Higher Math for Condensed Matter

[Kitaev_Model]

So, I am an incoming MS student interested in going into condensed matter theory. I have always been more comfortable running C++ or FORTRAN simulations or crunching contour integrals than doing math proofs. I only gained the interest in CMT relatively late in my undergraduate career when I discovered what a horrible experimentalist I'd make(I'm hellishly clumsy), I began coding, and I took graduate level SS physics, meaning I didn't really have the time to go for a second degree in math. Thus, I neglected proof-based math courses throughout my undergraduate period and tended to instead learn stuff on my own, including in graduate level physics courses. This means while I am not completely ignorant about certain parts of "higher" mathematics that can be applied to physics-basic group theory, for example-I lack a truly rigorous background in things like algebra.

Due to the rather unique nature of my upcoming years for a US student seeking a physics Phd, I will have the opportunity, if I choose to do so, to take undergrad classes in things like algebra, topology, and real analysis. I'm wondering how useful that would be. On one hand, you can never have enough math and I really do want to improve my mathematics skills. But I'm also wondering if it would be better to simply pick up the useful parts as I go by myself or in physics courses-my graduate CM professor says that he will be teaching a fair amount of differential geometry in class, for example. Graduate QM this semester included some basic group theory as we went through Chapter 3 in Sakurai that helped cement a lot of my self study. A lot of courses in pure math are going to have a lot of things that are irrelevant to physics, and I've never been a very proof-anal guy. That being said, I tried that as an undergraduate, and while it did work to an extent, I always felt somewhat lacking compared to many. I will confess that I'd rather focus on doing well in graduate level physics courses and if I have spare time, sharpening my computational skills.

My research interests are relatively well defined for a guy fresh out of undergrad(I'm pretty set on condensed matter theory/computation), but I'm far from decided on what I'm going to do an eventual Phd thesis on, assuming (God willing) I get the chance to. I tend to lean towards strongly correlated electrons and many-body physics, which is very computational in nature, so I'm theoretically good to go for that. However there are branches of condensed matter theory that are more analytical-topological phenomena, for instance-and I don't want to be unable to understand or choose to do a Phd thesis in that direction, should I decide to do so.

So, anybody got advice?

 

[Arsenic&Lace]

Opinions vary widely on this topic. Some say that the pure math is useful and you should take pure math courses. Others say the pure math is useful but you don't need that much detail and you can get by with review papers and textbooks for physicists. I happen to think it's probably a waste of time from a scientific standpoint, but not from a "can I understand what this guy's work is about?" standpoint. Nobody knows for sure though.

My observations are that the most intense pure math in CMT seems to be found in work by people like Alexei Kitaev on topological matter. Another thing to notice is that this field is quite new and has barely contacted experiment. Look at quantum gravity or topological quantum computing, and you'll find that the papers are typically vastly more mathematical than papers in fields which have contacted experiment, even something like QCD or classical gravity. To me this is an indication that, without experimental guidance, people trend towards Ptolemy: adding epicycles on epicycles and creating bewilderingly intricate theories because they haven't made the correct postulates, which would result in a rather more simple theory.

Even if the above conjecture is true however, you will still have an advantage if you can read the papers written by the more mathematically oriented, so I personally have invested a bit of time in review papers very specific to the mathematics used in the applications; I took courses previously but found it to be far too time consuming.

 

[Kitaev_Model]

Yeah, I've noticed that, and it's why I'm asking. Again, I'm not looking to become a mathematician, what I am looking to do is to be able to understand others work and if I desire, work in the field. I've gotten different answers from different people even after talking about my specific interests. If that is true, since the knowledge that I'll need will be far more specialized than the classes will offer, I'm better off learning on an "as needed" basis on my own?

Yes, I've noticed that the more mathematical topics tend to be things like what Kitaev does. That's pretty cool and I don't want to rule that out as a Phd thesis, but if I had to choose now, I'd do something different.

 

[Arsenic&Lace]

I suspect the "as needed" basis will suffice, simply because working through a couple major reviews will give you the foundation and then years of work will allow you to slowly build upon it. But somebody with more experience than me (I'm in your position academically, as I'm not starting grad school until the fall) in CMT might have some contradictory views.

 

[radium]

It wouldn't hurt to have taken maybe one or two math courses like topology or analysis just to get a perspective, but all of the math in CMT (which ranges from fields like topology/geometry to group theory, representation theory) can be learned on an as needed basis. A really good book for this is Nakahara's Topology, Geometry, and Physics and others like it.

Basically everyone I have spoken with in the field has this opinion, even some of the most mathematical theorists (basically just as mathematical as Kitaev).

Topological order has actually been around for quite some time and has been observed experimentally. It is most definitely not a new field, it has just exploded over the past decade and gone in new directions (some very abstract). The first system was the FQHE experimentally discovered in the 1980s. Laughlin and Tsui got the nobel prize for this I'm 1998. Symmetry protected topological order has also been experimentally detected in the quantum spin hall effect, topological insulators, topological crystalline insulators.

 

[Kitaev_Model]

It wouldn't hurt to have taken maybe one or two math courses like topology or analysis just to get a perspective, but all of the math in CMT (which ranges from fields like topology/geometry to group theory, representation theory) can be learned on an as needed basis. A really good book for this is Nakahara's Topology, Geometry, and Physics and others like it.

Basically everyone I have spoken with in the field has this opinion, even some of the most mathematical theorists (basically just as mathematical as Kitaev).

Topological order has actually been around for quite some time and has been observed experimentally. It is most definitely not a new field, it has just exploded over the past decade and gone in new directions (some very abstract). The first system was the FQHE experimentally discovered in the 1980s. Laughlin and Tsui got the nobel prize for this I'm 1998. Symmetry protected topological order has also been experimentally detected in the quantum spin hall effect, topological insulators, topological crystalline insulators.

OK. This kind of confirms my latent opinion. I was able to understand topological order and the FQHE in grad level Solid State physics last year without a topology or real analysis class, so I guess I should be fine.

 

[Dale]

Please post technical discussions in the technical forums.

 

[Arsenic&Lace]

Please post technical discussions in the technical forums.

So instead of deleting an interesting discussion, could you fork it to a new thread?