Ising model

The Ising model, named after physicists Ernst Ising and Wilhelm Lenz, is a mathematical model that helps us understand how materials act when they become magnetic. It uses simple ideas to show the tiny parts of a material that act like tiny magnets, called "spins." These spins can point in one direction or the opposite, like tiny compass needles.

In the Ising model, these spins are arranged in a pattern, often in a grid. Each spin interacts only with its nearby neighbors. When neighboring spins point in the same direction, they have less energy, which is a stable state. Heat can change this stability, letting the material change its overall structure. This model shows how a material can change from non-magnetic to magnetic, known as a phase transition.

The Ising model was created by Wilhelm Lenz, who gave it to his student, Ernst Ising, as a problem to solve. Ernst solved the simplest version of the model. A more complex version for a two-dimensional grid was solved much later by Lars Onsager. This model stays important in physics because it helps scientists understand real magnetic materials and has uses in other areas, such as computer science and neural networks.

Definition

The Ising model is a way to understand how tiny parts of a material can act like tiny magnets. Imagine a grid where each point can be either "+1" (pointing one way) or "-1" (pointing the opposite way). These points are called "spins."

Each spin can affect its nearby neighbors. Scientists use this model to study how materials can become magnetic. The model looks at how these spins arrange themselves under different conditions, helping us learn about real magnetic materials.

Main article: Ising model
Further information: Hamiltonian function, magnetic moment, pseudo-Boolean function, Boltzmann distribution, inverse temperature, partition function

Basic properties and history

The Ising model helps us understand how tiny parts of materials, called spins, work together to create magnetic effects. These spins can point in one direction or the opposite, like tiny magnets.

In a line of spins, the order stays the same. But when spins are arranged in a grid, they can change between ordered and disordered states depending on temperature. Scientists first showed this in the 1930s and solved it for a square grid in the 1940s while studying how materials behave at different temperatures.

Main article: Griffiths inequality

Main article: FKG inequality

Historical significance

In the early twentieth century, scientists were unsure if atoms really existed. Some, like James Clerk Maxwell and Ludwig Boltzmann, showed that atoms could explain how gases behave. But statistical mechanics, which describes how atoms work together, didn’t explain everything about liquids, solids, or very cold gases.

When quantum mechanics was developed, it helped prove that atoms were real. However, some people still had questions about statistical mechanics. One reason was that in very large systems, tiny changes can cause big differences in behavior—something that doesn’t happen in small systems.

No phase transitions in finite volume

Some scientists thought that phase transitions—like water turning to ice—couldn’t happen in small systems. They believed this because, in small systems, the math always gave smooth results, without sudden jumps. But for very large (or endless) systems, these jumps could happen. This idea was first shown clearly using the Ising model by Rudolf Peierls.

Peierls droplets

After Wilhelm Lenz and Ernst Ising created the Ising model, Peierls showed that phase transitions could happen in systems spread out over two dimensions. He looked at what happens at very high and very low temperatures. At very high temperatures, spins (which represent tiny magnetic moments) are random. At lower temperatures, these spins start to group together.

Peierls studied whether, at very low temperatures, a system where most spins point in one direction could change to mostly pointing the opposite way. He found that small groups, or “droplets,” of opposite spins could form. This work helped show how phase transitions could occur.

Yang–Lee zeros

Main article: Lee–Yang theorem

After a solution by Onsager, Yang and Lee studied how the math describing the model changes when the temperature gets very close to a special critical point. They looked at how the partition function, which sums up all possible states of the system, changes near this point.

Applications

Magnetism

The Ising model helps explain ferromagnetism — how materials like iron can become magnets. Inside iron atoms, tiny particles called electrons act like tiny magnets. These electron "spins" line up in the same direction, creating a strong magnetic field. The Ising model studies how these spins influence each other to see if they can all align together.

Lattice gas

The Ising model can also describe how atoms move and arrange themselves. Imagine space as a grid where each spot can either have an atom or be empty. Atoms attract their neighbors, so it’s easier for nearby spots to also have atoms. By changing how likely it is to add an atom, scientists can control how dense the material is. This helps us understand patterns in materials and biological systems, like how molecules stick to receptors on cells.

Neuroscience

The Ising model is useful for studying how neurons — the brain’s nerve cells — work together. Each neuron can be active (sending signals) or inactive at any moment. The model helps scientists understand how groups of neurons influence each other, forming patterns of activity. This is important for learning how the brain processes information and makes decisions.

Spin glasses

Spin glasses are special materials where magnetic spins are arranged randomly. The Ising model helps describe these materials, where some neighbors attract while others repel. This creates complex behavior that is still being studied.

Artificial neural network

The Ising model inspired early designs of artificial neural networks, which are computer systems that mimic how the brain learns. These networks adjust their connections over time to remember information, much like how the Ising model shows how spins influence each other.

Sea ice

The Ising model can also describe sea ice, where each part of the ice can be either water or solid ice. This helps scientists study how ice forms and melts.

Cayley tree topologies and large neural networks

The Ising model has been studied on special tree-like structures called Cayley trees, which can represent large neural networks. These studies help understand how connections in the brain might affect its overall behavior, such as switching between sleeping and waking states. Scientists are interested in how the structure of these networks influences their function.

Numerical simulation

The Ising model can be hard to study when there are many possible states. Imagine a grid where each point, or "spin," can be either +1 or -1. With L spins, there are 2^L possible ways they can align! To handle this, scientists use special computer methods called Monte Carlo methods.

One key idea is the Hamiltonian, a formula that measures the energy of the system.

The most common way to simulate this is the Metropolis algorithm. It works by randomly picking a spin to flip and deciding whether to keep the change based on how it affects energy. If flipping lowers the energy, it’s always accepted. If it raises the energy, it’s accepted only sometimes. This process repeats until the system settles into a stable pattern.

Solutions

The Ising model helps us understand how tiny magnetic bits, called "spins," interact with each other in a material. These spins can point in one of two directions, like tiny magnets. When arranged in rows and columns, we can see how they behave under different conditions.

In one dimension, like a straight line of spins, the model shows that there’s no sudden change in behavior at any temperature above absolute zero. This means the spins don’t suddenly align together to create a strong magnet. However, at absolute zero, where all movement stops, the spins can align perfectly.

In two dimensions, like a flat grid, the Ising model does show a sudden change at a specific temperature. This is called a phase transition, where the spins start to align together below this temperature, creating a magnetized state. This was solved exactly by a scientist named Onsager.