From Plus and Minus Signs to Phase Transitions: A Century of the Ising Model

Why does a magnet lose its spontaneous magnetization near the Curie temperature when it is heated? Why does water, under standard atmospheric pressure, undergo a liquid-gas transition near 100°C? The microscopic mechanisms and the types of phase transition are not the same, but they share a central word: phase transition. The constituents of the system have not changed, yet the macroscopic state can change qualitatively near a phase boundary or a critical point.

The charm of the Ising model is that it captures this change with remarkably few rules. Each lattice site carries only a plus sign or a minus sign, and neighboring sites interact through a simple energy. Once the number of sites is large enough, however, order, disorder, critical fluctuations, and self-similarity across scales all begin to emerge.

It is not a miniature copy of a real magnet, but a minimal skeleton for thinking about collective phenomena.

1. From the Curie temperature to Ising's lattice

In 1895, Pierre Curie systematically studied how magnetism varies with temperature. When a ferromagnetic material is heated above a certain critical temperature, spontaneous magnetization disappears; this temperature is now called the Curie temperature. A natural question follows: what rules among a huge number of microscopic magnetic moments can produce such a sharp macroscopic change?

Black-and-white portrait of Pierre Curie — **Pierre Curie** · 1859–1906 Image: Nobel Foundation / Wikimedia Commons (public domain)

In 1920, Wilhelm Lenz proposed a bold simplification. Imagine the material as a regular lattice, and keep only two possible states at each site:

\[ \sigma_i \in \{-1,+1\}. \]

Ising spin configuration on a square lattice, with red up arrows and blue down arrows — **An Ising spin configuration on the square lattice** · red up arrows represent (+1), blue down arrows represent (-1), and gray lines connect nearest-neighbor sites Drawing: Yuguang XIAO

This binary variable should not be read as a full quantum description of an electron spin. It records only the two possible orientations of a local magnetic moment along a chosen axis. For the classical nearest-neighbor ferromagnetic model, the Hamiltonian is

\[ H(\sigma) =-J\sum_{\langle i,j\rangle}\sigma_i\sigma_j -h\sum_i\sigma_i, \qquad J>0. \]

The first term favors neighboring spins that point in the same direction, while the second term describes an external magnetic field. Temperature enters through the Gibbs-Boltzmann weight: at low temperature, low-energy configurations are favored; at high temperature, many more disordered configurations become visible.

Write down the energy

Neighboring spins lower the energy when they agree; the energy pushes the system toward large aligned regions.

Let temperature choose

The probability of observing a configuration \(\sigma\) is

\[ \mathbb P_T(\sigma)=\frac{1}{Z(T)} \exp\!\left(-\frac{H(\sigma)}{k_B T}\right). \]

Sum over all configurations

The normalizing constant \(Z(T)\) is the partition function:

\[ Z(T)=\sum_{\sigma} \exp\!\left(-\frac{H(\sigma)}{k_B T}\right). \]

The difficulty appears immediately: a system with \(N\) sites has \(2^N\) configurations. Even with only 256 sites, the number of states is about \(1.16\times10^{77}\). We cannot enumerate them one by one, yet we still want to read from this combinatorial explosion macroscopic quantities such as magnetization, specific heat, and correlation length.

2. The failure in one dimension, the answer in two

Wilhelm Lenz gave this problem to his student Ernst Ising. In his 1924 doctoral thesis, Ising solved the one-dimensional nearest-neighbor model, and published the result in 1925: at any positive temperature, the one-dimensional short-range chain has no spontaneous magnetization.

Black-and-white portraits of Wilhelm Lenz and Ernst Ising side by side — Left: **Wilhelm Lenz**; right: **Ernst Ising** Image: Westlake University popular article reposted in this Zhihu discussion

The reason can be understood in terms of domain walls. To flip a segment of up spins into down spins, one only pays a finite energy cost at the two ends; but the longer the chain is, the more places a domain wall can appear. As soon as the temperature is positive, entropy eventually wins, domain walls proliferate, and long-range order is broken into pieces.

In two dimensions, the situation is completely different. In 1944, Lars Onsager rigorously computed the free energy of the square-lattice Ising model at zero external field and proved that it does have a nonzero critical temperature. For isotropic coupling, the critical point satisfies

\[ \sinh\!\left(\frac{2J}{k_B T_c}\right)=1, \qquad \frac{k_B T_c}{J} =\frac{2}{\ln(1+\sqrt 2)} \approx 2.269. \]

Below \(T_c\), macroscopic magnetization can remain stable; above \(T_c\), thermal fluctuations destroy this order. In 1952, Chen-Ning Yang gave the exact formula for the spontaneous magnetization of the two-dimensional square-lattice model:

\[ m(T)= \left[1-\sinh^{-4}\!\left(\frac{2J}{k_B T}\right)\right]^{1/8}, \qquad T<T_c. \]

The exponent \(1/8\) is not decorative. It describes how the magnetization disappears near the critical point, and it is one of the most famous critical exponents of the two-dimensional Ising universality class.

3. Kramers-Wannier duality

Before Onsager obtained the exact two-dimensional solution, Hendrik Kramers and Gregory Wannier discovered in 1941 that the high-temperature expansion of the square-lattice model can be matched with a low-temperature expansion on the dual lattice. If one further assumes that the phase transition point is unique, this correspondence determines the critical temperature.

This is high-low temperature duality. Consider the partition function \(Z(\beta)\) at zero external field, where \(\beta = 1/(k_B T)\):

\[ Z(\beta)=\sum_{\sigma} \exp\!\left(\beta J\sum_{\langle i,j\rangle}\sigma_i\sigma_j\right). \]

At low temperature, where \(\beta\) is large, the system fluctuates mainly around the fully aligned ground states. Flipping a region of spins creates an island, whose boundary forms a contour or domain wall. Summing over such contours gives an expansion in the low-temperature parameter \(e^{-2\beta J}\).

At high temperature, where \(\beta\) is small, one expands the Boltzmann factor using \(e^{\beta J\sigma_i\sigma_j}=\cosh(\beta J)\bigl(1+\sigma_i\sigma_j\tanh(\beta J)\bigr)\). After summing over spins, only those terms survive in which each vertex has an even number of incident selected edges. These are configurations made of closed loops. Thus the high-temperature partition function can be written as a loop expansion in \(\tanh(\beta J)\).

The dual lattice of the square lattice is again a square lattice. Interpreting the closed loops in the high-temperature expansion as low-temperature contours on the dual lattice gives the relation between temperature parameters:

\[ e^{-2\beta J}=\tanh(\beta^*J). \]

Here we ignore multiplicative factors depending on the lattice size and the boundary conditions; with periodic boundary conditions, one must also handle different topological sectors. If the system has only one transition point, then it must be a fixed point of the duality transformation, namely \(\beta_c=\beta_c^*\). Therefore

\[ \sinh\!\left(\frac{2J}{k_B T_c}\right)=1. \]

This agrees with the exact result Onsager later obtained. High-low temperature duality links two apparently opposite regimes, and became an early model for many duality ideas in modern statistical physics and quantum field theory.

4. From exact solutions to universality

The formulas of Onsager and Yang answer key questions about the two-dimensional model exactly, but they also push a broader question to the front: why do systems with very different microscopic details display the same power laws near criticality?

Far from the critical temperature, correlations between spins usually extend only over a finite distance. Near \(T_c\), the correlation length grows rapidly: small clusters sit inside larger clusters, and these larger clusters sit inside even larger structures. The system no longer prefers a characteristic scale, and after zooming in one sees statistically similar textures.

This is universality. Critical behavior is often not determined by the lattice spacing or the chemical details of the atoms, but mainly by spatial dimension, symmetry, and the range of interaction. The magnetization exponent \(\beta=1/8\), the correlation-length exponent \(\nu=1\), and the susceptibility exponent \(\gamma=7/4\) together describe a whole two-dimensional Ising universality class.

In the 1970s, Kenneth G. Wilson developed the renormalization group into a systematic framework for understanding universality. Looking at the system at larger and larger scales is equivalent to integrating out short-scale degrees of freedom. Parameters from different microscopic models may flow to the same fixed point, while only a few relevant directions determine the critical exponents. Universality is therefore not just an empirical coincidence, but a consequence of common structure under changes of scale.

Black-and-white portrait of Lars Onsager — Left: **Lars Onsager**, who found the exact solution of the two-dimensional square-lattice Ising model and received the 1968 Nobel Prize in Chemistry; right: **Kenneth G. Wilson**, founder of the renormalization-group framework for critical phenomena and recipient of the 1982 Nobel Prize in Physics Images: left, Nobel Foundation / Wikimedia Commons (public domain); right, Nobel Foundation / Wikimedia Commons (CC BY-SA 4.0)

Black-and-white portrait of Kenneth G. Wilson — Left: **Lars Onsager**, who found the exact solution of the two-dimensional square-lattice Ising model and received the 1968 Nobel Prize in Chemistry; right: **Kenneth G. Wilson**, founder of the renormalization-group framework for critical phenomena and recipient of the 1982 Nobel Prize in Physics Images: left, Nobel Foundation / Wikimedia Commons (public domain); right, Nobel Foundation / Wikimedia Commons (CC BY-SA 4.0)

5. Conformal structure and rigorous probability

In two dimensions, scale invariance has a stronger form. In 1970, Alexander Polyakov proposed using conformal symmetry to understand critical fluctuations. A conformal transformation can change lengths locally while preserving angles, providing a sharper mathematical language for the statement that a critical point has no characteristic length scale.

Later, work by Stanislav Smirnov, Dmitry Chelkak, and others rigorously established conformally invariant scaling limits for several observables and interfaces of the planar critical Ising model. Another rigorous route studies high-dimensional phase transitions through random currents, correlation inequalities, and random geometry. Hugo Duminil-Copin received the 2022 Fields Medal for solving long-standing problems in the probabilistic theory of phase transitions in statistical physics, especially in dimensions three and four.

Outdoor portrait of Stanislav Smirnov — Left: **Stanislav Smirnov**, 2010 Fields Medalist; right: **Hugo Duminil-Copin**, 2022 Fields Medalist. Their work represents two major directions in the rigorous probability theory of critical models Images: Wikipedia / Wikimedia Commons. Left, Renate Schmid / Oberwolfach Photo Collection, see Stanislav Smirnov image page (CC BY-SA 2.0 DE); right, see Hugo Duminil-Copin entry

Hugo Duminil-Copin standing in front of a whiteboard covered with mathematical formulas — Left: **Stanislav Smirnov**, 2010 Fields Medalist; right: **Hugo Duminil-Copin**, 2022 Fields Medalist. Their work represents two major directions in the rigorous probability theory of critical models Images: Wikipedia / Wikimedia Commons. Left, Renate Schmid / Oberwolfach Photo Collection, see Stanislav Smirnov image page (CC BY-SA 2.0 DE); right, see Hugo Duminil-Copin entry

6. Three dimensions: rigorous results and the modern conformal bootstrap

The question “has the three-dimensional Ising model been solved?” easily becomes a linguistic trap. As of 2026, we still do not know a closed formula comparable to Onsager's two-dimensional solution that gives the free energy and correlation functions of the three-dimensional nearest-neighbor model. But rigorous statistical mechanics asks not only whether one can write down such a formula; it also asks whether phase transitions exist, when Gibbs states are unique, and how order parameters behave at criticality.

Existence of a phase transition

Rudolf Peierls's contour argument shows that in nearest-neighbor ferromagnetic models in two and higher dimensions, multiple Gibbs states can exist at sufficiently low temperature. Cluster expansions and uniqueness arguments give the opposite picture at high temperature, thereby establishing the existence of a phase transition in a rigorous sense.

Continuity at the critical point

Michael Aizenman, Hugo Duminil-Copin, and Vladas Sidoravicius used the random-current representation in infinite volume to prove that the spontaneous magnetization of the three-dimensional nearest-neighbor Ising model vanishes continuously at the critical point. In other words, the transition is continuous:

\[ \lim_{\beta\downarrow\beta_c}m^*(\beta)=0. \]

Analytic problems that remain open

The three-dimensional model still lacks a complete closed form for the free energy, arbitrary-distance correlation functions, and a rigorous derivation of a critical structure comparable to the two-dimensional case. One central difficulty is to derive conformal invariance of the three-dimensional scaling limit directly from the lattice model.

The random-current representation rewrites spin correlation functions as sums over integer-valued currents with parity constraints, connecting analytic questions to connectivity in random geometry. This is one reason the Ising model remains so fertile: the same model keeps generating new conversations between algebra, combinatorics, probability, and analysis.

The conformal bootstrap in the continuum limit

Rigorous probabilistic methods start from the lattice model. The modern conformal bootstrap takes a different route. It assumes that the continuum limit of the three-dimensional Ising critical point is described by a unitary conformal field theory. Instead of enumerating lattice configurations or running Monte Carlo simulations, it studies the consistency conditions that local operators and their correlation functions must satisfy.

The same four-point function can be expanded by first fusing the first two operators, or by choosing a different fusion channel. Associativity of the operator product expansion requires the two expansions to agree exactly; this is crossing symmetry. For the four-point function of the spin operator \(\sigma\), the constraint can be schematically written as

\[ \sum_{\mathcal O} \lambda_{\sigma\sigma\mathcal O}^{2} F_{\Delta,\ell}(u,v)=0. \]

Here \(u,v\) are conformal cross-ratios, \(\Delta\) and \(\ell\) are respectively the scaling dimension and spin of an intermediate operator, and \(\lambda_{\sigma\sigma\mathcal O}\) is an OPE coefficient. Unitarity further imposes \(\lambda^2\geq0\) and lower bounds on operator dimensions. Thus an apparently infinite-dimensional field-theoretic problem can be turned into convex optimization and semidefinite programming: if a proposed set of critical data cannot satisfy these constraints, it is excluded.

In 2008, Riccardo Rattazzi, Slava Rychkov, and collaborators made the numerical conformal bootstrap in higher dimensions computationally effective. In 2012, Sheer El-Showk and collaborators identified the characteristic kink associated with the three-dimensional Ising model on the boundary of the allowed region. In 2015, David Simmons-Duffin's semidefinite-programming solver SDPB greatly improved the computational power of the method. In 2016, Filip Kos and collaborators combined several correlators involving the spin operator \(\sigma\) and the energy operator \(\varepsilon\), shrinking the allowed region to the famous Ising island.

A representative high-precision result from this mixed-correlator analysis is

\[ \Delta_\sigma\approx0.5181489, \qquad \Delta_\varepsilon\approx1.412625. \]

These are scaling dimensions, not anomalous dimensions themselves. In three dimensions, they can be converted into the usual critical exponents as follows:

\[ \begin{aligned} \eta&=2\Delta_\sigma-(d-2)\approx0.03630,\\ \nu&=\frac{1}{d-\Delta_\varepsilon}\approx0.62997, \qquad d=3. \end{aligned} \]

The conformal bootstrap does not give a closed analytic formula like the two-dimensional free energy. It shows another way in which a model can be “solved”: using only symmetry, associativity, and unitarity in the continuum limit, one obtains extremely strong and systematically improvable numerical constraints on three-dimensional critical data.

7. From spins to heights: the emergence of interface models

In the phase-coexistence region at zero external field and \(0<T<T_c\), the plus phase and the minus phase may coexist. To study the geometry of the boundary between them, one shifts attention from the spins themselves to the phase interface.

In a finite box, one can impose boundary conditions named after Roland Dobrushin: for instance, fix \(+1\) spins above a reference plane and \(-1\) spins below it, forcing the system to form an interface across the middle. Such boundary conditions turn the questions “where is the interface?” and “how does it fluctuate?” into rigorous mathematical problems.

Two-dimensional random interface between red plus spins and blue minus spins under Dobrushin boundary conditions — **A two-dimensional Ising interface under Dobrushin boundary conditions** · the red upper region represents \(+1\) spins, the blue lower region represents \(-1\) spins, and a random horizontal interface forms between the two phases Image: Vincent Beffara, provided by Yuguang XIAO

At sufficiently low temperature, droplets created by thermal fluctuations and overhangs of the interface are suppressed. If one further adopts a no-overhang approximation, so that the interface has a single height at each horizontal position, it can be described by an integer-valued height function \(h_x\). The original spin model then leads to an effective class of height models.

The most common effective interface energy depends only on height differences between neighboring positions:

\[ H_p(h)=K\sum_{\langle x,y\rangle}|h_x-h_y|^p, \qquad K>0. \]

when \(p=1\), one obtains the classical solid-on-solid (SOS) model;
when \(p=2\), one obtains the discrete Gaussian model.

This correspondence is a low-temperature effective description, not an unconditional identity for the original Ising interface. It discards droplets, overhangs, and more complicated local topology, while retaining the most important degrees of freedom for studying large-scale interface fluctuations.

Dimension and interface fluctuations

The interface of the two-dimensional Ising model corresponds to a one-dimensional height function. The associated one-dimensional height model has random-walk-type fluctuations at any positive effective temperature. Translating back to the Ising model, this interface picture applies in the coexistence region \(0<T<T_c\). In the height model, the typical behavior is

\[ \bigl\langle(h_x-h_0)^2\bigr\rangle\asymp |x|. \]

The interface width grows with the observation scale. In this sense, the two-dimensional Ising interface is rough, rather than localized near a fixed straight line.

The interface of the three-dimensional Ising model corresponds to a two-dimensional random surface. In 1972, Roland Dobrushin proved that at sufficiently low temperature, the interface is localized near the reference plane, and the typical height fluctuation at a fixed position remains of order \(O(1)\). This is low-temperature rigidity in a rigorous sense.

The passage from spins to heights turns a phase-transition problem into a problem about random surfaces. It is also a basic starting point for the study of crystal surfaces, wetting transitions, and the shapes of interfaces under long-range interactions. Once the distance scale of the interaction is changed, we enter the world of long-range Ising models.

8. Long-range interactions: an exception in one dimension

The nearest-neighbor model assumes that each spin interacts directly only with nearby sites. This is an extraordinarily successful idealization, but in many systems the influence extends over longer distances. The long-range Ising model therefore replaces the Hamiltonian by

\[ H(\sigma)=-\sum_{i<j}J_{ij}\sigma_i\sigma_j, \qquad J_{ij}\asymp\frac{1}{|i-j|^{d+\sigma}}, \quad \sigma>0, \]

Here \(d\) is the spatial dimension, and the parameter \(\sigma\) controls how fast the interaction decays. The smaller \(\sigma\) is, the stronger the influence of distant spins; the larger \(\sigma\) is, the closer the model becomes to a short-range one.

The most striking change occurs in one dimension. If the coupling is written as \(J(r)\sim r^{-\alpha}\), Freeman Dyson proved in 1969 the existence of a low-temperature phase transition for \(1<\alpha<2\). The endpoint \(\alpha=2\) is subtler: in the same year, David J. Thouless analyzed long-range order and anomalous critical behavior for inverse-square coupling; only in 1982 did Jürg Fröhlich and Thomas Spencer rigorously establish the phase transition at this endpoint. In the notation above, these cases correspond to \(0<\sigma<1\) and \(\sigma=1\), respectively. Long-range bonds therefore change the conclusion that one-dimensional short-range models have no positive-temperature phase transition.

Black-and-white portrait of David J. Thouless — **David J. Thouless** · 2016 Nobel Prize in Physics Image: Mary Levin / University of Washington, via Wikimedia Commons (CC BY-SA 4.0)

As \(\sigma\) varies, the model passes through regimes of mean-field behavior, genuinely long-range critical behavior, and crossover toward the short-range universality class. How the critical exponents change, and how long-range interactions modify interface geometry, remain active research directions and are among the topics I have recently been studying.

9. Disorder and learning: two branches of Ising-type energy

Once one leaves the uniform nearest-neighbor ferromagnet, the Ising-type energy can remain, but the meaning of the couplings changes. Two particularly important theoretical branches lead to disordered systems and neural networks.

The first branch is spin glasses. If the couplings \(J_{ij}\) are no longer all positive, but have random signs, local preferences conflict with each other, producing frustration and an extremely complex structure of Gibbs states. Giorgio Parisi constructed the replica-symmetry-breaking scheme for the mean-field model introduced by David Sherrington and Scott Kirkpatrick, using a hierarchical order parameter to describe this disorder. Later work by Francesco Guerra and Michel Talagrand provided a rigorous mathematical foundation for Parisi's formula for the free energy.

Color portrait of Giorgio Parisi — **Giorgio Parisi** · 2021 Nobel Prize in Physics Image: Lorenza Parisi / Wikimedia Commons (CC BY-SA 4.0)

The second branch leads to neural networks. The Hopfield network, named after John J. Hopfield, writes the states of binary neurons as an Ising-type energy function; stable memories correspond to attractors in the energy landscape. The Boltzmann machine developed by Geoffrey Hinton and collaborators incorporates thermal fluctuations into this framework, allowing the network to represent data by a Gibbs distribution. Both lines of work borrow the central language of statistical physics: how many simple variables form global structure through interaction.

Color portrait of John J. Hopfield — Joint recipients of the 2024 Nobel Prize in Physics Images: left, bhadeshia123 / Wikimedia Commons (CC BY 3.0); right, Christopher Michel / Wikimedia Commons (CC BY-SA 4.0)

Color portrait of Geoffrey Hinton — Joint recipients of the 2024 Nobel Prize in Physics Images: left, bhadeshia123 / Wikimedia Commons (CC BY 3.0); right, Christopher Michel / Wikimedia Commons (CC BY-SA 4.0)

Conclusion: simple rules, complex worlds

The Ising model is not a “theory of everything.” Its enduring value lies in offering an exceptionally clear way of thinking: how do local interactions compete with randomness? How can small rules at finite scale create macroscopic order in the thermodynamic limit? Why do distant locations suddenly become correlated near a critical point?

The figures in this story are not merely a list of prizes. Pierre Curie identified the macroscopic phenomenon that required explanation. Wilhelm Lenz and Ernst Ising compressed the problem into a lattice model. Hendrik Kramers and Gregory Wannier revealed the duality between high and low temperatures. Lars Onsager solved the two-dimensional model exactly. Kenneth G. Wilson explained universality through changes of scale.

Later, Roland Dobrushin turned interface rigidity into a rigorous problem, while Stanislav Smirnov and Hugo Duminil-Copin advanced the probabilistic theory of critical models. David J. Thouless, Giorgio Parisi, John J. Hopfield, and Geoffrey Hinton showed how interaction range, disorder, and energy landscapes can carry the same kind of mathematical structure into new questions.

From the failure of the one-dimensional chain, to the exact two-dimensional solution, high-low temperature duality, conformal invariance, the three-dimensional conformal bootstrap, random interfaces, long-range interactions, and disordered systems, this model made only of plus and minus signs has already travelled through a century. It keeps reminding us that microscopic rules can be almost naive in their simplicity, while the worlds they generate need not be simple at all.