Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition

1944/02/01 by Lars Onsager · 6,524 citations
Chemistry · Mathematics · Physics and Astronomy · #Combinatorics #Condensed matter physics #Crystal (programming language) #Energy (signal processing) #Graph theory and applications #Mathematical physics #Mathematics #Molecular spectroscopy and chirality #Order (exchange) #Partition function (quantum field theory) #Physics #Quantum mechanics #Theoretical and Computational Physics

paper · doi:10.1103/physrev.65.117

published in Physical Review 65(3-4), 117-149 (American Institute of Physics)

openalex publication_date 1944/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

The partition function of a two-dimensional "ferromagnetic" with scalar "spins" (Ising model) is computed rigorously for the case of vanishing field. The eigenwert problem involved in the corresponding computation for a long strip crystal of finite width (n atoms), joined straight to itself around a cylinder, is solved by direct product decomposition; in the special case n=\ensuremath∞ an integral replaces a sum. The choice of different interaction energies (\ifmmode±\else\textpm\fiJ,\ifmmode±\else\textpm\fiJ^\ensuremath') in the (0 1) and (1 0) directions does not complicate the problem. The two-way infinite crystal has an order-disorder transition at a temperature T=T_c given by the condition sinh(\frac2JkT_c) sinh(\frac2J^\ensuremath'kT_c)=1. The energy is a continuous function of T; but the specific heat becomes infinite as \ensuremath-log |T\ensuremath-T_c|. For strips of finite width, the maximum of the specific heat increases linearly with log n. The order-converting dual transformation invented by Kramers and Wannier effects a simple automorphism of the basis of the quaternion algebra which is natural to the problem in hand. In addition to the thermodynamic properties of the massive crystal, the free energy of a (0 1) boundary between areas of opposite order is computed; on this basis the mean ordered length of a strip crystal is (exp ((2J)/(kT)) tanh(\frac2J^\ensuremath'kT))ⁿ.

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• Interplay between consensus and coherence in a model of interacting opinions
• Random matrix theory and classical statistical mechanics: Spin models
• Random cyclic matrices
• Dynamical response functions in the quantum Ising chain with a boundary
• Fine corrections in the effective string describing SU(2) Yang-Mills theory in three dimensions
• The Critical Z-Invariant Ising Model via Dimers: Locality Property
• Exact results of a mixed spin-1/2 and spin-S Ising model on a bathroom tile (4–8) lattice: Effect of uniaxial single-ion anisotropy
• Tensor network analysis of critical coupling in two dimensional ϕ4 theory
• Ising model in planar lacunary and fractal lattices: A path counting approach
• Determination of the Origin and Magnitude of Logarithmic Finite-Size Effects on Interfacial Tension: Role of Interfacial Fluctuations and Domain Breathing
• Ising Correlations and Elliptic Determinants
• On the computational complexity of Ising spin glass models
• An Algebraic Derivation of the Eigenspaces Associated with an Ising-Like Spectrum of the Superintegrable Chiral Potts Model
• Yang-Lee Zeros of the Q-State Potts Model in the Complex Magnetic Field Plane
• Generalization of the Peierls-Griffiths theorem for the Ising model on graphs
• Dynamic scaling behavior in the presence of a periodic magnetic driving across Ising continuous transitions
• Superfluid helium
• Thermal percolation for interacting monomers adsorbed on square lattices
• Topology and phase transitions II. Theorem on a necessary relation
• Phase diagrams and crossover in spatially anisotropicd=3Ising,XYmagnetic, and percolation systems: Exact renormalization-group solutions of hierarchical models
• Phase diagram for intermittent maps
• Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation
• Statistical Theory of Equations of State and Phase Transitions. II. Lattice Gas and Ising Model
• Hard-square lattice gas
• Hard hexagons: exact solution
• The random geometry of equilibrium phases
• Effects of boundary conditions on magnetization switching in kinetic Ising models of nanoscale ferromagnets
• Implications evinced by the phase diagram, anisotropy, magnetic penetration depths, isotope effects and conductivities of cuprate superconductors
• Ising model [wikipedia]
• Square lattice Ising model [wikipedia]
• Timeline of thermodynamics [wikipedia]
• Markers of criticality in phase synchronization. [europepmc]
• Scale invariance in natural and artificial collective systems: a review. [europepmc]
• Hard magnetic properties in nanoflake van der Waals Fe 3 GeTe 2 . [europepmc]
• Understanding dynamics of coherent Ising machines through simulation of large-scale 2D Ising models. [europepmc]
• Suppression of magnetic ordering in XXZ-type antiferromagnetic monolayer NiPS 3 . [europepmc]
• Heuristic recurrent algorithms for photonic Ising machines. [europepmc]
• Cell death and survival due to cytotoxic exposure modelled as a two-state Ising system. [europepmc]
• Deep learning on the 2-dimensional Ising model to extract the crossover region with a variational autoencoder. [europepmc]
• Entropy and Mutability for the q -State Clock Model in Small Systems. [europepmc]
• The Emergence of Integrated Information, Complexity, and 'Consciousness' at Criticality. [europepmc]
• Benchmarking Attention-Based Interpretability of Deep Learning in Multivariate Time Series Predictions. [europepmc]
• Electronic phase-crossover and room temperature ferromagnetism in a two-dimensional (2D) spin lattice. [europepmc]
• The Magnetic Genome of Two-Dimensional van der Waals Materials. [europepmc]
• Functional Heterogeneity of Cell Populations Increases Robustness of Pacemaker Function in a Numerical Model of the Sinoatrial Node Tissue. [europepmc]
• Observation of a phase transition within the domain walls of ferromagnetic Co 3 Sn 2 S 2 . [europepmc]
• Modeling convergent scale-by-scale skin color patterning in multiple species of lizards. [europepmc]
• Direct Magnetic Evidence, Functionalization, and Low-Temperature Magneto-Electron Transport in Liquid-Phase Exfoliated FePS 3 . [europepmc]
• Kagome superconductors AV 3 Sb 5 (A = K, Rb, Cs). [europepmc]
• Magnetism of two-dimensional chromium tellurides. [europepmc]
• Quorum sensing as a mechanism to harness the wisdom of the crowds. [europepmc]
• Enantiomers Self-Sort into Separate Counter-Twisted Ribbons of the Fddd Liquid Crystal─Antiferrochirality and Parachirality. [europepmc]
• Investigating structural and functional aspects of the brain's criticality in stroke. [europepmc]
• Strain Engineering of Intrinsic Ferromagnetism in 2D van der Waals Materials. [europepmc]
• From Stoner to local moment magnetism in atomically thin Cr 2 Te 3 . [europepmc]
• Ferromagnetism on an atom-thick & extended 2D metal-organic coordination network. [europepmc]

Related

• Some generalized order-disorder transformations
• Disorder solutions for generalized 2D Ising Model with multi-spin interaction
• Disorder solutions for the partition functions of the two-dimensional Ising-like models
• The Thermodynamic Stability of Two Dimensional Crystals with an Extended Coupling Scheme
• Quantitative disorder effects in low-dimensional spin systems