Yesterday I submitted a new paper to the arxiv. This is is my favourite curve from it. Some of you who follow the blog will recognize the blue circle-dots and guess that this is the output of the dot-generation I’ve been tinkering away at (and reporting on somewhat cryptically) since April (see e.g. here, here, and here). Correct. There are many reasons why that is the case. One of them might well be because it looks like a very comfy chair, and by time I’d submitted the paper, I was rather tired. I’d pulled an all-nighter to finish the paper because I wanted to submit it by noon yesterday, and the night before I had to spend several hours at a social event.
So once it was an appropriate time to leave the place I was at, I said my goodbyes, jumped on my bike, pedalled home, put on some coffee, some Ana Tijoux (through headphones, so as not to wake anyone. Why her? Kinetic energy was what I needed at that moment – her vocal style is full of that. Try “La Bala” or “1977”. It is in Spanish, but that’s just fine.), and from 10:30pm to about 12 hours later, ground out the paper. I had to do this since I took some time away from the research project for a week, and then on Tuesday evening noticed the title and abstract of a new paper on the arxiv that suggested some overlap with what I was doing. So I had no choice but to gather all the results I’d been gathering the last several weeks and write them up and get them out, putting off reading the other paper until afterwards, so as to remain independent. Hence the all-nighter to finish it all. It was a pretty easy paper to write since I’ve had the results for a while, knew what I wanted to say, and it was just a matter of pulling everything together and writing a lot of background to set the scene for the results. A fair amount of the time was spent fiddling with things like how to generate figures from Matlab that embed nicely into the text, and so forth. Technical tedium.
The physics? Another reason I like the above curve is because it examines physics from an old favourite phase transition I co-discovered almost 14 – gosh yes, cvj, fourteen! – years ago. To my knowledge it is perhaps the earliest example of a second order phase transition in a gauge theory described holographically using a gravity dual. On the gravity side it is physics at a special point for a family of charged black holes. What I did in this new paper was map out how the entanglement entropy (an important and interesting quantity with a wide range of applications in many fields) behaves near a number of phase transitions from that old paper, and to my mind the behaviour near the second order point is the most enchanting. That’s the curve, plotting entropy, [tex]S [/tex], vs inverse temperature [tex] \beta [/tex]. Now this is all interesting me now because there’s a lot of effort these days in several fields to understand the physics of lots of phases (novel and otherwise) of matter (both theoretical and in experimental condensed matter) that often arise after phase transitions, sometimes second order, like superconducting transitions. Entanglement entropy is a useful probe of the physics (helping keep track of things like when the phase transition has taken place), but it is a hard thing to compute using traditional methods. So I got interested in looking for examples of second order points where one can actually compute the entanglement using gravity methods. It seemed prudent to look at the earliest example of such a second order point I could think of, and it was the one I found back in 1999. It was good nostalgic fun to revisit my old work on the subject, and bring aspects of it back into current work.
Another reason this is all interesting is because it allowed me to compare the physics to another mapping of entanglement entropy that I was involved in a year ago, for a second order point representing a kind of superconducting phase transition. I blogged about that here. There, I found with Tameem Albash (by studying a rare example of a complete (in a certain sense – fully back-reacted is a better term) gravity dual of a superconductor) that the entanglement entropy also changes its slope as it goes through the second order point, but there’s a kink – the slope changes abruptly. The example I presented yesterday shows that the entropy goes through a lovely smooth inflection point to change its slope through the transition. Interesting, and worth understanding further.
In any case, there it is. There’s a lot more in the paper than just that matter, such as the fact that the whole context is studying entanglement entropy in finite volume. This is where I think I overlap a bit with whatever’s in the paper that came out on Tuesday. Now I’ve had a good night’s sleep (yay!) I need to go and read that now and see what’s in there.