  The effective spin-1/2 antiferromagnetic Heisenberg-Ising chain materials,
ACo$_2$V$_2$O$_8$, A = Sr, Ba, are a rich source of exotic fundamental
phenomena and have been investigated for their model magnetic properties both
in zero and non-zero magnetic fields. Here we investigate a new member of the
family, namely PbCo$_2$V$_2$O$_8$. We synthesize powder and single crystal
samples of PbCo$_2$V$_2$O$_8$ and determine its magnetic structure using
neutron diffraction. Furthermore, the magnetic field/temperature phase diagrams
for magnetic field applied along the c, a, and [110] crystallographic
directions in the tetragonal unit cell are determined via magnetization and
heat capacity measurements. A complex series of phases and quantum phase
transitions are discovered that depend strongly on both the magnitude and
direction of the field. Our results show that \pcvo is an effective spin-1/2
antiferromagnetic Heisenberg-Ising chain with properties that are in general
comparable to those of SrCo$_2$V$_2$O$_8$ and BaCo$_2$V$_2$O$_8$. One
interesting departure from the results of these related compounds, is however,
the discovery of a new field-induced phase for the field direction $H\|$[110]
which has not been previously observed.
