My question is: Is there a topological invariance/winding number for non-translation invariance system? For example, if we modify the interacting parameter in SSH model, such that it depends on the lattice, can we still use winding number as a topological invariant of the system?
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2$\begingroup$ Consider to spell out acronyms. $\endgroup$– Qmechanic ♦Commented Apr 14 at 15:34
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1$\begingroup$ Look at the papers of Emil Prodan. In particular arxiv.org/abs/1010.0595 andProdan E, Hughes T and Bernevig B 2010 Phys. Rev. Lett. 105 115501 which work with disordered systems. $\endgroup$– mike stoneCommented Apr 14 at 15:57
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$\begingroup$ Thank you so much for your guidance, prof. Stone! $\endgroup$– feng linCommented Apr 15 at 9:11
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There are several index formulas you can use. In addition to papers by Emil Prodan, you should look at the book he wrote with Schulz-Baldes, "Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics."
If you fiddle with the strength of the hopping terms to much, you will create lots of phase changes and zero modes. In that case, you need to switch to a local invariant. See the arxiv preprint 2402.07224 and its references.
