# Violation of the Holographic Viscosity Bound

in a Strongly Coupled Anisotropic Plasma

###### Abstract

We study the conductivity and shear viscosity tensors of a strongly coupled super-Yang-Mills plasma which is kept anisotropic by a parameter that depends linearly on one of the spatial dimensions. Its holographic dual is given by an anisotropic axion-dilaton-gravity background and has recently been proposed by Mateos and Trancanelli as a model for the pre-equilibrium stage of quark-gluon plasma in heavy-ion collisions. By applying the membrane paradigm which we also check by numerical evaluation of Kubo formula and lowest lying quasinormal modes, we find that the shear viscosity purely transverse to the direction of anisotropy saturates the holographic viscosity bound, whereas longitudinal shear viscosities are smaller, providing the first such example not involving higher-derivative theories of gravity and, more importantly, with fully known gauge-gravity correspondence.

###### pacs:

11.25.Tq, 11.10Wx, 12.38.Mh^{†}

^{†}preprint: TUW-11-28

Introduction. Hydrodynamic simulations of heavy-ion collisions suggest Romatschke:2007mq that the produced quark-gluon plasma is behaving like an almost perfect fluid with a ratio of shear viscosity over entropy density not far from the famous result associated with the membrane paradigm of black holes KPM:BHMP and which holographic gauge-gravity duality maps to the corresponding quantity of maximally supersymmetric Yang-Mills theory in the limit of infinite color number and infinite ’t Hooft coupling Aharony:1999ti ; Policastro:2001yc . This value has been conjectured to form the lower bound for any realistic matter Kovtun:2004de . It was found to be saturated universally Buchel:2003tz ; Iqbal:2008by in dual theories involving an isotropic horizon described by Einstein gravity. Values above this bound are obtained when corrections due to finite coupling strength are included Buchel:2008ae , but it has been shown that values violating the bound can arise in higher-derivate gravities Brigante:2007nu , although so far no complete gauge-gravity correspondence has been established for finite violations.

The possibility of nonuniversal shear viscosity within two-derivative gravity has been considered before in gravity duals of noncommutative gauge theories Landsteiner:2007bd and p-wave superfluids Natsuume:2010ky , where shear modes in addition to the universal purely transverse one exist. For p-wave superfluids, nonuniversal values above the bound have been recently obtained in Erdmenger:2010xm .

Anisotropies are of particular importance to the nonequilibrium dynamics of the quark gluon plasma in heavy-ion collisions. At weak coupling, they give rise to non-Abelian plasma instabilities Mrowczynski:1988dz and turbulent behavior Arnold:2005ef that has been argued to effectively reduce the shear viscosity Asakawa:2006tc , which is parametrically large in a weakly coupled plasma Arnold:2003zc . More directly accessible experimental signatures such as anisotropic photon and dilepton emission Schenke:2006yp , momentum broadening of jets Romatschke:2006bb as well as quarkonium dissociation Burnier:2009yu have been studied in the hard-loop effective theory Mrowczynski:2004kv for a weakly coupled anisotropic plasma where on sufficiently small time scales the anisotropy can be taken as stationary.

While holographic models of the evolution of strongly coupled quark-gluon plasma towards isotropy are of immediate interest and have been studied extensively Hubeny:2010ry , further insight may be obtained by models where the anisotropy is temporarily fixed. The first such attempt was presented by Janik and Witaszczyk in Janik:2008tc , where the gravity dual involves a naked singularity. In Rebhan:2011ke we have studied electromagnetic signatures of this model which are qualitatively similar to weak-coupling results at high frequencies, but which involve vanishing conductivities in the limit of zero frequency and absence of hydrodynamic behavior.

Anisotropic axion-dilaton-gravity solution. Recently, Mateos and Trancanelli Mateos:2011ix ; Mateos:2011tv have adapted the spatially anisotropic duals of Lifshitz-like fixed points of Azeyanagi:2009pr to a setup with AdS boundary conditions and finite temperature. This provides an anisotropic version of an super-Yang-Mills plasma where the anisotropy is kept stationary by deforming the gauge theory by

(1) |

with depending linearly on one of the spatial dimensions. The constant is related to the density of D7-branes (homogeneously distributed along ) that are dissolved in the bulk of the dual theory (they do not extend to the holographic boundary). The dual bulk theory is effectively 5-dimensional axion-dilaton gravity with negative cosmological constant and bulk action

(2) |

where and axion . Here we have dropped a factor of the ten-dimensional solution and set its radius . The remaining five-dimensional geometry which asymptotes to AdS for can be given by a line element of the form

(3) | |||||

The functions and can be written in terms of the dilaton , which itself has to satisfy a third-order nonlinear differential equation in Mateos:2011tv . For nonvanishing anisotropy parameter there is an anisotropic horizon with , which is completely regular. This will allow us to apply the membrane paradigm along the lines of Iqbal:2008by for studying conductivities and shear viscosities, both of which have two independent components at nonzero anisotropy.

The thermodynamics of this setup and its instabilities are discussed in Mateos:2011tv . Upon holographic renormalization which brings in a reference scale , the stress tensor of the gauge theory has the form

(4) |

with a conformal anomaly . The entropy density is given by

(5) |

with . Unlike the strictly positive which continues to be given by a quarter of the horizon area over spatial volume, and the pressure components depend separately on and and can become negative in various regions of the phase diagram. When is increased from the isotropic limit , the pressure anisotropy first becomes oblate (), while prolate anisotropies can be obtained for larger . For small there is a qualitative similarity to the plasma instabilities in an anisotropic weakly coupled plasma in that the homogeneous phase is unstable against filamentation of the distribution of the dissolved D7 branes along Mateos:2011tv . For sufficiently large the instabilities against filamentation however disappear, and also various components of the stress tensor (but not the entropy) can become negative.

Conductivities. Including a standard Maxwell action for a U(1) gauge field in the bulk with coupling constant leads to a conjugate momentum with respect to evolution in the coordinate that defines the current

(6) |

As in Iqbal:2008by , regularity at the horizon in terms of the Eddington-Finkelstein coordinate for infalling observers defined by

(7) |

relates linearly to the electric field , so that the horizon acts as a conductor. The anisotropic metric (3) however leads to two different conductivities

(8) | |||||

(9) |

where . In the limit of vanishing frequency and wave vector we have which allows us to conclude that give the DC conductivities of the boundary theory. The numerical results as a function of the anisotropy parameter are given in Fig. 1.

Because are determined by geometric data, they do not depend on and thus only on the ratio . Whether a given value corresponds to oblate or prolate anisotropy however depends on Mateos:2011tv . Our finding that is always smaller than one may seem surprising, but we have in fact found an extreme version of this phenomenon in the singular geometry of Janik:2008tc . While in this case both DC conductivities vanish, the ratio of AC conductivities as for both oblate and prolate anisotropies (see Fig. 11 of Rebhan:2011ke ).

Following the procedure of Kovtun:2003wp , one can also obtain two related diffusion constants with common charge susceptibility

(10) |

We have checked RSprep the results for by numerical calculation of the lowest quasi-normal mode in the diffusive channel, thus verifying that the membrane paradigm is indeed valid for conductivities of an anisotropic horizon.

Shear viscosities. In an anisotropic fluid with axial symmetry, the viscosity tensor LL:VII has five independent components, two of which are shear viscosities which we denote as

(11) |

and which can be extracted by Kubo formulas from the corresponding retarded correlation functions of the stress tensor. In the gravity dual we need to consider metric fluctuations , for which we adopt the gauge . Expanding the action (2) to second order in we obtain

(12) | |||||

where we introduced

(13) |

Considering first Fourier modes and their conjugate momenta we find that requiring regularity at the horizon as above for vector perturbations gives the purely transverse shear viscosity

(14) |

which reproduces the universal value for Einstein gravity with isotropic horizons. Because the radial evolution becomes trivial as ,

(15) |

the (momentum independent) result of the membrane paradigm gives the purely transverse shear viscosity of the boundary theory at , exactly as in the universality proof of Iqbal:2008by .

Turning to nontransverse metric fluctuations we find that the radial flow is still trivial in the limit for the case of because

(16) |

for any (while would have a nontrivial flow ). We can therefore use the membrane paradigm to calculate the longitudinal shear viscosity from

(17) | |||||

for , which is plotted in Fig. 2 as a function of . Note that the ratio turns out to be identical to .

We have also checked (17) by a direct numerical evaluation of the Kubo formula

(18) |

Momentum diffusion. As a further corroboration, we have also studied the diffusive behavior of the relevant shear modes. The longitudinal shear viscosity is related to the diffusive pole of the shear mode at small but nonvanishing momenta, where is oriented in the -direction in order to preserve O(2) symmetry which avoids a coupling to axion modes (the general case will be discussed elsewhere RSprep ). We can use the trick of Kovtun:2003wp of Kaluza-Klein dimensional reduction in the -direction, which maps the problem to that of U(1) gauge fields with a -dependent effective coupling. This yields the diffusion constant

We have also checked this result by calculating numerically the lowest lying quasinormal mode of the retarded propagator of the shear mode which satisfies the ordinary differential equation

(20) |

We can now compare (Violation of the Holographic Viscosity Bound in a Strongly Coupled Anisotropic Plasma) with the analogous calculation of the shear mode with momentum oriented in the direction, yielding (the breaking of SO(2) invariance in this case does not lead to complications from axion modes)

(21) | |||||

where the last equality follows from (14) together with (5), but is also reproduced numerically. Taking the ratio of Eqs. (Violation of the Holographic Viscosity Bound in a Strongly Coupled Anisotropic Plasma) and (21) leads again to the result

(22) |

Conclusion. As mentioned above, a result for that deviates from the universal result of Einstein gravity with isotropic horizons has been recently obtained in a dual model for p-wave superfluids Erdmenger:2010xm , however one that is above . To our knowledge, the result (22) is the first example of a shear viscosity that falls below this value without recourse to higher-derivative gravity and moreover with fully known gauge-gravity correspondence.

Parts of the phase diagram in and of this theory exhibit instabilities against filamentation into inhomogeneous mixed configurations with and some , while for sufficiently large the anisotropic homogeneous phase is thermodynamically stable Mateos:2011tv . Strictly speaking, the above calculation of hydrodynamical transport coefficients is limited to stable homogeneous phases, but we have numerically verified their existence on lines of constant .

It is not clear what implications, if any, our result of for a strongly coupled stationary anisotropic plasma would have for real nonequilibrium evolution. Intrinsically anisotropic hydrodynamic descriptions of quark-gluon plasma have recently been proposed in Florkowski:2010cf , however without allowing for different shear viscosity components. It would be interesting to study the effects of different shear viscosities in a hydrodynamical evolution, and also what ratios a weakly coupled anisotropic non-Abelian plasma would give rise to.

Acknowledgments. We would like to thank Alex Buchel, Johanna Erdmenger, Patrick Kerner, Andreas Schmitt, Aleksi Vuorinen, and Hansjörg Zeller for discussions. This work was supported by the Austrian Science Fund FWF, project no. P22114.

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