Anisotropic plasma at finite U(1) chemical potential Xian-Hui Ge 葛先辉 Department of Physics, Shanghai University with L. Cheng and S. J. Sin arXiv: Phys. Lett. B 734, arXiv: JHEP 中国引力和相对论天体物理学术年会

Introduction t_out<t<t_iso

Motivations QGP can carry a non-zero chemical potential The U(1) symmetry plays an important role in many condensed matter systems The future application of the anisotropic black brane to condensed matter system calls for a U(1) chemical potential

Motivations prolate spheroid oblate spheroid

Main contents Anisotropic black brane solution: prolate and oblate Type IIB supergravity in Einstein frame 5-dimensional effective action

Equations of motion Einstein equation Dilaton equation Maxwell equation Axion

The metric is assumed to be Anisotropic direction

Numerics: prolate solution Charge increases

Oblate solution Charge increases

Analytic solution

Metric functions

Temperature and entropy: numerical prolate oblate

Analytic temperature prolateoblate

Analytic entropy prolate oblate

Zero temperature limit 1. prolate case: extremal configuration cannot be reached (agree with arXiv: ) 2. oblate case: zero temperature, non-vanishing entropy T=0 requires

Stress tensor counter terms to the axion-dilaton gravity theory The conformal anomaly The stress tensor

Energy density and pressure

Thermodynamics: prolate case The anisotropic constant a measures the number of D7-brane per unit length We can introduce a chemical potential conjugate to “a”. The first law of thermodynamics The grand canonical thermodynamic potentials are given by

In the absence of anisotropy, we recover the relation The grand thermodynamic potential is scheme-dependent pressure rescaling

The entropy density, temperature and U(1) chemical potential is scheme-independent. They only depend on their horizon values. The local thermodynamic stability requires

Phase Structure The black hole solution found in asymptotically AdS space shows its rich phase structure Hawking-Page phase transition neutral black hole with spherical horizon (Ren Zhao et al 2014) Van der Waals liquid-gas behavior RN-AdS with spherical horizon ( R.G.Cai 2000; X.N. Wu 2010,J.X. Lu, et al 2014) Planar black brane solution Thermodynamic stable For example: Schwarzschild-AdS, RN-AdS, Lifshitz (neutral and charged)

Scheme-independent instability

For a given temperature, there are two branches of black brane solution The smaller branch with smaller horizon radius is unstable, yielding a negative specific heat The instability uncovered here is due to the competing effect between the horizon radius and the anisotropy It is renormalization schemes independent

Oblate case The oblate black brane solution is thermodynamically stable.

For example

The “chemical potential” with respect to the charge “a” is given by The U(1) chemical potential is scheme-independent

Two components (a and rho) two phases (isotropic & anisotropic) Five regions metastable unstable

Summary on the thermodynamic variables

η/s Tensor perturbation Shear viscosity stisfies the KSS bound

Tensor type perturbation Shear viscosity

Conductivity

Real part of conductivity Imaginary part of conductivity Conductivity with momentum dissipation? Yes

Summary Future application in holographic QCD and Condensed Matter Systems

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