Ising Model

Introduction

The Ising model is the simplest and most influential model of a magnet, and one of the most widely studied models in all of statistical mechanics. Picture a regular lattice of sites, with an arrow (“spin”) at each site that can point only up or down — nothing in between. Each spin wants to align (or, depending on the sign of the interaction, anti-align) with its nearest neighbors, while thermal noise constantly tries to randomize the spins. The competition between these two effects — order from the interactions, disorder from temperature — is what makes the model so rich: at low temperature the spins lock into a common orientation, while at high temperature they fluctuate independently, and somewhere in between the system undergoes a genuine phase transition.

The model is named after Ernst Ising, who solved the one-dimensional case in his 1925 doctoral thesis (Ising (1925)) — a problem suggested to him by his advisor Wilhelm Lenz. Ising found that in one dimension there is no phase transition at any nonzero temperature: thermal fluctuations always eventually destroy the order. It took another two decades before Lars Onsager solved the two-dimensional model exactly in 1944 (Onsager (1944)), showing that a true phase transition does occur in 2D — one of the most celebrated results in theoretical physics, and the reason the Ising model remains a standard testbed for new methods and algorithms to this day. See the Wikipedia article for a broader overview and history.

The Hamiltonian of the classical Ising model is

where:

• $S_i^z, S_j^z \in \{+1, -1\}$ are classical variables representing an “up” or “down” spin at lattice sites $i$ and $j$ — not quantum operators, since this is a classical statistical-mechanics model,
• $J$ is the interaction strength between neighboring spins: the coupling is antiferromagnetic if $J > 0$ (neighboring spins prefer to point opposite ways) and ferromagnetic if $J < 0$ (neighboring spins prefer to align),
• $h$ is an external magnetic field that favors one spin direction over the other,
• the sum $\langle i,j \rangle$ runs over nearest-neighbor pairs of spins on the lattice.

Replacing the classical variables $S_i^z$ with quantum spin-1/2 operators, and adding a field along $x$, gives the Transverse Field Ising Model — a closely related but genuinely quantum-mechanical model.

Phenomena

The Ising model has been applied to a remarkably wide range of physical systems and phenomena, well beyond its original motivation of magnetism.

• Ferromagnetism: For $J < 0$, spins tend to align in the same direction, leading to a spontaneous, macroscopic magnetization at low temperatures, even in zero field.
• Antiferromagnetism: For $J > 0$, spins tend to align in alternating directions, resulting in no net magnetization but strong local (staggered) ordering.
• Phase transitions: In two or more dimensions, the model exhibits a phase transition at a critical temperature $T_c$, from a disordered (paramagnetic) phase at high temperature — where the average magnetization $m = \langle S_i^z \rangle$ vanishes — to an ordered phase at low temperature with $m \neq 0$. This transition is continuous (second-order): $m$ grows smoothly from zero as the temperature is lowered through $T_c$, rather than jumping discontinuously.
• Dimensionality matters: The 1D chain has no finite-temperature phase transition at all — true order only appears at $T=0$. The 2D square-lattice model does have a phase transition and was solved exactly by Onsager; no exact solution is known in 3D, where the critical exponents must instead be obtained numerically (e.g. by Monte Carlo) or via the renormalization group.
• Universality: Very different physical systems — magnets, binary alloys undergoing an order-disorder transition, and even the liquid-gas critical point (via the equivalent “lattice gas” formulation) — share exactly the same critical exponents as the Ising model near their respective transitions. This universality of critical behavior is one of the most important ideas to emerge from the study of the Ising model.

Methods

The Ising model without a magnetic field can be solved exactly in 1D and 2D (see above), but away from these special cases, and to study finite-size systems directly, numerical methods are essential. Below is a summary of the key numerical techniques available in ALPS:

ALPS’s classical Monte Carlo application, spinmc, implements both local and cluster updates for the Ising model and related classical spin models — see the Classical Monte Carlo pages for the algorithms in detail.

Because no simulation can study an infinite lattice, extracting $T_c$ and the critical exponents in practice relies on finite-size scaling. Near the transition, a quantity like the susceptibility or the correlation length would formally diverge in an infinite system, but on a finite lattice of linear size $L$ it is instead cut off once the correlation length grows to be comparable to $L$ itself — so the apparent peak in, say, the susceptibility is rounded off and shifted away from $T_c$, by an amount that shrinks in a universal, predictable way as $L$ grows. By simulating several lattice sizes and comparing how such quantities (magnetization, susceptibility, the Binder cumulant) depend on $L$ near the apparent transition, one can extrapolate to the thermodynamic limit ($L \to \infty$) and read off both the critical temperature and the critical exponents precisely, even though every individual simulation is necessarily finite.

Three tutorials walk through simulating the 2D Ising model with spinmc from start to finish:

• MC-01(a): Classical Monte Carlo simulations and autocorrelations — introduces the model and shows how autocorrelation times blow up near $T_c$ with local updates
• MC-01(b): Classical Monte Carlo simulations and equilibration/convergence — diagnoses whether a run has thermalized and converged
• MC-07: Phase transition in the Ising model — uses finite-size scaling to extract the critical temperature and critical exponents

For an overview of the other models in ALPS, see Models in ALPS.