Wilson Loops

A Wilson loop follows an isolated band subspace around a closed momentum-space path. Its eigenphases are invariant under changes of basis within that subspace and can be tracked as a transverse momentum varies. Numerically, overlap matrices and seam conventions are more reliable than derivatives of individual eigenvectors.

Helpful background. Berry-curvature numerics fixes the connection sign; non-Abelian Berry phase explains the geometric construction.

Overlap matrices and gauge covariance

Let $V_j$ contain orthonormal columns spanning the selected $r$ bands at successive points. Define forward overlaps $M_j=V_j^\dagger V_{j+1}$. A frame change $V_j\mapsto V_jG_j$ gives

$$M_j\mapsto G_j^\dagger M_jG_{j+1}.$$

For a closed path with its final frame sewn to the initial frame,

$$W=M_0M_1\cdots M_{N-1} \quad\mapsto\quad G_0^\dagger WG_0.$$

Hence its eigenvalues are gauge invariant. Product order matters: reversing it generally changes a non-Abelian loop. The last overlap must include the physical reciprocal-lattice sewing transformation rather than an arbitrary numerical identification.

Fix its direction explicitly: let $S_G$ map coefficient vectors at $k_0$ to the equivalent endpoint $k_0+G$, so that $H(k_0+G)=S_GH(k_0)S_G^\dagger$. On a mesh containing $k_0$ but omitting the duplicate endpoint, set $V_N=S_GV_0$ and use

$$M_{N-1}=V_{N-1}^\dagger S_GV_0.$$

The endpoint frame change is then the same $G_0$ as at the start, giving the stated conjugation law. As an algebraic seam test, a constant scalar frame with sewing $S_G=e^{-2\pi i\nu}$ has $W=e^{-2\pi i\nu}$ and $\gamma=2\pi\nu$ modulo $2\pi$. Replacing that sewing by identity loses the entire phase.

These overlaps presume that the vectors are expressed in a common orthonormal ambient basis along the open path. If the orbital basis changes with momentum, include its cross-momentum overlap matrix between the two coefficient frames. The reciprocal sewing fixes the endpoint identification; it does not replace the basis connection on interior links.

With $\mathcal A=iV^\dagger\nabla V$, forward overlaps approximate $\exp(-i\mathcal A\cdot\Delta k)$. Thus for one band the Berry phase is $\gamma=-\operatorname{Arg}W$ modulo $2\pi$. Keep this sign when converting phases to hybrid Wannier centers.

Polar transport

Finite-step overlaps are not exactly unitary because neighboring subspaces differ. Take an SVD

$$M_j=L_j\Sigma_jR_j^\dagger,\qquad Q_j=L_jR_j^\dagger,$$

and multiply $Q_j$ in place of $M_j$ to obtain a unitary discrete transport. This removes contraction of norms from the overlap magnitudes while retaining their unitary part.

Polar transport does not fix a singular or poorly resolved overlap. Inspect the smallest singular value in every $\Sigma_j$ and refine the mesh when subspaces change rapidly. The raw product and polar product should approach compatible eigenphases under refinement.

For nonsingular $M_j$, $Q_j=M_j(M_j^\dagger M_j)^{-1/2}$ is its unique unitary polar factor. Under endpoint frame rotations it transforms as $Q_j\mapsto G_j^\dagger Q_jG_{j+1}$, since the positive square root transforms by conjugation. Thus unitarizing each link preserves the closed loop’s gauge covariance, not merely its numerical norm.

For a loop around a reciprocal period $G=2\pi/a$, one-band center conventions can be written $\bar x/a=\gamma/(2\pi)$ modulo one. The absolute center depends on orbital embedding and unit-cell conventions; compare only after those conventions match.

Dimerized-chain phase benchmark

For the two-band dimerized chain with real hoppings and a fixed periodic orbital basis, an occupied lower band winds around the equator of its Bloch sphere when $|t_2|>|t_1|$. Its Berry phase is $\pi$ modulo $2\pi$; when $|t_1|>|t_2|$ it is zero.

Compute the loop product at a sequence of mesh sizes and randomize every eigenvector phase independently. The phase should remain unchanged. At $|t_1|=|t_2|$, the gap closes and the isolated occupied-band loop is not defined through the touching point.

This test checks phase quantization but cannot establish an absolute polarization without orbital positions and ionic/background conventions. That physical interpretation belongs to polarization and pumping.

Tracking spectra and winding

For several occupied bands, sort eigenphases only for display. Near phase crossings, independent sorting can create false discontinuities. Track eigenvalues on the unit circle using overlap information or a minimum-distance matching, and allow crossings that reflect degeneracy.

Unwrap only after resolving the mesh. A jump from near $+\pi$ to near $-\pi$ can be a branch-cut crossing rather than a physical discontinuity. The winding of the total phase $\arg\det W$ is often easier to validate than individual branch labels and connects to the total Chern number with the chosen path orientation.

For loops directed along increasing $k_x$ and swept along increasing $k_y$, the present forward-overlap convention gives

$$C=\frac1{2\pi}\Delta_{k_y} \operatorname{unwrap}\operatorname{Arg}\det W_x(k_y).$$

Indeed $\gamma_x=-\arg\det W_x$, while integrating $\Omega_{xy}=\partial_x\mathcal A_y-\partial_y\mathcal A_x$ over a strip gives minus the change in $\gamma_x$, including its seam terms. Compare this winding with a two-dimensional plaquette calculation. Reverse either loop orientation or transverse sweep and the sign reverses. A phase increment exceeding $\pi$ between adjacent transverse samples makes ordinary unwrapping ambiguous and calls for refinement.

Exercise

Why is multiplying only the diagonal overlaps insufficient for a degenerate occupied subspace?

Solution

An arbitrary unitary rotation mixes the occupied eigenvectors at each momentum, changing which diagonal entry represents a given “band.” Full overlap matrices transform covariantly and their closed product has gauge-invariant eigenvalues. Diagonal-only products discard this mixing and depend on arbitrary eigensolver choices.

References

• Gresch, D., et al., “Z2Pack: Numerical implementation of hybrid Wannier centers for identifying topological materials,” Physical Review B 95, 075146 (2017), arXiv:1610.08983.
• Vanderbilt, D., Berry Phases in Electronic Structure Theory, Cambridge University Press (2018), doi:10.1017/9781316662205.