Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This question is inspired by another one about the simplest model of topological insulator, where 4tnemele showed a nice two band model in the answer.

I read that and am wondering if we and push that to one dimension.

For example, by analogy to the graphene case, if we have a Hamiltonian in 1D (say x) as $H(k_x)=(k_x-k_0)+m$ for $k_x>0$. When $k_x=k_0$, one has $m>0$. $H(k_x)=(k_x+k_0)+m$ for $k_x<0$. When $k_x=-k_0$, one has $m<0$. A smooth connection in between, we will have a conductive edge (two ends in the 1D structure), right?

If I want to make a intuitive picture like below, is it correct? enter image description here

Any suggestion for real materials show this behavior?

share|cite|improve this question
I can't say anything too insightful as an answer to your actual question, but I think it's interesting to note that novel "edge modes" on the free ends of 1D systems are actually quite generic, two beautiful examples being emergent spin-1/2 excitations at the tips of S=1 Heisenberg magnets (see also the AKLT chain) or Majorana fermion modes on the ends of the Kitaev chain. – wsc Feb 26 '11 at 4:01
up vote 9 down vote accepted

Topological insulators are gapped states of free fermions with particle number conservation and time-reversal symmetry. According to the K-theory classification, there is no Topological insulator in 1D.

However, 1D interacting fermions with time-reversal symmetry do have non-trivial symmetry protected topological phases if the particle number is conserved only mod n. The result can be obtained from group cohomology theory arXiv:1106.4772 of Chen, Gu, Liu, and Wen.

share|cite|improve this answer
Dear Xiao-Gang Wen, if you cite yourself, it would be good if you could say so explicitly in your answers, not just in the links, cf. Physics.SE policy and SE policy. – Qmechanic May 30 '12 at 17:07
Hey Prof. Wen, how to understand the particle number conservation in topological insulators. Normally people only say it is time-reversal invariant. Thank you! – Timothy Oct 16 '12 at 13:02
@Qmechanic I thought "Chen, Gu, Liu, and Wen" explicitly contains my name Wen. Jeremy: Yes, normally people only say it is time-reversal invariant. But "insulator" implies particle number conservation. – Xiao-Gang Wen Oct 18 '12 at 5:34
Dear @Xiao-Gang Wen: What is meant is, that it would be best, if you could refer to yourself using a 1st person pronoun. – Qmechanic Oct 18 '12 at 5:44
@Qmechanic OK, Thanks – Xiao-Gang Wen Oct 18 '12 at 6:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.