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Universal Scaling Behaviors and Cross-over Behavior of Non-Fermi to Fermi Liquid in High Temperature Superconductivity Sung-Ho Salk Pohang University of Science and Technology and Korean Academy of Science and Technology Strong Coulomb repulsion between electrons sets forth the long-range anti-ferromagnetic order displaying a Mott insulator at half-filling or near half-filling and allows the d-wave superconductivity over a wide range of hole doping. The two distinctive energy scales of the dome-shaped superconducting transition temperature, TC * and the monotonously decreasing pseudogap temperature T above TC universally appear showing nearly the same value of optimal doping, x 0.15 in all the observed phase diagrams of hole doped high temperature superconducting cuprate oxides. Additional subjects of prime importance to be resolved are the observed universal scaling behaviors in the ratios of T (Ref. 1) and Eres (Ref. 2) respectively. Here TC TC E res is the magnetic resonance peak energy. Resorting to our proposed slave-boson theory (Ref. 3) we explain the cause of these universal scaling behaviors regardless of cuprate samples. Following the universal phase diagrams we will discuss the cross-over behavior of non-Fermi to Fermi liquid. In agreements with observations this theory has been able to consistently reproduce not only the phase diagrams of the dome-shaped TC and the * monotonously decreasing T but also other physical properties of importance such as the optical conductivity, spectral function, superfluid weight, magnetic susceptibility and specific heat (see Refs.: 4 and 5). We attribute this success to the pivotal role of the spin-charge coupling manifested in the rigorous derivation (Ref. 1) of the U(1) (SU(2)) slave-boson theory of the t-J Hamiltonian. In passing we stress that this theory is markedly different from other earlier proposed slave-boson theories which are based on the separation of spin and charge degrees of freedom, i.e. the spin-charge separation in their effective mean field Lagrangians. Owing to the presence of the spin-charge coupling in our approach the resulting effective mean field Lagrangian is no longer decoupled between the spin (spinon) and charge (holon) sectors. Time permitting, we plan to discuss how other recent studies (see Refs.: 4 through 6) of non-Fermi liquid behavior based on gauge/gravity duality (AdS/CFT correspondence) can be correlated with our physical findings including the cross-over behavior from the non- Fermi to Fermi liquid. In short, we find that the spin-charge coupling is essential to reproduce all of the aforementioned physical properties. To put it otherwise, we demonstrate that coupling between the charge pairing order and the spin singlet paring order, i.e. the spin fluctuations of the shortest possible correlation length plays a key role for the observed high temperature superconductivity including the universal scaling behaviors mentioned above. On the other hand, defining a proper supercharge operator we find that a supersymmetric quantum mechanical condition is met only in the vanishing limit of the spin-charge coupling or both the spin and charge pairing orders, at which hole doping a quantum critical point may appear in the phase diagram of high TC cuprates. References : 1. M. Oda et. al., J. Phys. Soc. Jpn. 69, 983 (2000); T. Nakano et. al., J. Phys. Soc. Jpn 67, 2622(1998). 2. S. Pailhes et al., Phys. Rev. Lett. 96, 257001 (2006); P. Bourges et al., Physica C 424, 45 (2005). 3. S. -S. Lee and S. -H. S. Salk Phys. Rev. B 64, 052501 (2001); Phys. Rev. B 66, 054427 (2002). 4. S. -S. Lee and S. -H. S. Salk, Phys. Rev. B 71, 134518 (2005); J. -H. Eom, S. -H. S. Salk, Phys. Rev. B 72, 064508 (2005); J. -H. Eom, S.-S. Lee, K. -S. Kim, S.-H. S. Salk, Phys. Rev. B 70, 024522 (2004); S. -S. Lee, J. -H. Eom, K. -S. Kim, S. -H. Suck Salk, Phys. Rev. B 66, 064520 (2002); S .-S. Lee and S. -H. Salk, J. Low.Temp. Phys.117, 295 (1999). 5. . S. J. Shin and S. -H. S. Salk, Journal of Superconductivity and Magnetism, In press; S. -S. Lee and S. -H. Salk, To be submitted; S. J. Seung, S. H. Salk and M. Aihara, To be submitted. 6. S. -S. Lee, arxiv.org.0809.3402v2 [hep-th] 23 Sept 2008; Phys. Rev. D 79, 086006 (2009). 7. T. Faulkner et. al., Science 329, 1043 (2010). 8. S. Sachdev, lanl.arxiv.org.1010.0682v2 [cond-mat.str-el] 6 Oct 2010.