2D Ising Model

Extension of 1D Ising Model to a 2D $L \times L$ grid. Energy:

H = - h \sum_{i = 0}^{N} σ_{i} - J \sum_{⟨ i, j ⟩} σ_{i} σ_{j}

where $⟨ i j ⟩$ sums over all nearest-neighbor pairs. In $d$ dimensions each spin has $2 d$ neighbors. Magnetization:

M = ⟨ σ_{i} ⟩ = \frac{\sum_{i} σ_{i}}{N}, Ω (m) = \frac{N!}{m! (N - m)!}, M = \frac{2 m}{N} - 1

where $m$ is the number of spin-up atoms. Boltzmann factor:

e^{- H / k T} = e^{\frac{1}{\hat{T}} \sum_{⟨ i j ⟩} s_{i} s_{j} + \frac{{\hat{h}}^{'}}{\hat{T}} \sum_{i} s_{i}}, \hat{T} = k T / J, {\hat{h}}^{'} = h / J

Exact critical temperature ( $h = 0$ ): ${\hat{T}}_{c} = \frac{2}{\ln (1 + \sqrt{2})} \approx 2.269$

Cases: effect of $H$ and $J$

Case	Energy minimum	Low- $T$ phase	Notes
$H = 0, J > 0$	all spins aligned, $σ_{i} σ_{j} = + 1$	$⟨ M ⟩ = \pm 1$ (either)	Spontaneous symmetry breaking; large barrier between $M = + 1$ and $M = - 1$ ; system freezes in one — ergodicity breaking
$H = 0, J < 0$	all spins alternating, $σ_{i} σ_{j} = - 1$	$⟨ M ⟩ = 0$ even at low $T$	Also spontaneous symmetry breaking; two ground states (checkerboard and its inverse); long-range order still present
$H \neq 0, J = 0$	all spins parallel to $H$	$⟨ M ⟩$ follows Maxwell-Boltzmann for a single spin	$N$ independent spins; no phase transition; $M$ changes smoothly with $T$
At high $T$ in all cases: entropy dominates, spins roughly equal up/down, $⟨ M ⟩ \approx 0$ .

Spontaneous symmetry breaking ( $H = 0, J > 0$ ): flipping all spins leaves energy unchanged — the system has a symmetry. But below $T_{c}$ it picks one of the two minima and stays. Ensemble average gives $⟨ M ⟩ = 0$ but time average gives $⟨ M ⟩ \neq 0$ — the two disagree, so the system is not ergodic.

Negative temperature ( $H \neq 0, J = 0$ , special case): as spins flip from all-parallel-to- $H$ toward half-up/half-down, entropy increases with energy. Then past the halfway point, entropy decreases with energy — meaning $\frac{1}{T} = \frac{\partial S}{\partial E} < 0$ , so $T < 0$ . Only possible because the Ising model has no kinetic energy; in real systems kinetic degrees of freedom make $S$ always increasing with $E$ .

Phase transition in $ρ (m)$

The full distribution $ρ (m)$ (not just $⟨ m ⟩$ ) characterizes the phase:

$T > T_{c}$ : single peak at $m = 0$ , width $\sim 1 / \sqrt{N}$ — disordered
$T = T_{c}$ : width becomes independent of $N$ — infinite correlation length; system cannot be decomposed into independent subsystems; self-similar (renormalization group)
$T < T_{c}$ : two peaks at $m = \pm m^{*}$ — two coexisting phases; as $N \to \infty$ transitions between them vanish

Phase transition: as $h$ crosses 0 for $T < T_{c}$ , magnetization jumps discontinuously (in the $N \to \infty$ limit).

Simulation: MCMC

Partition function $Z (T, J, h) = \sum_{{s_{i}}} e^{- H / k T}$ has $2^{N}$ terms — exact sum intractable. Use Markov Chain Monte Carlo to sample from the Boltzmann distribution instead.

Two algorithms:

Metropolis: flips individual spins; can get trapped in metastable states near $T_{c}$ — good for visualising real-time-like dynamics
Wolff: flips correlated clusters; escapes metastable states in one step — better for finding true equilibrium near $T_{c}$

MCMC "time steps" are not physical time — only guarantees convergence to Boltzmann distribution, not real dynamics.

Ising model as a lattice gas (liquid-gas mapping)

Coarse-grain a 2D molecular gas to grid cells of size $\sim r_{0}$ (molecular diameter): each cell is either occupied ( $s_{i} = + 1$ ) or empty ( $s_{i} = - 1$ ). Nearest-neighbor interaction $J \sim U_{0}$ (attractive well depth).

Number of molecules: $n = \sum_{i} (1 + s_{i}) / 2$ . The field $h$ term becomes:

- h \sum_{i} s_{i} = - 2 h n + const ⟹ e^{- h / k T \sum s_{i}} \to e^{- 2 h / k T \cdot n}

So $h$ plays the role of chemical potential: $μ = g = 2 h$ . System is in the grand canonical ensemble — fixed $μ$ , not fixed $N$ .

Density: $ρ = \frac{n}{N} \cdot \frac{1}{v_{0}}$ , related to magnetization by $ρ = \frac{m + 1}{2} \cdot \frac{1}{v_{0}}$ .

Phase diagram in $(h, T)$ maps to $(μ, T)$ , and qualitatively to $(p, v)$ — isotherms below $T_{c}$ show the same discontinuous volume jump as Van der Waals. High-density phase ( $⟨ m ⟩ > 0$ ) = liquid; low-density phase ( $⟨ m ⟩ < 0$ ) = gas.

Universality

Near $T_{c}$ , many different microscopic models (Ising, Van der Waals, real fluids) become quantitatively identical at large length scales — they share the same critical exponents. This is universality. Explaining it requires renormalization group (beyond this course).

Cases: effect of H and J

Phase transition in ρ(m)

Simulation: MCMC

Ising model as a lattice gas (liquid-gas mapping)

Universality

Cases: effect of $H$ and $J$

Phase transition in $ρ (m)$