Math for Condensed Matter or Materials Science Theory

[redrum419_7]

I am a Physics major/ Math minor and would like to know what sort of undergrad upper-level math classes would be the most useful for Condensed Matter or Materials Science Theory?

 

[Physics Monkey]

In my opinion, reasonable answers to your question vary considerably within the wide world of condensed matter physics. Do you have any more specific idea of what you might want to study e.g. nanophysics or quantum computing?

 

[redrum419_7]

Quantum computing

 

[Physics Monkey]

On the more theoretical side you might try looking for classes in combinatorics, algorithms, complexity (classical). In my experience it helps to speak some of these languages in the theoretical quantum computing field. Advanced linear algebra e.g. numerical methods, abstract vector spaces, etc. is certainly fundamental to quantum computing research.

If you're more interested in physical realizations of quantum computers then the picture is different. Some familiarity with the above subjects may be helpful, but one will have to work a lot more closely with experiments (for which advanced mathematical training is often less helpful).

If you provide more information about your interests, then we can probably get a better idea of what might be a good fit for you mathwise.

 

[redrum419_7]

I was thinking more on the theoretical side of quantum computers that deals with condensed matter theory and, as you said, the physical realizations of them. I am already taking a Numerical methods and linear algebra course this summer. I was thinking that maybe QFT would help. Does QFT play a role in quantum computer research? I definitely want the math minor though so would topology or differential geometry help? I know they play some role in QFT.

 

[chill_factor]

On the more theoretical side you might try looking for classes in combinatorics, algorithms, complexity (classical). In my experience it helps to speak some of these languages in the theoretical quantum computing field. Advanced linear algebra e.g. numerical methods, abstract vector spaces, etc. is certainly fundamental to quantum computing research.

If you're more interested in physical realizations of quantum computers then the picture is different. Some familiarity with the above subjects may be helpful, but one will have to work a lot more closely with experiments (for which advanced mathematical training is often less helpful).

If you provide more information about your interests, then we can probably get a better idea of what might be a good fit for you mathwise.

quantum information theory seems to have very little to do with materials science.

what level of math is needed to even take a graduate level condensed matter physics class for physicists, never mind go into theory?

 

[homeomorphic]

I definitely want the math minor though so would topology or differential geometry help? I know they play some role in QFT.

There's one approach to quantum computing in which topology plays a big role. Microsoft has a big group of physicists and mathematicians working on this:

http://stationq.ucsb.edu/

However, that's only one approach. I'm a topology grad student who is interested in this sort of thing. It's my sense that you probably won't see very much topology while studying condensed matter, though. Occasionally, it might be relevant. But it's hard to say.

I like the idea of just studying whatever interests you to some extent. Like if you are trying to learn guitar, you want to play songs that you really like because that will motivate you. There isn't a clear-cut right answer of what to study. Just general guidelines, at most. If I ask my adviser if I should take this or that class, he'll always say it depends on what you want to do.

 

[cgk]

Does QFT play a role in quantum computer research? I definitely want the math minor though so would topology or differential geometry help? I know they play some role in QFT.

QFT in the high energy QFT's people sense is not used in condensed matter. There is condensed matter quantum field theory, but it deals with entirely different problems and approaches to dealing with them than the high-energy QFT stuff. Knowing high-energy QFT is unlikely to help you with the condensed-matter theory.

About topology: While topology does play a role in these branches of physics (particularly in 2D systems), the level and type of topology is again very different from what mathematicans study as "topology". You will never encounter a "topological space" without any superstructure or any of the associated theorems anywhere in physics. And even if your topology class is about what one would more likely associate with the word "topology" (tori, holes, etc), there is no need to study it from a mathematical perspective if you are willing to simply believe a few intuitive fundamental theorems instead of spending one or two years working towards their proofs.

Differential geometry might come in more handy (e.g., for the E&M aspects), but again it is not so clear.

What you *really* want to learn is (a) linear algebra (in all its forms. You will need all of them, and need to know them inside out), (b) numerics (again, as much as possible), (c) basic computer science (algorithms and data structures), (d) programming. If you are not a good programmer, you will not be a good researcher in condensed matter theory. And becoming a good programmer takes time (and in order to be a good programmer, you need to have at least a decent understanding of computer science).

quantum information theory seems to have very little to do with materials science.

It does have to do something with many-body physics. Many modern many-body approaches aimed for model problems in condensed matter argue in terms of information theory (say, density matrix renormalization grop, tensor networks, multiscale entanglement renormalization, general fermionic circuits). If you study many-body methods (which are at the heart of many branches of condendes matter theory), there is a good chance you'll come across things like area laws and all kinds of strange information entropies.

 

[homeomorphic]

About topology: While topology does play a role in these branches of physics (particularly in 2D systems), the level and type of topology is again very different from what mathematicans study as "topology". You will never encounter a "topological space" without any superstructure or any of the associated theorems anywhere in physics. And even if your topology class is about what one would more likely associate with the word "topology" (tori, holes, etc), there is no need to study it from a mathematical perspective if you are willing to simply believe a few intuitive fundamental theorems instead of spending one or two years working towards their proofs.

I'm so far on the topology side of this that I can't say much authoritatively, but it's my sense that, while this may be true for condensed matter students, at some point, more topology could possibly prove to be worthwhile. For example, Mike Freedman, at station Q won a Fields Medal for his work in topology. I am pretty sure his topology background has been useful to him at station Q. If you look at some of the papers in the area of topological quantum computing, I think some of them use some reasonably sophisticated topology. If you are making use of 3-manifolds, modular or fusion categories, mapping class groups, then knowing topology well helps. And I've seen all of those things come up in topological quantum computing papers.

So, I stick by my position that it's more a matter of preference in what you want to study than anything else. However, I would delay studying topology until you need it because if you aren't using what you learn, it gets too much like random topics that are hard to keep track of.

 

[Physics Monkey]

quantum information theory seems to have very little to do with materials science.

This is mostly true, although the state of affairs is slowly changing.

what level of math is needed to even take a graduate level condensed matter physics class for physicists, never mind go into theory?

Traditional condensed matter physics is not that complex mathematically. A solid grounding in complex analysis and Fourier analysis will get you pretty far. It also helps to have a solid numerical grounding. I think people tend to find that such courses are much harder physically than mathematically, that is one must integrate many different physical components to understand a material. But of course, there is arbitrarily complicated mathematics in condensed matter physics ...

 

[Physics Monkey]

Regarding topology in quantum computing, it is true that there is an approach inspired by topological ideas, but knowing quite a bit of this approach, I want to emphasize two things.

1. While Mike Freedman and the math Q guys understand formal topology, most of the physicists (even at Q) really don't bother with the formal stuff. Furthermore, the topology one needs is pretty elementary e.g. knot stuff and is pictorially obvious. There are interesting open questions e.g. universal braiding-only gates, but these questions also require a knowledge of all the other subjects I mentioned.

2. The actual physical implementation of such topological schemes has basically nothing to do with topology. It's all about the stuff that the pretty topology fails to capture e.g. finite timescales, finite lengthscales, finite temperature, ...

Besides topological quantum computing field theory has very little do with quantum computing right now.

 

[Physics Monkey]

Topology is quite useful in condensed matter physics, but you have to pick your problems and again it often appears at a rather elementary level e.g. homotopy groups or homology of simple spaces. That being said, things can get quite complex so that, for example, some of the formal tools of algebraic topology are useful for calculations. However, these kinds of questions tend to move further and further from experiment and into the land of string theory and mathematical physics.

If one really loves topology then string theory or certain other types of condensed matter theory are probably a better fit than quantum computing.

 

[redrum419_7]

Thanks everyone for the advice