--- title: "The magnets that cannot satisfy all their neighbours" description: "What did Parisi discover in the apparently disordered states of a spin glass?" canonical_url: "https://sandboxphysics.com/stories/thermal-complexity/parisi-spin-glass/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/thermal-complexity/parisi-spin-glass/" last_modified: "2026-10-24" --- [How order survives, changes and gets frustrated](https://sandboxphysics.com/stories/thermal-complexity/#order) Heat · Chance · Collective behaviour # The magnets that cannot satisfy all their neighbours What did Parisi discover in the apparently disordered states of a spin glass? Sandbox Physics · 2026-10-24 Nobel contribution: 2021-2 ## Three preferences can already be impossible to satisfy Put three little magnets at the corners of a triangle. Give each one only two possible directions, up or down, and suppose every neighbouring pair prefers opposite directions. Set the first up and the second down. The third cannot oppose both of them. Whatever it does, at least one pair has an unsatisfied preference. Physicists call this conflict frustration. A spin glass contains many such competing demands, with their strengths and signs distributed irregularly. Here a spin means the magnetic degree of freedom of an atom, pictured as a small directional magnet. In certain alloys, magnetic atoms are scattered through a nonmagnetic host. Different pairs can favour alignment or opposition, making the best collective arrangement difficult to find. Sources: [1] [2] ## The disorder stays while the spins try to rearrange Two different kinds of variation enter this problem. The spins can turn as temperature changes or time passes. The locations of the magnetic atoms, and the resulting interaction pattern in a model, are held fixed during that rearrangement. This persistent background is called quenched disorder. Thermal fluctuations explore the possibilities allowed by that particular background. At high temperature, thermal motion readily disrupts local preferences. At low temperature, the spin pattern can evolve very slowly and remember how the sample was prepared. An overall magnetization near zero gives little description of this behaviour: cancellation can occur in both a rapidly changing disordered state and a slowly frozen, irregular one. Experiments on alloys made these questions concrete. Cannella and Mydosh’s 1972 study of gold–iron alloys examined how the low-field magnetic response changed with temperature and concentration. Features such as a sharp cusp in the response pointed to collective behaviour that required a richer account than all spins simply choosing one common direction. Sources: [2] [3] ## One pair of states did not describe the landscape In a simple ferromagnet at low temperature and zero applied field, two natural ordered choices are predominantly up and predominantly down. A spin glass offers a much more crowded problem. Many arrangements can make good compromises among incompatible interactions. The system’s statistical description has to represent how these possibilities contribute. Sherrington and Kirkpatrick introduced an idealized model in 1975 in which every spin interacts with every other through random couplings. The couplings are scaled as the number of spins grows so the thermodynamic problem remains meaningful. This mean-field model removes the detailed geometry of a particular alloy and makes the collective mathematical problem sharply defined. Early calculations treated the possible equilibrium states too uniformly and produced an unphysical low-temperature result. In 1979, Giorgio Parisi found a way to organize their differences. His solution needed a richer measure of order than the single overall magnetization used for an ordinary magnet. Sources: [4] [5] ## Compare arrangements by how much they overlap Independently draw two equilibrium configurations at the same temperature from the same fixed interaction sample, and compare the direction at every site. If most sites agree, the two arrangements have a large overlap. If many differ, the overlap is smaller. The crucial question is the distribution of these similarities across the equilibrium possibilities, not merely one comparison chosen by eye. Parisi’s construction described a hierarchy of relationships among states. Within the calculation, replicas are mathematical copies of a system used to handle the average over disorder. Allowing those copies to have structured, unequal relationships is called replica symmetry breaking. The phrase names the mathematical method; the physical content concerns the organization of equilibrium states. This made a hidden kind of order expressible. A sample could have little net magnetization and still have a highly structured collection of possible arrangements. Parisi’s hierarchy supplied the structure needed for a thermodynamic solution of the mean-field model. Sources: [5] [1] ## A conjectured solution became a proved thermodynamic formula The mathematical justification came much later. Michel Talagrand’s 2006 paper proved the Parisi free-energy formula for the Sherrington–Kirkpatrick model, building on Francesco Guerra’s interpolation method. The result established the thermodynamic formula for that specified model. Connecting all details of short-range, finite-dimensional spin glasses remains a distinct physical and mathematical problem. Parisi received half of the 2021 Physics Prize for discovering the interplay of disorder and fluctuations in physical systems. Manabe and Hasselmann shared the other half for climate work. Spin glasses helped reveal how a fixed pattern of competing interactions can support a rich organization of states, an idea that also became useful in the statistical physics of associative memory. Sources: [6] [1] ## Sources and reading notes 1. [Royal Swedish Academy of Sciences, Scientific Background for the Nobel Prize in Physics 2021](https://www.nobelprize.org/uploads/2021/10/sciback_fy_en_21.pdf) Read the spin-glass section for frustration, overlap and the distinction between Parisi’s contribution and the climate contribution. 2. [S. F. Edwards and P. W. Anderson, Theory of Spin Glasses (1975)](https://doi.org/10.1088/0305-4608/5/5/017) Follow the distinction between spin fluctuations and a fixed disordered interaction background. 3. [V. Cannella and J. A. Mydosh, Magnetic Ordering in Gold-Iron Alloys (1972)](https://doi.org/10.1103/PhysRevB.6.4220) Read the low-field response against temperature and concentration, keeping the measured alloy distinct from idealized models. 4. [David Sherrington and Scott Kirkpatrick, Solvable Model of a Spin-Glass (1975)](https://doi.org/10.1103/PhysRevLett.35.1792) Identify the infinite-range random couplings that define this mean-field model. 5. [Giorgio Parisi, Infinite Number of Order Parameters for Spin-Glasses (1979)](https://doi.org/10.1103/PhysRevLett.43.1754) Read the hierarchy of order parameters in its stated mean-field and replica-theory setting. 6. [Michel Talagrand, The Parisi Formula, Annals of Mathematics 163, 221–263 (2006)](https://annals.math.princeton.edu/2006/163-1/p04) Check the theorem’s model and thermodynamic quantity: the Sherrington–Kirkpatrick free-energy formula. 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