  We study the ground state and thermodynamic properties of the spin-half XXZ
model, with an Ising interaction $J_z$ and a transverse exchange interaction
$J_{x}$, on a pyrochlore tube obtained by joining together elementary cubes in
a one-dimensional array. Periodic boundary conditions in the transverse
directions ensure that the bulk of the system consists of corner-sharing
tetrahedra, with the same local geometry as the pyrochlore lattice. We use
exact diagonalization, the density matrix renormalization group (DMRG), and
minimally entangled typical thermal states (METTS) methods to study the system.
When $J_z$ is antiferromagnetic ($J_{z}>0$) and $J_x$ is ferromagnetic
($J_{x}<0$), we find a transition from a spin liquid to an XY ferromagnet,
which has power-law correlations at $T=0$. For $J_{z}<0$ and $J_{x}>0$,
spin-two excitations are found to have lower energy than spin-one at the
transition away from the fully polarized state, showing evidence for incipient
spin-nematic order. When both interactions are antiferromagnetic, we find a
non-degenerate ground state with no broken symmetries and a robust energy gap.
The low energy spectra evolve smoothly from predominantly Ising to
predominantly XY interactions. In the spin-liquid regime of small $|J_{x}|$, we
study the confinement of monopole-anti-monopole pairs and find that the
confinement length scale is larger for $J_x<0$ than for $J_{x}>0$, although
both length scales are very short. These results are consistent with a local
spin-liquid phase for the Heisenberg antiferromagnet with no broken symmetries.
