The increased ability to engineer two-dimensional (2D) systems, either using materials, photonic lattices, or cold atoms, has led to the search for 2D structures with interesting properties. One such property is the presence of flat bands. Typically, the presence of these requires long-ranged hoppings, fine-tuning of nearest neighbor hoppings, or breaking time-reversal symmetry by using a staggered flux distribution in the unit cell. We provide a prescription based on carrying out projections from a parent system to generate different flat band systems. We identify the conditions for maintaining the flatness and identify a path-exchange symmetry in such systems that cause the flat band to be degenerate with the other dispersive ones. Breaking this symmetry leads to lifting the degeneracy while still preserving the flatness of the band. This technique does not require changing the topology nor breaking time-reversal symmetry as was suggested earlier in the literature. The prescription also eliminates the need for any fine-tuning. Moreover, it is shown that the subsequent projected systems inherit the precise fine-tuning conditions that were discussed in the literature for similar systems, in order to have and isolate a flat band. As examples, we demonstrate the use of our prescription to arrive at the flat band conditions for popular systems like the Kagome, the Lieb, and the Dice lattices. Finally, we are also able to show that a flat band exists in a recently proposed chiral spin-liquid state of the Kagome lattice only if it is associated with a gauge field that produces a flux modulation of the Chern-Simons type.
Pronounced 2/3 magnetization plateau in a frustrated $S$ = 1 isolated spin-triangle compound: Interplay between Heisenberg and biquadratic exchange interactions
Electric-field driven flat bands in the distorted sawtooth chain via the Katsura-Nagaosa-Balatsky mechanism
Yerevan State University（埃里温国立大学）；CANDLE, Synchrotron Research Institute（CANDLE 同步辐射研究所）；Institute of Applied Problems of Physics（应用物理问题研究所）；University of Augsburg（奥格斯堡大学）；Andronikashvili Institute of Physics, Javakhishvili Tbilisi State University（贾瓦希什维利第比利斯国立大学安德罗尼卡什维利物理研究所）
