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Anomalous Transport from Kubo Formulae

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Strongly Interacting Matter in Magnetic Fields

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

Abstract

Chiral anomalies have profound impact on the transport properties of relativistic fluids. In four dimensions there are different types of anomalies, pure gauge and mixed gauge-gravitational anomalies. They give rise to two new non-dissipative transport coefficients, the chiral magnetic conductivity and the chiral vortical conductivity. They can be calculated from the microscopic degrees of freedom with the help of Kubo formulae. We review the calculation of the anomalous transport coefficients via Kubo formulae with a particular emphasis on the contribution of the mixed gauge-gravitational anomaly.

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Notes

  1. 1.

    See however the very recent attempts to establish non-renormalization theorems in [37] and [38].

  2. 2.

    A recent comprehensive review on BRST symmetry is [48].

  3. 3.

    In D=4k+2 dimensions also purely gravitational anomalies can appear [52].

  4. 4.

    If dynamical gauge fields are present, such as the gluon fields in QCD even the one point function of the charge does decay over (real) time due to non-perturbative processes (instantons) or at finite temperature due to thermal sphaleron processes [54]. Even in this case in the limit of large number of colors these processes are suppressed and can e.g. not be seen in holographic models in the supergravity approximation.

  5. 5.

    It is possible to define a generalized formalism to make any choice for the gauge field A 0=ν, so that one recovers formalism (A) when ν=μ and formalism (B) when ν=0 as particular cases (see [55] for details).

  6. 6.

    Notice that h 0y can also be understood as the so-called gravito-magnetic vector potential A g , which is related to the gravito-magnetic field by . This allows to interpret σ V not only as the generation of a current due to a vortex in the fluid, i.e. the chiral vortical effect, but also as a chiral gravito-magnetic conductivity giving rise to a chiral gravito-magnetic effect, see [56] for details.

  7. 7.

    The chemical potential is introduced as the energy needed to introduce an unit of charge from the boundary to behind the horizon A(∞)−A(r H) which corresponds to the prescription (B) in Table 17.1. Observe that we have left the source value A(∞)=ν as an arbitrary constant for reasons we will explain later.

  8. 8.

    The complete system of equations depending on frequency and momentum is showed in Appendix 2. The system consists of six dynamical equations and two constraints.

  9. 9.

    is coming from the counterterms of the theory.

  10. 10.

    In principle A 0 could be gauged away for the symmetric case and in consequence observables should not depend on its value. For example look at [45] to see how in presence of a U(1) V ×U(1) A symmetry with only the U(1) V conserved, propagators do not depend on the specific value of the zero component of the vector gauge source V 0.

  11. 11.

    For a four dimensional holographic model with gravitational Chern-Simons term and a scalar field this has also been shown in [65].

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Acknowledgements

This work has been supported by Plan Nacional de Altas Energías FPA2009-07908, FPA2008-01430 and FPA2011-25948, CPAN (CSD2007-00042), Comunidad de Madrid HEP-HACOS S2009/ESP-1473. E. Megías would like to thank the Institute for Nuclear Theory at the University of Washington, USA, and the Institut für Theoretische Physik at the Technische Universität Wien, Austria, for their hospitality and partial support during the completion of this work. The research of E.M. is supported by the Juan de la Cierva Program of the Spanish MICINN. F.P. has been supported by fellowship FPI Comunidad de Madrid.

Author information

Authors and Affiliations

  1. Instituto de Física Teórica UAM-CISC, Univ. Autónoma de Madrid, C/ Nicolás Cabrera 13-15, 28049, Madrid, Spain

    Karl Landsteiner & Francisco Peña-Benitez

  2. Grup de Física Teòrica and IFAE, Departament de Física, Universitat Autònoma de Barcelona, Bellaterra, E-08193, Barcelona, Spain

    Eugenio Megías

  3. Departamento de Física Teórica, Univ. Autónoma de Madrid, 28049, Madrid, Spain

    Francisco Peña-Benitez

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  1. Karl Landsteiner
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  2. Eugenio Megías
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Correspondence to Karl Landsteiner .

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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

Appendices

Appendix 1: Boundary Counterterms

The result one gets for the counterterm coming from the regularization of the boundary action of the holographic model in Sect. 17.4 is

(17.128)

where hat on the fields means the induced field on the cut-off surface and

$$ P = \frac{1}{6}\widehat{R}, \qquad P^i_j = \frac{1}{2} \bigl[ \widehat{R}^i_j - P \delta^i_j \bigr]. $$
(17.129)

As a remarkable fact there is no contribution in the counterterm coming from the gauge-gravitational Chern-Simons term. This has also been derived in [72] in a similar model that does however not contain S CSK .

Appendix 2: Equations of Motion for the Shear Sector

These are the complete linearized set of six dynamical equations of motion,

(17.130)
(17.131)
(17.132)

and two constraints for the fluctuations at ω,k≠0

(17.133)

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Landsteiner, K., Megías, E., Peña-Benitez, F. (2013). Anomalous Transport from Kubo Formulae. 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_17

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