  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.
