Correctness#

Correctness is checked at several independent levels: primitive operator algebra, sector dimensions, Hamiltonian matrix elements, analytic few-body spectra, sparse versus matrix-free action, and eigensolver residuals.

Operator algebra and sectors#

Small occupation spaces are compared directly with the canonical fermionic and bosonic relations,

\[ \{c_p,c_q^\dagger\}=\delta_{pq}, \qquad [b_p,b_q^\dagger]=\delta_{pq}. \]

Fixed-particle-number dimensions are checked against

\[ D_F(M,N)=\binom{M}{N}, \qquad D_B(M,N)=\binom{M+N-1}{N}, \]

and symmetry-constrained sectors are checked against the corresponding products of combinatorial factors. These tests establish the basis and primitive operator action before any model-specific Hamiltonian is constructed.

Analytic Hubbard dimer#

For the symmetric half-filled Hubbard dimer,

\[ E_0=\frac{U-\sqrt{U^2+16t^2}}{2}, \]

and the total double occupancy is

\[ \langle D\rangle =\frac12\left(1-\frac{U}{\sqrt{U^2+16t^2}}\right). \]

The numerical values coincide with the analytic curve across the plotted interaction range.

Analytic and exact-diagonalization double occupancy of the Hubbard dimer — Total Hubbard-dimer double occupancy. The analytic curve and values obtained from the finite Fock-space Hamiltonian coincide across the plotted interaction range.#

The singlet-triplet gap provides a second check involving the low-energy spectrum,

\[ \Delta_{ST} =\frac{\sqrt{U^2+16t^2}-U}{2} \xrightarrow[U/t\gg1]{}\frac{4t^2}{U}. \]

Analytic and exact-diagonalization singlet-triplet gap of the Hubbard dimer — Hubbard-dimer singlet-triplet gap. The finite-matrix eigensolution follows the closed-form result and approaches the strong-coupling superexchange scale.#

The complete derivation and operator definitions are given in Hubbard dimer: charge fluctuations and superexchange.

Constrained Schwinger-boson spin space#

For one Schwinger boson on each of two sites, the constrained bosonic Fock space is four dimensional and represents two spin-\(1/2\) degrees of freedom. The Heisenberg Hamiltonian

\[ H=J\mathbf S_0\cdot\mathbf S_1 \]

has

\[ E_{S=0}=-\frac{3J}{4}, \qquad E_{S=1}=\frac{J}{4}. \]

Analytic singlet and triplet energies represented with constrained Schwinger bosons — Constrained Schwinger-boson Heisenberg dimer. The four-state bosonic sector reproduces the analytic singlet and triplet branches.#

An independently constructed total-spin operator gives \(\langle S_{\mathrm{tot}}^2\rangle=0\) for the singlet and \(2\) for each triplet.

Total spin squared for the Schwinger-boson Heisenberg-dimer eigenstates — Total-spin expectation values at \(J=1\). The result identifies one singlet and the three states of the triplet manifold.#

The construction is shown in Schwinger bosons for a spin-1/2 Heisenberg dimer.

Independent Hamiltonian actions#

For a test vector \(x\), explicit sparse construction and matrix-free application are compared through

\[ \epsilon_\infty =\frac{\|H_{\mathrm{CSC}}x-H_{\mathrm{mf}}x\|_\infty} {\max(1,\|H_{\mathrm{CSC}}x\|_\infty)}. \]

Real and complex operator families are tested in both statistics namespaces, including symmetry-constrained sectors. Compiled execution paths are compared against the NumPy implementation using the same represented Hamiltonian.

Eigensolver residuals#

Each returned Ritz pair can be checked with

\[ r_n=\|Hv_n-E_nv_n\|_2. \]

Analytic agreement and small residuals test different failure modes: the former checks the model and operator algebra against an external result, while the latter checks convergence of the numerical eigenproblem actually solved.