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The optical conductivity of the 2D $t-J$ model and the origin of electron incoherence in the high-T$_{c}$ cuprate superconductors: a variational study

Understanding the origin of electron incoherence is the first step toward a theoretical description of the non-Fermi liquid behavior of the high-T$_{c}$ cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude behavior of the optical response of the system and the anomalous density fluctuation behavior in the long wave length limit. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system in the normal state and the superfluid stiffness in the superconducting state. It is found that such electron incoherence remains significant even in the clean limit and at low temperature and thus must be attributed to the strong electron correlation effect in the cuprate superconductors. Here we study such an intrinsic effect in the 2D $t-J$ model through the variational calculation of its optical conductivity $σ(ω)$. We assume a resonating valence bond ground state as our starting point and find that a significant portion of the total optical spectral weight remains incoherent throughout the phase diagram. The optical absorption is found to extend all the way to an energy of the order of the bare band width. We find that both the total optical weight $\bar{K}$ and the integrated incoherent optical weight $I$ increase monotonically with doping, with their ratio $R_{incoh}=I/\bar{K}$ decreasing monotonically with doping. Our results indicate that the majority part of electron incoherence in the 2D $t-J$ model can be attributed to the electron fractionalization mechanism assumed in such a treatment. We also find that the Drude weight deduced from $D=\bar{K}-I$ scales linearly with hole doping, without any sign of a non-monotonic behavior in the overdoped regime.