Landau quantization evolution of the spinon pair spectrum in weak Mott insulator La$_{0.15}$Sm$_{0.85}$MnO$_{3+\delta}$ with spinon Fermi surface with increasing temperature and magnetic field strength

Measurements of the magnetization temperature dependences of La$_{0.15}$Sm$_{0.85}$MnO$_{3+\delta}$ in the temperature range 4.2 – 100 K revealed a threshold magnetization feature near the temperature $T_{d}\cong$ 50 K, associated with the existence of a small pseudogap $\Delta_e$ in the electron spectrum, which is characteristic of a weak Mott dielectric. An increase in the strength $H$ of the external magnetic field leads to suppression of the dielectric pseudogap $\Delta_e$, an increase in the density of states of free charge/ spin carriers at EF, and transformation of the charge/spin density waves fragments. A quantization of the pairs spectrum of the low-energy magnetic excitations of a $Z_2$ quantum spin liquid in the form of spinon-gauge field composite quasiparticles was found in the temperature range 4.2 – 12 K. Formation of a continuous spectrum of the quantum spin liquid excitations in the regime of “weak magnetic fields” $H$ = 100 Oe, 350 Oe, 1 kOe is explained within the framework of the Landau quantization models of the spectrum of composite quasiparticles with fractional values of the factor $\nu$ filling of three overlapping Landau zones. In the regime of a “strong external magnetic field” $H$ = 3.5 kOe, the new quantum oscillations of the temperature dependences of the magnetization of an incompressible spinon liquid were founded in the form of three narrow steps (plateaus) corresponding to the complete filling of non-overlapping Landau zones with integer values of the filling factor by spinons.