Mean-field and cluster methods

Approximation techniques that replace interacting systems with effective fields or finite clusters to make their collective behavior tractable. They estimate collective behavior while simplifying correlations among system components.

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Mean-field theory: An approximation that replaces each component’s interactions with an average effective field produced by the rest of the system. The single-site limit captures the effective-field strategy shared by these methods.
Statistical mechanics: The framework connecting microscopic states and probabilities to macroscopic physical behavior. Mean-field and cluster approximations estimate macroscopic behavior from statistical-mechanical models.
Ising model: A lattice model of interacting binary spins used to study magnetism and phase transitions. Its mean-field solution gives a clear benchmark for neglected critical fluctuations.
Monte Carlo method: A computational method that estimates quantities by drawing samples from probability distributions. Sampling can retain fluctuations that deterministic mean-field equations suppress.
Landau theory: A phenomenological framework that describes phase transitions through an expansion of free energy in an order parameter. Mean-field predictions often share Landau theory’s neglect of long-wavelength fluctuations.
Cluster variation method: A variational approximation that estimates a system’s free energy using probabilities on overlapping clusters of sites. Overlapping clusters systematically restore correlations omitted by single-site mean-field theory.
Partition function: A sum over states that encodes a system’s thermodynamic properties at a specified temperature and set of parameters. Approximation schemes often target the partition function or its free energy.
Hubbard model: A lattice model describing particles that hop between sites and interact through on-site repulsion. Dynamical mean-field theory makes its correlated-electron phases tractable.
Exact diagonalization: A numerical technique that finds eigenvalues and eigenvectors of a finite system’s Hamiltonian matrix. It treats a small system exactly, providing benchmarks for approximate cluster calculations.
Critical exponent: A scaling exponent that characterizes how a physical quantity behaves near a continuous phase transition. Mean-field approximations predict characteristic exponents that can differ from observed values.