Formation of gapless $Z_2$ spin liquid phase manganites in the (Sm$_{1-y}$Gd$_{y}$)$_{0.55}$Sr$_{0.45}$MnO$_{3}$ system in zero magnetic field: Topological phase transitions to states with low and high density of 2D-vortex pairs induced by the magnetic field

The evolution of the ground state of the manganese spin ensemble in the (Sm$_{1-y}$Gd$_{y}$)$_{0.55}$Sr$_{0.45}$MnO$_{3}$ in the case of isovalent substitution of rare-earth samarium ions with large radii with gadolinium ions with significantly smaller radii is studied. The measured temperature dependences of the $ac$ magnetic susceptibility and the field dependences of the $dc$ magnetizations are analyzed using the Heisenberg–Kitaev model describing the transition from the ordered spin state with classical isotropic AFM exchange to the frustrated spin state with quantum highly anisotropic FM exchange. A continuous transition from the 3D ferromagnetic state of manganese spins in the initial sample with $y$ = 0 to zigzag AFM ordering of CE-type spins in ab planes for $y$ = 0.5, coexisting in samples with $y$ = 0.5, 0.6, and 0.7 at temperatures below $T_{\mathrm{N}}\cong$ 48.5 K with a disordered phase such as a quantum Griffiths phase is identified. As the gadolinium concentration further increases, the CE-type zigzag AFM structure is molten, which leads to the appearance of an unusual phase in Gd$_{0.55}$Sr$_{0.45}$MnO$_{3}$ in the temperature range close to the absolute zero. This phase has characteristic features of a gapless $Z_2$ quantum spin liquid in zero external magnetic field. The step changes in the magnetization isotherms measured at 4.2 K in the field range of $\pm$75 kOe are explained by quantum phase transitions of the $Z_2$ spin liquid to a phase with topological order in weak magnetic fields and a polarized phase in strong fields. The significant difference between critical fields and magnetization jumps in isotherms indicates the existence of hysteretic phenomena in quantum spin liquid magnetization–demagnetization processes caused by the difference between localization–delocalization of 2D vortex pairs induced by a magnetic field in a quantum spin liquid with disorder.