MeanFieldTheories.jl

MeanFieldTheories.jl is a Julia package for studying quantum many-body systems using mean-field theory and related methods. It provides a complete workflow from constructing many-body Hamiltonians to obtaining self-consistent ground states and calculating collective excitation spectra.

See documents: https://Quantum-Many-Body.github.io/MeanFieldTheories.jl

Features

• Fully customizable quantum system Degrees of freedom (site, sublattice, spin, orbital, valley, …) are freely defined by the user via SystemDofs, with user-specified constraints.

• High flexibility for generating operator representations DOF index constraints can be applied directly to generate_onebody and generate_twobody to select only the desired terms on each bond.

• Highly free forms of interaction Two-body interaction allows four different site index $(i,j,k,l)$. The creation-annihilation ordering of the operator string is also arbitrary and handled automatically.

• Unrestricted Hartree-Fock in both real and momentum space. All four Wick contraction channels (Hartree and Fock, both pairs) are kept open with no preset symmetry breaking.

• Complete post-HF excitation spectrum. On top of the mean-field ground state, collective modes are accessible, yielding dynamic structure factors and excitation gaps directly.

Documentation Contents

• Quantum System
• • 1. Degrees of Freedom
  • 2. Lattice
  • 3. Operators
  • Summary
• Introduction to the Hartree-Fock Approximation
• • 1. The Many-Body Problem in Condensed Matter
  • 2. The Variational Ansatz: Slater Determinants
  • 3. Mean-Field Decoupling via Wick's Theorem
  • 4. The Self-Consistent Field (SCF) Iteration
  • 5. Geometry and Dynamics of the SCF Map
  • 6. Unrestricted Hartree-Fock and Symmetry Breaking
  • References
• Real-Space Hartree-Fock Approximation
• • Theory
  • Code Implementation
• Momentum-Space Hartree-Fock Approximation
• • Theory
  • Code Implementation
• Bethe-Salpeter Equation, Tamm-Dancoff and Random Phase Approximation
• • 1. The Bethe-Salpeter Equation
  • 2. Hierarchy of Approximations
  • 3. TDA and RPA: Two Approximations Within the BSE
  • 4. Implementation in MeanFieldTheories.jl
  • References
• Particle-Hole Excitation Theory on Top of the Hartree-Fock Mean Field
• • 1. Parametrization of the Particle-Hole Excited State
  • 2. Variational Principle and the Eigenvalue Problem
  • 3. Complete Derivation of the Effective Hamiltonian $\mathcal{H}^{n_0 n,\, n_0' n'}_{\mathbf{k}\mathbf{p}}(\mathbf{q})$
• Particle-Hole and Hole-Particle Excitation Theory on Top of the Hartree-Fock Mean Field
• • 1. Full Random Phase Approximation (RPA)
  • 2. RPA Eigenvalue Problem: Derivation from Equation of Motion
  • 3. Derivation of the $\mathcal{B}$ Matrix
  • 4. Hermiticity and Inter-Block Relations for Implementation
  • 5. Derivation of the $\mathcal{D}$ Matrix
• Visualization
• • Lattice Visualization
  • Magnetization Visualization
• Example: SDW-CDW Phase Diagram of Extended Hubbard Model
• • Physical Model
  • Method
  • Code
  • Running the Example
  • Results
  • References
• Example: AFM Transition in Honeycomb Lattice Hubbard Model
• • Physical Model
  • Method
  • Code
  • Running the Example
  • Results
  • References
• Example: Finite-Temperature AFM on Simple Cubic Lattice
• • Physical Model
  • Method
  • Code
  • Running the Example
  • Results
  • References
• Example: Kane-Mele-Hubbard Model — SDW Phase Boundary
• • Physical Model
  • Method
  • Code
  • Running the Example
  • Results
  • References
• Example: AFM Magnon Dispersion on Square Lattice
• • Physical Model
  • Method
  • Code
  • Running the Example
  • Results
• Example: Ferromagnetic Excitations in the Tasaki Model
• • Physical Model
  • Method
  • Code
  • Running the Example
  • Results
  • References
• API Reference
• • Quantum System
  • Real-Space Hartree-Fock
  • Momentum-Space Hartree-Fock
  • Analysis
  • Visualization
  • Excitations
  • Utilities