Ising Model

In the first lectures of this series, we discussed a problem where one particle experienced a complicated motion, and we learnt methods to solve the equation of motion numerically. In the past two lectures we had a first encounter with systems containing many particles, concentrating on random processes. This necessitated some first applications of statistical methods for their description. However, in all cases considered so far - diffusion, percolation, cluster growth - the interaction among the particles was of no concern. This will change in the last two lectures - we will consider a system where the interactions between the physical degrees of freedom play an essential role. We will also see that these interactions can trigger phase transitions, like the one we encountered, as a sketch of the real thing in percolation. Here, with interactions between the particles, we will see a much richer structure. Examples for such phase transitions are the condensation of a gas into a liquid or the transition to ferro-magnetism in some materials such as iron. In both cases the concept of temperature is in the centre of the phenomenon, and accordingly we will finally enter the realm of thermodynamics and statistical physics.

In this lecture we will rely even more on a stochastic approach by simulating the interactions within the system through random numbers. This approach is known as Monte Carlo method, and it is widely used in all fields of physics, especially when complicated interactions or geometries or a huge number of particles are involved and analytical methods cannot be used. Here we will discuss a comparably simple system, namely the Ising model, which we employ to qualitatively understand ferro-magnetism. Along this discussion we will encounter phase transitions again - this time, however, we will be able to make the connection to physics. We will formulate this through the canonical ensemble, a tool central for statistical physics.

Discussion on the topics covered here can be found in chapter 8 sections 1-4 in Giordano & Nakanishi

The model

Let us start by stating that magnetism is an inherently quantum mechanical phenomenon - Niels Bohr, even before the advent of quantum mechanics, proved that ferro-magnetism cannot be described by classical physics. In the following years it became apparent that the electrons' spin and the associated magnetic moment is a key ingredient to ferro-magnetism, which emerges when many such spins conspire to align, such that all spins point in the same direction. Then the macroscopic magnetisation that can be measured builds up from the individual spins with their comparably small magnetic moment (In fact, there is also a second effect contributing, namely the magnetic moment related to the orbital angular momentum of the electrons, but this is far beyond the scope of our discussion here, and omitting it keeps the language and physics simple). Thus it is central to understand how the spin conspiracy for alignment takes place, even more so, since this is a temperature-dependent effect (typically ferro-magnetism vanishes at sufficiently large temperatures). The model discussed here, known as the Ising model, is very simple and far from being able to quantitatively describe real materials; it is, however, perfectly suited to discuss qualitatively the dynamics.

In this model, spins are distributed over a lattice, in our case a square lattice in d dimensions, such that there is exactly one spin at each lattice site. The spins can point only into the positive or negative z- direction, i.e. either up or down. Other orientations are not allowed in our version of the model. Therefore, each lattice site, i, is associated with one spin, which can take the values s_i=±1. In principle now, all spins could and would interact with each other, in practice the strength of these interactions falls off so quickly with distance that it is a sensible approximation to consider only the interactions between the nearest neighbours. In a ferromagnet, this interaction would then favour alignment of the interacting spins. With this motivation, the Ising model interaction reads

Having said that, the story would in principle be over, if there was not the disordering effect of temperature. Temperature in principle induces motion into the system, allowing some spins to flip "out of phase" with their neighbours. Therefore, to keep the story interesting we will assume that the system is in equilibrium with a heat bath at a temperature T, which we can steer from the outside. In other words, we assume that there are means to externally define the temperature of the system. Then the behaviour of the system is defined by the canonical ensemble, a concept central to statistical physics. We will use it very casually here, for a deeper and more profound discussion of such concepts please take a look into the literature. However, one way to view the effect of temperature is that over time the spins flip back and forth and the overall system will thereby move into different spin configurations. Similar to the example of the milk in the tea, it will explore all possibilities, i.e. all possible spin configurations. In contrast to the milk example, here not all possibilities of the system have equal probability (we had no notion of energy, temperature or similar there), but rather the probability of finding the system in a certain state α is proportional to the Boltzmann factor

Here, k_B again is the Boltzmann constant, T is the temperature of the heat bath and thus of the system, and E_α is the energy of the state, given by the equation below. From this we learn that, with rising temperatures, states with increasing energies become more and more populated. Each of those states α is a micro-state of the system, characterised by a specific configuration of spins. In a lattice with N lattice sites, there are therefore 2^N micro-states, a number which makes the system difficult to solve, especially for large N, the case that interests us most.

While on the microscopic level, the system goes from one micro-state to another by spins flipping and thereby exchanging energy with the heat bath, any macroscopic measurement of the systems properties, such as magnetisation, will effectively average over the many micro-states the system visits during the measurement process. Therefore, identifying M_α with the magnetisation corresponding to a micro-state α,

the total, macroscopic magnetisation M is given by

with the probabilities given by the Boltzmann factors and the sum including all micro-states visited. In such a way, all macroscopic observables can be expressed by their counterparts on the level of micro-states and corresponding probabilities. It is therefore the name of the game to calculate these probabilities.

In the next section we will discuss how to calculate some of these observables analytically, with some approximation, namely the magnetisation. Obviously the difficulty in doing this lies in the huge number of states, which must be dealt with. Despite the fact that there are extremely powerful analytical tools, due to the difficulties with applying them to such large systems there are only very few exact results known. This makes simulation extremely attractive for this and similar problems, and we will spend quite some time on this later on.

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