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@@ -98,11 +98,11 @@ unknown, and modern theories \cite{kambe_odd-parity_2018, haule_arrested_2009, harima_why_2010, thalmeier_signatures_2011, tonegawa_cyclotron_2012, rau_hidden_2012, riggs_evidence_2015, hoshino_resolution_2013, ikeda_theory_1998, chandra_hastatic_2013, harrison_hidden_nodate, -ikeda_emergent_2012} propose a variety of possibilities. Our work here seeks to unify two experimental observations: one, the $\Bog$ ``nematic" elastic susceptibility $(c_{11}-c_{12})/2$ softens anomalously from room temperature down to T$_{\mathrm{HO}}=17.5~$ K \brad{find old citations for this data}; and two, a $\Bog$ nematic distortion is observed by x-ray scattering under sufficient pressure to destroy the \ho\ state \brad{cite johan}. +ikeda_emergent_2012} propose a variety of possibilities. Our work here seeks to unify two experimental observations: one, the $\Bog$ ``nematic" elastic susceptibility $(c_{11}-c_{12})/2$ softens anomalously from room temperature down to T$_{\mathrm{HO}}=17.5~$ K \brad{find old citations for this data}; and two, a $\Bog$ nematic distortion is observed by x-ray scattering under sufficient pressure to destroy the \ho\ state \cite{choi_pressure-induced_2018}. Recent \emph{resonant ultrasound spectroscopy} (\rus) measurements examined the thermodynamic discontinuities in the elastic moduli at T$_{\mathrm{HO}}$ \cite{ghosh_single-component_nodate}. The observation of discontinues only in compressional, or $\Aog$, elastic moduli requires that the point-group representation of \ho\ is one-dimensional. This rules out a large number of order parameter candidates \brad{cite those ruled out} in a model-free way, but still leaves the microscopic nature of \ho~ undecided. -Recent X-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure \brad{cite Johan}. Above \brad{whatever pressure they find it at...}, \urusi\ undergoes a $\Bog$ nematic distortion. While it is still unclear as to whether this is a true thermodynamic phase transition, it may be related to the anomalous softening of the $\Bog$ elastic modulus---$(c_{11}-c_{12})/2$---that occurs over a broad temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by these results, hinting at a $\Bog$ strain susceptibility associated with the \ho\ state, we construct a phenomenological mean field theory for an arbitrary \op\ coupled to strain, and the determine the effect of its phase transitions on the elastic response in different symmetry channels. +Recent X-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure \cite{choi_pressure-induced_2018}. Above \brad{whatever pressure they find it at...}, \urusi\ undergoes a $\Bog$ nematic distortion. While it is still unclear as to whether this is a true thermodynamic phase transition, it may be related to the anomalous softening of the $\Bog$ elastic modulus---$(c_{11}-c_{12})/2$---that occurs over a broad temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by these results, hinting at a $\Bog$ strain susceptibility associated with the \ho\ state, we construct a phenomenological mean field theory for an arbitrary \op\ coupled to strain, and the determine the effect of its phase transitions on the elastic response in different symmetry channels. We find that only one \op\ symmetry reproduces the anomalous $(c_{11}-c_{12})/2$ elastic modulus, which softens in a Curie-Weiss like manner from room temperature, but which cusps at T$_{\mathrm{HO}}$. That theory associates \ho\ with a $\Bog$ \op\ \emph{modulated along the $c$- axis}, the \afm\ state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. Besides the agreement with ultrasound data across a broad temperature range, the theory predicts uniform $\Bog$ strain at high pressure---the same distortion which was recently seen in x-ray scattering experiments \cite{choi_pressure-induced_2018}. This theory strongly motivates future ultrasound experiments under pressure approaching the Lifshitz point, which should find that the $(c_{11}-c_{12})/2$ diverges once the uniform $\Bog$ strain sets in. |