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The Kagome-lattice-based material, Volborthite, $\mathrm{Cu_3 V_2 O_7 (OH)_2\cdot 2 H_2 O}$, has been considered as a promising platform for discovery ofunusual quantum ground states due to the frustrated nature of spininteractions. Here we explore possible quantum spin liquid and magneticallyordered phases in a two-dimensional non-symmorphic lattice described by $p2gg$layer space group, which is consistent with the spatial anisotropy of the spinmodel derived from density functional theory (DFT) for Volborthite. Using theprojective symmetry group (PSG) analysis and Schwinger boson mean field theory,we classify possible spin liquid phases with bosonic spinons and investigatemagnetically ordered phases connected to such states. It is shown, in general,that only translationally invariant mean field states are allowed intwo-dimensional non-symmorphic lattices, which simplifies the classificationconsiderably. The mean field phase diagram of the DFT-derived spin model isstudied and it is found that possible quantum spin liquid phases are connectedto two types of magnetically ordered phases, a coplanar incommensurate $(q,0)$spiral order as the ground state and a closely competing coplanar commensurate$(\pi,\pi)$ spin density wave order. In addition, periodicity enhancement ofthe two-spinon continuum, a signature of symmetry fractionalization, is foundin the spin liquid phases connected to the $(\pi,\pi)$ spin density wave order.We discuss relevance of these results to recent and future experiments onVolborthite.

Quantum spin liquid and magnetic order in a two-dimensional non-symmorphic lattice: considering the distorted Kagome lattice of Volborthite