Spectral localizers

FORCE_EVAL / PROPERTIES / SPECTRAL_LOCALIZER evaluates a local topological index and protection gap from the converged GPW/GAPW Hamiltonian and AO overlap. It is independent of Wannier90 and Kubo transport. No additional library is required.

Finite systems

For CELL PERIODIC NONE and POISSON PERIODIC NONE, the operators are the analytic Cartesian AO moments. For the ordered plane \((X,Y)\), the covariant localizer is

\[\begin{split} L=\begin{pmatrix} H-ES & \kappa[(X-xS)-i(Y-yS)]\\ \kappa[(X-xS)+i(Y-yS)] & -(H-ES) \end{pmatrix},\qquad B=\operatorname{diag}(S,S). \end{split}\]

Here \(H\), \(S\), \(X\) and \(Y\) are AO matrices, \((x,y)\) and \(E\) are the query position and energy, and \(\kappa\) has units of energy per length. For class A, the index is half the signature of \(L\). The gap is the smallest absolute generalized eigenvalue of \((L,B)\), not a pivot magnitude. LAPACK pivoted LDL factorization is checked against the eigenvalue result.

&PROPERTIES
  &SPECTRAL_LOCALIZER
    FORMULATION FINITE
    INVARIANT CHERN
    PLANE XY
    POSITION [bohr] 0 0 0
    ENERGY [hartree] 0.0
    KAPPA [hartree*bohr^-1] 0.01
  &END
&END

Choose the energy and scale for the system. Repeated positions and lists of energies and scales produce scans. A real scalar Hamiltonian is not expected to have a nonzero class-A Chern index.

Spin-orbit coupling and periodic systems

SOC T adds GTH spin-orbit coupling after a restricted SCF calculation in the complete AO spinor space. It needs SOC pseudopotential parameters and is not self-consistent noncollinear DFT. INVARIANT Z2 additionally requires TIME_REVERSAL T. The code checks this symmetry and uses an orientation-sensitive Pfaffian sign in two dimensions. DIMENSION 3 uses the class-AII chiral-block determinant construction and all three position directions.

FORMULATION PERIODIC uses analytic sine/cosine AO integrals on a complete Born-von-Karman torus. Set MP_GRID to its replication factors and use ETA (energy units) instead of KAPPA. Nonperiodic directions must have size one. For two dimensions, PLANE must agree with the two periodic directions in both CELL and POISSON. Three dimensions require PERIODIC XYZ. The torus is built at the frozen SCF potential, independently of the SCF mesh and its symmetry reduction. Finite Cartesian coordinates are not substituted for periodic position operators.

Numerical controls

SOLVER DENSE is the reference implementation. MAX_AO bounds the complete scalar AO space including torus cells. Spinors double that space, and dense localizers require quadratic storage. EPS_METRIC, MATRIX_TOLERANCE and EPS_GAP control metric conditioning, operator checks and gap resolution. Unresolved gaps do not yield an index. SPECTRAL_FLATTENING optionally replaces the shifted Hamiltonian by its metric-covariant matrix sign after checking an electronic gap. Its scale and gap are distinct from the final localizer gap.

DFT / PRINT / AO_MATRICES / POSITION exports the finite AO moments in the same ordering as overlap and Hamiltonian matrices. SOC exports the three real antisymmetric GTH components, whose physical orbital operators include a factor of \(i\). Both exports are restricted to isolated systems.

These are finite-basis local diagnostics. Check stability against basis, system size, query position and scale before interpreting a material index. Broader examples are available in TopologicalCP2K.