Ising Simulation

The Ising is a tool used in statistical physics to study phase transitions and collective dynamics in systems with many interacting parts. Initially designed to represent ferromagnetism, it models spins on a lattice that can take one of two states, interacting with their neighbors. Its applications extend to areas like neural network activity, protein folding and quantum cluster states. The model's simplicity allows for analytical solutions in certain cases and computational approaches in others, making it broadly applicable for exploring the connection between microscopic interactions and macroscopic phenomena.

Here I'm making a small monte carlo simulation of an Ising Model with Glauber dynamics.

1. Energy of the Ising Model:

The energy of a configuration of spins $\{ \sigma_i \}$ (where $\sigma_i = \pm 1$ ) on a lattice is given by:

E = -J \sum_{\langle i,j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i

Where:

$J$ is the interaction strength between neighboring spins (positive for ferromagnetic, negative for antiferromagnetic),
$\langle i,j \rangle$ denotes summation over nearest neighbors,
$h$ is an external magnetic field, and
$\sigma_i$ is the spin at site $i$ .

2. Glauber Dynamics Update Rule:

The Glauber dynamics updates the spin at site $i$ based on the local energy configuration. The probability of a spin $\sigma_i$ flipping to $\sigma_i'$ is:

P(\sigma_i \to \sigma_i') = \frac{1}{1 + \exp\left( \frac{\Delta E}{k_B T} \right)}

Where:

$\Delta E = E(\sigma_i' \text{ flipped}) - E(\sigma_i)$ is the change in energy when the spin flips,
$k_B$ is the Boltzmann constant,
$T$ is the temperature.

For the case of a single spin flip $\sigma_i \to \sigma_i' = -\sigma_i$ , the energy change $\Delta E$ for the flip is:

\Delta E = 2 \sigma_i \left( \sum_{\langle i,j \rangle} J \sigma_j + h \right)

This means that the probability of flipping a spin depends on the interaction with neighboring spins and the external field. At high temperatures ( $T \to \infty$ ), flips occur more freely, while at low temperatures ( $T \to 0$ ), the system tends to settle into aligned configurations.

Note the critical temperature for an ising model should lie around

T_c = \frac{2J}{ln(1 + \sqrt{2})} \approx 2.27 J

in our sim J is always 1.0, so the critical temp is $\approx$ 2.27.

Lattice state plot

Here we plot the lattice state for various temperatures and time steps.

The simulation calculates the lattice states using Glauber dynamics in the update_step_numba function. At each step, a random site is picked, and the change in energy ( $\delta E$ ) due to flipping that spin is computed from the local interactions (the sum of the nearest neighbors’ spins and any external field). The flip is then accepted with probability $p_{flip} = \frac{1}{1+e^{\delta E / T}}$ . Repeating this process for many steps yields a time evolution of the spin configuration.

At low temperatures, we often see large, stable clusters where spins align and rarely flip, indicating an ordered phase. Near critical temperature ( $T_c$ ), these clusters form and dissolve in a more balanced manner, and we catch glimpses of interesting domain structures. At high temperatures, spins will flip frequently and produce a more disordered or “noisy” lattice with low magnetization.

Identifying the critical point

We can plot the following graphs: magnetization, average_energy, susceptibility, and specific_heat.

For magnetization we should see magnetization drop at the critical point

For specific heat we should see the center of an $S$ curve at the critical point.

For susceptibility we should see a peak at the critical point.