High-field magnetization and magnetic phase diagram of $alpha$-Cu$_{2}$V$_{2}$O$_{7}$

High-field magnetization of the spin-$1/2$ antiferromagnet $alpha$-Cu$_2$V$_2$O$_7$ was measured in pulsed magnetic fields of up to 56 T in order to study its magnetic phase diagram. When the field was applied along the easy axis (the $a$-axis), two distinct transitions were observed at $H_{c1}=6.5$~T and $H_{c2}=18.0$~T. The former is a spin-flop transition typical for a collinear antiferromagnet and the latter is believed to be a spin-flip transition of canted moments. The canted moments, which are induced by the Dzyaloshinskii-Moriya interactions, anti-align for $H_{c1}<H<H_{c2}$ due to the anisotropic exchange interaction that favors the antiferromagnetic arrangement along the $a$-axis. Above $H_{c2}$, the Zeeman energy of the applied field overcomes the antiferromagnetic anisotropic interaction and the canted moments are aligned along the field direction. Density functional theory was employed to compute the exchange interactions, which were used as inputs for quantum Monte Carlo calculations and then further refined by fitting to the magnetic susceptibility data. Contrary to our previous report in Phys. Rev. B {bf 92}, 024423, the dominant exchange interaction is between the third nearest-neighbor spins, which form zigzag spin-chains that are coupled with one another through an intertwining network of the nonnegligible nearest and second nearest-neighbor interactions. In addition, elastic neutron scattering under the applied magnetic fields of up to 10 T reveals the incommensurate helical spin structure in the spin-flop state.