# Ising model The **Ising model** is a lattice model of [[Statistical_mechanics|statistical mechanics]] in which each site carries a variable with two values, s = ±1, and neighbouring sites are coupled so that agreement costs less energy than disagreement. Its energy is `H = -J*sum_<ij> s_i*s_j - B*sum_i s_i`, the first sum running over nearest-neighbour pairs and the second over sites in an external field.[^anag-hamiltonian] Nothing in that rule mentions long-range order, and yet in two or more dimensions the model orders spontaneously below a sharp temperature. It is the simplest system known to have a genuine [[Phase_transition|phase transition]], and it has become the reference case against which theories of [[Critical_exponent|critical behaviour]] are tested. In the microsim below the reader drives a 256 × 256 lattice with a single control, the inverse temperature β, over the range 0.2 to 0.7. Spins are updated by the [[Metropolis–Hastings_algorithm|Metropolis]] rule `A = min(1, exp(-beta*dE))`; at small β the lattice is a grey fizz, and as β is raised the domains coarsen, percolate and lock into one sign across the critical value `beta_c = ln(1 + sqrt(2))/2 = 0.4407`.[^anag-betac] Live readouts give ⟨|m|⟩ and the energy per bond, baked ⟨|m|⟩(β) curves for L = 16, 32, 64 and 128 carry a marker at the reader's β, and the Onsager curve `m = (1 - sinh(2*beta)^(-4))^(1/8)` is dashed, passing through 0.749 at β = 0.45.[^onsager-m] Equilibrium snapshots swap in on a slider move, so the reader never waits out critical slowing down. On the [[Physics]] flagship this article serves Part IV — Branches and fields, section *Condensed matter* (row P62), and it is also placed on [[Materials_science]], where the same lattice describes order–disorder in an [[Alloy|alloy]]. This page owns the cluster's C33 root sim, the `TwoStateLattice`; [[Ferromagnetism]] is its sibling and reuses this lattice for its hysteresis loop. ## Definition A configuration assigns s_i = ±1 to every site; on a square lattice of side L there are N = L² sites and N_l = 2N nearest-neighbour bonds under periodic or helical boundaries.[^anag-hamiltonian] J sets the energy scale and B the external field. With J > 0 the model is ferromagnetic, since aligned neighbours lower the energy, and the aligned ground state has `E0 = -(2*J + B)*N`.[^anag-hamiltonian] All thermodynamics follows from the [[Partition_function_(statistical_mechanics)|partition function]], the sum of exp(−βH) over configurations — of which there are 2^N, so that even a 5 × 5 lattice already has about 3.4 × 10⁶ terms.[^anag-partition] The questions asked of that sum are statistical. Does a spontaneous magnetization survive as the field is taken to zero, and below what temperature? How do the magnetization, the [[Heat_capacity|specific heat]] and the susceptibility behave as it is approached? The answers are given as [[Critical_exponent|critical exponents]], which depend on the dimension and the spin symmetry but not on lattice geometry or on J — the property called universality, and the reason so crude a model predicts real materials near their critical points. ### Discussion The model's interest lies in what it leaves out. There is no [[Quantum_mechanics|quantum mechanics]] in it: the spins commute and the model is classical. There is no lattice dynamics, no [[Crystal_structure|crystal structure]] beyond the connectivity, no interaction beyond nearest neighbours. What remains is a competition between energy, which prefers agreement, and [[Entropy|entropy]], which prefers disorder, mediated by [[Temperature|temperature]] — and that alone produces an ordered phase, a disordered phase and a singular point between them. ### Simplifications Each simplification buys something. Restricting the spin to two values removes the continuous symmetry that would otherwise permit soft modes and change the physics in low dimensions. Restricting the coupling to nearest neighbours makes the energy change of a flip local: `dE = 2*s_k*sum_nn s`, so on a square lattice ΔE takes only the values 0, ±4 and ±8, and a flip changes the magnetization by `dM = -2*s_k`.[^anag-flip] Only 2d + 1 terms are touched, which is what makes simulation cheap.[^likharev-local] A regular lattice gives the translational symmetry the exact solutions exploit. ### Connection to graph maximum cut Finding the ground state of an Ising model with arbitrary couplings is the same problem as cutting a [[Graph_theory|graph]] into two parts so as to maximise the weight of the edges crossing between them. A spin assignment partitions the vertices; an antiferromagnetic bond contributes negatively when its endpoints agree, so minimising the energy maximises the cut. The correspondence runs both ways, and is why the Ising Hamiltonian has become a standard encoding for [[Combinatorial_optimization|combinatorial optimisation]] on annealing hardware. Disorder in the couplings is what makes the problem hard. ## Basic properties and history Wilhelm Lenz proposed the model in 1920 and set it as a thesis problem to Ernst Ising, who solved the one-dimensional case in 1925 and found no phase transition.[^lenz1920][^ising1925] Because that answer was negative, the model was widely taken to be a failure for two decades, until exact and rigorous work showed the result to be an accident of dimension. ### No phase transition in one dimension In one dimension the model is solved exactly by a transfer matrix, with eigenvalues `lambda_+- = exp(J/T)*[cosh(h/T) +- sqrt(sinh(h/T)^2 + exp(-4*J/T))]`, an order parameter `eta = sinh(h/T)/sqrt(sinh(h/T)^2 + exp(-4*J/T))` and a zero-field susceptibility `chi(0) = exp(2*J/T)/T`, which diverges only as T → 0: the critical temperature is exactly zero.[^likharev-1d] The reason is a counting argument. One flipped bond in a chain costs a fixed energy 2J but can sit anywhere along it, so its entropy grows with the length and disorder wins at any positive temperature. A finite chain of N spins behaves as though it had a critical temperature 2J/ln N, which drifts to zero as the chain grows — slowly enough that a short chain looks ordered and a long one does not, which is the clearest warning available that a simulation of a small system can show a transition that is not there.[^likharev-walls] Mean-field theory, applied to the same chain, predicts a transition at 2J, and is simply wrong.[^likharev-table] ### Phase transition and exact solution in two dimensions In two dimensions a domain boundary is a closed curve rather than a point, and its energy grows with its length, so order survives at finite temperature. Onsager's 1944 solution of the zero-field square lattice made this exact: `beta_c = ln(1 + sqrt(2))/2 = 0.4406867935`, that is T_c = 2J/ln(1 + √2) ≈ 2.269 J.[^onsager1944][^anag-betac][^likharev-table] An estimate from domain-wall entropy, counting three directions at each step, gives 2J/ln 3 ≈ 1.82 J — the right mechanism, low by about 20 %.[^likharev-walls] The exact exponents are α = 0, β = 1/8, γ = 7/4, δ = 15, ν = 1 and η = 1/4, none of them the mean-field values, and all of them shared with every other system of the same symmetry and dimension.[^anag-exponents] Onsager's result was the first demonstration that a phase transition could be produced by a short-range interaction in a model simple enough to be solved exactly, and it settled the question that Ising's negative one-dimensional answer had left open for nineteen years. ## Historical significance The model's importance is not that it describes a substance but that it settled questions of principle: whether a transition can occur at all in a short-range classical system, and how one can be seen in a finite calculation. ### No phase transitions in finite volume On a finite lattice the partition function is a finite sum of positive analytic terms, so nothing can be singular; a true transition exists only in the infinite-size limit.[^anag-exponents] Simulation therefore never shows a sharp transition, only a rounded one, and the rounding is itself measurable: a thousand spins already smear the critical temperature by about 3 %.[^likharev-walls] The microsim shows this, since its baked ⟨|m|⟩(β) curves for L = 16, 32, 64 and 128 steepen toward the Onsager curve as L grows; that finite-size drift is the standard route to the infinite-lattice values. ### Peierls droplets Rudolf Peierls gave the first proof in 1936 that the two-dimensional model orders at low temperature, by bounding the probability of large droplets of reversed spin.[^peierls1936] A droplet boundary of length ℓ costs energy proportional to ℓ, so its Boltzmann weight falls exponentially in ℓ, while the count of such boundaries grows exponentially at a smaller rate. Reversed regions are rare and small, and the bulk magnetization survives. The argument is the rigorous form of the domain-wall counting above, and it is what tells the reader that the ordered patch spanning the microsim's lattice at large β is not an artefact of the finite box. ### Kramers–Wannier duality Hendrik Kramers and Gregory Wannier showed in 1941 that the two-dimensional square lattice at low temperature maps onto the same model at high temperature, ordered and disordered regimes exchanging places.[^kramers1941] If the transition is unique it must sit at the self-dual point, which fixes the critical temperature before any exact solution — and the value so obtained is the one Onsager confirmed. Duality remains a standard tool in [[Condensed_matter_physics|condensed matter physics]]. ## Applications The model is applied wherever a system has two local states and a preference for agreement. With disordered couplings it defines spin glasses, in which competing bonds frustrate any single ordered state; with asymmetric couplings it underlies associative-memory models of [[Neural_network|neural networks]].[^edwards1975][^hopfield1982] In [[Materials_science|materials science]] the binary version describes order–disorder in a [[Solid_solution|solid solution]], the two states being the species on a site, and its boundaries feed the [[Phase_diagram|phase diagrams]] used to plan [[Annealing_(materials_science)|annealing]]. ### Magnetism In its original reading J is an exchange coupling between atomic moments and the two states are the orientations of a spin, so below T_c the lattice carries a spontaneous [[Magnetization|magnetization]] with no applied field. The model reproduces the qualitative facts of [[Ferromagnetism|ferromagnetism]] — a [[Curie_temperature|Curie temperature]], a diverging susceptibility above it, and coexisting regions of opposite sign that are its version of [[Magnetic_domain|magnetic domains]] — but not the anisotropy and dipolar interaction that set the size and shape of real domains, so a snapshot pictures the mechanism, not a [[Microstructure|microstructure]]. ### Lattice gas Reinterpreting s = +1 as an occupied cell and s = −1 as empty converts the model into a lattice gas, J becoming an attraction between neighbouring particles, B a chemical potential, and the magnetization the density. Under that dictionary the ferromagnetic transition is the condensation of a gas into a liquid, and the two systems share one [[Critical_point_(thermodynamics)|critical point]]. The correspondence is exact, and explains why the exponents measured in a fluid match those of a magnet: the two share a symmetry and a dimension, which is all that need be shared. ## Numerical simulation Since the partition function cannot be summed for any useful lattice, the standard approach is importance sampling: generate configurations with probability proportional to their Boltzmann weight and average over them.[^anag-partition] This is what the microsim does at interactive speed. ### Metropolis algorithm The Metropolis rule, introduced in 1953 for a hard-sphere fluid, proposes flipping one spin and accepts with probability `A = min(1, exp(-beta*dE))`, always accepting a move that lowers the energy.[^metropolis1953][^anag-metropolis] Detailed balance is what makes the sampled distribution the Boltzmann one, and the locality of ΔE is what makes each attempt cost a handful of operations.[^anag-metropolis] Since ΔE takes only the values 0, ±4 and ±8, the exponentials are tabulated once per temperature as `prob[i] = exp(-2*beta*i)` for i = 2 and 4 — 0.4317 and 0.1864 at β = 0.21 (derived) — and the exponential never appears in the inner loop.[^anag-lookup][^derived] This page owns the cluster's C33 root sim. Its lattice object, `TwoStateLattice`, runs one to three sweeps per frame over 65,536 sites, and the sibling page [[Ferromagnetism]] reuses it for a swept-field hysteresis loop. Two conventions from the Portal Book are built into the readouts: the magnetization is averaged as ⟨|m|⟩ rather than ⟨m⟩, because a finite lattice flips sign as a whole from time to time and the signed average tends to zero; and the energy is per bond, `e = E/(2*N)`, with `c = beta^2*N*(<e^2> - <e>^2)` and `chi = beta*N*(<m^2> - <m>^2)` from fluctuations.[^anag-observables] ### As a Markov chain The sequence of configurations is a [[Monte_Carlo_method|Markov chain]] whose stationary distribution is the Boltzmann distribution, so averages are correct only after the chain has forgotten its starting configuration; successive sweeps are correlated, so the statistical error falls as one over the square root of the number of *independent* samples rather than of sweeps.[^anag-observables] The correlation time diverges at the critical point — critical slowing down — which is the practical reason the microsim ships pre-equilibrated snapshots at thirty-two values of β: a reader dragging the slider through β_c would otherwise read the magnetization off a lattice that has not equilibrated. The baked curves use of order 10⁵ sweeps per point after burn-in.[^anag-observables] Two conventions matter for reproducibility: the boundary condition, since helical and periodic wrapping differ measurably on small lattices, and the pseudorandom generator, since no published run reproduces without its seed — which is why the microsim is tested against invariants such as the energy of a torus changing only in multiples of four.[^anag-metropolis][^derived] ## Solutions Exact results exist in one and two dimensions and in the limit of high dimension; three dimensions has resisted exact solution and is known numerically and by renormalization-group methods to high precision. The four cases below are best read as a sequence in the dimension, because what changes with dimension is not the rule — which is identical throughout — but how much room a domain boundary has, and that alone decides whether order survives. ### One dimension The transfer matrix solves the chain in closed form for any field and temperature, and gives T_c = 0: no ordered phase at positive temperature, as Ising found.[^likharev-1d][^ising1925] Mean-field theory predicts T_c = 2Jd = 2J here, which is not merely inaccurate but qualitatively wrong — the clearest demonstration that mean field can invent a transition that does not exist.[^likharev-table] ### Two dimensions The zero-field square lattice is solved exactly, with β_c = ln(1 + √2)/2 and the spontaneous magnetization `m = (1 - sinh(2*beta)^(-4))^(1/8)`, which gives 0.749 at β = 0.45, 0.911 at β = 0.5 and 0.974 at β = 0.6.[^onsager1944][^onsager-m] The exponent β = 1/8 in that formula is the one the microsim's dashed curve exhibits near threshold, and it is far from the mean-field value of 1/2. Mean-field theory places T_c at 4J against the exact 2.269 J, an overestimate of 76 %.[^likharev-table] ### Three dimensions No exact solution is known. Monte Carlo work gives the transition to six figures, J/T_c = 0.221650 ± 0.000005, that is T_c ≈ 4.513 J against the mean-field 6 J, an overestimate of 33 %.[^likharev-table] The exponents are close to, but measurably different from, the mean-field values, and the cubic lattice is the three.js variant of this page's sim. ### Four dimensions and above The Levanyuk–Ginzburg criterion shows that fluctuations dominate only below four dimensions, so four is the upper critical dimension: at and above it the mean-field exponents β = 1/2, γ = 1, δ = 3 and ν = 1/2 become exact, up to logarithmic corrections in exactly four.[^likharev-mf] The progression from d = 1, where mean field invents a transition, through d = 2 and 3, where it is qualitatively right and quantitatively poor, to d ≥ 4, where it is exact, is the model's most economical lesson. ## See also - [[Phase_transition]] - [[Critical_exponent]] - [[Metropolis–Hastings_algorithm]] - [[Ferromagnetism]] - [[Condensed_matter_physics]] - [[Lev_Landau]] - [[Monte_Carlo_method]] ## Footnotes Explanatory notes are given inline; all citations, including the derived values, are collected under *References* below. ## References [^anag-hamiltonian]: Anagnostopoulos, Konstantinos (2016). *Computational Physics: A Practical Introduction to Computational Physics and Scientific Computing (using C++)*, 2nd ed. Ch. 15, "Simulation of the d = 2 Ising Model," pp. 522–523 (the Hamiltonian on N = L² sites with N_l = 2N bonds, periodic and helical boundaries, and the ground-state energy E₀ = −(2J + B)N). Portal Book 061. https://open.umn.edu/opentextbooks/textbooks/computational-physics-a-practical-introduction-to-computational-physics-and-scientific-computing-using-c [^anag-partition]: Anagnostopoulos, *Computational Physics* (2016), pp. 523–524 (the partition function has 2^N terms; a 5 × 5 lattice has 2²⁵ ≈ 3.4 × 10⁶ configurations). Portal Book 061. [^anag-betac]: Anagnostopoulos, *Computational Physics* (2016), p. 524 (β_c = ½·ln(1 + √2) = 0.4406867935 for J = 1, B = 0). Portal Book 061. [^anag-exponents]: Anagnostopoulos, *Computational Physics* (2016), p. 525 (the two-dimensional exponents α = 0, β = 1/8, γ = 7/4, δ = 15, ν = 1, η = 1/4; a finite lattice always rounds the transition). Portal Book 061. [^anag-flip]: Anagnostopoulos, *Computational Physics* (2016), pp. 527, 532, 534 (ΔE = 2·s_k·Σ_nn s with |ΔE| ≤ 8, and ΔM = −2·s_k). Portal Book 061. [^anag-metropolis]: Anagnostopoulos, *Computational Physics* (2016), pp. 527–529 (the Metropolis acceptance A = exp(−β·ΔE) for ΔE > 0 and 1 otherwise, derived from detailed balance). Portal Book 061. [^anag-lookup]: Anagnostopoulos, *Computational Physics* (2016), pp. 536–537 (the acceptance lookup table prob[i] = exp(−2·β·i) for i = 2, 4). Portal Book 061. [^anag-observables]: Anagnostopoulos, *Computational Physics* (2016), pp. 533–534 (e = E/(2N), m = |M|/N averaged as ⟨|m|⟩ rather than ⟨M⟩; c = β²·N·(⟨e²⟩ − ⟨e⟩²); χ = β·N·(⟨m²⟩ − ⟨m⟩²); successive sweeps are correlated, so errors fall as one over the root of the number of independent samples; runs of order 2 × 10⁵ sweeps). Portal Book 061. [^likharev-local]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part SM: Statistical Mechanics*. Ch. 4, pp. 135–136 (only 2d + 1 terms of the energy change when one spin is flipped). Portal Book 075. https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics [^likharev-1d]: Likharev, *Part SM: Statistical Mechanics* (2013), pp. 130–132 (the exact one-dimensional transfer-matrix solution, the order parameter, χ(0) = exp(2J/T)/T, and T_c = 0). Portal Book 075. [^likharev-walls]: Likharev, *Part SM: Statistical Mechanics* (2013), pp. 133–134 (domain-wall entropy: a finite chain of N spins has a quasi-critical temperature 2J/ln N; the two-dimensional wall estimate 2J/ln 3 ≈ 1.82 J against Onsager's 2J/ln(1 + √2) ≈ 2.269 J; a lattice of about a thousand spins smears T_c by roughly 3 %). Portal Book 075. [^likharev-table]: Likharev, *Part SM: Statistical Mechanics* (2013), p. 137 (Table 4.2, T_c/J as mean field / exact — d = 1: 2/0; d = 2: 4/2.269; d = 3: 6/4.513 — and the three-dimensional Monte Carlo value J/T_c = 0.221650 ± 0.000005). Portal Book 075. [^likharev-mf]: Likharev, *Part SM: Statistical Mechanics* (2013), pp. 118–122 (mean-field exponents β = ½, γ = 1, δ = 3, ν = ½, α = 0 as a jump; the Levanyuk–Ginzburg criterion, by which mean field fails near T_c for d < 4). Portal Book 075. [^onsager-m]: The Onsager spontaneous magnetization m = (1 − sinh(2β)⁻⁴)^(1/8) is a standard closed form; it is not printed in the Portal Book extracts read for this page (page to pin). Its values at β = 0.45, 0.5 and 0.6 — 0.749, 0.911 and 0.974 — are recorded in sub-manual 04 §5.6 as derived checks. [^derived]: Values marked *derived* were computed for Wikitube from the cited equations and are not printed in the books. At β = 0.21 the acceptance table gives prob[2] = exp(−4 × 0.21) = 0.4317 and prob[4] = exp(−8 × 0.21) = 0.1864. From Table 4.2, mean-field theory overestimates T_c by 4/2.269 − 1 = 76 % in two dimensions and 6/4.513 − 1 = 33 % in three. [^lenz1920]: Lenz, Wilhelm (1920). "Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern." *Physikalische Zeitschrift* 21: 613–615. [^ising1925]: Ising, Ernst (1925). "Beitrag zur Theorie des Ferromagnetismus." *Zeitschrift für Physik* 31 (1): 253–258. [^onsager1944]: Onsager, Lars (1944). "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition." *Physical Review* 65 (3–4): 117–149. [^peierls1936]: Peierls, Rudolf (1936). "On Ising's model of ferromagnetism." *Mathematical Proceedings of the Cambridge Philosophical Society* 32 (3): 477–481. [^kramers1941]: Kramers, H. A.; Wannier, G. H. (1941). "Statistics of the Two-Dimensional Ferromagnet. Part I." *Physical Review* 60 (3): 252–262. [^metropolis1953]: Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. (1953). "Equation of State Calculations by Fast Computing Machines." *The Journal of Chemical Physics* 21 (6): 1087–1092. [^edwards1975]: Edwards, S. F.; Anderson, P. W. (1975). "Theory of spin glasses." *Journal of Physics F: Metal Physics* 5 (5): 965–974. [^hopfield1982]: Hopfield, J. J. (1982). "Neural networks and physical systems with emergent collective computational abilities." *Proceedings of the National Academy of Sciences* 79 (8): 2554–2558. ## External links - [Computational Physics](https://open.umn.edu/opentextbooks/textbooks/computational-physics-a-practical-introduction-to-computational-physics-and-scientific-computing-using-c), the Open Textbook Library record for Portal Book 061, whose Chapter 15 is a complete Metropolis implementation - [Part SM: Statistical Mechanics](https://open.umn.edu/opentextbooks/textbooks/part-sm-statistical-mechanics), the Open Textbook Library record for Portal Book 075 - The Wikipedia pair's external links list further lecture notes and applet collections <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Ising_model.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Ising model* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Ising_model.html" data-title="Ising model"></div> *Built from `MICROSIM_GUIDE/specs/sims/Ising_model.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Ising_model) : [Wikitube](https://en.wikitube.io/wiki/Ising_model) · pinned revision [1352146297](https://en.wikipedia.org/w/index.php?oldid=1352146297) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]], [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P62 · sim pending (matter/Ising_model).*