The Ising Model and Exact Results

Cooperative ordering — a magnet acquiring a spontaneous moment, a liquid condensing from its vapor, an alloy segregating into ordered sublattices — needs a model simple enough to solve yet rich enough to have a transition. The Ising model is that model. It places a two-valued spin on each site of a lattice and couples neighboring spins so that alignment lowers the energy. Whether the resulting competition between energy and entropy produces long-range order depends on the dimension of the lattice, and the answer can be computed exactly in one dimension and, for the zero-field case, in two.

The Ising Hamiltonian

To each site $i$ of a lattice assign a spin $s_i\in \{+1,-1\}$. Neighboring spins interact, and an external field $h$ couples to each spin:

where $⟨ij⟩$ runs over nearest-neighbor pairs and $J$ is the coupling constant. For $J>0$ aligned neighbors ($s_i{s}_j=+1$) lower the energy, favoring ferromagnetic order; for $J<0$ the model favors antialignment. The partition function is the sum over all $2^N$ spin configurations,

The same Hamiltonian carries three physical readings, and every exact result transfers among them without change.

• Uniaxial ferromagnet — $s_i$ is a spin that points along or against an easy axis, $h$ is an applied magnetic field, and the order parameter is the magnetization per spin $m=⟨s_i⟩$.
• Lattice gas — set $n_i=(1+s_i)/2\in \{0,1\}$ for an empty or occupied cell. Nearest-neighbor attraction between occupied cells maps onto the spin coupling, and the field maps onto the chemical potential. The liquid-gas critical point becomes the Ising critical point.
• Binary alloy — $s_i=\pm 1$ labels which of two metals sits at site $i$, and $J$ measures whether like or unlike neighbors are preferred, controlling order-disorder transitions in the alloy.

The one-dimensional chain by transfer matrix

On a ring of $N$ sites the energy is a sum of identical nearest-neighbor terms, which lets the Boltzmann weight factorize into a product of two-site pieces. Symmetrize the field between the two sites of each bond and define a $2\times 2$transfer matrix whose rows and columns are indexed by the spin values,

Summing the product of bond weights around the ring is a matrix trace:

with ${\lambda }_{\pm }$ the two eigenvalues of $\mathsf{T}$. In the thermodynamic limit the larger eigenvalue dominates, $Z\to {\lambda }_{+}^N$, and the free energy per spin is

At zero field the eigenvalues are ${\lambda }_{\pm }=e^{\beta J}\pm e^{-\beta J}$, that is ${\lambda }_{+}=2\cosh \beta J$ and ${\lambda }_{-}=2\sinh \beta J$, so

This function is analytic for every $T>0$: $\cosh \beta J$ is smooth and strictly positive, so no derivative of $f$ ever diverges at finite temperature. The one-dimensional Ising chain has no phase transition except at $T=0$.

The eigenvalue ratio fixes the correlation length. The connected correlation of two spins $r$ sites apart decays as $⟨s_0{s}_r{⟩}_c=({\lambda }_{-}/{\lambda }_{+}{)}^r\equiv e^{-r/\xi }$, so

As $T\to 0$, $\beta J\to \infty$ and $\coth \beta J\to 1$, driving $\xi \to \infty$: order becomes long-ranged only in the zero-temperature limit. At any positive temperature $\xi$ is finite and the chain is disordered on scales beyond it.

Why one dimension cannot order

The absence of a transition in one dimension has a direct physical cause, Landau's domain-wall argument. Start from the perfectly ordered ground state, all spins up. Introduce a single domain wall: everything to the left of one bond stays up, everything to the right flips down. Only the one bond across the wall is now unsatisfied, costing energy

The wall can be placed on any of the $N$ bonds of the chain, so inserting it raises the entropy by

The free-energy change of adding one wall is therefore

For any fixed $T>0$, the entropy term grows without bound as $N\to \infty$, so $\Delta F\to -\infty$: creating domain walls always lowers the free energy. Walls proliferate, the chain breaks into arbitrarily many domains, and long-range order is destroyed. The order survives only at exactly $T=0$, where the entropy term vanishes.

The same accounting explains why two dimensions differ. A domain wall in two dimensions is a closed loop of length $\ell$, with energy $2J\ell$. The number of loops of length $\ell$ grows roughly as ${\mu }^{\ell }$ with a lattice-dependent constant $\mu$ (near $3$ on the square lattice), giving entropy $k_B\ell \ln \mu$. The free-energy change per unit wall length,

stays positive for $T<2J/(k_B\ln \mu )$: at low enough temperature long walls are suppressed, and the ordered phase is stable. This is Peierls's argument, and it establishes that a genuine finite-temperature transition exists in two dimensions. The competition that one dimension resolves in favor of disorder, two dimensions resolves in favor of order below a critical temperature.

Onsager's exact two-dimensional results

Onsager solved the two-dimensional Ising model on the square lattice at zero field, and the result is the benchmark against which every approximate theory is measured. The critical temperature separating ordered from disordered phases is fixed by the condition $\sinh (2{\beta }_{c}J)=1$, giving¹

The heat capacity does not jump at $T_c$; it diverges logarithmically on both sides,

a symmetric spike with no finite peak height. This is the exponent $\alpha =0$ in its logarithmic realization, and it already contradicts the finite discontinuity that the Ehrenfest scheme and mean-field theory both predict. Below $T_c$ the spontaneous magnetization rises as a power of the reduced temperature,

where here $\beta$ denotes the magnetization critical exponent, not the inverse temperature. The value $1/8$ differs sharply from the mean-field $1/2$ of the next lesson, and the discrepancy is the experimental fingerprint that mean-field theory misses the physics of a two-dimensional critical point.

Universality across the three interpretations

The three physical systems packed into one Hamiltonian share not only the model but its critical numbers. The magnet, the lattice gas, and the binary alloy all have the same $T_c$ structure, the same logarithmic heat-capacity divergence, and the same magnetization exponent $1/8$ in two dimensions, because the exponents depend only on the dimensionality and the symmetry of the order parameter, not on the microscopic details of the coupling. This is the first concrete instance of universality: the liquid-gas critical point of a real fluid and the Curie point of a real magnet fall into the same class as the two-dimensional (or, in nature, three-dimensional) Ising model and share its exponents.

The exact solutions bound what any theory must reproduce: in one dimension no transition at positive temperature, in two dimensions a transition at $k_B{T}_c/J=2.269$ with a logarithmic heat capacity and $\beta =1/8$. The mean-field approximation of the next lesson gets the existence of a transition right in high enough dimension but the exponents wrong, and reconciling the two is the task of Landau theory and the renormalization group.

Summary

• The Ising Hamiltonian $H=-J{\sum }_{⟨ij⟩}{s}_i{s}_j-h{\sum }_i{s}_i$ models a uniaxial ferromagnet, a lattice gas (hence the liquid-gas critical point), and a binary alloy, with the same exact results in each reading.
• The one-dimensional chain is solved by the transfer matrix: $Z={\lambda }_{+}^N+{\lambda }_{-}^N$, free energy $f=-k_{B}T\ln (2\cosh \beta J)$, analytic for all $T>0$, so there is no finite-temperature transition; the correlation length ${\xi }^{-1}=\ln \coth \beta J$ diverges only as $T\to 0$.
• Landau's domain-wall argument gives the reason: in one dimension a wall costs fixed energy $2J$ but gains entropy $k_{B}T\ln N$, so walls always proliferate; in two dimensions a wall is a loop whose free energy per length stays positive below a critical temperature, allowing order.
• Onsager's two-dimensional solution gives $k_B{T}_c/J=2/\ln (1+\sqrt{2})\approx 2.269$, a logarithmically divergent heat capacity, and magnetization exponent $\beta =1/8$ — exponents shared by every system in the same universality class.