Exact Diagonalization

Exact diagonalization solves a specified finite Hamiltonian without a many-body wavefunction ansatz. “Exact” describes the finite-model formulation, not infinite numerical precision or the thermodynamic limit. The main task is to separate eigensolver error, Hilbert-space truncation, and finite-size effects.

Helpful background. Many-body assembly constructs the operator; symmetry sectors reduces it.

Full spectra and selected eigenpairs

For Hilbert dimension $D$, dense diagonalization needs $O(D^2)$ storage and typically $O(D^3)$ work. It is suitable when many eigenstates are required, including high-temperature thermodynamics. Sparse Lanczos-type methods access a few extremal states through matrix-vector products, but do not provide the missing spectrum for free.

A normalized approximate eigenpair $(E,v)$ has residual

$$r=Hv-Ev.$$

For Hermitian $H$, some exact eigenvalue lies within $\|r\|_2$ of $E$, as derived in matrix diagonalization. Near degeneracy, individual eigenvectors can still rotate strongly while the residual remains tiny. Compare projectors or observable matrices within the nearly degenerate subspace.

If every eigenvalue outside a target subspace lies at distance at least $\Delta>0$ from $E$, its projector $P$ obeys $\|(I-P)v\|\leq\|r\|/\Delta$. This bounds the unwanted component of the computed state. The spectral separation must be established independently; the residual alone does not supply it or identify which level was found.

For $H=\operatorname{diag}(0,g)$ and $v=(1,1)^T/\sqrt2$, the Rayleigh energy is $g/2$ and residual norm $g/2$, yet the fidelity with either eigenvector is only $1/2$. Taking $g$ arbitrarily small makes the residual small without improving either individual eigenvector. This is why near-degenerate calculations should report subspace accuracy and physically relevant observables.

Choose tolerances relative to the scale of the intended conclusion. Resolving a small splitting requires residuals and roundoff effects well below that splitting, even if the absolute energies are extensive.

Heisenberg dimer from basis to observable

For $H=J\boldsymbol S_0\cdot\boldsymbol S_1$, $J>0$, the spectrum is a singlet at $-3J/4$ and three triplets at $J/4$. The ground state has

$$\langle S_0^zS_1^z\rangle=-\frac14,\qquad \langle\boldsymbol S_0\cdot\boldsymbol S_1\rangle=-\frac34.$$

Diagonalize the full basis and each magnetization block. Their spectra must agree after accounting for multiplicities. Check the correlation directly from the eigenvector, rather than inferring it from the energy alone. This independently tests observable assembly.

For a longer chain, reproduce the $J=0$ field-only limit and a small full-space result before relying on a large symmetry-reduced calculation. More sites do not compensate for an incorrect local operator.

Finite-size sequences

Record size, boundary conditions, conserved sector, and normalization per site. Compare like with like: a gap from one sector to another differs from the first excitation within a fixed sector. Determine which excitations a physical probe can access before choosing the gap.

A smooth sequence $e_0(L)$ can support an extrapolation only with a motivated form and enough sizes. Open boundaries often add $1/L$ surface contributions absent from periodic chains. Even/odd effects and level crossings can invalidate a naive polynomial fit.

Show the raw finite-size data and stability under removing the smallest size or changing the fit window. A small least-squares residual measures agreement with the fit form; it does not prove that form is asymptotically correct.

Observables beyond eigenvalues

Static observables require contractions with normalized eigenvectors. Dynamic correlations require the operator-applied state and relevant target sectors; see correlation functions. Thermal quantities require complete spectra or controlled thermal approximations; see partition functions.

Store basis conventions and sector identities with eigenvectors. Without them, a vector is not a portable physical result. For parameter sweeps, track subspaces by overlaps instead of comparing arbitrary eigenvector phases.

Exercise

An eigenpair has residual $10^{-10}J$, while two low energies differ by $10^{-12}J$. What can be concluded?

Solution

The residual places an eigenvalue near each approximate energy at the $10^{-10}J$ scale, but does not resolve a $10^{-12}J$ splitting. Tighten the solve, inspect orthogonality and precision, and compare the low-energy subspace. The present error bound is too large to identify the small difference as a physical gap.

References

• Saad, Y., Numerical Methods for Large Eigenvalue Problems, revised edition, SIAM (2011), doi:10.1137/1.9781611970739.
• Weinberg, P., and Bukov, M., “QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems. Part I: spin chains,” SciPost Physics 2, 003 (2017), arXiv:1610.03042.