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 \maketitle
 
-The study of phase transitions is a central theme of condensed matter physics.
-In many cases, a phase transition between different states of matter is marked
-by a change in symmetry.  In this paradigm, the breaking of symmetry in an
-ordered phase corresponds to the condensation of an order parameter (\op) that
-breaks the same symmetries. Near a second order phase transition, the physics
-of the \op\ can often be described in the context of Landau--Ginzburg mean field
-theory. However, to construct such a theory, one must know the symmetries of
-the \op, i.e. the symmetry of the ordered state.
-
-A paradigmatic example where the symmetry of an ordered phase remains unknown
-is in \urusi.  \urusi\ is a heavy fermion superconductor in which
-superconductivity condenses out of a symmetry broken state referred to as
-hidden order (\ho) \cite{hassinger_temperature-pressure_2008}, and at
-sufficiently large hydrostatic pressures, both give way to local moment
-antiferromagnetism (\afm).  Despite over thirty years of effort, the symmetry of the
-\ho\ state remains unknown, and modern theories \cite{kambe_odd-parity_2018,
-  haule_arrested_2009, kusunose_hidden_2011, kung_chirality_2015,
-  cricchio_itinerant_2009, ohkawa_quadrupole_1999, santini_crystal_1994,
-  kiss_group_2005, 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.  Many of these theories rely on the formulation of a microscopic
-model for the \ho\ state, but without direct experimental observation of the
-broken symmetry, none have been confirmed. 
-
-One case that does not rely on a microscopic model is recent work
-\cite{ghosh_single-component_nodate} that studies the \ho\ transition using
-resonant ultrasound spectroscopy (\rus).  \Rus\ is an experimental technique
-that measures mechanical resonances of a sample. These resonances contain
-information about the full strain stiffness tensor of the material. Moreover,
-the frequency locations of the resonances are sensitive to symmetry breaking at
-an electronic phase transition due to electron-phonon coupling
+The study of phase transitions is central to condensed matter physics.  Phase
+transitions are often accompanied by a change in symmetry whose emergence can
+be described by the condensation of an order parameter (\op) that breaks the
+same symmetries. Near a continuous phase transition, the physics of the \op\
+can often be qualitatively and sometimes quantitatively described by
+Landau--Ginzburg mean field theories. These depend on little more than the
+symmetries of the \op, and coincidence of their predictions with experimental
+signatures of the \op\ is evidence of the symmetry of the corresponding ordered
+state.
+
+A paradigmatic example of a material with an ordered state whose broken
+symmetry remains unknown is in \urusi.  \urusi\ is a heavy fermion
+superconductor in which superconductivity condenses out of a symmetry broken
+state referred to as \emph{hidden order} (\ho)
+\cite{hassinger_temperature-pressure_2008}, and at sufficiently large
+hydrostatic pressures, both give way to local moment antiferromagnetism (\afm).
+Despite over thirty years of effort, the symmetry of the \ho\ state remains
+unknown, and modern theories \cite{kambe_odd-parity_2018, haule_arrested_2009,
+  kusunose_hidden_2011, kung_chirality_2015, cricchio_itinerant_2009,
+  ohkawa_quadrupole_1999, santini_crystal_1994, kiss_group_2005,
+  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.  Many of these
+theories rely on the formulation of a microscopic model for the \ho\ state, but
+since there has not been direct experimental observation of the broken
+symmetry, none can been confirmed. 
+
+Recent work that studied the \ho\ transition using \emph{resonant ultrasound
+spectroscopy} (\rus) was able to shed light on the symmetry of the ordered
+state without the formulation of any microscopic model
+\cite{ghosh_single-component_nodate}.  \Rus\ is an experimental technique that
+measures mechanical resonances of a sample. These resonances contain
+information about the sample's full strain stiffness tensor. Moreover, the
+frequency locations of the resonances are sensitive to symmetry breaking at an
+electronic phase transition due to electron-phonon coupling
 \cite{shekhter_bounding_2013}.  Ref.~\cite{ghosh_single-component_nodate} uses
-this information to place strict thermodynamic bounds on the symmetry of the
-\ho\ \op, again, independent of any microscopic model. Motivated by these
-results, in this paper we consider a mean field theory of an \op\ coupled to
-strain and the effect that the \op\ symmetry has on the elastic response in
-different symmetry channels. Our study finds that a single possible \op\
-symmetry reproduces the experimental strain susceptibilities and fits the
-experimental data well. The resulting theory associates \ho\ with $\Bog$ order
-\emph{modulated along the rotation axis}, \afm\ with uniform $\Bog$ order, and
-a Lifshitz point with the triple point between them.
-
-We first present a phenomenological Landau--Ginzburg mean field theory of
-strain coupled to an \op. We examine the phase diagrams predicted by this
-theory for various \op\ symmetries and compare them to the experimentally
-obtained phase diagram of \urusi.  Then we compute the elastic response to
-strain, and examine the response function dependence on the symmetry of the
-\op.  We compare the results from mean field theory with data from \rus\
-experiments.  We further examine the consequences of our theory at non-zero
-applied pressure in comparison with recent x-ray scattering experiments
-\cite{choi_pressure-induced_2018}. Finally, we discuss our conclusions and the
-future experimental and theoretical work motivated by our results.
+this information to place strict thermodynamic bounds on the dimension of the
+\ho\ \op\ independent of any microscopic model.
+
+Motivated by these results, 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 features of the
+experimental strain stiffness. That theory associates the \ho\ state with a
+$\Bog$ \op\ \emph{modulated along the rotation axis}, the \afm\ state with
+uniform $\Bog$ order, and the triple point between them with a Lifshitz point.
+Besides the agreement with \rus\ data in the \ho\ state, the theory predicts
+uniform $\Bog$ strain in the \afm\ state, which was recently seen in x-ray
+scattering experiments \cite{choi_pressure-induced_2018}. The theory's
+implications for the dependence of the strain stiffness on pressure and doping
+strongly motivates future \rus\ experiments that could either further support
+or falsify it.
 
 The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
 respect this symmetry. We will introduce a phenomenological free energy density