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Topological Invariants

PhoXonic.jl provides functions to compute topological invariants of photonic and phononic band structures.

Zak Phase (1D)

The Zak phase is the Berry phase acquired by a Bloch state when the wave vector k traverses the full Brillouin zone. For systems with inversion symmetry, the Zak phase is quantized to 0 or π.

lat = lattice_1d(1.0)
geo = Geometry(lat, Dielectric(1.0), [(Segment(0.2, 0.8), Dielectric(4.0))])
solver = Solver(Photonic1D(), geo, 128; cutoff=20)

result = compute_zak_phase(solver, 1:4; n_k=100)
result.phases  # Zak phase for each band
Per-band gauge ambiguity

Per-band Zak phases computed via the open Wilson line have an inherent ±π ambiguity due to endpoint gauge dependence in the truncated plane-wave basis. The multi-band total sum(result.phases) is gauge-invariant and reliable.

Unit Cell Choice

Different choices of unit cell origin yield different Zak phases, but the difference between bands remains invariant.

Wilson Loop Spectrum (2D)

The Wilson loop spectrum reveals band topology through the winding of eigenvalue phases. Non-zero winding indicates non-trivial topology.

lat = square_lattice(1.0)
geo = Geometry(lat, Dielectric(1.0), [(Circle([0.0, 0.0], 0.3), Dielectric(9.0))])
solver = Solver(TMWave(), geo, (32, 32); cutoff=5)

# Compute Wilson spectrum for bands 1-2
result = compute_wilson_spectrum(solver, 1:2; n_k_path=41, n_k_loop=50)

# Extract winding number
w = winding_number(result, 1)  # 0 = trivial, non-zero = topological

Parameters

ParameterDescription
n_k_pathNumber of k-points along the scanning direction
n_k_loopNumber of k-points for Wilson loop integration
loop_directionDirection of Wilson loop (:b1 or :b2, default :b2)

API Reference

See Advanced API - Topological Invariants.

References

  • Zak, J. (1989). Berry's phase for energy bands in solids. Phys. Rev. Lett. 62, 2747.
  • Yu, R. et al. (2011). Equivalent expression of Z2 topological invariant for band insulators using the non-Abelian Berry connection. Phys. Rev. B 84, 075119.