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In the book A Short Course on Topological Insulators (https://arxiv.org/abs/1509.02295) the authors start with a simple toy model, the SSH-Chain, which is a bipartite one-dimensional lattice with alternating hopping amplitudes $v$ and $w$. Its second quantized Hamiltonian is \begin{equation} \hat{H} = \sum_j v (b_j^\dagger a_j + a_j^\dagger b_j) + w (a_{j+1}^\dagger b_j + b_j^\dagger a_{j+1}), \end{equation} where $a_j$ resp. $b_j$ annihilate a particle at site $j$ of sublattice A resp. B. In momentum representation we may write this Hamiltonian as \begin{equation} \hat{H} = \sum_k \begin{pmatrix} a_k^\dagger & b_k^\dagger \end{pmatrix} \underbrace{\begin{pmatrix} 0 & v + w e^{ik} \\ v + w e^{-ik} & 0 \end{pmatrix}}_{=H(k)} \begin{pmatrix} a_k \\ b_k \end{pmatrix}. \end{equation} The Bloch Hamiltonian $H(k) = d(k)\cdot \sigma$, as a 2x2 matrix, is a linear combination of Pauli matrices $\sigma = (\sigma_x, \sigma_y, \sigma_z)$. This leads to the definition of the $d$-vector. In this case \begin{equation} d(k) = \begin{pmatrix} v + w \cos k & w \sin k & 0 \end{pmatrix}. \end{equation} The authors of the above mentioned book continue by defining a topological invariant, which is the winding number $W$ of the $d$-vector around the origin and show numerically that for $W \neq 0$ the open chain has localized edge states, i.e. is in a topological phase. In the figure below we can see the band structure (the eigenvalues of $H(k)$) and the curve traced by the $d$-vector for different parameters v and w.

enter image description here

My questions:

  1. The figure shows that the trivial insulator has exactly the same band structure as the topological insulator (when switching v and w). In popular science you often read that the topology is a property of the energy bands. In this case it is obviously not true. Is it true in any case? Or is the topological nontriviality always found in the eigenstates of $H(k)$?

  2. It seems like topological invariants are always defined in terms of the Bloch Hamiltonian $H(k)$, which is obtained from the Hamiltonian in position representation by Fourier transform. This is also the case for the Quantum Hall Effect and Quantum Spin Hall Effect. But going to the momentum representation is only a change in the basis. So we should be able to see that a system is topologically nontrivial also in the position representation. Is this actually possible?

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  • $\begingroup$ "In popular science you often read that the topology is a property of the energy bands". Do you have a reference for this? $\endgroup$ Nov 29, 2021 at 11:17
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    $\begingroup$ 1) I think you are confusing bulk and boundary band structure. The bulk band-structure of a trivial and topological band insulator might look the same, as they are by definition insulators (gapped system). But there are subtle topological differences between the corresponding Bloch waves that the bands represent, which are captured by the topological invariants. If you look at the same systems with a boundary, the topology difference might become apparent in the way bands connect (creating gapless edge modes). $\endgroup$
    – Heidar
    Nov 29, 2021 at 13:03
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    $\begingroup$ 2) In position space you can also trace the topological nature of the system. From $k$-space Bloch waves, you can construct $R$-space Wannier functions. Usually you can create Wannier states that are localized, but it turns out that this is not possible for topological bands. The 1D example above is a bit special though. The Wannier states in both phases are localized, but at different Wykoff points (it's a 1D artifact). $\endgroup$
    – Heidar
    Nov 29, 2021 at 13:04
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    $\begingroup$ "It is a property of the energy bands" ≠ "it is a property of the energy of the energy bands". The topology is a property of the band, which is a manifold in itself. The energy diagrams you've plotted are essentially functions from the band manifolds to the energy axis. But there's more to the bands than just their energy. $\endgroup$ Nov 30, 2021 at 10:42
  • $\begingroup$ @Heidar Are you able to point to a source discussing the localization of Wannier functions/continuity of bloch functions and its relation to topological bands in more detail? I know of the general idea that topological bands -> discontinuous bloch functions -> non-localized Wannier functions, however I don't understand why the SSH model is a "special case". All notes/reviews on the SSH model I've found agree with you in that they always chose continuous bloch functions, but don't mention that this is in general not possible for topological bands. (see also my comment on J Murrays answer) $\endgroup$
    – 1MegaMan1
    Mar 1 at 15:13

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I suspect part of the issue lies in your assumption that "band structure" refers to the curves on the $(E,k)$ diagrams, which isn't true. Rather, the band structure refers broadly to the map $k \mapsto H(k)$ where we recall that for a system with translational symmetry we can write $$H = \int_{\mathrm{BZ}} H(k) |k \rangle\langle k|$$ via Fourier transform. For the SSH model, one may use the semi-discrete Fourier transform to obtain $$H(k) = \pmatrix{0 & v+we^{-ik}\\v+we^{ik} & 0}$$ where $k\in \mathrm{S}^1$, and so in the two extreme limits we have $$w=0,v=1\implies H(k) = \pmatrix{0 & 1\\1&0} \qquad u_+(k) = \frac{1}{\sqrt{2}}\pmatrix{1\\1} \qquad u_-(k) = \frac{1}{\sqrt{2}}\pmatrix{1\\-1}$$ $$w=1,v=0\implies H(k) = \pmatrix{0 & e^{-ik}\\e^{ik}&0} \qquad u_+(k) = \frac{1}{\sqrt{2}}\pmatrix{e^{ik}\\1} \qquad u_-(k) = \frac{1}{\sqrt{2}}\pmatrix{e^{ik}\\-1}$$

The eigenvalues $E(k)$ in both of these limits are $\pm 1$ for every value of $k$; this is what is plotted on the $(E,k)$ diagrams. However, the critical difference between them is related to how the phase "wraps around" as $k$ varies from $-\pi$ to $\pi$.

In order for the $u$'s to be continuous functions$^\dagger$ of $k\in \mathrm S^1$, they must be periodic functions of $k$ on the interval $[-\pi,\pi]$. In general, the Bloch functions in this model can be expressed as $$u_{\pm} = \frac{1}{\sqrt{2}} \pmatrix{e^{i\varphi(k)}\\\pm 1}$$ In order for this to be periodic, then we must have that $e^{i\varphi(\pi)}=e^{i\varphi(-\pi)} \implies e^{i\big(\varphi(\pi)-\varphi(-\pi)\big)}=1 \implies \varphi(\pi)-\varphi(-\pi)=2\pi \nu$ for some integer $\nu$. For the $(w,v)=(0,1)$ case, this occurs because $\varphi(k)=0$ is trivially periodic. For the $(w,v)=(1,0)$ case, $\varphi(k)=k$ is no longer periodic but $\varphi(\pi)-\varphi(-\pi)=2\pi$.

This integer $\nu$ can be extracted as follows:

$$\nu = \frac{\varphi(\pi)-\varphi(-\pi)}{2\pi} = \frac{1}{2\pi}\int_{-\pi}^\pi \varphi'(k) \mathrm dk = \frac{1}{\pi} \int_{-\pi}^\pi \langle u|(-i\partial_k) |u\rangle \mathrm dk$$

The integrand is the so-called Berry connection.

$^\dagger$As an important note, it is generally not possible to find continuous Bloch functions of this kind. It turns out that the Chern number corresponding to a given band is an indicator of whether we can do this. For a 1D Brillouin zone, however, the Berry curvature (and by extension the Chern number) is trivially zero, and we can always find a continuous Bloch function.

In popular science you often read that the topology is a property of the energy bands. In this case it is obviously not true. Is it true in any case? Or is the topological nontriviality always found in the eigenstates of $H(k)$?

Popular science can't touch the ideas of eigenvalues and phase factors, for the most part. The band structure is the map $k\mapsto H(k)$, and it is this map and its associated ones (the Bloch functions, etc) which possess topological properties which may be trivial or non-trivial. The eigenvalues themselves do not inherently possess these properties.

That being said, there are many situations in which the eigenvalues might suggest underlying topological nontriviality; when e.g. spin-orbit coupling causes band inversion, you get something like this (source):

enter image description here

The bulk (solid) curves on the right exhibit a tell-tale sign of band inversion, which is in this case accompanied by non-trivial topology which is manifested in the surface (dashed) energy curves.

So we should be able to see that a system is topologically nontrivial also in the position representation. Is this actually possible?

For a truly infinite system, not really. In the SSH model, clearly the topological nontriviality of the bulk Hamiltonian is determined by the relative magntiudes of the intercell and intracell hopping amplitudes, but that's about it. The real magic of topological materials comes when you consider finite systems with boundaries, at which point the bulk-boundary correspondence associates the topological non-triviality of the bulk Hamiltonian to the existence of special edge states.

In particular, for a finite SSH chain, if the intra-cell hopping $v$ is zero and the intercell hopping $w$ is not, its clear that there will be zero energy states localized to the ends because there is no way for an electron on the first or last site to hop anywhere. The existence of these zero-energy boundary states for a finite chain (which is obvious from the position-space Hamiltonian, in which the edge sites don't appear) is determined by the nontriviality of the momentum-space band structure for an infinite chain. This remains true even when we leave the $v=0$ limit, which is a manifestation of the bulk-boundary correspondence.

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  • $\begingroup$ In all the lecture notes/reviews I've found which treat the SSH model and its topological properties, they are always able to choose continuous and periodic eigenfunctions for both the trivial and non-trivial phase. For example in this review they chose for the (topologically non-trivial) $(w,v)=(1,0)$ limit eigenfunctions of $(\pm e^{i k}, 1)^\intercal$ which can easily be seen to be continuous and periodic eigenfunctions of the 2x2 matrix. Not being able to chose continuous/periodic bloch functions (contd.) $\endgroup$
    – 1MegaMan1
    Mar 1 at 14:56
  • $\begingroup$ (contd.) seems to me goes directly against the comments made by @Heidar on the original question, where they state that the Wannier functions in both trivial and topological phases are localized. Localization of Wannier functions is only possible when the bloch functions are continuous and periodic. $\endgroup$
    – 1MegaMan1
    Mar 1 at 14:58
  • $\begingroup$ @1MegaMan1 Thanks for you comment. To be candid, I'm not sure what was going on in my head when I wrote that section of my answer, but it has been corrected. To answer your questions, one can choose a continuous global gauge if and only if the Chern number (related to the first Chern class) of the corresponding band vanishes. When the Brillouin zone is 1D, then this number trivially vanishes, and in that sense the SSH model is a special case. $\endgroup$
    – J. Murray
    Mar 1 at 19:02
  • $\begingroup$ @1MegaMan1 Your question about the localization of Wannier functions is intimately related to this as well. One can show that exponential localization is possible only if the Bloch functions are smooth, which is possible only if the Chern number vanishes. For e.g. $\mathbb Z_2$ topological insulators with vanishing Chern number, it is possible to obtain exponentially localized Wannier representations despite the phase not being topologically trivial. $\endgroup$
    – J. Murray
    Mar 1 at 19:28

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