Native topology across sliding geometries

The topology_phasons executable evaluates physical cross-geometry overlaps and first/second Chern pairings directly from CP2K STATE_EXPORT snapshots. It uses CP2K’s Gaussian cossin integrals and native LAPACK kernels. Neither Python, Z2Pack nor Wannier90 is a runtime dependency. Build with the CMake target topology_phasons; the executable suffix follows the CP2K build, for example topology_phasons.psmp.

The SCF calculations and the topology analysis are separate. Generate converged electronic states at the prescribed geometries, export the requested post-SCF k-points, then analyze these files. Changing the analysis mesh must not change the underlying SCF potential at a fixed geometry. The selected bands, spin convention, cell and atom count must remain consistent along the family.

Workflow

1. Choose a physically closed parameter family, a selected isolated subspace and explicit atom maps at its periodic boundaries. Two parameters give C1. Four parameters, for example (kx, ky, phi_x, phi_y), give the second Chern character pairing C2. A varying-atom-count twist scan is not such a family.

2. Enable STATE_EXPORT T in DFT/PRINT/WANNIER90 with NNKP or WILSON points. Keep the exported version-1 files, including all computed eigenvalues.

3. Write a mesh manifest as specified below and run, using one process:

   OMP_NUM_THREADS=4 topology_phasons.psmp mesh coarse.mesh > coarse.log
   

4. Repeat with every axis refined by an integer factor of at least two. To check a pair of meshes against an absolute invariant tolerance:

   OMP_NUM_THREADS=4 topology_phasons.psmp converge coarse.mesh fine.mesh 0.01
   

   This requires compatible dimensions, ranks, reciprocal periods and initial physical subspaces, and checks both the change and the finer estimate’s distance to an integer. The estimates themselves are never rounded. Supply the same physical family on both meshes; matching metadata alone cannot prove that all intermediate SCF states represent the intended branch. Additional refinements, shifted grids and ordinary electronic-structure convergence remain important. Numerical convergence does not certify a bulk spectral gap.

Physical mesh manifest, version 1

The following records appear in order. There are no inline comments:

1. CP2K_PHASON_MESH 1

2. Parameter dimension (2 or 4), selected rank.

3. Number of intervals along each axis, each at least 3.

4. Positive metric tolerance, direct-gap tolerance (hartree), minimum link singular value, maximum principal plaquette phase (radians, less than pi). Recommended starting values: 1e-7 1e-7 1e-8 1.5707963267948966.

5. Three fractional reciprocal coordinates for the origin.

6. One three-component reciprocal-period vector per parameter axis. Components must be integers; a pure displacement axis has vector 0 0 0. The geometry files, not these vectors, define the corresponding displacement.

7. Atom count and logical common-energy-reference assertion, e.g. 1 F.

8. One atom permutation per axis, using one-based indices. End atom i corresponds to start atom permutation(i), modulo periodic lattice translations. Each map must be bijective and the declared maps must commute.

9. One frame record for every index 0..n_1, ..., 0..n_d, including endpoints, with the first index varying fastest. A record contains a quoted snapshot filename, its one-based k-point index and an additive energy offset in hartree. Use absolute paths for portable invocation from another working directory.

For a mixed (kx, phi_x) mesh, the reciprocal-period rows are 1 0 0 and 0 0 0. At an endpoint kx=1, the manifest may reference the stored kx=0 state: the native evaluator retains the reciprocal seam phase. At a displacement endpoint, supply the corresponding endpoint geometry/state, not an arbitrary copy chosen only to force closure.

At every used frame the code checks AO-metric normalization and sampled selected/excluded-band separation. At a seam it verifies the atom mapping, contracted basis, geometry modulo allowed lattice translations and closure of the physical selected subspace. The endpoint polar sewing matrix is included in the link product. Independent eigenvector gauges are allowed.

common_energy_reference=T asserts that the supplied offsets put spectra on a justified common scale. For a lowest-band prefix the code then reports min(E[N+1])-max(E[N]). Otherwise it reports the direct isolation and explicitly omits the indirect-gap interpretation. It never independently zeros each Fermi energy. The global-gap assertion is not required to define an isolated band bundle.

Output and conventions

• CHERN_RAW: unrounded C1 or C2, selected by the parameter dimension.

• MINIMUM_LINK_SINGULAR_VALUE: conditioning before polar unitarization.

• MAXIMUM_PLAQUETTE_PHASE: phase-resolution diagnostic.

• Metric, seam, direct-gap and, when justified, indirect-gap diagnostics.

• REFINEMENT_CHANGE and INTEGER_RESIDUAL in convergence mode. An unsuccessful check terminates with nonzero status rather than printing a converged invariant.

The physical link is C_a^dagger O_ab C_b. O_ab includes displaced Gaussian centres, periodic images, the nonorthogonal metric and Bloch/reciprocal-seam phases. Both SOC spinor components are summed. No Hamiltonian, SOC term or magnetic field is added by the postprocessor.

For links directed from x to x+mu the plaquette is U_mu(x) U_nu(x+mu) U_mu(x+nu)^dagger U_nu(x)^dagger. C1 is minus the summed principal determinant phases divided by 2*pi, matching CP2K’s native Wilson winding convention. In four dimensions the full matrix logarithms F are retained:

C2 = -sum Tr(F12 F34 - F13 F24 + F14 F23)/(4*pi^2).

F is anti-Hermitian and includes the plaquette area. This is the physics second Chern character pairing, not Z2 and not generally the second Chern class. The estimator is not exactly quantized on a finite mesh. Unresolved phases, singular links and nonunitary plaquettes are rejected.

Reference links and individual overlaps

For independent model checks, topology_phasons.psmp links model.links reads: CP2K_TOPOLOGY_LINKS 1, dimension/rank, mesh intervals, the same four tolerances, then complex raw matrices (real/imaginary pairs) in vertex, direction, column-major-matrix order. Vertices have no repeated endpoints and the first mesh coordinate varies fastest. Physical sewing must already be included in these supplied links. converge-links coarse.links fine.links tolerance compares two such model meshes. This mode cannot validate a model’s basis or geometry.

topology_phasons.psmp overlap left.topology 1 right.topology 1 0 0 0 prints a single physical overlap. The final three integers shift the right reciprocal coordinate. Frame normalization and sampled isolation are checked first.

Memory, parallelism and validation

Screened atom-row contractions use OpenMP and never construct the full dense AO operator. At most a few selected-state frames and rank-sized working matrices are retained. Link matrices are streamed through an anonymous scratch file and plaquette curvatures are processed one vertex at a time, not stored as a dense four-dimensional tensor. Scratch disk use scales as 16 * rank^2 * dimension * product(mesh) bytes. Thread-local rank-sized matrices and text snapshot parsing can still be expensive for large occupied spaces. Use one process, not mpiexec -n >1; this auxiliary program does not distribute its file traversal over MPI ranks. No persistent link cache is trusted or reused.

The registered native unit tests cover moving normalized Gaussians, adjoint and seam-gauge checks, singular/underresolved links, C1 determinant phases, C2 gauge covariance and refinement of the rank-two 4D Dirac model with exact C2=-1. The existing Wilson/C1/Z2 tests remain unchanged. Material predictions require system-specific basis, SCF, mesh and size convergence in addition to these tests.

Method references: Fukui, Hatsugai and Suzuki (2005), Mochol-Grzelak et al. (2018), and Rosa, Ruzzene and Prodan (2021).