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.



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