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@@ -115,12 +115,14 @@ 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, +rau_hidden_2012, riggs_evidence_2015, hoshino_resolution_2013, +ikeda_theory_1998, chandra_hastatic_2013, harrison_hidden_nodate, ikeda_emergent_2012} propose associating any of a variety of broken symmetries -with \ho. This work proposes yet another, motivated by two experimental -observations: first, the $\Bog$ ``nematic" elastic susceptibility -$(C_{11}-C_{12})/2$ softens anomalously from room temperature down to +with \ho. This work analyzes a phenomenological model of order parameters of +general symmetry, linearly coupled to strain. We identify yet another mechanism +that is best compatible with two experimental observations: first, the $\Bog$ +``nematic" elastic susceptibility $(C_{11}-C_{12})/2$ softens anomalously from +room temperature down to $T_{\text{\ho}}=17.5\,\K$;\cite{de_visser_thermal_1986} and second, a $\Bog$ nematic distortion is observed by x-ray scattering under sufficient pressure to destroy the \ho\ state.\cite{choi_pressure-induced_2018} @@ -137,15 +139,14 @@ model-independent way, but doesn't differentiate between those that remain. 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 remains 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.\cite{wolf_elastic_1994, kuwahara_lattice_1997} Motivated by -these results---which hint 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 then determine the effect of its phase -transitions on the elastic response in different symmetry channels. +(depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion, +which might 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.\cite{wolf_elastic_1994, kuwahara_lattice_1997} Motivated by these +results---which hint 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 then determine the effect of its phase transitions on +the elastic response in different symmetry channels. We find that only one \op\ representation reproduces the anomalous $\Bog$ elastic modulus, which softens in a Curie--Weiss-like manner from room @@ -161,7 +162,7 @@ which should find that the $(C_{11}-C_{12})/2$ modulus diverges as the uniform $\Bog$ strain of the high pressure phase is approached. -\section{Model} +\section{Model \& Phase Diagram} The point group of \urusi\ is \Dfh, and any 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 @@ -292,7 +293,7 @@ to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$. With the strain traced out, \eqref{eq:fo} describes the theory of a Lifshitz point at $\tilde r=c_\perp=0$.\cite{lifshitz_theory_1942, lifshitz_theory_1942-1} The properties discussed in the remainder of this -section can all be found in a standard text, e.g., Chaikin \& +section can all be found in a standard text, e.g., in chapter 4 \S6.5 of Chaikin \& Lubensky.\cite{chaikin_principles_2000} For a one-component \op\ ($\Bog$ or $\Btg$) and positive $c_\parallel$, it is traditional to make the field ansatz $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$, @@ -319,10 +320,10 @@ 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 \urusi, whose \ho\ phase is bounded by a line of first order transitions at high pressure, and so we will henceforth neglect the possibility of a multicomponent order -parameter. The schematic phase diagrams for this model are shown in +parameter. Schematic phase diagrams for both the one- and two-component models are shown in Figure~\ref{fig:phases}. -\section{Results} +\section{Susceptibility \& Elastic Moduli} We will now derive the effective elastic tensor $C$ that results from the coupling of strain to the \op. The ultimate result, found in \eqref{eq:elastic.susceptibility}, is that $C_\X$ differs from its bare value @@ -388,7 +389,7 @@ on the configuration of the strain. Since $\eta_\star$ is a functional of $\epsi alone, only the modulus $C_\X$ will be modified from its bare value $C^0_\X$. Though the differential equation for $\eta_\star$ cannot be solved explicitly, we -can use the inverse function theorem to make us \eqref{eq:implicit.eta} anyway. +can use the inverse function theorem to make use of \eqref{eq:implicit.eta} anyway. First, denote by $\eta_\star^{-1}[\eta]$ the inverse functional of $\eta_\star$ implied by \eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$ corresponding to each solution of \eqref{eq:implicit.eta} it receives. This we @@ -481,9 +482,7 @@ corresponding modulus. \eqref{eq:static_modulus}. The fit gives $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $D_\perp q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and - $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of an additional - parameter to fit the standard bare modulus\cite{varshni_temperature_1970} - led to poorly constrained fits. (c) $\Bog$ modulus data and the fit of the + $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic term in $C^0_\Bog$ was here not needed for the fit.\cite{varshni_temperature_1970} (c) $\Bog$ modulus data and the fit of the \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and the fit transformed by $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from \eqref{eq:static_modulus} to equal $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, e.g., |