Calculating topological invariants under different conventions of tight-binding models

Question

There are two widely used conventions to construct the Bloch-like basis in a tight-binding model [1].

Convention I:
$$
\psi_\mathbf{k}=\frac{1}{\sqrt{N}}\sum_{\mathbf{R},j}c_j(\mathbf{k})e^{i\mathbf{k}\cdot(\mathbf{R}+\mathbf{a}_j)}|\phi_{\mathbf{R},j}\rangle
$$
where $j$ labels the orbitals $|\phi_{\mathbf{R},j}\rangle$ and $\mathbf{a}_j$ denotes the center position of $j$-th orbital. The corresponding tight-binding Hamiltonian and eigen-equation in k-space:
$$
\begin{gather}
H_{ij}(\mathbf{k})=\frac{1}{{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot(\mathbf{R}+\mathbf{a}_j-\mathbf{a}_i)}\langle\phi_{\mathbf{R},i}|\hat{H}|\phi_{\mathbf{R},j}\rangle\\
H_{ij}(\mathbf{k}) c_j(\mathbf{k})=E_\mathbf{k}\delta_{ij}c_j(\mathbf{k})
\end{gather}
$$
If the real-space Bloch functions obey periodic gauge $\psi_\mathbf{k}=\psi_{\mathbf{k}+\mathbf{G}}$, the tight-binding hamiltonian and eigenstates are not periodic with respect to reciprocal basis in this convention but satisfy
$$
c_j(\mathbf{k}+\mathbf{G})=e^{-i\mathbf{G}\cdot\mathbf{a}_j}c_j(\mathbf{k}),\qquad
H_{ij}(\mathbf{k}+\mathbf{G})=e^{i\mathbf{G}\cdot(\mathbf{a}_i-\mathbf{a}_j)}H_{ij}(\mathbf{k}).
$$

Convention II:
$$
\tilde{\psi}_\mathbf{k}=\frac{1}{\sqrt{N}}\sum_{\mathbf{R},j}\tilde{c}_j(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{R}}|\phi_{\mathbf{R},j}\rangle
$$
The corresponding tight-binding Hamiltonian and eigen-equation in k-space:
$$
\begin{gather}
\tilde{H}_{ij}(\mathbf{k})=\frac{1}{{N}}\sum_{\mathbf{R}}e^{i\mathbf{k}\cdot\mathbf{R}}\langle\phi_{\mathbf{R},i}|\hat{H}|\phi_{\mathbf{R},j}\rangle\\
\tilde{H}_{ij}(\mathbf{k}) \tilde{c}_j(\mathbf{k})=E_\mathbf{k}\delta_{ij}\tilde{c}_j(\mathbf{k})
\end{gather}
$$
If $\psi_\mathbf{k}$ respects periodic gauge, the hamiltonian and eigenstates in convention II are periodic functions in the reciprocal space:
$$
\tilde{c}_j(\mathbf{k}+\mathbf{G})=\tilde{c}_j(\mathbf{k}),\qquad
\tilde{H}_{ij}(\mathbf{k}+\mathbf{G})=\tilde{H}_{ij}(\mathbf{k}).
$$

The difference of the two conventions is that the information of the spatial distribution of orbitals is involved in Convention I but not in Convention II. Prof. Vanderbilt showed in his book [1] that the eigenstates $c_j(\mathbf{k})$ and $\tilde{c}_j(\mathbf{k})$ in the two conventions correspond, respectively, to the cell-periodic Bloch function $u_\mathbf{k}(r)=e^{-i\mathbf{k}\cdot\mathbf{r}}\psi_\mathbf{k}(\mathbf{r})$ and the original Bloch function $\psi_\mathbf{k}(\mathbf{r})$.  Although the two conventions can give exactly the same Chern number, the local Berry curvatures take different values in the two conventions, and different choices of unit cells result in different Berrry curvatures in Convention II. Refs.[2,3] show that only Convention I can give the "physical" Berry curvatures whose distribution respects all the real-space geometric symmetries of the system. So it seems that we should use Convention I to analyze the symmetries and to calculate topological invariants in general.

However,  in many tight-binding models, maybe we have to use Convention II to calculate topological invariants. The simplest example is the SSH model. The Hamiltonians in the two conventions are, respectively,
$$
\begin{align}
\text{Convention I:}&\quad H(\mathbf{k})=\mathbf{p}(\mathbf{k})\cdot\vec{{\sigma}}=\begin{pmatrix}
0 & v\,e^{ik(a_2-a_1)}+w\,e^{ik(a+a_2-a_1)}\\
v\,e^{ik(a_1-a_2)}+w\,e^{ik(-a+a_1-a_2)} & 0
\end{pmatrix},\\[15pt]
\text{Convention II:}&\quad \tilde{H}(\mathbf{k})=\tilde{\mathbf{p}}(\mathbf{k})\cdot\vec{{\sigma}}=
\begin{pmatrix}
0 & v+w\,e^{ika}\\
v+w\,e^{-ika} & 0
\end{pmatrix},
\end{align}
$$
where $a_1$, $a_2$ are the coordinates parallel to the periodic direction of the A,B atoms in the unit cell. Both $H(\mathbf{k})$ and $\tilde{H}(\mathbf{k})$ respect chiral symmetry, however, only $\tilde{H}(\mathbf{k})$ defined in convention II can give an integer winding number of $\tilde{\mathbf{p}}(\mathbf{k})$ thanks to its periodicity $\tilde{\mathbf{p}}(\mathbf{k}+\mathbf{G})=\tilde{\mathbf{p}}(\mathbf{k})$.
If we calculate the Zak phase (Berry phase traversing the whole BZ), Convention II can always give a quantized result ($0$ or $\pi$), but Convention I can not give a quantized result unless the selected unit cell has either mirror or inversion symmetry such that $a_1=-a_2$.

To summarize, my question is, are the two conventions both applicable for calculating any topological invariants? Say, if we use convention I to describe SSH model, what is the proper way to obtain a quantized winding number? Or do we have to use different conventions for calculating different quantities?

[1] Vanderbilt D., Berry phases in electronic structure theory, (Cambrige, 2018).

[2]  Dobardzic V. et at., Generalized Bloch theorem and topological characterization, Phys. Rev. B 91, 125424 (2015).

[3] Fruchart M. et al., Parallel transport and band theory in crystals, EPL 106,  60002 (2014).