Filling-Dependent Intertwined Electronic and Atomic Orders in the Flat-Band State of 1T-TaS₂

The delicate interplay among the complex intra/inter-layer electron-electron and electron-lattice interactions is the fundamental prerequisite of these exotic quantum states, such as superconductivity, nematic order, and checkerboard charge order. Here, we explore the filling-dependent multiple stable intertwined electronic and atomic orders of the flat-band state of 1T-TaS₂ encompassing hole order, phase orders, coexisting left- and right-chiral orders, and mixed phase/chiral orders via scanning tunneling microscopy (STM). Combining first-principles calculations, the emergent electronic/atomic orders can be attributed to the weakening of electron-electron correlations and stacking-dependent interlayer interactions. Moreover, achiral intermediate ring-like clusters and nematic charge density wave (CDW) states are successfully realized in intralayer chiral domain wall and interlayer heterochiral stacking regions through chiral overlap configurations. Our study not only deepens the understanding of filling-dependent electronic/atomic orders in flat-band systems but also offers perspectives for exploring exotic quantum states in correlated electronic systems.