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@@ -126,15 +126,15 @@ 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 -in three parts: that of the strain, the \op, and their interaction. -The most general quadratic free energy of the strain $\epsilon$ is +respect this symmetry. Our phenomenological free energy density contains three +parts: the free energy for the strain, the \op, and their interaction. The +most general quadratic free energy of the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare strain -stiffness tensor $C$ tensor is constrained by both that $\epsilon$ is a -symmetric tensor and by the point group symmetry \cite{landau_theory_1995}. The -latter can be seen in a systematic way. First, the six independent components -of strain are written as linear combinations that behave like irreducible -representations under the action of the point group, or +stiffness tensor $C$ tensor is constrained by both the index symmetry of the +strain tensor and by the point group symmetry \cite{landau_theory_1995}. The +six independent components of strain can written as linear combinations that +each behave like irreducible representations under the action of the point +group, or \begin{equation} \begin{aligned} \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth} @@ -145,13 +145,13 @@ representations under the action of the point group, or \end{aligned} \label{eq:strain-components} \end{equation} -Next, all quadratic combinations of these irreducible strains that transform -like $\Aog$ are included in the free energy as +All quadratic combinations of these irreducible strains that transform like +$\Aog$ are included in the free energy, \begin{equation} f_\e=\frac12\sum_\X C_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)}, \end{equation} where the sum is over irreducible representations of the point group and the -stiffnesses $C_\X^{(ij)}$ are +bare stiffnesses $C_\X^{(ij)}$ are \begin{equation} \begin{aligned} &C_{\Aog}^{(11)}=\tfrac12(C_{1111}+C_{1122}) && @@ -162,20 +162,22 @@ stiffnesses $C_\X^{(ij)}$ are C_{\Eg}^{(11)}=C_{1313}. \end{aligned} \end{equation} -The interaction between strain and the \op\ $\eta$ depends on the -representation of the point group that $\eta$ transforms as. If this -representation is $\X$, then the most general coupling to linear order is +The interaction between strain and an \op\ $\eta$ depends on the representation +of the point group that $\eta$ transforms as. If this representation is $\X$, +the most general coupling to linear order is \begin{equation} - f_\i=b^{(i)}\epsilon_\X^{(i)}\eta + f_\i=b^{(i)}\epsilon_\X^{(i)}\eta. \end{equation} -If $\X$ is a representation not present in the strain there can be no linear -coupling, and the effect of $\eta$ going through a continuous phase transition -is to produce a jump in the $\Aog$ strain stiffness \cite{luthi_sound_1970, -ramshaw_avoided_2015, shekhter_bounding_2013}. We will therefore focus our -attention on \op\ symmetries that produce linear couplings to strain. Looking -at the components present in \eqref{eq:strain-components}, this rules out all -of the u-reps (odd under inversion) and the $\Atg$ irrep as having any -anticipatory response in the strain stiffness. +If the representation $\X$ is not present in the strain there can be no linear +coupling, and the effect of the \op\ condensing at a continuous phase +transition is to produce a jump in the $\Aog$ strain stiffness if $\eta$ is +single-component \cite{luthi_sound_1970, ramshaw_avoided_2015, +shekhter_bounding_2013}, and jumps in other strain stiffnesses if +multicompenent \cite{ghosh_single-component_nodate}. Because we are interested +in physics that anticipates the phase transition, we will focus our attention +on \op\ symmetries that can produce linear couplings to strain. Looking at the +components present in \eqref{eq:strain-components}, this rules out all of the +u-reps (which are odd under inversion) and the $\Atg$ irrep. If the \op\ transforms like $\Aog$, odd terms are allowed in its free energy and any transition will be abrupt and not continuous without |