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Strongly Interacting Matter in Magnetic Fields pp 591–624Cite as

A Review of Magnetic Phenomena in Probe-Brane Holographic Matter

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Part of the book series: Lecture Notes in Physics ((LNP,volume 871))

Abstract

Gauge/gravity duality is a useful and efficient tool for addressing and studying questions related to strongly interacting systems described by a gauge theory. In this manuscript we will review a number of interesting phenomena that occur in such systems when a background magnetic field is turned on. Specifically, we will discuss holographic models for systems that include matter fields in the fundamental representation of the gauge group, which are incorporated by adding probe branes into the gravitational background dual to the gauge theory. We include three models in this review: the D3–D7 and D4–D8 models, that describe four-dimensional systems, and the D3–D7’ model, that describes three-dimensional fermions interacting with a four-dimensional gauge field.

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Notes

  1. 1.

    However note that the scale-invariance is broken at the quantum level since the beta function is proportional to N f /N c and therefore non-vanishing. In the limit N c →∞ with N f being fixed, the beta function is approximately zero, i.e. we may treat the theory as being scale invariant also at the quantum level.

  2. 2.

    For simplicity, we will consider the single flavor case with one D8-brane and one anti-D8-brane. This does not affect any of the results qualitatively.

  3. 3.

    Note that, although the boundary value is non-zero, this is a normalizable mode since the field strength is normalizable. Ordinarily, boundary values of bulk fields correspond to parameters in the boundary theory. But in this case there is a possible ambiguity, since the boundary value of \(a^{A}_{\mu}\) can also describe a non-trivial gradient of the pseudoscalar field.

  4. 4.

    For N f >1 the baryons correspond to instantons in the non-abelian D8-brane theory [11, 45]. This reproduces the known description of baryons as Skyrmions in the chiral Lagrangian. In this description the baryon charge comes from the CS term coupling the U(1) V field to the instanton density in the SU(N f ) V part.

  5. 5.

    The axial symmetry is broken by an anomaly. However this is a subleading effect at large N c which we can neglect. In particular, we will assume that the one-flavor pseudoscalar η′ is massless. For a discussion of the U(1) A anomaly and the η′ mass in the context of the Sakai-Sugimoto model see [11, 47].

  6. 6.

    We would like to stress that this is a gauge invariant definition. The standard boundary condition on the gauge field in AdS/CFT fixes the value of a M (u→∞). In this case only the transformations that vanish at u→∞ are gauged in the bulk. In particular, these transformations do not change the asymptotic value of a 0.

  7. 7.

    At non-zero baryon number density the magnetic field can actually induce an inverse magnetic catalysis in this model [53, 54].

  8. 8.

    Other boundary terms at u→∞ could affect the on-shell action, and therefore the currents. See for example [60, 61].

  9. 9.

    This is also consistent with the fact that there are no quarks in this phase to carry such a current.

  10. 10.

    Strictly speaking, at zero temperature the theory is in the confining (and broken chiral symmetry) phase. We are considering the meta-stable state obtained by adiabatically reducing the temperature.

  11. 11.

    In [15] this term was derived by demanding invariance of the CS term ∫C 4∧F∧F under gauge transformations of the RR field and then fixing c(∞)=0. However it can also be obtained by canceling the surface term in the variation of the CS term, when we present it as ∫F 5∧A∧F.

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Acknowledgements

O.B. and G.L. would like to thank Matt Lippert and Niko Jokela, who were an integral part in all our work on the models reviewed in Sects. 22.3 and 22.4. O.B. also thanks the Aspen Center for Physics, where this work was completed, for its hospitality. J.E. would like to thank her collaborators Martin Ammon, Matthias Kaminski, Patrick Kerner, René Meyer, Jonathan Shock and Migael Strydom, involved in the joint work presented in Sect. 22.2. This work was supported in part by the Israel Science Foundation under grant No. 392/09, and in part by the US-Israel Binational Science Foundation under grant No. 2008-072.

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Authors and Affiliations

  1. Technion, Israel Institute of Technology, Haifa, 32000, Israel

    Oren Bergman

  2. Max Planck Institute for Physics, 80805, Munich, Germany

    Johanna Erdmenger

  3. Department of Mathematics and Physics, University of Haifa at Oranim, Tivon, 36006, Israel

    Gilad Lifschytz

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  1. Oren Bergman
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  2. Johanna Erdmenger
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Correspondence to Oren Bergman .

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Editors and Affiliations

  1. Department of Physics, Stony Brook University, Nicolls Rd 100, Stony Brook, 11794, New York, USA

    Dmitri Kharzeev

  2. Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, Nicolas Cabrera 13-15, Madrid, 28049, Spain

    Karl Landsteiner

  3. Institut für Theoretische Physik, Technische Universität Wien, Wiedner Haupstr. 8, Wien, 1040, Austria

    Andreas Schmitt

  4. Physics Department (MC 273), University of Illinois at Chicago, West Taylor Street 845Rm 2236, Chicago, 60607, Illinois, USA

    Ho-Ung Yee

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Bergman, O., Erdmenger, J., Lifschytz, G. (2013). A Review of Magnetic Phenomena in Probe-Brane Holographic Matter. In: Kharzeev, D., Landsteiner, K., Schmitt, A., Yee, HU. (eds) Strongly Interacting Matter in Magnetic Fields. Lecture Notes in Physics, vol 871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37305-3_22

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