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We investigate the phase diagram of the cross-coupled Heisenberg spin ladderwith antiferromagnetic couplings. For this model there have been conflictingresults for the existence of the columnar dimer phase, which was predicted onthe basis of weak coupling field theory renormalisation group arguments. Thenumerical work on this model has been based on various approaches, includingexact diagonalization, series expansions and density-matrix renormalizationgroup calculations. Using the recently developed tensor network states andground-state fidelity approach for quantum spin ladders we find no evidence forthe existence of the columnar dimer phase. We also provide an argument based onthe symmetry of the Hamiltonian which suggests that the phase diagram forantiferromagnetic couplings consists of a single line separating therung-singlet and Haldane phases.

The antiferromagnetic cross-coupled spin ladder: quantum fidelity and tensor networks approach