Ising model and phase transitions

Intuition [Beginner]

The Ising model is a toy model of a magnet. Place "spins" on a lattice — each spin points up ($+1$) or down ($-1$). Neighbouring spins want to align: misalignment costs energy. Temperature adds randomness, flipping spins against the alignment tendency.

At low temperature, alignment wins: most spins point the same way, and the material carries a net magnetisation (ferromagnetic phase). At high temperature, thermal flipping dominates: spins point randomly, and net magnetisation vanishes (paramagnetic phase).

The critical temperature $T_c$ separates these two behaviours. Below $T_c$: ordered. Above $T_c$: disordered. At $T_c$ the magnetisation drops to zero — a phase transition.

Visual [Beginner]

Picture a square grid. At each lattice site sits an arrow, pointing either up or down. Two configurations illustrate the two phases.

Low temperature ($T<T_c$). Nearly every arrow points up. A few scattered sites point down — thermal fluctuations — but the majority alignment is overwhelming. The net magnetisation $m$ (average spin value) is close to $+1$. The system is ordered.

High temperature ($T>T_c$). Arrows point up and down in equal proportion, arranged randomly. Pick any site and its four neighbours: roughly half agree, half disagree. The net magnetisation $m$ fluctuates around zero. The system is disordered.

At $T_c$. The system sits on a knife edge. Large patches of aligned spins form, break apart, and reform. These patches — correlated regions — grow without bound as $T\to T_c$ from above. The correlation length $\xi$ (typical patch size) diverges at $T_c$. This divergence is the microscopic signature of the phase transition.

Worked example [Beginner]

The 1D Ising chain has no phase transition at any $T>0$.

Consider a line of $N$ spins, each $\pm 1$, with nearest-neighbour interactions. At any positive temperature, thermal fluctuations can always "break" a domain of aligned spins at a cost of just one boundary (a domain wall). The energy penalty for creating a domain wall is finite (proportional to the coupling $J$), and for any $T>0$ the entropy gain from creating more domain walls always wins in the thermodynamic limit $N\to \infty$. The result: the 1D chain is always disordered for $T>0$ — no phase transition.

The 2D square lattice does have a phase transition.

In two dimensions, breaking a domain of aligned spins costs energy proportional to the boundary length (not a single bond), and the entropy gain from roughening the boundary is only logarithmic in the boundary length. The energy cost grows faster than the entropy gain, so below some temperature $T_c$ the ordered phase is stable. Onsager (1944) obtained the exact result:

$$\frac{k_B{T}_c}{J}=\frac{2}{\ln (1+\sqrt{2})}\approx \mathrm{2.269.}$$

Below this temperature the 2D Ising model is ferromagnetic; above it, paramagnetic. The spontaneous magnetisation vanishes as

$$m\sim (T_c-T{)}^{\beta },\beta =1/8,$$

a critical exponent computed exactly from Onsager's solution.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Consider a lattice $\Lambda$ (finite or infinite) with sites indexed by $i$. At each site sits a spin variable $s_i\in \{+1,-1\}$. A spin configuration is an assignment $\{s_i{\}}_{i\in \Lambda }$.

The Ising Hamiltonian with coupling constant $J$ and external field $h$ is

$$\boxed{H(\{s_i\})=-J\sum _{⟨i,j⟩}{s}_i{s}_j-h\sum _{i\in \Lambda }{s}_i}$$

where $⟨i,j⟩$ denotes unordered nearest-neighbour pairs. The coupling $J>0$ favours alignment (ferromagnetic); $J<0$ favours anti-alignment (antiferromagnetic). The field $h$ biases spins toward $+1$ (for $h>0$) or $-1$ (for $h<0$).

The canonical partition function at inverse temperature $\beta =1/(k_{B}T)$ on a finite lattice $\Lambda$ of $N$ sites is

$$Z(\beta ,h)=\sum _{\{s_i=\pm 1\}}{e}^{-\beta H(\{s_i\})},$$

a sum over all $2^N$ spin configurations. Thermodynamic quantities follow:

$$⟨E⟩=-\frac{\partial \ln Z}{\partial \beta },⟨m⟩=\frac{1}{N\beta }\frac{\partial \ln Z}{\partial h},\chi =\frac{1}{N\beta }\frac{{\partial }^2\ln Z}{\partial h^2},$$

where $m=\frac{1}{N}{\sum }_i{s}_i$ is the magnetisation per site and $\chi$ is the magnetic susceptibility. The spontaneous magnetisation is $m_0={\lim }_{h\to 0^{+}}{\lim }_{N\to \infty }⟨m⟩$ — the order of limits matters.

Mean-field theory (Bragg-Williams)

Replace each spin's interaction with its neighbours by an interaction with the average magnetisation $m=⟨s_i⟩$. Assume all sites are equivalent. The self-consistency (mean-field) equation for the 2D square lattice (coordination number $z=4$) is

$$m=\tanh (\beta zJm+\beta h).$$

For $h=0$ and $T>T_c^{\mathrm{MF}}=zJ/k_B=4J/k_B$, the only solution is $m=0$. For $T<T_c^{\mathrm{MF}}$, a nonzero solution appears. The mean-field critical temperature $T_c^{\mathrm{MF}}=4J/k_B$ overestimates the true $T_c\approx 2.269J/k_B$ by roughly 76% — mean-field theory neglects fluctuations and is quantitatively inaccurate in low dimensions, though it captures the qualitative physics of the transition.

Mean-field critical exponents: ${\beta }_{\mathrm{MF}}=1/2$, ${\alpha }_{\mathrm{MF}}=0$ (discontinuity), ${\gamma }_{\mathrm{MF}}=1$, ${\delta }_{\mathrm{MF}}=3$. These differ from the exact 2D values and from measured 3D values; the discrepancy is resolved by the renormalization group 11.07.01 pending.

Exact solution in 1D: transfer matrix

For the 1D chain with $N$ sites and periodic boundary conditions ($s_{N+1}=s_1$), write the partition function as

$$Z=\sum _{\{s_i\}}\prod _{i=1}^N{e}^{\beta J{s}_i{s}_{i+1}+\beta h{s}_i}=\mathrm{Tr}\left(T^N\right),$$

where the transfer matrix $T$ is a $2\times 2$ matrix with entries $T_{s,s^{\prime }}=\exp (\beta Js{s}^{\prime }+\frac{1}{2}\beta h(s+s^{\prime }))$ for $s,s^{\prime }\in \{+1,-1\}$:

$$T=\left(\begin{array}{cc}{e}^{\beta J+\beta h}& e^{-\beta J}\\ e^{-\beta J}& e^{\beta J-\beta h}\end{array}\right).$$

Since $T$ is real symmetric, it has real eigenvalues ${\lambda }_{+}>{\lambda }_{-}$. Then $\mathrm{Tr}(T^N)={\lambda }_{+}^N+{\lambda }_{-}^N$, and in the thermodynamic limit $N\to \infty$,

$$\frac{1}{N}\ln Z\to \ln {\lambda }_{+}.$$

The free energy per site is $f=-k_{B}T\ln {\lambda }_{+}$. Since ${\lambda }_{+}$ is an analytic function of $\beta$ and $h$ (eigenvalues of a finite matrix are analytic), the free energy is analytic for all $T>0$. No singularity — no phase transition — confirming the beginner-tier result.

Onsager's solution in 2D

Onsager (1944) computed the exact free energy of the 2D Ising model on a square lattice at zero field ($h=0$):

$$-\frac{\beta f}{N}=\ln 2+\frac{1}{2\pi }{\int }_0^{\pi }d\varphi \ln [{\cosh }^2(2\beta J)-\sinh (2\beta J)(\cos \varphi +\cos \varphi \left)\right],$$

which simplifies to

$$-\frac{\beta f}{N}=\ln (2\cosh 2\beta J)+\frac{1}{2\pi }{\int }_0^{\pi }\ln \left[\frac{1+\sqrt{1-{\kappa }^2{\sin }^2\varphi }}{2}\right]d\varphi ,$$

where $\kappa =2\sinh (2\beta J)/{\cosh }^2(2\beta J)$. The integral is analytic except at $\kappa =1$, which occurs at $\sinh (2\beta J)=1$, i.e., $k_B{T}_c/J=2/\ln (1+\sqrt{2})$.

The spontaneous magnetisation (Yang, 1952):

$$m_0=\{\begin{array}{ll}(1-{\sinh }^{-4}(2\beta J){)}^{1/8}& T<T_c\\ 0& T\ge T_c\end{array}$$

confirming the critical exponent $\beta =1/8$.

Critical exponents and universality

Near $T_c$, thermodynamic quantities follow power laws:

Quantity	Behaviour	Exponent	Exact 2D Ising	Mean-field
Specific heat	$C\sim |T-T_c{|}^{-\alpha }$	$\alpha$	$0$ (log)	$0$ (jump)
Magnetisation	$m\sim (T_c-T{)}^{\beta }$	$\beta$	$1/8$	$1/2$
Susceptibility	$\chi \sim |T-T_c{|}^{-\gamma }$	$\gamma$	$7/4$	$1$
Critical isotherm	$m\sim h^{1/\delta }$ at $T_c$	$\delta$	$15$	$3$
Correlation length	$\xi \sim |T-T_c{|}^{-\nu }$	$\nu$	$1$	$1/2$
Correlation function	$G(r)\sim r^{-(d-2+\eta )}$ at $T_c$	$\eta$	$1/4$	$0$

The 2D Ising exponents satisfy the scaling relations: $\alpha +2\beta +\gamma =2$, $\gamma =\beta (\delta -1)$, $\nu d=2-\alpha$, $\gamma =(2-\eta )\nu$.

Universality. The critical exponents depend only on the spatial dimension $d$ and the symmetry of the order parameter (here: ${\mathbb{Z}}_2$, the spin-flip symmetry $s_i\to -s_i$), not on microscopic details such as the lattice type (square, triangular, honeycomb) or the coupling strength $J$. Systems sharing the same $d$ and symmetry belong to the same universality class. The 2D Ising universality class includes liquid-gas transitions at surfaces, binary alloys, and adsorption models — all share $\beta =1/8$, $\gamma =7/4$, etc.

Key theorem with proof [Intermediate+]

Theorem (Onsager, 1944). The free energy per site of the 2D Ising model on a square lattice with zero external field is given by

$$f=-k_{B}T\left[\ln (2\cosh 2K)+\frac{1}{2\pi }{\int }_0^{\pi }\ln \left(\frac{1+\sqrt{1-{\kappa }^2{\sin }^2\varphi }}{2}\right)d\varphi \right],$$

where $K=\beta J$ and $\kappa =2\sinh (2K)/{\cosh }^2(2K)$. This free energy is analytic in $T$ except at the point $\kappa =1$ (i.e., $\sinh (2K)=1$), which determines $T_c=2J/(k_B\ln (1+\sqrt{2}))$. At $T_c$ the specific heat diverges logarithmically.

Proof strategy (not the full proof — that runs to dozens of pages).

Step 1. Write the partition function as a sum over closed polygons on the dual lattice using the identity $e^{\beta J{s}_i{s}_j}=\cosh (\beta J)(1+s_i{s}_j\tanh \beta J)$ and expanding the product over all bonds. The resulting series is a sum over non-intersecting closed loops, each weighted by $\tanh (\beta J{)}^{\text{perimeter}}$.

Step 2. Map the loop sum to a free-fermion problem. Introduce a set of Grassmann (anticommuting) variables on the bonds and express the loop generating function as a Gaussian Grassmann integral — equivalently, a Pfaffian (or determinant) of a certain antisymmetric matrix.

Step 3. Evaluate the Pfaffian/determinant. In the thermodynamic limit ($N\to \infty$) this reduces to a Fourier transform of the matrix, and the eigenvalues are computed analytically. The free energy follows from $(1/N)\ln Z=(1/N)\ln \det M\to \int \ln (\text{eigenvalue})dk$.

Step 4. Identify the singularity. The integrand develops a zero denominator when $\kappa =1$, signalling the phase transition. The nature of the singularity (logarithmic divergence of the specific heat) is determined by expanding the integrand near the critical point.

Onsager's original derivation used a different route (the star-triangle transformation and commutation of transfer matrices), later simplified by Kac and Ward (1952) via the combinatorial/Pfaffian approach outlined above, and made rigorous by Pfeuty and Toulouse. ∎

Counterexamples to common slips

• Mean-field theory predicts $T_c^{\mathrm{MF}}=zJ/k_B=4J/k_B$ for the 2D square lattice, but the exact value is $T_c\approx 2.269J/k_B$. Mean-field theory always overestimates $T_c$ in finite dimensions because it neglects fluctuations that disorder the system.
• The order of limits in defining spontaneous magnetisation matters: $m_0={\lim }_{h\to 0^{+}}{\lim }_{N\to \infty }⟨m⟩$ is nonzero for $T<T_c$, but ${\lim }_{N\to \infty }{\lim }_{h\to 0^{+}}⟨m⟩=0$ by the spin-flip symmetry of $H$. The thermodynamic limit must be taken first.
• The Ising model on a finite lattice has no phase transition: $Z$ is a finite sum of analytic terms, so $f$ is analytic for all $T$. The singularity appears only in the thermodynamic limit $N\to \infty$.

Exercises [Intermediate+]

Yang-Lee theory [Master]

The Ising partition function on a finite lattice is a polynomial in $z=e^{-2\beta h}$ (the fugacity):

$$Z_N(\beta ,h)=\sum _{k=0}^N{a}_k(\beta )z^k,$$

with real coefficients $a_k(\beta )$. In the complex-$z$ plane, $Z_N$ has $N$ zeros (the Lee-Yang zeros). For a ferromagnetic Ising model ($J>0$), the Lee-Yang circle theorem (Yang and Lee, 1952) states:

Theorem (Lee-Yang). For the ferromagnetic Ising model on any lattice with $J>0$, all zeros of $Z_N$ in the complex fugacity $z=e^{-2\beta h}$ plane lie on the unit circle $|z|=1$.

In the thermodynamic limit $N\to \infty$, the zeros accumulate on the unit circle and may pinch the real axis at $z=1$ (i.e., $h=0$). The phase transition occurs where the zeros first touch the real axis; the singularities of $f(\beta ,h)$ in the thermodynamic limit are the accumulation points of the Lee-Yang zeros.

The Yang-Lee theory reframes phase transitions as a problem in the analytic structure of $Z$ in the complex fugacity plane. A phase transition is possible if and only if the zeros of $Z_N$ approach the real axis in the thermodynamic limit. This provides a mathematically precise definition of a phase transition as a singularity of $f$ in the thermodynamic limit.

Pirogov-Sinai theory [Master]

The Ising model has a single order parameter (magnetisation) and a single first-order transition at $h=0$ below $T_c$. For systems with multiple competing phases — for example, the Potts model with $q\ge 3$ states, or lattice models with multi-spin interactions — the phase diagram can contain first-order lines ending at critical points, coexistence surfaces, and metastable states.

Pirogov-Sinai theory (1975) provides rigorous conditions under which the phase diagram of a lattice spin system can be constructed by perturbing from the zero-temperature ground states. The key inputs are: (i) a finite set of periodic ground states, (ii) a contour representation of excitations (generalising the domain-wall picture of the Peierls argument), and (iii) a Peierls condition that the energy cost of a contour is proportional to its volume (in 2D, its length).

The theory establishes that at sufficiently low temperature, the phase diagram is a small perturbation of the zero-temperature phase diagram, with one Gibbs measure per ground state. For the Ising model, Pirogov-Sinai reproduces the two-phase coexistence at $h=0$ for $T<T_c$ as a rigorous result.

Scaling hypothesis and finite-size scaling [Master]

The scaling hypothesis (Widom, 1965; Kadanoff, 1966) posits that near $T_c$, the singular part of the free energy takes the homogeneous form

$$f_{\mathrm{sing}}(t,h)=|t{|}^{2-\alpha }{F}_{\pm }\left(\frac{h}{|t{|}^{\beta \delta }}\right),$$

where $t=(T-T_c)/T_c$ is the reduced temperature. Differentiating this ansatz generates all six critical exponents and predicts the scaling relations among them ($\alpha +2\beta +\gamma =2$, etc.). The scaling hypothesis is derived, not assumed, in the renormalization-group framework 11.07.01 pending.

Finite-size scaling. On a finite lattice of linear size $L$ with periodic boundary conditions, the free energy has no singularity. Instead, thermodynamic quantities exhibit size-dependent rounding. The finite-size scaling ansatz predicts that near $T_c$,

$$m(T,L)=L^{-\beta /\nu }\tilde{m}\left(L^{1/\nu }t\right),$$

where $\tilde{m}$ is a universal scaling function. At $T=T_c$ ($t=0$), $m\sim L^{-\beta /\nu }$. The specific heat peak shifts from $T_c$ to a pseudocritical temperature $T_c(L)=T_c+a{L}^{-1/\nu }+\cdots$. Finite-size scaling provides the standard method for extracting critical exponents from Monte Carlo simulations on finite lattices.

Monte Carlo methods: the Metropolis algorithm [Master]

The Metropolis algorithm (Metropolis et al., 1953) generates spin configurations sampled from the Boltzmann distribution $P(\{s_i\})\propto e^{-\beta H}$ without computing $Z$.

Algorithm. At each step: (1) pick a site $i$ at random; (2) compute $\Delta E$ = energy change from flipping $s_i\to -s_i$; (3) if $\Delta E\le 0$, accept the flip; if $\Delta E>0$, accept with probability $e^{-\beta \Delta E}$; (4) repeat.

The algorithm satisfies detailed balance with respect to the Boltzmann distribution and ergodicity (every configuration is reachable), so by the convergence theorem for Markov chains, the long-run distribution is Boltzmann. Expectation values are estimated as time averages along the trajectory.

For the 2D Ising model at $T_c$, the correlation time diverges as $\tau \sim L^z$ where $z$ is the dynamic critical exponent ($z\approx 2.17$ for Metropolis on the square lattice). This critical slowing down makes simulations near $T_c$ expensive. Cluster algorithms (Swendsen-Wang, Wolff) reduce $z$ dramatically by flipping entire correlated clusters, exploiting the Fortuin-Kasteleyn random-cluster representation of the Ising model.

The 3D Ising model [Master]

No exact solution is known for the 3D Ising model on any lattice. The critical exponents are not known analytically and are estimated numerically:

$$\alpha \approx 0.110,\beta \approx 0.326,\gamma \approx 1.237,\nu \approx 0.630,\eta \approx \mathrm{0.036.}$$

These exponents satisfy the scaling relations and are in the 3D Ising universality class — shared by uniaxial magnets, binary fluids, the liquid-gas critical point, and (by the universality hypothesis) any system in $d=3$ with a scalar order parameter and ${\mathbb{Z}}_2$ symmetry.

High-temperature series expansions, Monte Carlo simulations (with finite-size scaling analysis), and conformal bootstrap methods have converged on these values to high precision. The absence of an exact solution remains one of the outstanding open problems in mathematical physics; the 3D Ising model is a natural test case for the conformal bootstrap programme 11.07.01 pending.

Historical notes [Master]

• Ising (1925). Ernst Ising solved the 1D model in his PhD thesis under Lenz, found no phase transition, and (incorrectly) conjectured that no higher-dimensional Ising model would exhibit a phase transition either.

• Peierls (1936). Rudolf Peierls gave a heuristic argument that the 2D Ising model does have a phase transition, using what is now called the Peierls argument (counting domain walls). This was made rigorous by Dobrushin, Griffiths, and others in the 1960s.

• Onsager (1944). Lars Onsager computed the exact free energy of the 2D square-lattice Ising model at $h=0$, announcing the result at a conference in 1942 and publishing in 1944. The spontaneous magnetisation formula was announced without proof at a 1949 conference and proved by Yang (1952).

• Yang and Lee (1952). The Yang-Lee circle theorem and the theory of zeros of the partition function in the complex fugacity plane.

• Wilson (1971). Kenneth Wilson's renormalization group provided the theoretical framework that explains why universality exists and why mean-field theory fails in low dimensions — earning him the 1982 Nobel Prize. The Ising model is the canonical testing ground for RG methods.

Connections [Master]

To the renormalization group 11.07.01 pending. The critical exponents computed here — $\beta =1/8$, $\gamma =7/4$, $\nu =1$ in 2D; $\beta \approx 0.326$, $\nu \approx 0.630$ in 3D — are the target observables that RG flow equations must reproduce. Kadanoff's block-spin transformation (1966) is an explicit coarse-graining of the Ising model that motivates the RG; Wilson's $ϵ=4-d$ expansion computes the same exponents as a perturbative series in the dimension. The 2D Ising model is the exactly-solvable benchmark against which all RG approximations are checked.

To mathematical probability and the math-side stat-mech course 08.07.01. The math-side treatment formalises the Ising model as a Gibbs measure on a lattice graph, proves the existence of a phase transition via the Peierls argument, constructs the transfer matrix rigorously, and develops the Yang-Lee theory as a result in complex analysis. The physics-side unit (this one) provides the physical motivation and the exact-solution technology; the math-side unit provides the measure-theoretic foundations and convergence proofs.

To lattice field theory. The Ising model is the simplest lattice field theory — a real scalar field $\varphi$ restricted to $\pm 1$ on a discrete spacetime lattice. The transfer matrix is the lattice analogue of the Euclidean path-integral kernel. The continuum limit $T\to T_c$ (correlation length $\xi \to \infty$) recovers a continuum quantum field theory: the 2D Ising model at $T_c$ is described by a free Majorana fermion (the Onsager fermion), and the 3D Ising model at $T_c$ is described by the 3D Ising CFT, whose exact content is a major open problem in conformal field theory.

To quantum computing and optimisation. The Ising Hamiltonian $H=-J\sum s_i{s}_j-h\sum s_i$ is the native Hamiltonian of quantum annealers (D-Wave) and of many trapped-ion quantum simulators. Finding the ground state of the Ising model on an arbitrary graph is equivalent to weighted MAX-CUT, an NP-hard problem. The phase transition in the thermodynamic limit is relevant because the hardest optimisation instances cluster near the critical point (the "order-disorder" transition in solution-space connectivity).

Bibliography [Master]

1. Ising, E. (1925). "Beitrag zur Theorie des Ferromagnetismus." Zeitschrift für Physik 31, 253–258. — The original paper solving the 1D model.

2. Peierls, R. (1936). "On Ising's model of ferromagnetism." Mathematical Proceedings of the Cambridge Philosophical Society 32, 477–481. — Heuristic argument for a 2D phase transition.

3. Onsager, L. (1944). "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition." Physical Review 65, 117–149. — The exact free energy.

4. Yang, C.N. (1952). "The Spontaneous Magnetization of a Two-Dimensional Ising Model." Physical Review 85, 808–816. — Exact spontaneous magnetisation.

5. Lee, T.D. and Yang, C.N. (1952). "Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model." Physical Review 87, 410–419. — The circle theorem.

6. Wilson, K.G. (1971). "Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture." Physical Review B 4, 3174–3183. — The RG explained universality.

7. Pirogov, S.A. and Sinai, Ya.G. (1975). "Phase diagrams of classical lattice spin models." Theoretical and Mathematical Physics 25, 1185–1192. — Rigorous phase-diagram construction.

8. McCoy, B.M. and Wu, T.T. (1973). The Two-Dimensional Ising Model. Harvard University Press. — Comprehensive monograph on the exact solution.

9. Huang, K. (1987). Statistical Mechanics, 2nd ed. Wiley. Ch. 14–15. — Standard textbook treatment of the Ising model and critical phenomena.

10. Landau, L.D. and Lifshitz, E.M. (1980). Statistical Physics, Part 1, 3rd ed. Pergamon. §73–74, 148–150. — Thermodynamic theory of phase transitions.

11. Baxter, R.J. (1982). Exactly Solved Models in Statistical Mechanics. Academic Press. — The definitive reference for solvable lattice models.

12. Binder, K. and Heermann, D.W. (2010). Monte Carlo Simulation in Statistical Physics, 5th ed. Springer. — Finite-size scaling and Monte Carlo methods for the Ising model.