Timeline for What is the topological space in “topological materials/phases of matter”?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 23, 2021 at 22:31 | comment | added | Ruben Verresen | @J.Murray Thanks for the encouragement. I have given it a go! I hope our two posts give a complementary picture :) (I think so) | |
| Sep 22, 2021 at 21:46 | vote | accept | WillG | ||
| Sep 22, 2021 at 20:57 | comment | added | J. Murray | Let us continue this discussion in chat. | |
| Sep 22, 2021 at 20:50 | comment | added | WillG | My question is, what's the precise formulation of "indexing by a parameter"? Perhaps I'm being difficult, but there must at least be some rules for this process. Otherwise I could "index" the space of pairs $(k_x,k_y)$ with just a single parameter, even bijectively and continuously with a space-filling curve. So of course we want a definition of $\Gamma$ that forbids this. | |
| Sep 22, 2021 at 20:44 | comment | added | J. Murray | @WillG A generic lattice is not square/cubic, of course. In the more general case, one first constructs the reciprocal lattice; from there, the Brillouin zone is defined to be the reciprocal lattice's Wigner-Seitz unit cell. In the general case it is homeomorphic to a torus in the sense that points on opposite boundaries are identified, but its explicit shape can be quite strange. | |
| Sep 22, 2021 at 20:39 | comment | added | J. Murray | @WillG I'm not sure what you mean about identifying parts of the spectrum. The $|k\rangle$'s are (generalized) eigenstates indexed by a parameter $k\in [-\pi,\pi]$, where $|k=-\pi\rangle$ and $|k=\pi\rangle$ are explicitly the same object (just plug $\pm \pi$ into the definition). That's why we identify the space of possible $k$'s as the circle $S^1$, not an interval. For a 2D square lattice, we would have two so-called crystal momentum components $k_x$ and $k_y$ each belonging to $S^1$, so the space of pairs $(k_x,k_y)$ is given by the 2-torus $S^1\times S^1 \simeq \mathbb T^2$. | |
| Sep 22, 2021 at 20:29 | comment | added | WillG | Is $\Gamma$ instead constructed from the Hamiltonian's eigenspace? You reference the "parameter space," which makes sense to me, yet I can't think of its precise definition. | |
| Sep 22, 2021 at 20:25 | comment | added | WillG | I am still a bit unclear on how the space $\Gamma$ is technically defined (in general). I see the intuition behind identifying the endpoints of the interval $[-\pi,\pi]\subset\mathbb R$ in the first example, but the idea of "identifying parts of the spectrum whose corresponding eigenvectors are the same" does not seem like it generalizes to higher dimensions (since the spectrum is still just a subset of $\mathbb R$). | |
| Sep 22, 2021 at 20:21 | comment | added | WillG | Thanks for this thorough and pedagogical answer! This is indeed the kind of introduction I was looking for. | |
| Sep 22, 2021 at 0:02 | comment | added | J. Murray | @RubenVerresen Sure, this is not a comprehensive survey of topological phenomena - just a more or less self-contained pedagogical introduction to some key ideas. My interpretation was that the OP was only passingly familiar with solid state physics, and I tried to bear that in mind when considering the breadth and scope of my answer. That being said, if you can write an answer which covers strongly interacting systems and intrinsic topological order subject to those constraints, I would love to read it. | |
| Sep 21, 2021 at 23:12 | comment | added | Ruben Verresen | This is a nice answer but talks exclusively about free fermion systems. This thus misses out on the concept of 'intrinsic topological order', which is a very rich and interesting concept, and much more powerful than the free-fermion notion of topological phases of matter (but also harder to realize). | |
| Sep 21, 2021 at 19:02 | history | edited | J. Murray | CC BY-SA 4.0 |
added 33 characters in body
|
| Sep 21, 2021 at 18:24 | history | edited | J. Murray | CC BY-SA 4.0 |
added 124 characters in body
|
| Sep 21, 2021 at 17:51 | history | edited | J. Murray | CC BY-SA 4.0 |
added 124 characters in body
|
| Sep 21, 2021 at 17:15 | history | edited | J. Murray | CC BY-SA 4.0 |
added 164 characters in body
|
| Sep 21, 2021 at 17:09 | history | answered | J. Murray | CC BY-SA 4.0 |