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@@ -125,12 +125,38 @@ 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 \cite{choi_pressure-induced_2018}. Above 0.13--0.5 $\GPa$ (depending on temperature), \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---$(c_{11}-c_{12})/2$ in Voigt notation---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 $\Bog$ 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. +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 \cite{choi_pressure-induced_2018}. Above 0.13--0.5 $\GPa$ +(depending on temperature), \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---$(c_{11}-c_{12})/2$ in Voigt notation---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 $\Bog$ 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. \emph{Model.} @@ -185,9 +211,12 @@ 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. -If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd terms are allowed in its free energy and any transition will be first order and not continuous without fine-tuning. Since the \ho\ phase transition is second-order \brad{cite something}, we will henceforth rule out $\Aog$ \op s as well. -For the \op\ representation $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general -quadratic free energy density is +If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd +terms are allowed in its free energy and any transition will be first order and +not continuous without fine-tuning. Since the \ho\ phase transition is +second-order \brad{cite something}, we will henceforth rule out $\Aog$ \op s as +well. For the \op\ representation $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the +most general quadratic free energy density is \begin{equation} \begin{aligned} f_\op=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2 @@ -275,9 +304,10 @@ between the two components of the \op. In this case the uniform ordered phase is only stable for $c_\perp>0$, and the modulated phase is now characterized by helical order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. The uniform to modulated transition is now continuous. This does not reproduce -the physics of \ho, which has a first order transition between \ho\ and \afm, and so we will henceforth neglect -the possibility of a multicomponent order parameter. The schematic phase -diagrams for this model are shown in Figure~\ref{fig:phases}. +the physics of \ho, which has a first order transition between \ho\ and \afm, +and so we will henceforth neglect the possibility of a multicomponent order +parameter. The schematic phase diagrams for this model are shown in +Figure~\ref{fig:phases}. \emph{Results.} We will now derive the \emph{effective elastic tensor} $C$ that results from @@ -295,10 +325,9 @@ The generalized susceptibility of a single component ($\Bog$ or $\Btg$) \op\ is \begin{equation} \begin{aligned} &\chi^\recip(x,x') - =\frac{\delta^2F[\eta,\epsilon_\star[\eta]]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\langle\eta\rangle} + =\frac{\delta^2F[\eta,\epsilon_\star[\eta]]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\langle\eta\rangle} =\big[\tilde r-c_\parallel\nabla_\parallel^2 \\ - &\qquad\qquad-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+12u\langle\eta(x)\rangle^2\big] - \delta(x-x'), + &\qquad\qquad-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+12u\langle\eta(x)\rangle^2\big]\delta(x-x'), \end{aligned} \label{eq:sus_def} \end{equation} @@ -308,7 +337,7 @@ where $\recip$ indicates a \emph{functional reciprocal} in the sense that \end{equation} Taking the Fourier transform and integrating over $q'$ we have \begin{equation} - \chi(q) + \chi(q) =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4 +12u\sum_{q'}\langle\tilde\eta_{q'}\rangle\langle\tilde\eta_{-q'}\rangle\big)^{-1}. \end{equation} @@ -316,10 +345,10 @@ Near the unordered to modulated transition this yields \begin{equation} \begin{aligned} \chi(q) - &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2 - +|\Delta\tilde r|} \\ - &=\frac1{D_\perp}\frac{\xi_\perp^4} - {1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2}, + &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2 + +|\Delta\tilde r|} \\ + &=\frac1{D_\perp}\frac{\xi_\perp^4} + {1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2}, \end{aligned} \label{eq:susceptibility} \end{equation} @@ -327,7 +356,7 @@ with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}=\xi_{\perp0}|t|^{-1/4}$ and $\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}=\xi_{\parallel0}|t|^{-1/2}$, where $t=(T-T_c)/T_c$ is the reduced temperature and $\xi_{\perp0}=(D_\perp/aT_c)^{1/4}$ and -$\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths \brad{needs a descriptor like "in and perpendicular to the x-y plane" or something like that}. +$\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths. Notice that the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde r|)^{-1}$ does not diverge at the unordered to modulated transition. Though it anticipates a transition with Curie--Weiss-like divergence at $\Delta\tilde |