving a non-zero total magnetic field at "human perceptible" distances.

The creation of a macroscopic magnetic field from a bunch of atom-sized mini-magnets results from a careful balance between two somewhat opposing principles in physics:

Energy Minimization
The interactions between the atomic-scale magnets (usually called "spins") is such that the lowest energy configuration with two spins has the two spin vectors pointing in the same direction. From the perspective of energy alone, the lowest energy state of a large chunk of matter would have all the little spin vectors aligned, giving huge total magnetic fields!

Entropy Maximization
The configuration in which all atomic magnets line up is one very special case out of an incomprehensibly large number of possible configurations. Unless there is a huge "energy cost" for an individual spin which is not lined up with its neighbors, the sheer number of possible unaligned configurations completely swamps the one unique "ground state", and a macroscopic-sized system shows no net magnetization. The randomness of the real configuration (and randomness is essentially what "entropy" measures) tends to wash out the large scale magnetism predicted by energy considerations alone.

To get some appreciation of the importance of randomness and entropy, consider the experiment of tossing a very biased coin with

P[ Heads ] = 0.999

P[ Tails ] = 0.001

Even though "Tails" is a rare event for any one coin, the probability that all coins in a set are heads becomes vanishingly small as the number of coins increases:

N[Coins] P[All Heads]
100 0.9048
1,000 0.3677
10,000 0.000045

That is, "rare events" (such as a biased coin showing tails or a spin state fluctuating into a high energy state) are not all that rare if there are enough different ways for the fluctuation to happen!

Temperature Holds The Key

Qualitatively, the existence of any macroscopic magnetic field from some chunk of matter depends on the relative importance of the energy minimization and entropy (randomization) components just noted. One of the most significant developments of 19-th century physics was the discovery of the appropriate probability function to characterize the relative importance of the (numerous) microscopic configurations.

Let "S" denote some particular state of a system (e.g., specify the particular state of each of N little atomic magnets). We assume that the system is large (e.g., the number N is very big) and that the system is in thermal equilibrium (i.e., the system and the surrounding environment exchange energy and are at a common temperature "T"). Then, the probability that the specified state (S) actually occurs is given by the Boltzmann probability distribution function:

where

To gain some insight into the importance of the temperature, consider two different states A and B, with E(A) < E(B). The relative probability that the system is in the two states is given by

At hight temperatures (i.e., for kT much larger than the energy difference |D|), the system becomes equally likely to be in either of the states A or B - that is, randomness and entropy "win". On the other hand, if the energy difference is much larger than kT, the system is far more likely to be in the lower energy state.

The Ising Model: A Simplified Picture For Magnets

In spite of the relative simplicity of the Boltzmann probability function, it would be hopeless to use it to try to calculate the the magnetic properties of a "real" iron magnet for several reasons:

As is often the case in physics, it is better to put aside the "ugly realities" of the real world, looking instead at a very simplified model. In place of real atoms of iron, the Ising model considers the interaction of elementary objects called "spins" which are located at sites in a simple, 2-dimensional array. These spins are simple objects, and a given spin can only exist in one of two states:

The Ising model thus describes a bunch of little arrows arranged on a checkerboard, with the important constraint that each arrow is pointing either up or down (no sideways allowed). This simplified universe is shown in the following picture.

Simple Ising Model Universe

A state in the Ising model is simply the specification of the spin (up or down) at each of the lattice sites. The energy of a state is usually written as a sum of two terms:

In the standard (and simplest form) of the Ising model, the interactions among the little spin vectors are restricted to Nearest Neighbors. That is, in the drawing above:


The Red Site Only Interacts With The
Four Immediately Adjacent Yellow Sites.

This leads to a fairly simple expression for the energy of any particular state in the Ising model:

where

At the "nuts and bolts" level, this completes the definition of the Ising Model.

Digression: Reflections On How Physics "Really Works"

We began the page thinking about magnetism in real matter and have ended up looking instead at a small, 2-dimensional universe with very simple interactions. This process of "Simplify, Simplify, Simplify ..." is, in fact, characteristic of the way in which many breakthroughs in physics were actually accomplished. It can be argued that the real "art" of physics (or any other science) comes in in finding clever approximations and models which discard as many "real world warts" as possible without also throwing away anything "significant". (Indeed, Nobel Prizes have been won by scientists who created suitable "toy models" for some complicated aspect of the real world.)

While the Ising Model picture is certainly simplistic from the perspective of "real iron", it does provide important insight into how real world magnetism actually happens. In particular, there is a "phase transition" in the Ising Model at a well-defined critical temperature. Above this temperature, random thermal motion wins, and then is no significant magnetization. Below this temperature, the "energy pay-off" from aligned spins wins, and the system exhibits macroscopic magnetization.

Real materials like iron also exhibit a magnetization phase transition at a specific temperature. That is, the simplified model got the "essential physics" right!

Monte Carlo: Doing Physics With Random Numbers

At first glance, it would seem that the Ising Model has been fully solved:

To clarify the meaning of "probability sums" consider the problem of calculating the (average) magnetization of the system. For any fixed state S, the magnetization is proportional to the "excess" number of spins pointing up:

To evaluate the average magnetization of the whole system, we simply sum up these values over all the possible states, weighting each state by the (Boltzmann) probability that the state actually occurs:

This looks simple enough, but the calculation is completely hopeless due to the number of terms in the sum. The relationship between the number of sites in the "Ising universe" and the number of different states of the system is

(two orientations at the first site time two orientations at the second site times ...).

The 2**N behavior for the number of states is a computational disaster! (Throughout this paragraph, "A**B" means "A raised to the B power".) For a small, 16x16 lattice (256 total sites), these are about 10**77 different states in the probability sum. Being generous, a really really fast computer might be able to evaluate about 10**10 individual terms per second, and the entire calculation would be completed in about 10**67 seconds. In contrast, the age of the universe is only about 10**20 seconds.

Conclusion:
The straightforward "brute force and ignorance" approach will not work.
Instead, we need to be clever.

One particularly clever thing to do is to try to pick out only the "important" states in the probability sum, and this can be done by, effectively, "pretending to be nature".

Consider a system of spins in equilibrium with an external heat source at temperature T. The system evolves through a number of independent, random spin flips. An individual spin flip has an associated change in the overall system energy:

And, according to the standard Boltzman distribution, the likelihood that this transition occurs is of the form:

Now, consider a computer algorithm which attempts to mimic nature by

After some number of passes through the lattice to "wash out" initial spin assignments, this procedure generates spin configurations which naturally follow the real probability function. Physical quantities (such as the total magnetization) can then be evaluated by simply averaging the values of that quantity over the simulated samples.

There are two qualitative reasons why this is a reasonable procedure for estimating average quantities for interacting systems:

  1. The procedure does not waste a whole lot of time/effort generating or examining improbable configurations.
  2. Rather than exploring "all" configurations - including enormous numbers of configurations which are extremely similar in terms of macroscopic quantities - the algorithm meanders along a modest number of representative paths.
The procedure randomly samples states according to the real underlying probability distribution. This, in turn, makes it possible to estimate system averages with reasonable sized simulation samples (e.g., a few thousand samples).

The name Monte Carlo refers to the famous casino in Monaco, and emphasizes the important role of random decisions within the algorithm. At essentially each step in the evolution of the calculation,

In a sense, we are actually "doing physics by tossing dice".

The basic Monte Carlo procedure of simulating "reality" through a number of random simple steps is often the only realistic way of performing calculations in large, complicated systems.