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Possible coexistence of short-range resonating-valence-bond and long-range stripe correlations in the spatially anisotropic triangular-lattice quantum magnet Cu$_2$(OH)$_3$NO$_3$

We show that short-range resonating-valence-bond correlations and long-range order can coexist in the ground state (GS) of a frustrated spin system. Our study comprises a comprehensive investigation of the quantum magnetism on the structurally disorder-free single crystal of Cu$_2$(OH)$_3$NO$_3$, which realizes the $s$ = 1/2 Heisenberg model on a spatially anisotropic triangular lattice. Competing exchange interactions determined by fitting the magnetization measured up to 55 T give rise to an exotic GS wavefunction with coexistence of the dominant short-range resonating-valence-bond correlations and weak long-range stripe order (ordered moment $M_0$ = $|\langle s_i^z\rangle|$ $\sim$ 0.02). At low temperatures, a first-order spin-flop transition is visible at $\sim$ 1-3 T. As the applied field further increases, another two magnetic-field-induced quantum phase transitions are observed at $\sim$ 14-19 and $\sim$ 46-52 T, respectively. Simulations of the Heisenberg exchange model show semi-quantitative agreement with the magnetic-field modulation of these unconventional phases, as well as the absence of visible magnetic reflections in neutron diffraction, thus supporting the GS of the spin system of Cu$_2$(OH)$_3$NO$_3$ may be approximate to a quantum spin liquid. Our study establishes structurally disorder-free magnetic materials with spatially anisotropic exchange interactions as a possible arena for spin liquids.