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The following is my rough understanding of topological phases of matter (please let me know if it is incorrect.) Topologically ordered phases of matter are topological in the sense that they are determined by their topological excitations, and specifically by the braiding and fusion of these. The topological data that classify topologically ordered phases are encoded in higher fusion categories. (There are also invertible topological orders, which have no topological excitations but have a non-trivial gravitational response.) SPT phases are topological in the sense of having topologically-protected, gapless edge modes, but are only non-trivial if we don't break a symmetry $G$. The topological data that classify SPT phases are encoded in generalized cohomology theories depending on the symmetry $G$.

Dirac and Weyl semimetals are also referred to as topological phases of matter and called topological semimetals. In what sense are topological semimetals topological? What are the topological data that characterize topological semimetal phases?

There are several other questions on topological semimetals and on the meaning of topological phases in general, but I think this question is not a duplicate because I am asking specifically what topological data are used to classify topological semimetals.

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  • $\begingroup$ Isn’t it the number of available energy states below the fermi level? $\endgroup$ Aug 13, 2020 at 6:28
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    $\begingroup$ If you can describe why that number is topologically protected and what mathematical object counts it (for example, the number of edge states in an integer quantum Hall system is the first Chern class of a vector bundle integrated over the Brillouin zone), I would accept that as an answer. $\endgroup$
    – d_b
    Aug 13, 2020 at 20:15

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Topological (semi)metals are topological in a sense which is very similar to topological insulators and superconductors, i.e., their topological invariant can be phrased in terms of the 'twist' of the (eigenstate) vector bundle over momentum space. The only difference is that for topological insulators, one considers the whole of momentum space, whereas for topological (semi)metals, one only considers a submanifold.

As an example, let us consider Weyl semimetals (which exist in 3+1D). These are characterized by certain gapless points in momentum space. Focus on one of these gapless points, and enclose it in momentum space by a two-sphere. The dispersion on this sphere now looks like a gapped two-dimensional system. Hence, one can calculate the Chern number, and one will find it to be $\pm 1$ (depending on which Weyl node one had chosen). In fact, this is the reason that there is a gapless Weyl node: since the Chern number for this sphere is nonzero, we know that the Weyl node is ungappable (otherwise we would be able to derive a contradiction by shrinking the sphere to a point).

More generally, the topological invariant associated to a topological (semi)metal whose Fermi surface has codimension $p$ in momentum space is given by the topological invariant of a topological insulator in $p-1$ dimensions (and one can study the symmetries of the topological semimetal to figure out which of the classes of the tenfold way one ends up in). In the above example of a Weyl semimetal, the codimension is $p=3$ (i.e., one has to tune three momentum parameters to find the gapless point), and we found that its invariant is that of a two-dimensional Chern insulator. Two other examples:

  1. Graphene in two dimensions: its gapless Dirac points have codimension $p=2$. These can be enclosed by a circle in momentum space for which one can calculate the Zak phase, giving a topological invariant for 1D topological insulators (which again protects the Dirac nodes, conditional on preserving time-reversal symmetry---closely related to how the Zak phase for 1D TI's indeed requires this symmetry).
  2. Conventional Fermi surfaces! In any spatial dimension $d$, we can consider a Fermi surface (which of course has codimension $p=1$). This is a bit of a limiting case, where the 'sphere' one uses to 'enclose the Fermi surface' is now $S^{p-1} = S^0$, which just consists of two disconnected points (*). Even this limiting case makes sense: the topological invariant is measuring the filled energy levels for (any) point strictly inside the Fermi sphere (i.e., $|\vec k| < k_F$) minus the filled energy levels for a point outside the Fermi sphere. This topological invariant stabilizes a Fermi surface in any dimension! (**)

A fun historical anecdote: The above explanation makes it seem as if the theory of topological (semi)metals was an afterthought/generalization of topological insulators. Amusingly, historically it was reversed: in the early 2000's, Volovik (in his book 'Universe in a Helium Droplet') and Horava (arxiv:hep-th/0503006) realized this topological stability of (semi)metals, identifying the aforementioned topological invariants. It was only later that this same mathematics was re-interpreted (***) to imply the existence and classification of topological insulators. (That being said, initially the topological edge modes of topological (semi)metals were not readily understood and only seemed to come after the discovery of topological insulators.)


(*) Alternatively, one can work in Euclidean space, considering the inverse Green's function $\mathcal G(\vec k, \varepsilon) = i \varepsilon - H_{\vec k}$. Using this extra euclidean direction, one can enclose the Fermi surface with a circle, $S^1$. Since away from the Fermi surface, $\mathcal G(\vec k, \varepsilon)$ is an invertible complex matrix, we say the the topological invariant characterizing a Fermi surface is thus $\pi_1(GL(n,\mathbb C))$ (for $n$ large, i.e., we want an invariant that is stable under adding additional bands). This fundamental group is indeed non-trivial: $\pi_1(GL(n,\mathbb C)) = \mathbb Z$, implying the topological stability of conventional Fermi surfaces.

(**) Of course this is predicated on the assumption of having translation symmetry and particle number conservation. Breaking these symmetries can destabilize a Fermi surface.

(***) The re-interpretation involves imagining that the extra momentum directions are physical tuning parameters, such that the above 'topological stability of Fermi surfaces' becomes 'topological stability of phase transitions between two distinct gapped insulators', implying that these insulators must indeed be topologically distinct. E.g., again consider graphene, and take two distinct 1D slices of momentum space which are separated by one of the Dirac cones: now interpret this as a family of 1D Hamiltonians which are separated by a Dirac CFT critical point.

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