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We can write the Hamiltonian for SSH model as

$H=\sum_i(t+\delta t)c_i^{\dagger} c_{i+1}+(t-\delta t)c_{i+1}^\dagger c_i+h.c$

We know that there are two topological phases

$N_1=0$ for $\delta t >0$

$N_1=1$ for $\delta t <0$

As far as I know, $N_1=1$ corresponds to zero energy states at both edges.

My questions are:

Do the zero energy states contribute to the closing of bulk energy gap and will the SSH chain conduct? Does $N_1$ correspond to conductance as in $2D$ topological insulators?

If $N_1$ doesn't correspond to conductance, what is significant or interesting in having these states?

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  • $\begingroup$ What is $N_1$? As far as I know, there are only topological quantities (i.e. solitons) at the domain walls in the SSH model. $\endgroup$
    – Xcheckr
    Commented May 5, 2015 at 2:11
  • $\begingroup$ @Xcheckr $N_1 $ topological invariant for non trivial topological phase $\endgroup$
    – lee
    Commented May 5, 2015 at 2:16
  • $\begingroup$ Where did you get that the topological invariant is 0 for $\delta$t>0 and 1 for $\delta$t<0? $\endgroup$
    – Xcheckr
    Commented May 5, 2015 at 2:18
  • $\begingroup$ @Xcheckr to my knowledge we can calculate $N_1$ by finding winding number for the hamiltonian using formula $N_1=\frac{1}{2\pi i}\oint dk h\partial_kh^{-1}$ $\endgroup$
    – lee
    Commented May 5, 2015 at 2:26
  • $\begingroup$ You have to be careful when using formulae like this. All that happens when $\delta$t changes sign is that there is a different configuration for the single and double bonds. There is only a non-zero winding number at a domain wall. Remember, polarization itself doesn't matter, only changes in polarization matter. See C. Kane's lecture for a good explanation: youtube.com/watch?v=si6ldpWeQ8c $\endgroup$
    – Xcheckr
    Commented May 5, 2015 at 3:15

2 Answers 2

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The zero energy states are localized at the boundary, and their wave functions decay exponentially in the bulk. So they are boundary modes (or edge states), which do not count as the bulk states, and do not contribute to the bulk gap closing. The only way to close the bulk gap in the SSH model is to tune $\delta t$ to zero. As long as $\delta t\neq 0$, the bulk gap (which is proportional to $|\delta t|$) is always open, no matter how many edges you cut open or how many edge modes you have.

$N_1$ does not corresponds to the Hall conductance in this 1D SSH model, because the 1D system simply does not have any notion of a Hall conductance. The Hall (or spin Hall) conductance is specialized to the 2D topological insulators, which can not be generalized to other dimensions. $N_1$ is also not related to the longitudinal conductance, because SSH chain has a bulk gap, so its conductance is simply zero (that is why topological insulator is an insulator in the first place).

Since the idea of the Hall conductance can not be generalized, then what is the physical meaning of the topological numbers in general dimensions? The answer is on the boundary: the topological number counts the number of boundary modes. In the 1D SSH model, $N_1$ counts the number of zero energy modes on the 0D boundary. In the 2D quantum Hall insulator, the topological number (Hall conductance) counts the number of chiral edge modes on its 1D boundary. In the 2D quantum spin Hall insulator, the topological number (spin Hall conductance, or $\mathbb{Z}_2$ number) counts the number of helical (counter-propagating) edge modes on its 1D boundary.

So what is the significance of having these symmetry protected topological (SPT) states? Well, a subject of condensed matter physics is to look for new states of matter. $N_1=1$ SSH chain is a new state of matter that can not be smoothly connected to the trivial chain with $N_1=0$. So it is interesting on the theory side. But is there any practical use of the zero energy modes? For SSH chain, I am not sure. However there is a very similar system, called the Majorana chain, which is basically a 1D topological superconductor with Majorana zero modes on the edge. The Majorana zero modes are associated with the topological ground state degeneracy of the superconducting chain, so that we can use this degeneracy to make topological quantum memories for quantum computers in the future.

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    $\begingroup$ What symmetry protects this topological phase? Is it the particle-hole or sub-lattice symmetry? I am sorry that I ask this question twice...(one in the question's comment section) $\endgroup$
    – an offer can't refuse
    Commented Jan 30, 2016 at 12:26
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    $\begingroup$ @buzhidao The SSH model describes a 1D fermionic SPT state in the symmetry class AIII, so the protecting symmetry is $U(1)\times Z_2^S$, i.e. charge conservation and sub-lattice (chiral) symmetry. The symmetry acts as $U(1):c_i\to e^{i\theta}c_i$ and $Z_2^S:c_i\to(-)^i\mathcal{K}c_i^\dagger$. The sub-lattice symmetry is an antiunitary symmetry that has the effects of particle-hole transform. On the single particle level, the sub-lattice symmetry acts as a unitary operator that anticommute with the Hamiltonian. $\endgroup$
    – Everett You
    Commented Feb 1, 2016 at 10:07
  • $\begingroup$ Does this share some resemblance of pristine graphene? The zigzag graphene also has gapless edge states and sublattice symmetry in the bulk... $\endgroup$
    – an offer can't refuse
    Commented Feb 1, 2016 at 11:36
  • $\begingroup$ @buzhidao No. Graphene is not a SPT. It is not gapped in the first place. Its edge state is not stable against disorder. $\endgroup$
    – Everett You
    Commented Feb 2, 2016 at 5:42
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    $\begingroup$ @L.K. You can think in this way that each time a Dirac fermion mass reversal will change the topological index $N_1$ by 1. So if we want to go from $N_1$ in the bulk to $N_1=0$ outside the bulk, we will have to close the bulk gap $N_1$ times, which results in $N_1$ zero modes on the boundary. A mathematically more rigorous statement is to use the index theorem (en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem). $\endgroup$
    – Everett You
    Commented Apr 15, 2017 at 17:23
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  1. There are some typos in the way you write the Hamiltonian. Should read $H = \sum_i (t+\delta t)c_{2i+1}^\dagger c_{2i}+ (t-\delta t)c_{2i+2}^\dagger c_{2i+1} + h.c.$

  2. $N_1$ does not correspond to conductance. Interestingly, in the superconducting counterpart of the SSH model, in topological superconductor wires, the bulk topological invariant does correspond to a quantized conductance value at 0 energy, the value at the zero-bias conductance peak.

  3. $N_1$ corresponds to a bulk sublattice polarization. It represents a value by which occupation of sublattice $A$ in the bulk is shifted with respect to sublattice $B$. This leads directly to a bulk-boundary correspondence, whereby $N_1$ also predicts a number of 0-energy, sublattice-polarized edge states. [https://arxiv.org/abs/1311.5233]

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