method-mf

name	method-mf
description	Use when a mean-field reproduction needs method-level routing — Hartree-Fock / unrestricted Hartree-Fock for lattice fermions, Weiss mean-field decoupling of spin models, self-consistent-field order parameters and phase diagrams.

Method MF

Mean field is the fast self-consistent baseline track: replace interactions with effective single-particle fields solved to self-consistency. Use it for quick order-parameter and phase-diagram estimates and for generating reference states that seed correlated methods (DMRG, VMC, QMC).

Route

1. Use mean field for Hartree-Fock / unrestricted Hartree-Fock on lattice fermion models, Weiss mean-field decoupling of spin models, and self-consistent-field order parameters or phase diagrams.
2. Treat the result as an upper bound on ordering: mean field overestimates order and misses quantum fluctuations (it can predict false transitions in 1D).
3. Do not use this track for DMFT (needs an impurity solver) or DFT (different formalism and stack) — both are separate method families.

Implementation

There is no dedicated tool skill for mean field. Implement the self-consistent-field loop directly on the canonical Julia stack (run /setup-julia if Julia is not usable). The method card owns the decoupling choice, the SCF update, the convergence criterion, and verification (SCF stability, an ED upper-bound check on small systems, and analytic limits).

Details

Self-consistent mean-field methods for quantum lattice models. Approximate the many-body problem by replacing interactions with effective single-particle fields determined self-consistently. Serves as a fast baseline and starting point for more accurate methods.

Scope

Use this card for:

• Hartree-Fock and unrestricted Hartree-Fock for lattice fermion models (Hubbard, t-J).
• Mean-field decoupling of spin models (Weiss mean-field theory).
• Self-consistent field iterations to convergence.
• Phase diagrams and order parameters at the mean-field level.
• Generating initial states or reference configurations for correlated methods (DMRG, VMC, QMC).

Do not use this card for:

• Dynamical mean-field theory (DMFT) — that requires an impurity solver and is a separate method family.
• Density functional theory (DFT) — different formalism and software stack.

Notation

• ⟨n_i⟩, ⟨S^z_i⟩: local expectation values (order parameters) determined self-consistently.
• SCF iteration: one cycle of solving the effective single-particle problem and updating the mean fields.
• Convergence criterion: change in total energy or order parameters between successive iterations.

Pitfalls

• Symmetry breaking: mean-field solutions can spontaneously break symmetries that the exact ground state respects. Compare energies of different symmetry-broken solutions.
• Overestimation of order: mean-field typically overestimates ordering tendencies (higher critical temperatures, larger order parameters). Treat as an upper bound on ordering.
• Missing correlations: by construction, mean-field misses quantum fluctuations. In 1D, mean-field can predict false phase transitions that do not exist (Mermin-Wagner).
• Multiple solutions: SCF can converge to different solutions depending on the initial guess. Scan initial conditions.

Verification

• SCF convergence: energy and order parameters should be stable across final iterations.
• Comparison to exact results: for small systems, compare mean-field energy to exact diagonalization. The mean-field energy should be an upper bound.
• Known limits: check against analytically solvable limits (e.g., non-interacting, fully polarized).

Citations

• Methodology reference (reproduction-grade algorithm, parameters, validation, gap analysis): references/mf-methodology.md
• Ashcroft & Mermin, Solid State Physics (1976) — Hartree-Fock for lattice models.
• Auerbach, Interacting Electrons and Quantum Magnetism (1994) — mean-field for spin systems.