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Diffstat (limited to 'main.tex')
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@@ -115,7 +115,12 @@ strongly motivates future ultrasound experiments under pressure approaching the \section{Model} The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally -respect this symmetry in the high-temperature phase. Our phenomenological free energy density contains three parts: the elastic free energy, the \op, and the interaction between strain and \op. The most general quadratic free energy of the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, where the six irreducible components of strain are +respect this symmetry in the high-temperature phase. Our phenomenological free +energy density contains three parts: the elastic free energy, the \op, and the +interaction between strain and \op. The most general quadratic free energy of +the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$. Linear +combinations of the six independent components of strain form five irreducible +components of strain as \begin{equation} \begin{aligned} \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth} @@ -147,13 +152,14 @@ The interaction between strain and an \op\ $\eta$ depends on the point group rep \begin{equation} f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta. \end{equation} -If the representation $\X$ is not present in the strain \brad{what does "present in the strain" mean?} 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 $\Aog$ elastic modului if $\eta$ is -single-component \cite{luthi_sound_1970, ramshaw_avoided_2015, -shekhter_bounding_2013}, and jumps in other elastic moduli 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 s that can produce linear couplings to strain. Looking at the +If there doesn't exist a component of strain that transforms like the +representation $\X$ 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 $\Aog$ +elastic modului if $\eta$ is single-component \cite{luthi_sound_1970, +ramshaw_avoided_2015, shekhter_bounding_2013}, and jumps in other elastic +moduli 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 s that can produce linear couplings to strain. Looking at the components present in \eqref{eq:strain-components}, this rules out all of the \emph{u}-reps (which are odd under inversion) and the $\Atg$ irrep. @@ -221,7 +227,7 @@ traditional to make the field ansatz $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$ \brad{Why is it traditional to ignore any in-plane modulation (x1, x2)?}. For $\tilde r>0$ and $c_\perp>0$, or $\tilde r>c_\perp^2/4D_\perp$ and $c_\perp<0$, the only stable solution is $\eta_*=q_*=0$ and the system is unordered. For $\tilde r<0$ there are free -energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has uniform order \brad{probably helpful to specify what kind of order here - uniform $\Bog$ order, correct?}. For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free +energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has uniform order of whatever . For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free energy minima for $q_*^2=-c_\perp/2D_\perp$ and \begin{equation} \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u} |