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This nice image of Dirac cones (from this article), in a ($E,\vec k$ graph) will be an introduction for several questions, in the realm of topological insulators.

1) Does the Dirac cone appears only at the surface ?

2) Is the shape (the cone) important ?

3) The Dirac cone is gapless, so is it only stable by symmetry-protection ?

4) Suppose a Dirac cone is opened, then closed, then re-opened. In the open situation, there is a energy gap, so there is a possible non-trivial topology. So it is possible to change the topology in the open->close->open process ?

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I hope somebody will provide more details, but let me just give some quick answers. 1) Yes, since topological insulators (TI) are per definition gapped in the bulk. The existence of gapless boundary modes can be heuristically argued for as follows: phase transition between to different TI's can only happen if the bulk gap closes. If one put two different TI's next to each other, the gap must close on the boundary between them such that there can be a transition. –  Heidar Jul 13 '13 at 11:34
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2) Depends on with respect to what it should be important. As far as stability of the edge modes are concerned, that's not important. However the shape will be important for questions about the detailed dynamics for example. If one takes higher energy/momentum contributions into account in the boundary low-energy effective theory, then the Dirac equation will get non-relativistic corrections in general. This is also clear from your picture. See for example arxiv.org/abs/0908.1418. Eq. (4) contain the first correction to the dispersion and thus the change of shape of the Dirac cone. –  Heidar Jul 13 '13 at 11:35
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3) Yes, the gaplessness of the boundary mode is protected by a symmetry (as is the case for all TI's). 4) I am not sure I understand this question. –  Heidar Jul 13 '13 at 11:36

1 Answer 1

Forewords: As Heidar remarked in the associated comments, my answers were not dedicated to the topological insulator situation. I'll try to correct myself in some edits I'll write [-> into brackets <-] and into answer-bis, but I let my answers about topological superconductors, since they may be helpful.

Short answers, please separate your questions if you want more detailed answered:

1) Dirac cone's on surface: Some emergent Dirac cones appears in the bulk of the $p$-wave chiral superconductor, see the book by Volovik for more details, available freely on his homepage at Aalto University. I'm not at ease with the notion of band structure on the surface. I have no idea what it means... That's just the closure of the gap which happens on the surface/edge for me. [-> Please see the Heidar comments for a clever discussion <-].

1-bis: the topological insulator situation. The topological insulator case is easier to discuss, since a bulk insulator has no closure of the gap by definition. Then, the Dirac-linear-closure can only happens at the edge. See also point 4 below, and the Heidar's comments about the Jackiw-Rebbi model below.

2) Shape of the cone: The shape, per se is not important. What you need is a linear dispersion relation with a crossing point. (NB: Without crossing, the dispersion corresponds to the Weyl fermion particles.) The cone structure is the simplest structure like this.

3) Symmetry protected topology: I don't know the full answer to this question. I would say no, not for the emergent Dirac cones in superconducting/superfluid phase: the cone can there be topologically protected as well. But the topology depends strongly on the symmetry for quadratic Hamiltonians, especially the three discrete ones of particle-hole $P$ such that $\left\{ P,H\right\} =0$ with $P^{2}=\pm1$, time-reversal $T$ such that $\left[T,H\right]=0$ with $T^{2}=\pm1$ (both $P$ and $T$ have anti-unitary representation, and $H$ is a representation of the Hamiltonian), and the chiral $C\equiv PT$ ones (a situation exists when $C$ is present without neither $P$ nor $T$). This is still troubling for me. I think it's essentially a matter of convention whether you want to call these discrete symmetries some kind of topology (whatever it means) or not. Topology for me means you've got a Chern number $\nu\neq0$, and you will keep it until you change one of the discrete symmetries I mentioned. But some Chern numbers are protected by symmetry as well, so it is a mess to disentangle all these notions at the end.

3-bis: the topological insulator situation. For the topological insulator once again, the situation is easier, since the topological classification is crystal clear: the topological characteristic are provided by symmetry. These symmetries are just the three discrete symmetries I discussed in point 3.

4) Opening <--> closure of the gap I think the answer to this question has been answered long ago by Volkov, and Pankratov, Two-dimensional massless electrons in an inverted contact JETP, 42 178 (1985) (article for free) or I misunderstood it. The answer is yes, and you get an instanton solution at the boundary, as in the Jackiw-Rebbi. Volkov and Pankratov discuss the Dirac dispersion relation, not a relativistic model.

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1) Since the question was in the context of topological insulators (TI), you cannot have gapless modes in the bulk per definition. If you have, then you are not in the phase of a TI. Its easy to get a Dirac cone in the bulk. Write down a simple model for a TI, say the one in physics.stackexchange.com/questions/3282/… . The low-energy theory will be a massive Dirac equation in the bulk. When the mass is zero and you have a Dirac cone in the bulk. That's the point of phase-transition, and therefore not a TI phase. –  Heidar Jul 13 '13 at 12:16
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Band structure on the surface actually do make sense. In the above mentioned model, I assumed translational symmetry and thus no edge and therefore $\mathbf k = (k_x,k_y)$ is a good quantum number. The eigenvalues of $H(\mathbf k)$ are the bulk band structure. Now assume that there is an edge at $x = 0$ and $x=L$. Now $k_x$ is not a good quantum number anymore but $k_y$ still is. Fourier transform $k_x$ to real space: $H(k_x,k_y)\rightarrow H_e(k_y)$. Our $2\times 2$ matrix is now turned into a $2L\times 2L$ matrix depending only on $k_y$. –  Heidar Jul 13 '13 at 12:24
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The eigenvalues of $H_e(k_y)$ (there will be $2L$ of them parametrized by $k_y$) is what you can call the edge band structure (although it also contain the bulk part). There one will see gapless bands, inside many gapped bands. Finding the eigenvector corresponding to the gapless modes, one will find that they are localized at the boundaries. Alternatively one a take the low-energy effective theory of the bulk and do the same, solving the diff equations one will get the boundary modes (similar to the Jakiw-Rebbi analysis). These are sometimes called Kaplan fermions in lattice gauge theory. –  Heidar Jul 13 '13 at 12:29
    
@Heidar Thanks for your comments. I've tried to correct correspondingly. Please tell me if there are still mistakes. Thanks a lot for the bulk-edge-band-structure discussion, too. –  FraSchelle Jul 13 '13 at 15:24
    
@Oaoa : +1 for the detailed answer –  Trimok Jul 15 '13 at 8:39

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