# Calculating topological invariants under different conventions of tight-binding models

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 Berry curvatures in Convention II. Refs.[2,3] shows 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, it seems that 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).

The two conventions are both applicable to calculate any topological invariants, convention II is way more practical because of the periodicity of the Hamiltonian in momentum space. As an example for using convention I for the SSH model, check out https://arxiv.org/abs/1212.0572 . Here, the authors say "We point out that the Zak phase of each dimerization is a gauge dependent quantity, i.e. it depends on the choice of origin of the unit cell, however, the difference of Zak phases of the two dimerizations is uniquely defined." This would be different if using convention II, when the Zak phases of both bands would be quantized.

The two conventions are related by a diagonal unitary transformation: $$$$\tilde c_{nj}(\mathbf k) = e^{i\mathbf k \cdot \mathbf a_j} c_{nj}(\mathbf k),$$$$ where $$n$$ is the band index and $$j$$ indexes internal degrees of freedom of the unit cell (e.g. orbital and sublattice degrees of freedom) with $$\mathbf a_j$$ the sublattice position within the unit cell. We find that $$$$\tilde H(\mathbf k) = U(\mathbf k)^\dagger H(\mathbf k) U(\mathbf k)$$$$ with $$U_{jj'}(\mathbf k) = \delta_{jj'} e^{i \mathbf k \cdot \mathbf a_j}$$. What are the implications on the Berry connection? It is straightforward to show that \begin{align} \tilde{\mathbf A_n}(\mathbf k) & = i \sum_j \tilde c_{nj}(\mathbf k)^* \nabla_{\mathbf k} \tilde c_{nj}(\mathbf k) \\ & = \mathbf A_n(\mathbf k) - \sum_j \mathbf a_j \left|c_{nj}(\mathbf k) \right|^2, \end{align} where $$n$$ is the band index. This is not a gauge transformation, unless all the $$\mathbf a_j$$ are equal, which is only the case for a Bravais lattice. Hence, the Berry phase in momentum space will in general be different for the two basis choices. However, for the SSH chain the difference of the Zak phase between the trivial and topological phase will remain the same because the extra term has the same value in both phases. For the SSH chain, this is easy to show since $$p_3=\tilde p_3=0$$, we have $$\left|c_{n1} \right| = \left|c_{n2} \right| = 1$$. When $$p_3$$ is nonzero, this does not work but in that case the Zak phase is no longer quantized in any basis.

Similarly, the Berry curvature, for example in two spatial dimensions, becomes $$$$\tilde F_{xy}(\mathbf k) = F_{xy}(\mathbf k) - \sum_j \left( a_j^y \partial_{k_x} - a_j^x \partial_{k_y} \right) \left|c_{nj}(\mathbf k) \right|^2.$$$$ When integrated over the whole Brillouin zone, the second term gives no contribution. This is easy to see for a square Brillouin zone, again because $$\left|c_{nj}(\mathbf k) \right|$$ is a periodic function, but it holds for any Brillouin zone.

So for (differences of) topological quantum numbers such as the Zak phase or the Chern number, the two conventions yield the same result.

EDIT

However, Convention I, which OP argued corresponds to the correct periodic part of the Bloch wave function, is still preferred, because there exist observables that are sensitive to the Berry curvature. For example, the Hall conductance of a metal is obtained by integrating the Berry curvature over only that part of the Brillouin zone which is occupied. Also, the Berry curvature plays an important role in the semiclassical equations of motion for crystals that either lack time-reversal or inversion symmetry.

Both conventions are equally valid. The key point here is to realize that operators whose matrix elements depend on the Bloch Hamiltonian, e.g., the current operator, will also be modified by the basis transformation. In the end, all observables should be the same in both conventions, though convention II may require more work, for example, due to the extra terms that popup in the Berry curvature.