Phase Transitions: Order from Disorder

The Ising model’s central phenomenon is a phase transition: a sharp, qualitative change in the system’s collective behavior as temperature crosses a critical value. Below the critical temperature, the system is ordered; above it, disordered. The transition is not gradual. In the thermodynamic limit (infinite system size), it is abrupt --- a discontinuity in the derivative of the free energy that signals a fundamental change in the system’s state.

The Critical Temperature

For the two-dimensional Ising model on a square lattice with nearest-neighbor coupling J and no external field, Onsager proved in 1944 that the critical temperature is:

T_c = 2J / (k ln(1 + sqrt(2))) approximately equal to 2.269 J/k

This is an exact result. Below T_c, the equilibrium state has nonzero magnetization. Above T_c, the equilibrium magnetization is zero. At T_c exactly, the system is at the boundary between these two regimes, and its behavior is qualitatively different from either.

In one dimension, Ising (1925) proved that no phase transition exists at any finite temperature. A one-dimensional chain of spins can always disorder through a local fluctuation that breaks the chain into two domains; the energy cost is 2J regardless of chain length, and at any finite temperature this cost is overcome by thermal fluctuations. The system cannot sustain long-range order.

In two dimensions, the energy cost of a domain wall scales with the system’s linear size L, because the wall must cut across the entire lattice. At low enough temperature, the cost of creating a system-spanning domain wall exceeds the entropy gain, and ordered domains are stable. This is the Peierls argument (1936), which proved the existence of a phase transition in two dimensions before Onsager derived the exact solution.

In three dimensions, no exact solution exists. The critical temperature is known from high-precision Monte Carlo simulations to be approximately T_c = 4.5115 J/k for the simple cubic lattice (Ferrenberg and Landau, 1991).

Spontaneous Symmetry Breaking

The Hamiltonian H = -J sum(s_i s_j) at h = 0 is invariant under global spin flip: replacing every s_i with -s_i leaves H unchanged. Both the all-up and all-down configurations are ground states with equal energy. There is nothing in the rules that distinguishes up from down.

Above T_c, this symmetry is manifest in the equilibrium state: the average magnetization M = <(1/N) sum(s_i)> is zero. The system fluctuates symmetrically around zero, spending equal time in states with net positive and net negative magnetization.

Below T_c, the symmetry breaks. The system settles into a state with M > 0 or M < 0, and the fluctuations are centered on this nonzero value. The system has “chosen” a direction even though the Hamiltonian did not specify one. This is spontaneous symmetry breaking: the equilibrium state has lower symmetry than the underlying rules.

In a finite system, spontaneous symmetry breaking does not strictly occur. A finite system at any temperature will eventually fluctuate between the up-magnetized and down-magnetized states, because the energy barrier between them is finite. The time required for this fluctuation, however, grows exponentially with system size: tau ~ exp(c * L^(d-1)), where c depends on temperature. For any macroscopic system (L ~ 10^8 lattice spacings), this time exceeds the age of the universe at temperatures well below T_c. Spontaneous symmetry breaking is a mathematical idealization of the thermodynamic limit, but it is an extraordinarily good description of any real finite system.

The Order Parameter and Its Divergences

The magnetization M is the order parameter of the Ising phase transition. It quantifies the degree of order in the system:

  • Above T_c: M = 0.
  • Below T_c: |M| > 0 and increases as temperature decreases, reaching M = 1 (full alignment) at T = 0.

Near T_c, the magnetization vanishes as a power law:

M ~ (T_c - T)^beta

where beta = 1/8 in two dimensions (Yang, 1952). This means the magnetization approaches zero continuously as T approaches T_c from below --- the transition is second-order (continuous).

Several other quantities diverge at T_c:

Susceptibility chi = dM/dh measures how sensitive the magnetization is to an applied field. It diverges as chi ~ |T - T_c|^(-gamma) with gamma = 7/4 in 2D. At T_c, the system is infinitely responsive to an infinitesimal field.

Correlation length xi measures the spatial extent of spin-spin correlations. Away from T_c, the correlation function <s_i s_j> - <s_i><s_j> decays exponentially with distance: ~ exp(-|i-j|/xi). At T_c, xi diverges: correlations extend across the entire system, decaying only as a power law ~ |i-j|^(-(d-2+eta)) with eta = 1/4 in 2D.

Specific heat C = dE/dT diverges logarithmically at T_c in 2D: C ~ -ln|T - T_c|. This logarithmic divergence is a special feature of two dimensions; in three dimensions, the specific heat diverges as a weak power law with exponent alpha approximately 0.110.

The divergence of the correlation length is the most physically significant. It means that at T_c, the system has no characteristic length scale. Fluctuations occur at every scale simultaneously --- from the lattice spacing to the system size. A snapshot of the spin configuration at T_c looks the same at every magnification, statistically. This scale-free behavior is what makes the critical point fundamentally different from the ordered and disordered phases.

Onsager’s Exact Solution

The two-dimensional Ising model is one of the very few interacting many-body systems with an exact analytical solution. Onsager published it in 1944 in Physical Review. The calculation uses the transfer matrix method: the partition function is expressed as the trace of a product of matrices, one for each row of the lattice, and the free energy in the thermodynamic limit is determined by the largest eigenvalue of the transfer matrix.

The key results:

Free energy per site in the thermodynamic limit is an explicit function of temperature involving elliptic integrals. The free energy is analytic everywhere except at T_c, where a logarithmic singularity appears in the specific heat.

Exact T_c: derived from the condition that the transfer matrix has a degenerate leading eigenvalue, giving T_c = 2J / (k ln(1 + sqrt(2))).

Specific heat: diverges logarithmically at T_c, a result that surprised physicists accustomed to mean-field theory, which predicts a finite discontinuity rather than a divergence.

Yang (1952) extended Onsager’s work to derive the spontaneous magnetization exactly: M = (1 - sinh^(-4)(2J/kT))^(1/8) for T < T_c. The 1/8 exponent was the first exact critical exponent and became one of the benchmarks against which universality would later be tested.

The three-dimensional Ising model has resisted exact solution despite more than seventy years of effort. The critical exponents are known to high precision from Monte Carlo simulation and from the conformal bootstrap method (El-Showk et al., 2014), but no closed-form solution for the free energy exists. Whether such a solution is possible remains an open question.

The significance of the exact solution is not merely technical. It provided the first rigorous demonstration that a phase transition can occur in a system with short-range interactions on a finite-dimensional lattice. Before Onsager, it was debated whether mean-field theory --- which predicts phase transitions but ignores spatial correlations --- was qualitatively correct or whether fluctuations would wash out the transition. Onsager showed that the transition survives, but with different exponents than mean-field theory predicts. This distinction ultimately led to the renormalization group.

Further Reading