Based on the analysis of the measurement data of angle-resolved photoemission spectroscopy (ARPES) and optics, we show that the charge transfer gap is significantly smaller than the optical one and is reduced by doping in electron doped cuprate super
conductors. This leads to a strong charge fluctuation between the Zhang-Rice singlet and the upper Hubbard bands. The basic model for describing this system is a hybridized two-band $t$-$J$ model. In the symmetric limit where the corresponding intra- and inter-band hopping integrals are equal to each other, this two-band model is equivalent to the Hubbard model with an antiferromagnetic exchange interaction (i.e. the $t$-$U$-$J$ model). The mean-field result of the $t$-$U$-$J$ model gives a good account for the doping evolution of the Fermi surface and the staggered magnetization.
We have proposed a novel numerical method to calculate accurately the physical quantities of the ground state with the tensor-network wave function in two dimensions. We determine the tensor network wavefunction by a projection approach which applies
iteratively the Trotter-Suzuki decomposition of the projection operator and the singular value decomposition of matrix. The norm of the wavefunction and the expectation value of a physical observable are evaluated by a coarse grain renormalization group approach. Our method allows a tensor-network wavefunction with a high bond degree of freedom (such as D=8) to be handled accurately and efficiently in the thermodynamic limit. For the Heisenberg model on a honeycomb lattice, our results for the ground state energy and the staggered magnetization agree well with those obtained by the quantum Monte Carlo and other approaches.
