# $\mathbb Z_2$ or $\mathbb Z$ invariant for the SSH model

+ 3 like - 0 dislike
107 views

I am trying to understand topological insulators and topological invariant. The SSH (Su-Schrieffer-Heeger) model is often invoked as a protoypical topological insulator in 1D that carries localized zero modes at the edge. In every single treatment I could find, people compute winding numbers or Zak phases that can have one of two possible values. Thus, they are $\mathbb Z_2$ invariants, right?

Then, often the classification of topological insulators from symmetries is often discussed, and a "periodic table" is presented. (For instance: https://topocondmat.org/w8_general/classification.html). The SSH model falls into class AIII or BDI, depending on whether one considers the electronic or the mechanical case (as in Kane & Lubensky 2013, Topological Boundary Modes in Isostatic Lattices). However, in $d=1$, these periodic tables predict a $\mathbb Z$ invariant, not a $\mathbb Z_2$ one!

So what is it that I am not understanding here? Is the invariant from the periodic table a different one? What is the $\mathbb Z$ invariant for the SSH model then? Or am I reading the table wrong?

This post imported from StackExchange Physics at 2018-06-19 08:54 (UTC), posted by SE-user henrikr

 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): Email me at this address if my answer is selected or commented on: 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$\hbar$ysicsOverfl$\varnothing$wThen 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.