9.1 Ising model

Ising model basics

The Ising model is a lattice-based model of interacting spins that captures the essential physics of phase transitions and collective behavior. Despite its simplicity, it reveals how microscopic interactions between neighboring elements can produce macroscopic phenomena like spontaneous magnetization. It's one of the few many-body models with exact solutions in certain dimensions, making it a central reference point across statistical mechanics.

Definition and components

The model consists of a lattice of fixed sites, each occupied by a spin variable that can take only two values: $s_i = +1$ (up) or $s_i = -1$ (down). Neighboring spins interact with each other, and the total energy of the system depends on the full spin configuration plus any external magnetic field applied to the system.

Lattice structure

• The lattice defines how spin sites are arranged in space. Common types include the linear chain (1D), square lattice (2D), and cubic lattice (3D).
• The coordination number $z$ is the number of nearest neighbors each spin has. For a square lattice, $z = 4$; for a simple cubic lattice, $z = 6$.
• Periodic boundary conditions are typically applied so that spins on opposite edges of the lattice interact as if the lattice wraps around. This minimizes finite-size effects and approximates an infinite system.
• Lattice dimensionality has a profound effect on the model's behavior. As you'll see, the 1D Ising model has no phase transition at finite temperature, while the 2D model does.

Spin states

The binary nature of the spins is what makes the model tractable. An up spin ($+1$) aligns with an applied external field, while a down spin ($-1$) opposes it. When many spins collectively align, the system develops a macroscopic magnetization.

At any nonzero temperature, thermal fluctuations cause individual spins to flip between states. The probability of finding the system in a given configuration is governed by the Boltzmann distribution: configurations with lower energy are exponentially more likely at low temperatures.

Hamiltonian formulation

The Hamiltonian encodes the total energy of the system as a function of all the spin variables. Everything you want to know about the thermodynamics follows from this energy function.

Energy calculation

The Ising Hamiltonian is:

$H = -J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i$

• The first term sums over all nearest-neighbor pairs $\langle i,j \rangle$. When two adjacent spins are parallel ($s_i s_j = +1$), the energy contribution is $-J$; when antiparallel ($s_i s_j = -1$), it's $+J$. So for $J > 0$, parallel alignment is energetically favored.
• The second term couples each spin to the external field $h$. It favors alignment of every spin with the field direction.

Coupling constant

The parameter $J$ controls the strength and nature of spin-spin interactions:

• $J > 0$: Ferromagnetic coupling. Neighboring spins prefer to align parallel.
• $J < 0$: Antiferromagnetic coupling. Neighboring spins prefer to align antiparallel.
• The magnitude of $J$ sets the energy scale of the problem and directly influences the critical temperature. Larger $|J|$ means stronger interactions and a higher temperature is needed to destroy order.

External magnetic field

The field $h$ represents an externally applied magnetic field. It tends to align all spins in one direction, competing with thermal disorder. When $h = 0$, the Hamiltonian has a $\mathbb{Z}_2$ symmetry (flipping all spins doesn't change the energy), and spontaneous symmetry breaking can occur below the critical temperature. A nonzero $h$ explicitly breaks this symmetry and can induce magnetization even above $T_c$.

Phase transitions

The Ising model's real power lies in its ability to exhibit a genuine phase transition between an ordered (magnetized) state at low temperature and a disordered (paramagnetic) state at high temperature.

Critical temperature

The critical temperature $T_c$ (also called the Curie temperature for ferromagnets) is the boundary between these phases. Below $T_c$, ordering interactions win over thermal fluctuations, and the system develops long-range order. Above $T_c$, thermal energy dominates and spins are randomly oriented on average.

At exactly $T_c$, the system exhibits remarkable behavior: the correlation length diverges (spin-spin correlations extend across the entire system), and the magnetic susceptibility also diverges, meaning the system becomes infinitely sensitive to small applied fields.

Spontaneous magnetization

Below $T_c$ and in the absence of an external field ($h = 0$), the system develops a nonzero average magnetization. This spontaneous magnetization is a signature of long-range order: even though no field is applied, the spins collectively choose a preferred direction.

The magnetization grows as temperature decreases below $T_c$ and exhibits non-analytic behavior near the transition, meaning you can't expand it as a simple Taylor series around $T_c$.

Order parameter

The order parameter for the Ising model is the average magnetization per spin:

$m = \frac{1}{N} \sum_i \langle s_i \rangle$

It equals zero in the disordered phase ($T > T_c$) and takes a nonzero value in the ordered phase ($T < T_c$). Near the critical point, it follows a power law:

$m \propto (T_c - T)^{\beta}$

where $\beta$ is a critical exponent. The value of $\beta$ depends on dimensionality but not on microscopic details like the exact value of $J$.

Mean field theory

Mean field theory is the simplest analytical approach to the Ising model. It replaces the complicated many-body problem with a much simpler single-spin problem, at the cost of ignoring fluctuations.

Approximation method

The core idea: instead of each spin interacting with its actual fluctuating neighbors, assume each spin feels an effective field produced by the average magnetization of all other spins. This decouples the problem so that each spin is independent, interacting only with the mean field.

The procedure works as follows:

1. Write the local field felt by spin $i$ as $h_{\text{eff}} = Jzm + h$, where $z$ is the coordination number and $m$ is the average magnetization per spin.
2. Compute the thermal average of a single spin in this effective field: $m = \tanh(\beta h_{\text{eff}}) = \tanh(\beta(Jzm + h))$.
3. Solve this self-consistency equation for $m$. At $h = 0$, the equation has a nonzero solution for $m$ only below the mean field critical temperature $T_c^{MF} = Jz/k_B$.

Limitations of mean field

Mean field theory gives the right qualitative picture but gets quantitative details wrong:

• It overestimates $T_c$ because it ignores the fact that nearby spins are correlated, not independent.
• It predicts classical (mean field) critical exponents (e.g., $\beta = 1/2$), which differ from the true values in low dimensions.
• It completely fails in 1D, where it predicts a phase transition that doesn't exist.
• The approximation improves as dimensionality increases. Above the upper critical dimension ($d = 4$ for the Ising model), mean field exponents become exact.

Exact solutions

Exact solutions are rare in statistical mechanics, which is why the solvable cases of the Ising model are so valuable. They provide rigorous benchmarks for testing approximations and simulations.

One-dimensional Ising model

Ernst Ising solved the 1D chain exactly in 1925. The key result: there is no phase transition at any finite temperature. The correlation length $\xi$ stays finite for all $T > 0$, and spontaneous magnetization only appears at $T = 0$. Physically, a single domain wall in 1D costs finite energy $2J$ but gains entropy proportional to $\ln N$, so at any nonzero temperature, domain walls proliferate and destroy long-range order.

This result highlights how dimensionality is crucial for phase transitions.

Two-dimensional Ising model

Lars Onsager solved the 2D square-lattice Ising model exactly in 1944, one of the great achievements in theoretical physics. Unlike 1D, the 2D model does exhibit a phase transition at a finite critical temperature:

$T_c = \frac{2J}{k_B \ln(1+\sqrt{2})} \approx \frac{2.269 J}{k_B}$

Near $T_c$, the spontaneous magnetization vanishes as:

$M \propto (T_c - T)^{1/8}$

giving the exact critical exponent $\beta = 1/8$. Compare this to the mean field prediction of $\beta = 1/2$; the difference is substantial and demonstrates that fluctuations matter enormously in 2D.

Onsager's solution

Onsager used the transfer matrix method, which reformulates the partition function as a product of matrices whose eigenvalues determine the free energy. The largest eigenvalue dominates in the thermodynamic limit.

His solution provided exact values for the free energy and specific heat (which diverges logarithmically at $T_c$), and confirmed that the critical exponents are non-classical. No exact solution exists for the 3D Ising model, which remains one of the outstanding unsolved problems in statistical mechanics.

Numerical methods

For systems without exact solutions (which is most of them, including the 3D Ising model), computational methods are essential.

Monte Carlo simulations

Monte Carlo methods use random sampling to estimate thermodynamic averages. Rather than summing over all $2^N$ spin configurations (impossible for large $N$), you generate a sequence of configurations sampled according to the Boltzmann distribution $P(\{s_i\}) \propto e^{-\beta H}$. Averages over this sequence converge to true ensemble averages.

Metropolis algorithm

The Metropolis algorithm is the most common way to implement Monte Carlo for the Ising model:

1. Pick a spin $s_i$ at random.
2. Compute the energy change $\Delta E$ that would result from flipping it.
3. If $\Delta E \leq 0$ (the flip lowers the energy), accept the flip.
4. If $\Delta E > 0$, accept the flip with probability $e^{-\Delta E / k_B T}$.
5. Repeat many times to generate a Markov chain of configurations.

This procedure satisfies detailed balance, guaranteeing convergence to the equilibrium Boltzmann distribution. Near $T_c$, the algorithm suffers from critical slowing down because the correlation length diverges and local spin flips become inefficient. Cluster algorithms like Wolff or Swendsen-Wang address this by flipping correlated groups of spins at once.

Critical exponents

Critical exponents describe how thermodynamic quantities diverge or vanish as the system approaches the critical point. They are among the most important quantities in the theory of phase transitions.

Definition of critical exponents

Near $T_c$, physical quantities follow power laws in the reduced temperature $t = (T - T_c)/T_c$:

• Order parameter: $m \propto |t|^{\beta}$ (for $T < T_c$)
• Susceptibility: $\chi \propto |t|^{-\gamma}$
• Correlation length: $\xi \propto |t|^{-\nu}$
• Specific heat: $C \propto |t|^{-\alpha}$

These exponents are not all independent. They satisfy scaling relations such as:

$\alpha + 2\beta + \gamma = 2 \quad \text{(Rushbrooke)}$ $\gamma = \nu(2 - \eta) \quad \text{(Fisher)}$

Universality classes

One of the deepest results in the theory of critical phenomena: systems with the same dimensionality, symmetry of the order parameter, and range of interactions share identical critical exponents, regardless of microscopic details. This grouping is called a universality class.

For example, the 3D Ising universality class ($\beta \approx 0.326$, $\gamma \approx 1.237$, $\nu \approx 0.630$) describes not just magnetic systems but also the liquid-gas critical point, binary fluid demixing, and certain structural transitions. The 2D Ising class has different exponents ($\beta = 1/8$, $\gamma = 7/4$, $\nu = 1$). Mean field exponents ($\beta = 1/2$, $\gamma = 1$, $\nu = 1/2$) apply above $d = 4$.

Applications

The Ising model's two-state structure maps onto many physical systems beyond simple ferromagnets.

Ferromagnetism

The most direct application. The Ising model describes how atomic magnetic moments in materials like iron and nickel collectively align below the Curie temperature, producing spontaneous magnetization. It captures the essential physics of magnetic domains, hysteresis, and the paramagnetic-to-ferromagnetic transition.

Antiferromagnetism

With $J < 0$, the Ising model describes materials where neighboring spins prefer antiparallel alignment, forming an alternating pattern on sublattices. The transition temperature is called the Néel temperature. Examples include chromium and many transition metal oxides like MnO. Antiferromagnetic ordering is also relevant to exchange bias in spintronic devices and plays a role in the physics of high-temperature superconductors.

Spin glasses

When the coupling constants $J_{ij}$ are random (varying in sign and magnitude from bond to bond), the Ising model describes spin glasses. These systems exhibit frustration (not all interactions can be simultaneously satisfied), leading to a complex energy landscape with many metastable states and extremely slow relaxation dynamics. The Sherrington-Kirkpatrick model is a mean field spin glass, and its mathematical structure connects to optimization problems and neural networks.

Extensions of Ising model

Several important models generalize the Ising framework by allowing more spin states or continuous spin variables.

Potts model

The Potts model generalizes the Ising model from 2 to $q$ possible spin states per site. The Ising model is the special case $q = 2$. For $q \leq 4$ in 2D, the transition is continuous; for $q > 4$ in 2D, it becomes first-order. Applications include modeling grain growth in polycrystalline materials and certain problems in computational biology.

XY model

In the XY model, each spin is a unit vector confined to a plane, so the spin variable is a continuous angle $\theta_i$. The 2D XY model does not have conventional long-range order (by the Mermin-Wagner theorem), but it exhibits the Kosterlitz-Thouless transition, a topological phase transition driven by the unbinding of vortex-antivortex pairs. It describes superfluid helium films and arrays of Josephson junctions.

Heisenberg model

The Heisenberg model allows spins to point in any direction in three-dimensional space ($\vec{s}_i$ is a 3-component unit vector). It has full $O(3)$ rotational symmetry, making it more realistic for isotropic magnetic materials. It belongs to a different universality class than the Ising model and supports spin-wave excitations (magnons) that the discrete Ising model cannot capture.

Ising model in statistical mechanics

The Ising model illustrates the full machinery of statistical mechanics: starting from a microscopic Hamiltonian, you construct the partition function, derive thermodynamic potentials, and extract measurable quantities.

Partition function

The partition function sums the Boltzmann weight over every possible spin configuration:

$Z = \sum_{\{s_i\}} e^{-\beta H(\{s_i\})}$

where $\beta = 1/(k_B T)$ and the sum runs over all $2^N$ configurations. From $Z$, you can compute any equilibrium thermodynamic quantity. The difficulty is that this sum grows exponentially with system size, which is why exact evaluation is only possible in special cases.

Free energy

The Helmholtz free energy is obtained from the partition function:

$F = -k_B T \ln Z$

At equilibrium, $F$ is minimized for given temperature and external field. Thermodynamic quantities follow as derivatives: the magnetization is $m = -\partial F / \partial h$, and the entropy is $S = -\partial F / \partial T$. Phase transitions show up as non-analyticities in $F$ or its derivatives (e.g., a discontinuity in the first derivative signals a first-order transition; a divergence in the second derivative signals a continuous transition).

Correlation functions

The two-point correlation function measures how the spin at one site is statistically related to the spin at another:

$G(r) = \langle s_i s_j \rangle - \langle s_i \rangle \langle s_j \rangle$

where $r = |r_i - r_j|$. In the disordered phase, correlations decay exponentially: $G(r) \sim e^{-r/\xi}$, where $\xi$ is the correlation length. At the critical point, $\xi \to \infty$ and correlations decay as a power law: $G(r) \sim r^{-(d-2+\eta)}$, where $\eta$ is another critical exponent. The fluctuation-dissipation theorem connects the susceptibility to the integral of the correlation function: $\chi = \beta \sum_r G(r)$.

Experimental realizations

Magnetic materials

Ferromagnets like iron and nickel, and antiferromagnets like MnO and $\text{FeF}_2$, are well-described by Ising-type models when the crystal field produces strong uniaxial anisotropy (restricting moments to point along one axis). Experimental techniques such as neutron scattering, magnetometry, and specific heat measurements have confirmed Ising critical exponents in these materials.

Binary alloys

In alloys like $\beta$-brass (Cu-Zn), each lattice site is occupied by one of two atom types. Assigning $s_i = +1$ for Cu and $s_i = -1$ for Zn maps the ordering problem directly onto the Ising model. These systems exhibit order-disorder transitions as a function of temperature, detectable through X-ray diffraction and resistivity measurements.

Lattice gas models

A lattice gas assigns $s_i = +1$ to an occupied site and $s_i = -1$ to a vacant one. This maps exactly onto the Ising model with a shifted chemical potential playing the role of the magnetic field. It describes adsorption of atoms on surfaces (e.g., hydrogen on palladium) and the liquid-gas transition. The critical point of the lattice gas belongs to the same universality class as the Ising model.