Ising model

A mathematical model of interacting spins on a lattice, each taking one of two values. It provides a simple setting for studying magnetism, statistical mechanics, and phase transitions.

Ising Hamiltonian: The energy function assigning costs to spin configurations through interactions and external fields. Its interaction and field terms specify which configurations the model favors.
Spin: An intrinsic form of angular momentum carried by quantum particles. The model idealizes each site with a two-valued spin variable.
One-dimensional Ising model: The Ising model on a chain of spins with interactions between neighboring sites. Its exact solution shows that finite-temperature ordering fails for short-range interactions.
Ferromagnetism: A form of magnetism in which neighboring moments align, producing spontaneous magnetization. The model's positive neighbor coupling captures the tendency of ferromagnets to align.
Potts model: A lattice model whose sites can occupy one of several discrete states interacting with neighbors. It generalizes the Ising model's two spin values to multiple states.
Magnetization: The net magnetic moment per unit volume or, in a spin model, the average spin. It serves as the order parameter distinguishing aligned from disordered phases.
Lattice (group): A discrete arrangement of points generated by integer combinations of basis vectors. Its sites define where spins sit and which pairs count as neighbors.
Two-dimensional Ising model: The Ising model on a two-dimensional lattice, with a finite-temperature critical point. Its square-lattice solution established a benchmark for critical phenomena.
Critical phenomena: Large-scale behavior near phase transitions, including divergent fluctuations and scaling laws. The model provides a tractable setting for studying universal critical behavior.
XY model: A lattice spin model in which each spin is a planar vector with a continuous angle. Its continuous spin direction contrasts with the Ising model's binary variables.

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