  We investigate the phase diagram of the cross-coupled Heisenberg spin ladder
with antiferromagnetic couplings. For this model there have been conflicting
results for the existence of the columnar dimer phase, which was predicted on
the basis of weak coupling field theory renormalisation group arguments. The
numerical work on this model has been based on various approaches, including
exact diagonalization, series expansions and density-matrix renormalization
group calculations. Using the recently developed tensor network states and
ground-state fidelity approach for quantum spin ladders we find no evidence for
the existence of the columnar dimer phase. We also provide an argument based on
the symmetry of the Hamiltonian which suggests that the phase diagram for
antiferromagnetic couplings consists of a single line separating the
rung-singlet and Haldane phases.
