Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

178 submissions , 140 unreviewed
4,360 questions , 1,685 unanswered
5,107 answers , 21,712 comments
1,470 users with positive rep
651 active unimported users
More ...

  Calculating topological invariants under different conventions of tight-binding models

+ 3 like - 0 dislike
105 views

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).

asked Jun 23 in Theoretical Physics by zrysky (20 points) [ revision history ]
edited Jun 23 by zrysky

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverf$\varnothing$ow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...