Gosh, I got a bit swamped there over the last week. Several things took me away from sitting down and doing a blog post, including teaching my class (more on that shortly), working on a film project (more on that longly) and doing my taxes (late this year – bah!) and the usual raft of committee meetings and so forth. But one of them was locking myself away for two days with my computer and a web connection and writing a paper from start to finish with Tameem Albash. We’d more or less completed the bulk of the project over two months ago, with some very interesting results that we’d talk about from time to time to try to understand what was going on, but I held things up, being distracted by several other things (some of which you know about from this blog). We decided two weeks ago that we’d just finish the thing once and for all, and then somehow ten days went by with me not getting to it. Then I decided to close the door and just do it. So from lunchtime on Friday through lunchtime on Sunday I became a recluse (kind of) and I wrote, in turn with Tameem, as we emailed and IMed back and forth, until we had a nice paper entitled “Holographic Studies of Entanglement Entropy in Superconductors”.
It is a project I’ve been hoping to do for a long time, but not all the pieces were on the market until late last year. The entanglement entropy has been a quantity of interest among physicists in various fields for years, especially in the condensed matter and quantum information community, and is regarded as a powerful probe of the physics of a system – potentially more powerful a probe that some traditional quantities that you might track. When you’re off into a strongly interacting regime, some of those traditional quantities might not even be available, and so you might not be able to know if your system has gone into a new phase, etc. The entanglement entropy is a quantity that is defined after dividing the system of interest into two disconnected pieces separated by a boundary of a fixed size, for the purposes of quantifying how much the degrees of freedom on each side of the division are are quantum entangled.
There are various strongly coupled phenomena in or near various quantum critical electronic systems of experimental interest that people would like to get to grips with more, and models of them are hard to construct using traditional methods. One of the very interesting things that’s been going on in a (growing) corner of string theory of late is constructing gravity duals (see below and earlier posts) of systems that have similar phenomena, with a view to having a means of understanding such physics in a setting where reliable computations are possible. Many of the papers we’ve written over the years have been notably full of physics from this area as a result of our interests in these issues, with titles containing phrases like Fermi surfaces, Landau levels, superconductors, and so on and so forth. The context has all been strongly coupled, where a lot of the critical behaviour has been somewhat non-traditional as a result. A lot has been learned in this subfield that people often call AdS/CMT, and it is in the spirit of the work I’ve described a lot here that has done similar things for applications to novel experimental results in nuclear physics. See several earlier posts, starting here.
The idea is to use the holographic correspondence techniques in superstring theory and M–theory I’ve described here many times to construct a model of gravity in higher dimensions that has the same physics as the non-gravitational physics in question. The holographic model of superconductivity in 2+1 dimensions is the thing of interest here. In fact, there’s only one completely known holographic (i.e. gravity) dual of a superconductor out there so far. Bobev et. al. wrote it down late last year. By completely known, I mean one needs to somehow take the effective model of gravity interacting with a Maxwell field and a charged scalar that Hartnoll et. al. wrote down in 2008, that gives you access to the neighborhood of the superconducting transition, and check that it can be made part of a complete theory of quantum gravity, getting control of all of its physics right down to being able to follow it down to its quantum critical ground state at zero temperature. Bobev et. al.’s model does that nicely, and it enjoys a stable (in N=8 supergravity in D=4, the biggest reduction of 11D supergravity) ground state, as shown here. (Gauntlett et. al. had done a nice job of constructing such an embedding in an earlier paper, but it was shown by Bobev et. al. in another paper that it [update: well, an infinite subfamily of the examples the embedding yields- see also this paper] suffers from an instability at zero temperature.)
So this is the model for which we wanted to study the entanglement entropy, and to show for the first time how the entanglement entropy behaves in such a non-trivial superconductivity setting. The other really great tool we had at our disposal is a nice geometrical method proposed by Ryu et. al. in 2006 for computing the entanglement entropy in a holographic setting, which works in several dimensions, allowing for easier computation than is usually possible in a field theory setting. So everything was ready for working this all out, and we set about doing it, completing the core studies late last year.
We found some really nice physics, including another example of a novel phase transition in the entanglement entropy that we discovered in an earlier study we did in 2010 when studying the evolution of the quantity after quenches of various types. What happens is that the entropy, as a function of the size of the entanglement boundary, suddenly changes its rate of growth at a given critical size. (See left – follow the part of the blue curve that is lowest for a given size.) This tells you that there’s an important length scale in the physics that the entanglement is sensitive to.
We fold this all into the whole story to work out the behaviour of the entropy as a function of temperature, at fixed boundary size, and so forth, giving diagrams like those shown below. You read them as follows. Coming down in temperature, you follow the red curve, which is the normal phase (represented by a certain type of black hole in the gravity dual), and at a critical temperature (vertical dotted line) there is a phase transition to the blue line where you’re now in the superconducting phase (a different kind of black hole in the gravity dual). You stay on the lowest part of the blue line toward lower temperature, all the way down to zero temperature (although we’ve not shown all the points all the way down – gets numerically messy).
You can see jumps and kinks in the entropy entropy curve as the temperature falls, telling you about things like reorganizations of the degrees of freedom as the phase changes, and so forth. Exciting. I’ve a feeling that while the entanglement entropy itself might not be something one can track directly in an experimental setting (people are actively discussing this issue), its derivatives might well be connected to some kind of experimental probe, and so I’m rather interested to see if one day this kind of physics can be seen in experiments on some of the novel materials people are working on.
Anyway, I ought to let you read what we wrote for more. Our new paper has the following abstract:
We present the results of our studies of the entanglement entropy of a superconducting system described holographically as a fully back-reacted gravity system, with a stable ground state. We use the holographic prescription for the entanglement entropy. We uncover the behavior of the entropy across the superconducting phase transition, showing the reorganization of the degrees of freedom of the system. We exhibit the behaviour of the entanglement entropy from the superconducting transition all the way down to the ground state at T=0. In some cases, we also observe a novel transition in the entanglement entropy at intermediate temperatures, resulting from the detection of an additional length scale.