Inter-chain Interactions, Multi-magnon condensation and Strain effect in chain compound NaVOPO$_4$

Employing first-principles modelling and many-body methods, the magnetic properties of spin-1/2 chain compound NaVOPO$_4$ are explored. The extensive first-principles calculations establish an intricate three-dimensionally coupled model that consists of weakly alternating $J$-$J^{\prime}$ antiferromagnetic chains running along cris-cross directions between two consecutive $ab$ planes, connected via two subleading couplings, a ferromagnetic exchange along the $c$ direction ($J_c$) and a weaker antiferromagnetic exchange ($J_a$) along the body diagonal direction. The exact diagonalization and density matrix renormalized group study has been carried out on a two-dimensional spin model with $J$-$J^{\prime}$-$J_c$ and effective $J_d$ couplings, constructed based on the full model, for numerical ease. The $J_c$-$J_d$ phase diagram is found to host a {\it disorder} phase with a finite spin gap for comparable values of $J_c$ and $J_d$, arising out of the competing nature of these two interactions, other than two ordered phases. The calculated thermodynamic properties of this model provide a fair description of experimentally measured data. The predominant manifestation of $J_c$ and $J_d$ in the disorder phase happens in the stabilisation of a multi-magnon condensed phase, upon gap closing by application of an external magnetic field. We further explore the effect of tensile uniaxial strain, which is found to drive the system from gapful to gapless ground state.