Not so accidental degeneracies: origin of dimensional-reduction in the Quantum Spin Ice Yb$_2$Ti$_2$O$_7$

Despite being the best-characterised example of a ``quantum spin ice'' [1], Yb$_2$Ti$_2$O$_7$ remains an enigma. One of its most striking, and puzzling, features are the diffuse, rod-like structures seen in quasi-elastic neutron scattering [2]. These suggest that spin fluctuations in Yb$_2$Ti$_2$O$_7$ decouple into independent Kagome planes, even though magnetic ions occupy a fully three-dimensional pyrochlore lattice [3]. Here, we use a combination of lattice gauge theory, spin-wave calculations and Monte Carlo simulation, to show how the dimensional-reduction seen in Yb$_2$Ti$_2$O$_7$ follows from a two-dimensional branch of excitations ``inherited'' from a nearby phase transition. This analysis sheds new light on ground state selection in a wide range of rare-earth pyrochlore oxides, including the model ``order-by-disorder'' system Er$_2$Ti$_2$O$_7$.\\[4pt] [1] Ross, Savary, Gaulin \& Balents, Phys. Rev. X {\bf 1}, 021002 (2011)\\[0pt] [2] Hodges \textit{et al.}, Phys. Rev. Lett. {\bf 88}, 077204 (2002)\\[0pt] [3] Ross \textit{et al.}, Phys. Rev. Lett. {\bf 103}, 227202 (2009)