User guide

Exact diagonalization replaces an interacting many-body problem by an exact eigenproblem in a finite Fock-space sector. The essential choices are physical: the local degrees of freedom, the conserved quantum numbers used to restrict the Hilbert space, the Hamiltonian, the observables, and the finite-size sequence needed for the question being studied.

Fixed-particle-number sectors, eigenvalue problems, and expectation values have the same mathematical role for fermions and bosons. Their local occupation rules and creation-annihilation algebra are treated separately where the statistics differ.

• Fock space, modes, and symmetry sectors
  • Mode labels and ordering
  • Fixed-particle-number sectors
  • Symmetry-constrained sectors
  • Fock states and basis coefficients
• Second-quantized operator algebra
  • Fermionic and bosonic algebra
  • Action on Fock states
  • Sums, products, and Hermitian conjugation
  • Bras, kets, and matrix elements
  • Common operator helpers
• Building Hamiltonians
  • Hopping and density interactions
  • Hubbard interactions
  • Spin exchange
  • Conserved quantities and sector choice
  • Complex hopping amplitudes
  • Sparse matrix structure
• Diagonalization and low-energy spectra
  • Complete spectra of small systems
  • Low-energy eigenpairs of larger sectors
  • Krylov-space controls
  • Basis-aware eigenstates
  • Residuals
• Observables and correlation functions
  • Expectation values
  • Real-space spin correlations
  • Connected charge correlations
  • Structure factors
  • One-body density matrix and natural occupations
  • Transition matrix elements
• Numerical considerations
  • Reduce the Hilbert space first
  • Sparse matrices and matrix-free operators
  • Matrix-free execution modes
  • Krylov-space memory
  • Numerical convergence of an eigenpair
  • Finite-size convergence