SpinLatticePhaseChanges

A simple package to simulate continuous phase changes in a 2 dimensional spin lattice with the Ising model.

Systems that exhibit phase changes are often complex and require immense amounts of information to fully parameterise. The Ising model is a toy model of a system of 2-state objects, most commonly a lattice of ferromagnetic spins, often used to investigate second order phase changes. Despite the Ising model's simplicity, it retains a number of important characteristics you would find in more complex systems.

In this implementation of the Ising model, we consider a lattice of electron spins that can be either spin-up or spin-down. The order parameter that characterises the phase change of this system is the average magnetization, $m$, of each site. It will be non zero only when in an ordered phase. By varying simulation parameters like the exchange energy between adjacent pairs of electrons and the applied magnetic field strength, many different systems can be investigated.

Core Structures

Spin The spin of an electron, which can either be pointing up (true) or down (false).

SpinGrid A 2 dimensional lattice of Spin structures. This represents the orientation of valence electron spins in a ferromagnetic metal, for instance.

Core Method

run_metropolis(spingrid::SpinGrid) Interactively run a simulation of the Ising model using the metropolis spin-flipping algorithm. Given a random spin lattice, running this method simulates the system coming into thermal equilibrium with its environment at temperature T.

Notes on Optimisation

Most performance gains have been made from optimising ΔE with segment methods to only consider parts of the spin lattice which would be affected from flipping spins. This was designed to be parallelised by calculating ΔE on each segment on a unique thread. SIMD Methods could also be used, splitting a SpinGrid into parts and broadcasting them to a cluster.

A SpinGrid typically consumes 15% more memory than a similarly sized Float64-Array and will need to be considered in your implementation.

Example

julia> spingrid = SpinGrid(100,100)
Spin[↓ ↑ … ↓ ↓; ↑ ↑ … ↓ ↑; … ; ↑ ↑ … ↑ ↑; ↓ ↑ … ↓ ↓]

julia> # Run a simulation at the critical point

julia> spingrid = run_metropolis(spingrid)

Temperature of the system (K) : 76

Heisenberg exchange energy (eV) : 3.5

Applied magnetic field strength (T) : 0

Direction of applied field (rad) : 0

Number of ising model flips (Int64) : 10000
Spin[↓ ↓ … ↓ ↓; ↓ ↓ … ↓ ↓; … ; ↓ ↓ … ↑ ↑; ↓ ↓ … ↓ ↑]

Technical Information

In this project I have used the following form of the Ising Hamiltonian, $\mathcal{H}_I$, where $S_i \in {±\frac{\hbar}{2}}$ is the spin of the electron at site $i$, $\mu_i$ its magnetic moment, $J_{ij}$ the Heisenberg exchange energy between electrons $i$ and $j$, and $\bar H$ is the applied magnetic field.

$$\mathcal{H}_I = -\sum_{i,j} J_{ij} \ \bar S_i \cdot \bar S_j - \sum_{i} \bar \mu_i \cdot \bar H$$

There is no standard form of the Ising model Hamiltonian in the literature, and various simplifications can be made. I regard this form as the clearest which retains the full generality of the system.

Credit

Ben Butterworth 2024

This was a fun little project and I am happy with how it's finished. One day I'd like to come back to it and develop it further.