Theory and Algorithms¶

This page summarizes the theoretical background and numerical algorithms implemented in ED2.

Exact Diagonalization overview¶

Exact Diagonalization (ED) solves the many-body eigenvalue problem

\[ H |\psi_n\rangle = E_n |\psi_n\rangle \]

by representing the Hamiltonian \(H\) as a matrix in a chosen basis and computing its eigenvalues and eigenvectors exactly (up to numerical precision).

Hilbert-space construction¶

Local basis¶

Each lattice site is associated with a local spin basis \(|m_i\rangle\), where

\[ m_i = -S, -S+1, \ldots, S . \]

The full Hilbert space for a system of \(L\) sites has dimension

\[ \dim \mathcal{H}_{\mathrm{full}} = (2S + 1)^L . \]

Symmetry decomposition¶

ED2 explicitly exploits lattice and internal symmetries to decompose the Hilbert space into independent symmetry sectors. Calculations are performed within selected sectors, leading to block-diagonal Hamiltonian matrices and substantial reductions in computational cost.

Translation symmetry¶

For translationally invariant lattices with periodic boundary conditions, ED2 can enforce translation symmetry and work in fixed crystal-momentum sectors. Basis states are classified according to their momentum quantum number \(k\), and only states belonging to the selected momentum sector are retained.

This symmetry reduces the effective Hilbert-space dimension by approximately a factor of the number of lattice sites.

Point-group symmetries¶

If the lattice geometry admits point-group symmetries (e.g., reflections or rotations), ED2 can project the Hilbert space onto irreducible representations of the corresponding point group.

This allows further block-diagonalization of the Hamiltonian and facilitates the classification of eigenstates by symmetry.

Spin-inversion symmetry¶

For spin-\(1/2\) systems without explicit symmetry-breaking fields, ED2 can exploit spin-inversion symmetry, defined by simultaneous inversion of all spins (\(S_i^z \rightarrow -S_i^z\)).

The Hilbert space is decomposed into even and odd spin-inversion sectors, which can be treated independently.

Truncated Hilbert space¶

In addition to symmetry decomposition, ED2 supports calculations in a restricted Hilbert space defined by the number of spin-down (or equivalent) excitations.

For spin-\(1/2\) systems, define

\[ N_{\downarrow} = \sum_i \frac{1 - \sigma_i^z}{2} . \]

The truncated space is defined by bounds

\[ N_{\downarrow}^{\min} \le N_{\downarrow} \le N_{\downarrow}^{\max} . \]

Only basis states satisfying these conditions are included.

Hamiltonian representation¶

The Hamiltonian is assembled as a sparse matrix in the chosen symmetry-adapted basis. Matrix elements are generated on-the-fly during Hamiltonian construction.

Typical Hamiltonian terms include exchange interactions and external fields.

Eigenvalue solvers¶

Full diagonalization¶

For very small Hilbert spaces, ED2 can perform full diagonalization using dense BLAS/LAPACK routines.

Lanczos algorithm¶

For larger systems, ED2 employs Lanczos-type iterative eigensolvers to compute a small number of extremal eigenvalues within a given symmetry sector.

Computational considerations¶

By combining symmetry decomposition, Hilbert-space truncation, and iterative solvers, ED2 enables calculations that would otherwise be infeasible in the full Hilbert space.

Relation to the CPC manuscript¶

This section corresponds to the algorithmic description presented in the accompanying journal manuscript.