Sparse exact diagonalization and symmetry sectors

A chain of L spin-one-half sites has $2^L$ product states. If total spin projection is conserved, a sector containing $N_\uparrow$ up spins has dimension $\binom L{N_\uparrow}$. A spinless-fermion lattice with fixed particle number has the same combinatorial count, although its matrix elements carry fermionic signs. For spinful fermions with independently conserved up and down counts, multiply the two binomial dimensions. Local boson occupation requires either an exact physical bound or an explicitly tested numerical cutoff.

A symmetry block is invariant only when its projector commutes with the Hamiltonian. A transverse field breaks conservation of spin projection along a different axis; restricting to that projection anyway removes genuine dynamics. Translation and reflection can reduce dimensions further when compatible with boundaries and each other. Their sectors are not always obtained by dividing a total dimension by the group size, because some product states have shorter symmetry orbits. The absolute ground state also requires comparing all relevant sectors, rather than assuming its quantum numbers.

A local lattice Hamiltonian is a sum of operators acting on short neighboring groups. In a product basis, each term connects only a limited set of configurations. One can enumerate those connected states and accumulate their amplitudes without storing every matrix entry. This matrix-free action is sufficient for Krylov methods. Sparse storage saves memory when nonzero entries are few, but the state vector still has one amplitude per basis state, preserving an exponential barrier in generic many-body problems.

Implementation checks should include Hermiticity, conservation of sector labels after every local move, and normalization of recovered eigenvectors. Fermionic operator order matters: exchanging occupied orbitals changes signs even when the allowed destination configuration is correct. For small systems, compare the matrix-free action against an independently assembled dense matrix. This is a useful verification because a wrong sign can leave the matrix Hermitian and all computed vectors normalized while changing the physical spectrum.

For a normalized approximate vector v, compute its Rayleigh energy and residual using the original Hamiltonian action. For Hermitian H, the residual norm bounds the distance from that energy to at least one exact eigenvalue. It does not by itself identify that eigenvalue as the ground state. Inferring an eigenvector overlap additionally needs spectral separation; a degenerate manifold permits rotations of its eigenvectors without changing their energies. Report a subspace when individual vectors are not uniquely defined.

After numerical convergence, vary lattice size, boundary conditions, and any local basis cutoff. These probe different errors from an eigensolver tolerance. A finite-system gap may be a physical excitation or merely momentum quantization; its limiting behavior needs a scaling argument. Small exact calculations are especially valuable as benchmarks for approximate methods, provided both use the same Hamiltonian and sector. The two-state lab illustrates normalized amplitudes only and has neither a lattice-size control nor a sparse Hamiltonian constructor.