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Dipolar-octupolar pyrochlore magnets in a strong external magnet fieldapplied in the [110] direction are known to form a `chain' state, withsubextensive degeneracy. Magnetic moments are correlated along one-dimensionalchains carrying effective Ising degrees of freedom which are noninteracting onthe mean-field level. Here, we investigate this phenomenon in detail, includingthe effects of quantum fluctuations. We identify two distinct types of chainphases, both featuring distinct subextensive, classical ground statedegeneracy. Focussing on one of the two kinds, we discuss lifting of theclassical degeneracy by quantum fluctuations. We map out the ground-state phasediagram as a function of the exchange couplings, using linear spin wave theoryand real-space perturbation theory. We find a hierarchy of energy scales in theground state selection, with the effective dimensionality of the system varyingin an intricate way as the hierarchy is descended. We derive an effectivetwo-dimensional anisotropic triangular lattice Ising model with only three freeparameters which accounts for the observed behavior. Connecting our results toexperiment, they are consistent with the observation of a disordered chainstate in Nd$_2$Zr$_2$O$_7$. We also show that the presence of two distincttypes of chain phases has consequences for the field-induced breakdown of theapparent $U(1)$ octupolar quantum liquid phase recently observed inCe$_2$Sn$_2$O$_7$.

Hierarchy of energy scales and field-tunable order by disorder in dipolar-octupolar pyrochlores