News From The Front, VIII: One Down…

work_snap…more to go. I’ve finished one of the papers I’ve been writing (this one co-authored with my student, Tameem) after delaying on it for months. I’m not sure how things got quite this backed up in terms of things I have to do, but they have. I meant to start on a new, long project last week, and all my efforts these days have been toward clearing away all those things I want to get done and dusted before focusing on that. It is taking time, but gradually the clearing is happening. Two more manuscripts to complete.

This paper reports on the continuation of the work we’ve been doing over the years in understanding the physics of various model systems in an applied magnetic field. This is in the context of holographic models of important strongly coupled phenomena that are of considerable interest in lots of fields of physics (particle physics, nuclear physics, condensed matter physics, atomic physics). (Since I don’t want to explain holography and so forth every time I talk about it, see a post I did about some of that here, and related posts in the list at the bottom of this one, if not sure what I’m talking about.) (Hmmmm, I see from my SPIRES listing that I’ve got seven papers mentioning magnetic field explicitly in the title in the last three years, and three or four more of the rest are occupied in large part with the issue too. No, really, I’m not obsessed.)

The issue here is the study of structures that suggest themselves as earmarks of Fermi surfaces in strongly coupled systems. It has been a goal for a long time in the context of gauge/gravity duals to understand what the signals of a Fermi surface would be. Would it be some geometrical object in the dual gravity theory, perhaps? Access to a computationally tractable description of such an object would be rather good to have in many situations of interest (including for lots of new physics being found in the lab, from high temperature superconductors to related phenomena), and this is what gravity duals have given us for many other phenomena over the last decade or so. Fermi surfaces (tons of information about what they are can be found on the web, so do google for it), and the whole technology (primarily due to Landau) of models of Fermi liquids are of great importance in condensed matter, for example, where complicated systems (from metallic conductors, insulators and semi-conductors to superconductors and other phases) get cast into a powerful, relatively simple language in terms of an effective field theory description. The Fermi surface is one of the central features of this description, and its fate in the context of models that have a strongly coupled description in terms of gravity (i.e. holography) is an important matter.

Well, I know I’ve lost 75% of the four of you who have read this far, but let me finish up by being a bit more detailed, for completeness.

Last year, a beautiful paper by Hong Liu, John McGreevy, David Vegh made a lot of progress in this area (building on work by Sung-Sik Lee). I like it a lot for several reasons (not the least because it is based on studying holographic aspects of charged black holes in anti-de Sitter spacetimes, an activity I helped kick off back in the old days of 1999 with Chamblin, Myers and Emparan). The extremal electrically charged black hole is dual to the strongly coupled (non-gravitational) system of interest at finite chemical potential [tex]\mu[/tex] and zero temperature and they construct and study the spectral function [tex]G_R(\omega,k)[/tex] of a fermion in the background. Just as in a condensed matter context, a pole in that function at some [tex](\omega_*,k_F)[/tex] represents quasiparticle excitations in the system and these are associated with Landau’s quasiparticle excitations at the edge of the Fermi surface at Fermi energy [tex]E_F=\omega_*+\mu[/tex] and momentum [tex]k_F[/tex]. Liu et., al., found such a pole at [tex]\omega_*=0[/tex] and non-zero [tex]k_F[/tex] and studied its properties.

Well, there’s a lot more to say about that, but let me not do that here. An obvious question to ask (besides all the ones about the related studies on holographic superconductivity that tell you that the system ought to have condensed to something else well before the black hole gets extremal – we chat about that in the introduction to our new paper) is how this all gets changed in the presence of magnetic field. An applied magnetic field is often a good probe (both theoretical and in the laboratory) of the physics of such systems and so a robust method for introducing one and studying its effects is good to have. Placing a magnetic charge on the black hole (making it a dyon) does that nicely (Tameem and I have used this before in the context of holographic superconductors here, here and here), and we started a study of that for this Fermi surface discussion last year in this paper, where we study the properties of the quasiparticle peak (at zeroth Landau level). (It actually moves to non-zero [tex]\omega_*[/tex] and splits into two peaks, with interesting properties: For example, the magnetic field can change the dispersion properties of the quasiparticles.)

We finished another project on this matter in October (and I’m embarrassed to say it’s only appearing tomorrow due to my tardiness). We extend the study to find and explicitly study the quasiparticle peaks for all Landau levels, uncovering a rather pretty story. (Landau levels are the discretely infinite tower of excitations you get for a fermion in a background magnetic field.) There’s a good deal of subtlety here, since there are two classes of solution to the problem.

One class is akin to the traditional separable solution in terms of Hermite functions that you find presented in the texts for free fermions in an external magnetic field. The separability extends to the whole of AdS, with the dependence on the AdS radial coordinate also separating out from the rest. (These are discussed in two papers that came out in the summer, here and here.) Interestingly, a sensible zero magnetic field limit of these solutions is only possible when the momentum is zero, and we find these solutions, as a result, less attractive for the Fermi surface issue than the second class (of which our first paper’s study was the zeroth member).

The second class has non-trivial radial behaviour, and is not separable, with simple Hermite functions only appearing at the black hole horizon. They have richer properties at arbitrary AdS radius. The zero magnetic field limit of these solutions place no restriction on the momentum, and it is for this reason we use them (in the spirit of the zero magnetic field study of Liu et. al.) to seek quasiparticles and Fermi surfaces, since one ought to be able to read off the momentum associated to the peaks as output of the computation, and not have to fix it by hand in order to have a good zero magnetic field limit.

Well, I’m getting carried away here. I ought to just let you read the paper. It appears in tomorrow’s listing (released tonight) of new papers on the arXiv, and I hope you find it interesting. (Update: It is here.) Here’s the title and the abstract saying what we do:

Landau Levels, Magnetic Fields and Holographic Fermi Liquids

We consider further a probe fermion in a dyonic black hole background in antiโ€“de Sitter spacetime, at zero temperature, comparing and contrasting two distinct classes of solution that have previously appeared in the literature. Each class has members labeled by an integer n, corresponding to the nth Landau level for the fermion. Our interest is the study of the spectral function of the fermion, interpreting poles in it as indicative of quasiparticles associated with the edge of a Fermi surface in the holographically dual strongly coupled theory in a background magnetic field H at finite chemical potential. Using a numerical study, we explicitly show how one class of solutions naturally leads to an infinite family of quasiparticle peaks, signalling the presence of a Fermi surface for each level n. We present some of the properties of these peaks, which fall into a well behaved pattern at large n, extracting the scaling of Fermi energy with n and H, as well as the dispersion of the quasiparticles.

Enjoy!

-cvj

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20 Responses to News From The Front, VIII: One Down…

  1. Michelle says:

    Sung-Sik Lee was my officemate!

  2. Blake Stacey says:

    Neat! I’ve been trying to catch up with the gauge/gravity duality literature, since lately I’ve been poking into the study of nonequilibrium systems with phase transitions in the directed percolation universality class. Various tricks exist to cast these systems in a QFT-like way (going back to Masao Doi in the 1970s, I think). The Lagrangian which comes out maps onto that for Reggeonic field theory, so who knows? AdS/CFT might be useful.

  3. Clifford says:

    Michelle: Wow. Small world!

    -cvj

  4. Per says:

    Hi Clifford

    How come its only available as pdf? This happen quite often and it is highly irritating because for some reason Acrobat and my computer is involved in a neverending conflict and dont work together.

  5. Clifford says:

    Hi,

    Sorry about that. Might be worth trying to resolve that conflict. Lots of pdf out there these days. You can try to generate ps from the source. Be sure to remove the third line. Not sure if the figures will embed right. You might have to turn them into eps first.

    Sorry.

    -cvj

  6. kim says:

    Clifford, when the universe cooled down why did the w and z bosons gain mass while the photons did not?

  7. Per says:

    Hi Clifford

    Its not your fault I assume? For the papers I submitted to arXiv I dont recall wetter it asks if you want .ps, .pdf or both. Or, perhaps, does it sort it out by itself? I noticed that you had several multi color graphs does that have anything to do with it?

    Anyways. Your field is somewhat, losely, connected to what I do so I could atleast understand the introduction. Very nicely written! Do you have any celebration rituals when you finish a paper? I usually go for dinner in some nice place. Food never tastes as good as the evening after you a paper appeared on the arXiv ๐Ÿ™‚

    Cheers Per

  8. Clifford says:

    It is our fault this time I think. We force pdf output in the source, in order to integrate the figures we produce the way we want it. Unified pdf and all that. Not neccessary, but that’s how we like it. Follow the steps above to do differently.

    Rituals. Celebration. On finishing paper. Huh. Used to, once upon a time, but these days not so much. Maybe too many projects overlapping? I don’t know. I still celebrate breakthroughs and particularly nice results, but these are often well before the paper is written as I no longer tend to race to get to print. Ultimately, it depends upon my mood, I think.

    -cvj

  9. kim says:

    Clifford did you see my post or did you miss it?

  10. Clifford says:

    I saw it. I am sorry, but I’m very tired of your use of the comments spaces to ask either overly personal questions or questions that you can mostly look up the answer to in a few seconds. I am happy to answer questions from time to time, but do try to do the initial legwork when a question occurs to you. I am not google, I’m afraid. Or maybe from time to time make some of the questions actually relevant to what I might have just spent hours writing? I would not mind it so much if you contributed to discussion from time to time, but instead it is just a relentless stream of random questions (although I do appreciate that they are friendly and polite). Very respectfully, I must insist that I am not a reference library.

    Best,

    -cvj

  11. kim says:

    wikipedia says: The fact that the W and Z bosons have mass while photons are massless was a major obstacle in developing electroweak theory. These particles are accurately described by an SU(2) gauge theory, but the bosons in a gauge theory must be massless. As a case in point, the photon is massless because electromagnetism is described by a U(1) gauge theory. Some mechanism is required to break the SU(2) symmetry, giving mass to the W and Z in the process..”

    so the answer is not that simple. Its not so much a question of whether they can be looked up, but that I wanted to engage in a discussion without it getting too technical because I dont have the background to understand the details.

  12. Clifford says:

    So basically what you’re saying is that every day (sometimes more than every day) I must promptly answer every random question on any subject that you come up with, whether it is relevant to anything I recently wrote about, within hours of you asking it? I’m sorry, but I simply can’t do it. Sometimes I will, sometimes I won’t.

    At some level, the answer to your specific _why_ question is probably “nobody knows”. I did not know that you were really asking the why question so much as the question about the Higgs mechanism itself. The unified electroweak force starts out with four massless force carriers, and the Higgs mechanism happens (the “breaking of symmetry” that is described a lot in many sources, besides wikipedia, – particles interact with the Higgs particle, sort of) which imparts mass to various particles, as well as giving mass to three of the massless force carriers and leaves one massless. _Why_ this happens is fundamentally not a question that can be answered at this level and nobody really knows the answer (any more than Einstein’s understanding of gravity in terms of the bending of spacetime geometry does not tell anyone fundamentally _why_ gravity is the way it is… further/deeper understanding may or may not shed light on both issues one day). That it _does_ happen and can be described by this mechanism (as has been experimentally verified) is remarkable in itself, and was of course a worthy Nobel prize or two. The upshot is that you end up with three massive force carriers that are identified with the Ws and Z and are the carriers of the known weak force, and one massless one that is identified with the photon, the carrier of the infinite range force we know and love, electromagnetism.

    Best Wishes.

    -cvj

  13. kim says:

    if Dr. Einsteins model does not tell us the answer to ‘why’, does that mean his theory of gravity is not unique? assuming it’s unique how was it shown to be? I’ll leave it there, there’s no demand for you to answer.

    cheers.

  14. Clifford says:

    Physics is mostly about how and not why. Why questions usually turn out to be how questions the next level down. And so on…

    As for your Einstein question: It is a very interesting issue… please read that book I recommended a day or two ago.

    -cvj

  15. Clifford says:

    Bah. Got to address the uniqueness issue. ๐Ÿ™‚ What does that really mean in a physics context? Now it _could_ mean that there is a unique theory that fits all the current data, but how would one even go about proving something like that. The best you can hope for is the simplest one that fits the data about the universe that you know _so far_ since you cannot rule out future experiments arbitrarily far in the future that go outside the validity of the theory you are working with, or simply show that it is wrong at that level of sensitivity and needs to be replaced by either the next simplest, or something completely different, and so on and so forth. This is how it works really – you only can fit what you’ve learned of the universe up to the time and accuracy of observation… further refinements of knowledge require further refinements of the theories you use to describe it.

    Best,

    -cvj

  16. Blake Stacey says:

    I noticed a minor typo in arXiv:1002.1120: reference 30 is missing a space in the title, “Rational solutions of Painlevéequations”.

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