Let me be clear. I like topology. And I like how it rears its head in materials.

It has led to the most striking advances in all of condensed matter (Fractional Quantum Hall) and has led to paradigms that allow us to understand things a lot better. BUT, at the same time: it has set ablaze a firestorm of buzz words with little coherency.

Here’s all the ways I know about topology in condensed matter: I’m going to rant a bit here.

Topological order Link to heading

A phenomenon that is ‘robust to all perturbations’ but even after forty years, it only appears as a super super clean, confined, 2D electron gas (Fractional Quantum Hall Effect). Sure, there are thoughts that it can come out of fractionally occupied Chern insulators but who has seen real evidence of that yet?

Topological insulators Link to heading

Often thought of as Symmetry Protected Topological phases- a step down from topological order that is ‘robust to all but a few perturbations’, but the surface states, predicted to be ‘dissipationless’ and ‘immune from Anderson localization’ usually die by the time you are at 1K or have disorder. So that ‘robustness’ is like ’eh’ in the face of temperature or disorder or those few pertubations.

Oh and these cool Higher Order TI’s (Bismuth or Bi4Br4). Super cool. Also though, WHY DO I NEED TO HAVE A PERFECTLY CUT CRYSTAL TO SEE these ’topologically protected’ surface states?? I don’t understand why you can use the term ’topologically protected’ surface states but also: ‘I need you to cut this crystal into a hexagonal prism’, or we aren’t going to see them. Like ????… Incrementally steps, I know, but frustrating nonetheless.

Topological spin textures Link to heading

This is when you have nontrivial ’topology’ in the sense of your spins can form skyrmions or other weird shapes that cannot be adiabatically wished away. Except with temperature…? These things are actually really cool and their microscopic origins are usually well understood so I like them. But this uses the word ’topology’ in a different way than the top two, which makes it super duper confusing.

Topological hall effect Link to heading

Something like the shape of the Bloch wavefunctions of an electron give you extra hall effect contributions? I actually don’t understand why this is so big yet. It seems like people predict a B linear term as the standard hall effect, a magnetization-dependence term attributed to the anomalous hall term and everything else is topological?

$$ \rho_{xy} = \rho_{Hall} + \rho_{M} + \rho_{top} $$

That magnetization-dependence is very finnicky and people don’t justify it as well as I’d like. It seems that topology is used as a ‘god of the gaps’ and I haven’t figured out what the microscopic origin is.

Summary Link to heading

I don’t understand gapless phases at all yet so don’t really want to comment on Topological metals like Dirac/Weyl semimetals. There’s also that whole business with topological superconductors which I’ve tried to dive into but haven’t been able to wrap my head around.

I want to say again that I like topology.. but the word is used way too much to get publications and it makes me hate the world sometimes.

One of the main issues is that theorists just talk about topological invariants and use huge machines to study materials at 0 Kelvin. But, the second you add temperature, a lot of the stuff they talk about is lost. As an experimentalist, my main focus are observables: not two-point correlation functions type observables, but response functions that can come out in an experiment. I’d love it if the community would focus on realizing finite temperature topological order. Dilution Fridges can get you down to those temperatures where things can be seen but WE’RE RUNNING OUT OF HELIUM SO UHHHHHHHHH.

Long gone are the days I was interested in room-temperature superconductors. Now I want room-temperature topological order! I can settle for room-temperature topological surface states as well ;)

Hopefully I can update this in the future on all the ways ’topology’ comes into Condensed Matter!

A few references for the interested:

  • Topological Phases of Matter by Moessner and Moore

  • Geometry, Topology, and Physics by Nakahara (See discussion on defects)

  • Quantum Info meets Quantum Matter by Duan Lu Zhou, Xie Chen, Xiao-Gang Wen, Bei Zeng

  • Topological Quantum by Steven H. Simon (Didn’t read this, but heard its really good)

P.S. I can attribute this post to a single graduate student in the Analytis group who told me they don’t like topological things. I didn’t get him at first but then they brought up some fairly good points that I’ve been thinking aobut for a while until I wrote this blog post. You know who you are.