We uncover a rich phenomenology of conducting honeycomb network superstructure, which plays a significant role in understanding the phase diagram topology and superconductivity of chargedensity wave materials such as 1T-TaS2. One of the most important theoretical questions about these materials is to understand the emerging superconductivity inside the nearly-commensurate charge-density wave state. Our key observation is that the conducting domain walls inside the charge-ordered state form a honeycomb network and the network magically supports a cascade of flat bands, whose unusual stability we thoroughly investigate. Furthermore, by combining the weak-coupling and strong-coupling approaches, we show that the superconductivity will be strongly enhanced in the network. This provides a natural mechanism for emerging superconductivity, and so explains the phase diagram topology of these charge density wave materials. Not only explaining emerging superconductivity, we show that abundant topological states including the corner states, which are closely related to that of the higher-order topology, appear.

Differentiable programming is a new programming paradigm which enables large scale optimization through automatic calculation of gradients also known as auto-differentiation. This concept emerges from deep learning, and has also been generalized to tensor network optimizations. Here, we extend the differentiable programming to tensor networks with isometric constraints with applications to multiscale entanglement renormalization ansatz (MERA) and tensor network renormalization (TNR). By introducing several gradient-based optimization methods for the isometric tensor network and comparing with Evenbly-Vidal method, we show that auto-differentiation has a better performance for both stability and accuracy. We numerically tested our methods on 1D critical quantum Ising spin chain and 2D classical Ising model. We calculate the ground state energy for the 1D quantum model and internal energy for the classical model, and scaling dimensions of scaling operators and find they all agree with the theory well.