Double Charge Ordering States and Spin Ordering State Observed in a RFe$_{2}$O$_{4}$ System

Charge, spin, and lattice degrees of orderings are of great interest in the layered quantum material RFe$_{2}$O$_{4}$ (R $=$ Y, Er, Yb, Tm, and Lu) system. Recently many unique properties have been found using various experimental methods. However so far the nature of the two-dimensional (2D) charge ordering (CO) state is not clear and no observation of its fine structure in energy has been reported. Here we report unambiguous observation of double 2D CO states at relatively high temperature in a polycrystalline Er$_{0.1}$Yb$_{0.9}$Fe$_{2}$O$_{4}$ using Raman scattering. The energy gaps between the 3D and the double 2D states are 170 meV (41.2 THz) and 193 meV (46.6 THz), respectively. We also observed a spin ordering (SO) state at below 210 K with characteristic energy of 45 meV (10.7 THz). Our investigation experimentally identified new fine structures of quantum orders in the system, which also extends the capability of optical methods in investigating other layered quantum materials.