Advanced Mean Field Solver Calculator

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Charts and diagnostics

Iteration history

Iter.	Input	Map	Output	Residual	Update	Free energy	Mixing	Method	Time

1. Mean-field model

Core equations

Model

Scalar Ising magnetization with mean-field coupling.

Unit mode

Energy-like inputs must share one unit system.

Interface mode

All controls remain included in project exports.

2. Physical parameters

Interaction J

Exchange or effective coupling.

On-site interaction U

Nearest-neighbor V

External field h

Temperature T

Boltzmann constant kB

Chemical potential μ

Coordination number z

Hopping amplitude t

Pairing interaction g

Energy cutoff

Density of states

Target filling

Spin magnitude

Degeneracy

Phase threshold

Boson occupation cutoff

Lattice sites

3. Order parameters and initial state

Initial guess mode

Initial values

Separate vector values using commas.

Custom dimension

Multiple-start count

Random seed

Symmetry restriction

Multiple solutions: nonlinear equations may contain stable, metastable, unstable, and symmetry-related fixed points. Multi-start mode compares distinct converged states.

4. Numerical solver

Method

Mixing α

Minimum mixing

Maximum mixing

Maximum update norm

Anderson memory

Broyden history

Line search

Trust radius

Oscillation action

5. Convergence and stopping

Maximum iterations

Minimum iterations

Absolute tolerance

Relative tolerance

Free-energy tolerance

Density tolerance

Eigenvalue tolerance

Successful consecutive steps

Residual norm

Divergence threshold

Stagnation window

Timeout seconds

6. Jacobian, derivatives, and stability

Jacobian source

Difference scheme

Derivative step

Stability mode

Condition warning

Derivative verification

7. Lattice and geometry

Spatial dimension

Lattice

Boundary

Sublattices

Twist angle

Defect fraction

Custom adjacency or hopping matrix

Stored in exports. Homogeneous built-in equations use z directly.

8. Thermal occupations and integration

Ensemble

Occupation

Chemical-potential mode

Smearing

Smearing width

Occupation cutoff

k-point grid

Integration points

Integration method

Energy minimum

Energy maximum

Symmetry reduction

9. Parameter sweeps and continuation

Enable

Sweep parameter

Start

End

Points

Spacing

Direction

Custom values

Use previous solution

Stop on failure

Mark transitions

10. Custom equation builder

Self-consistency equations

Separate vector equations with semicolons or line breaks. Use x0, x1, beta, and parameter names.

Optional free-energy expression

Allowed functions include tanh, exp, log, sqrt, abs, min, max, pow, fermi, bose, and langevin. Arbitrary PHP code is never executed.

11. Performance and execution

Matrix storage

Sweep workers

This portable build runs sequentially.

Cache mode

Preview resolution

Displayed history rows

Execution target

12. Presets and examples

13. Import and export

Import JSON

Ready. Select a model and solve.

Formula used

The application solves a fixed-point equation x = F(x; p). It reports the residual R(x) = F(x; p) − x. Convergence requires small residual and update norms.

Linear mixing uses x_n+1 = x_n + αR(x_n). Newton methods solve a numerical linearization. Stability comes from the map Jacobian and available free-energy curvature.

Curie–Weiss: m = tanh[(J z m + h)/(kB T)]

Antiferromagnet: mA = tanh[(h − J z mB)/(kB T)]

BCS: Δ = 2 g N(0) ∫ Δ tanh(E/2kBT)/(2E) dε

Hubbard closure: nσ = f(μ ± h − U n−σ)

How to use this calculator

Select a built-in physical model or custom equations.
Enter all physical parameters using consistent units.
Choose one initial state or enable multiple starts.
Use adaptive mixing for a robust first calculation.
Set tolerances, iteration limits, and derivative controls.
Enable continuation sweeps for phase boundaries and hysteresis.
Inspect convergence history, free energy, and stability warnings.
Export the complete configuration and numerical results.

Important: numerical convergence does not guarantee the globally stable physical phase. Compare free energies, Hessian estimates, multiple starts, and forward versus reverse sweeps.

Implementation notes and limitations

Numerical guidance

Reduce mixing when iterates oscillate. Increase integration points for BCS convergence tests. Increase the boson cutoff until Bose–Hubbard values stop changing.

Newton and Broyden methods are sensitive to starting values. Continuation usually provides better guesses near phase boundaries.

Physical scope

Mean-field theory neglects many fluctuations and correlations. The Hubbard option is an atomic homogeneous closure. The Bose option is a finite single-site Gutzwiller calculation.

Controls marked “documented” are retained for project planning but are not separate numerical kernels in this portable file.

Glossary

Order parameter: A quantity distinguishing phases, such as magnetization or a superconducting gap.
Residual: The difference between the calculated mean field and the current mean field.
Mixing: Damping that combines old and newly calculated values.
Continuation: Using a nearby solved point as the next initial state.
Metastability: A locally stable solution that is not necessarily the global free-energy minimum.
Spectral radius: The largest Jacobian eigenvalue magnitude. Values below one support local fixed-point stability.
Hessian: The matrix of second free-energy derivatives. Positive eigenvalues suggest a local minimum.