Wilson loops: gauge covariance, phase conventions and topological tests

Scope

A Wilson loop describes transport around a closed path within a specified band subspace. Its eigenvalues survive unitary changes of basis inside that subspace; individual matrix entries do not. Use a fixed-rank, isolated projector and a consistent boundary identification. An insulator's occupied bands and a fully gapped Bogoliubov Hamiltonian's negative-energy bands are different physical constructions. A computed phase spectrum alone does not demonstrate superconductivity, a measured bulk gap or Majorana modes. R1 R6

The loop can organize Chern winding, time-reversal topology and hybrid Wannier centers. Those interpretations require additional assumptions. A closed-loop phase is not generally quantized, and an attractive crossing pattern is not sufficient evidence for a topological invariant. Zak's original abstract emphasizes symmetry-dependent quantization; the complete original derivation has not been reviewed here. R2 R4

Mathematical assumptions

Let the columns of $U_j$ form an orthonormal frame. Define $\mathcal A=iU^\dagger dU$ and forward overlaps $M_j=U_j^\dagger U_{j+1}$. In a periodic model basis, close the final link with $U_N=U_0$ and order $W=M_0M_1\cdots M_{N-1}$ from left to right along the path. Since $M_j=I-i\mathcal A_j\Delta k+O(\Delta k^2)$, the corresponding ordered exponential has $-i\mathcal A$, not $+i\mathcal A$. For one band, $\arg W=-\gamma$ modulo $2\pi$, where $\gamma=\oint\mathcal A$. The inverse-loop convention is also valid if every phase and orientation is translated consistently. R1

Under $U_j\mapsto U_jG_j$, with unitary $G_j$, each link becomes $G_j^\dagger M_jG_{j+1}$ and the closed product becomes $G_0^\dagger WG_0$. Interior factors cancel. This algebra explains both spectral gauge invariance and the need for closure. Actual Bloch bases can require a reciprocal-lattice sewing operator and orbital-position information instead of a naive endpoint identification. A different orbital embedding is not automatically a harmless gauge rotation within the same physical projector. R1 R3

Finite-step overlaps are generally not unitary. For $M_j=V_j\Sigma_j Z_j^\dagger$, use its polar unitary factor $Q_j=V_jZ_j^\dagger$ when the overlap is nonsingular. This construction preserves gauge covariance and yields a unitary discrete product. Record the smallest singular values, subspace dimension, direct gap and mesh convergence. Polar projection cannot repair a singular overlap, an incorrectly selected subspace or an unresolved band touching. R2 R5

An independent rank-two example combines spinors $(\cos(\vartheta/2),e^{iwt}\sin(\vartheta/2))$ in separate two-dimensional blocks. Their analytic forward phases are $2\pi w\sin^2(\vartheta/2)$ modulo $2\pi$. Meshes of 32, 64 and 128 points approach these phases while raw-product eigenvalue magnitudes remain below one. Random $U(2)$ frame rotations, shifted basepoints and reversed paths pass the covariance and conjugation checks. Omitting the closing link deliberately breaks spectral gauge invariance. These are synthetic checks, not a material calculation. R1 R2

For an increasing-$k_x$ loop and positive $(k_x,k_y)$ orientation, the continuously unwrapped phase $\Theta(k_y)=\arg\det W(k_y)$ winds by $2\pi C$ as $k_y$ traverses its period in this convention. Four lower-band examples of $h=(\sin k_x,\sin k_y,m+\cos k_x+\cos k_y)\cdot\boldsymbol\sigma$ give $C=0,+1,-1,0$ at $m=-3,-1,+1,+3$. Three meshes agree with separate analytic-curvature quadratures, and reversing the loop reverses the sign. The critical masses $-2,0,2$ have explicit gap zeros and are rejected. Integer output alone is not a convergence certificate. R5

Experimental criteria

In a two-dimensional band insulator with time reversal satisfying $\mathcal T^2=-1$, the parity of Kramers-partner exchange over half the transverse Brillouin zone diagnoses the $\mathbb Z_2$ class. Preserve the complete occupied projector, the bulk gap and symmetry constraints. Sorting principal phases independently at every point can invent discontinuities at a branch cut; follow spectral connectivity or a validated gap-tracking procedure. The local checks above do not implement or certify a general $\mathbb Z_2$ classifier. R1 R2

For a single band with the stated Bloch convention, a hybrid center satisfies $\bar x/a=\gamma/(2\pi)=-\arg W/(2\pi)$ modulo one. In multiple bands, Wilson eigenphases correspond to centers in the appropriate parallel-transport or maximally localized hybrid construction, not arbitrary individual Wannier functions. Physical electronic polarization also needs electron charge, occupations, cell volume and a consistent origin; total polarization includes ionic contributions. Insulating adiabatic changes can be meaningful while an absolute branch remains ambiguous by a polarization quantum. A BdG phase spectrum cannot simply be substituted into the ordinary electron-polarization formula. R2 R3 R6

An experimental comparison needs an explicit map from the measured observable to the modeled projector, including sample state, orbital basis, energy window and relevant symmetry. Band inversion, a calculated loop or an isolated boundary feature does not by itself verify the full pairing state. No experimental transition temperature or material certification is assigned to the synthetic examples on this page. R1 R6

Limits and disputes

An isolated band manifold can have a well-defined Wilson loop even when a material is metallic. Its invariant is then not automatically the invariant of all physically occupied states: a Fermi surface makes that projector change rank. Likewise, band disentanglement selects a model subspace whose validity needs separate evidence. For BdG systems, respect particle-hole and time-reversal constraints and avoid interpreting Nambu duplication as independent electron occupations. Strong interactions, disorder and nodal systems can require different formulations. R5 R6

Complete scientific acceptance remains open. Selected author and published passages, two original polarization pages and independent numerical examples have been checked. The full Yu equivalence proof, full Zak derivation, final-version comparisons, realistic orbital sewing, material fits and a general time-reversal classifier have not been reproduced. The numerical checks certify only the stated bounded constructions. R1 R2 R3 R4

Discovery implications

Archive the Hamiltonian or projector, basis and embedding, symmetry operators, loop orientation, sewing rule and phase convention with every topology claim. Keep raw singular values and mesh history alongside the plotted spectrum. Compare a simple limit with an independent curvature or symmetry calculation before interpreting a complicated model. This makes a Wilson-loop result a reproducible hypothesis about a specified system rather than an unsupported label attached to a candidate superconductor. R1 R2 R5 R6

Related topics

Berry curvature, Chern number, time-reversal-invariant topology, and Majorana zero modes.

Material cases

No sample-specific material records are included in this article. Model results are not experimental transition temperatures.