# 1D topological insulator

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?

Any suggestion for real materials show this behavior?

• 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

## 1 Answer

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.

• 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
• @Jeremy (sorry for commenting on such an old question). The reason why people don't mention particle conservation is that they are using the standard complex Hamiltonians of the form $H = \sum_{ij}H_{ij}a^\dagger_ia_j$ which explicitly conserves particle number. Except if the Hamiltonian has particle-hole symmetry, which means it describes a superconductor in the mean-field approximation (using Nambu spinors). (Continued) – Heidar Dec 8 '12 at 17:50