1. 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:1404.1994 Phys. Lett. B 734, 116 2014 arXiv:1404.5027 JHEP 2014 2014 中国引力和相对论天体物理学术年会

2. Introduction t_out<t<t_iso

3. 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

4. Motivations prolate spheroid oblate spheroid

5. Main contents Anisotropic black brane solution: prolate and oblate 2016-7-1 Type IIB supergravity in Einstein frame 5-dimensional effective action

6. Equations of motion Einstein equation Dilaton equation Maxwell equation Axion

7. The metric is assumed to be 2016-7-1 Anisotropic direction

8. Numerics: prolate solution 2016-7-1 Charge increases

9. Oblate solution 2016-7-1 Charge increases

10. Analytic solution 2016-7-1

11. Metric functions

12. Temperature and entropy: numerical 2016-7-1 prolate oblate

13. 2016-7-1 Analytic temperature prolateoblate

14. 2016-7-1 Analytic entropy prolate oblate

15. Zero temperature limit 1. prolate case: extremal configuration cannot be reached (agree with arXiv:1204.3008) 2. oblate case: zero temperature, non-vanishing entropy 2016-7-1 T=0 requires

16. Stress tensor counter terms to the axion-dilaton gravity theory 2016-7-1 The conformal anomaly The stress tensor

17. 2016-7-1 Energy density and pressure

18. 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 2016-7-1

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

20. The entropy density, temperature and U(1) chemical potential is scheme-independent. They only depend on their horizon values. The local thermodynamic stability requires 2016-7-1

21. 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) 2016-7-1

22. Scheme-independent instability 2016-7-1

23. 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 2016-7-1

24. Oblate case 2016-7-1 The oblate black brane solution is thermodynamically stable.

25. 2016-7-1 For example

26. The “chemical potential” with respect to the charge “a” is given by The U(1) chemical potential is scheme-independent 2016-7-1

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

28. 2016-7-1 Summary on the thermodynamic variables

29. η/s Tensor perturbation Shear viscosity 2016-7-1 stisfies the KSS bound

30. Tensor type perturbation Shear viscosity 2016-7-1

31. Conductivity 2016-7-1

32. Real part of conductivity Imaginary part of conductivity Conductivity with momentum dissipation? Yes 2016-7-1

33. Summary 2016-7-1 Future application in holographic QCD and Condensed Matter Systems

34. 2016-7-1 Thank You !