Ising Model

The Ising model is the workhorse of statistical mechanics — a minimal model of ferromagnetism that reveals deep truths about phase transitions and collective behaviour.

Setup

Consider a 1D chain of $N$ spins, each taking the value $s_i \in \{+1, -1\}$ (spin-up or spin-down). Spins interact with their nearest neighbours, and the system uses periodic boundary conditions ($s_{N} \equiv s_0$).

Energy

The Hamiltonian (total energy) is:

$E = -J \sum_{i=0}^{N-1} s_i \, s_{i+1}$

• $J > 0$: ferromagnetic (aligned spins are favoured, lowering energy)
• $J < 0$: antiferromagnetic (anti-aligned spins are favoured)

Magnetisation

The magnetisation per spin measures the average alignment:

$M = \frac{1}{N} \sum_{i=0}^{N-1} s_i$

$M = +1$ is fully aligned up, $M = -1$ is fully aligned down, $M = 0$ is disordered.

Analytical Results (1D)

In one dimension, there is no phase transition at finite temperature. The analytical energy per spin is:

$\langle e \rangle = -J \tanh\!\left(\frac{J}{k_B T}\right)$

(setting $k_B = 1$ for simplicity).

Metropolis Algorithm

The Metropolis Monte Carlo algorithm samples spin configurations at temperature $T$:

1. Pick a random spin $s_i$
2. Compute the energy change $\Delta E$ if $s_i$ is flipped
3. Accept the flip with probability $\min(1,\, e^{-\Delta E / T})$

The acceptance probability as a standalone function is:

$A(\Delta E, T) = \begin{cases} 1 & \text{if } \Delta E \leq 0 \\ e^{-\Delta E / T} & \text{if } \Delta E > 0 \end{cases}$

Your Task

Implement four functions:

• ising_energy(spins, J=1.0) — compute total energy with periodic boundary conditions
• ising_magnetization(spins) — compute magnetisation per spin
• ising_energy_analytical(T_K, J=1.0) — return analytical energy per spin ($k_B = 1$)
• metropolis_acceptance(delta_E, T_K) — return the Metropolis acceptance probability