21 haves differently for different orders of phase transitions: for 1st-order phase transition, it rises sharply and show a divergent behavior, for the 2nd-order phase transition, it shows an upward cusp at T$_{c}$ , for the case of crossover, the cusp becomes smooth. It is discussed that the sharp rising bulk viscosity will lead to the breakdown of hydro- dynamics and affect the hadronization. Therefore the critical end point might be located through the observ- ables which are sensitive to the ratio of bulk viscosity over entropy density. instability persists and induces spatial non-uniform phase separation state. This explains why imbalanced cold atom experiments did not observe LOFF state rather showed strong evidence of phase separation. For the 2-flavor quark matter being considered, the electric Coulomb interaction is not strong enough to compete with the Higgs instability. I have introduced several topics of QCD phase struc- ture at high temperature and high density: the properties of strongly interacting quark gluon plasma, searching for the critical end point and the gapless color superconduc- tor. I give a brief introduction on the discovery of sQGP. It has been believed for more than 30 years that the QGP created at heavy-ion collisions should be weakly inter- acting gas system. However, a very small shear viscosity over entropy density ratio η/s is required to fit the RHIC data of elliptic flow v$_{2}$ . It is in contrary to the large value of η/s given by perturbative QCD calculation. This is a strong evidence that the deconfined matter created at RHIC is strongly interacted. The AdS/CFT duality gives a lower bound η/s = 1 / 4 π . Therefore, it is conjectured that the sQGP created at RHIC might be the most per- fect fluid observed in nature. However, a perfect fluid should have both vanishing shear and bulk viscosities. The property of the bulk viscosity of sQGP need to be studied before one draws the final conclusion. The status of gapless color superconductor is reported. The chromomagnetic instability, the Sarma instability and Higgs instability are clarified. In the gapless color su- perconducting phase, both the phase part and magnitude part of the order parameter will develop instabilities: The phase part develops into the chromomagnetic instability, which induces the plane-wave state; The magnitude part develops the Sarma instability and Higgs instability, the Sarma instability can be competed with charge neutral- ity condition, while the Higgs instability cannot be cured by Coulomb interaction, and induces the inhomogeneous state. I thank the collaboration with J.W.Chen, B.C.Li, H.Mao, D.L.Yang for the sQGP part, and I.Giannakis, D.Hou, H.C.Ren and I. Shovkovy for the gapless color superconductor part. The work is supported by CAS program ”Outstanding young scientists abroad brought- in”, CAS key project KJCX3-SYW-N2, NSFC10735040, NSFC10875134, and the K.C.Wong Education Founda- tion, Hong Kong. Recent studies show that the bulk viscosity over en- tropy density ratio ζ/s rises up near phase transitions. The result from QCD effective models show that ζ/s be- V. ACKNOWLEDGEMENTS IV. CONCLUSION Rev. C 29 , 2067(1984); R.K. Su, F.M. Lin Phys. Rev. C 39 , 2438(1989). [14] U.W. Heinz, P. R. Subramanian, H. Stocker and W. Greiner, J. Phys. G 12 , 1237 (1986). [15] S.C. Frautschi, Asymptotic freedom and color super- conductivity in dense quark matter, Proc.of the Work- shop on Hadronic Matter at Extreme Energy Density, N. Cabibbo (Editor), Erice, Italy (1978). [16] F. Barrois, Nucl. Phys. B129 (1977),390. [17] J. Bardeen, L.N. Cooper, J.R. Schrieffer Phys. Rev. 106:162 (1957); Phys. Rev. 108:1175 (1957). [1] D.J. Gross and F. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; H.D. Politzer, Phys. Rev. Lett. 30 (1973) 1346. [2] E. Klempt, hep-ph/0404270. [3] T.D. Lee and G.C. Wick, Phys. Rev. D 9 , 2291 (1974). [4] J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34 , 1353 (1975). [5] G. Baym and S.A. Chin, Phys. Lett. B 62 , 241 (1976). [6] E.V. Shuryak, Phys. Lett. B 78 , 150 (1978). [7] F. Karsch and E. Laermann, Phys. Rev. D 50 , 6954 (1994); F. Karsch, Nucl. Phys. A 698 , 199 (2002). [8] E.V. Shuryak and I. Zahed, Phys. Rev. C 70 , 021901 (2004); E. Shuryak, Prog. Part. Nucl. Phys. 53 , 273 (2004); E.V. Shuryak and I. Zahed, Phys. Rev. D 70 , 054507 (2004); E.V. Shuryak, Nucl. Phys. A 750 , 64 (2005). [9] D.H. Rischke, Prog. Part. Nucl. Phys. 52 , 197 (2004). [10] M. Gyulassy and L. McLerran, Nucl. Phys. A 750 , 30 (2005). [11] P. Jacobs and X.N. Wang, Prog. Part. Nucl. Phys. 54 , 443 (2005). [12] U.W. Heinz, hep-ph/0407360. [13] H.R. Jaqaman, A.Z. Mekjian, and L. Zamick, Phys. [20] K. Rajagopal and F. Wilczek, hep-ph/0011333; D.K. Hong, Acta Phys. Polon. B 32 , 1253 (2001); M. Al- ford, Ann. Rev. Nucl. Part. Sci. 51 , 131 (2001); G. Nar- dulli, Riv. Nuovo Cim. 25N3 , 1 (2002); T. Sch¨ afer, hep-ph/0304281; M. Buballa, Phys. Rept. 407 , 205 (2005); H.C. Ren, hep-ph/0404074; M. Huang, Int. J. Mod. Phys. E 14 , 675 (2005); I. A. Shovkovy, Found. [19] R. Rapp, T. Sch¨ afer,E.V. Shuryak and M. Velkovsky, Phys.Rev.Lett. 81 ,53(1998); M. Alford, K. Rajagopal and F. Wilczek, Phys.Lett. B 422 ,247(1998). [18] D. Bailin and A. Love, Phys. Rep. 107 , 325(1984).