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@@ -116,7 +116,7 @@ Kusunose_2011_On, Kung_2015, Cricchio_2009, Ohkawa_1999, Santini_1994, Kiss_2005, Harima_2010, Thalmeier_2011, Tonegawa_2012_Cyclotron, Rau_2012_Hidden, Riggs_2015_Evidence, Hoshino_2013_Resolution, Ikeda_1998_Theory, Chandra_2013_Hastatic, Harrison_2019_Hidden, Ikeda_2012} propose -associating any of a variety of broken symmetries with \ho. This work analyzes +associating any of a variety of broken symmetries with \ho. Motivated by the anomalous temperature dependence of one of the elastic moduli , this work analyzes a family of phenomenological models with order parameters of general symmetry that couple linearly to strain. Of these, only one is compatible with two experimental observations: first, the $\Bog$ ``nematic" elastic susceptibility @@ -152,9 +152,9 @@ temperature and then cusps at $T_{\text{\ho}}$. That theory associates \ho\ with a $\Bog$ \op\ modulated along the $c$-axis, the high pressure state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. In addition to the agreement with the ultrasound data across a broad -temperature range, the theory predicts uniform $\Bog$ strain at high +temperature range, our model predicts uniform $\Bog$ strain at high pressure---the same distortion that was recently seen in x-ray scattering -experiments.\cite{Choi_2018} This theory strongly motivates future ultrasound +experiments.\cite{Choi_2018} This work strongly motivates future ultrasound experiments under pressure approaching the Lifshitz point, which should find that the $(C_{11}-C_{12})/2$ modulus diverges as the uniform $\Bog$ strain of the high pressure phase is approached. @@ -238,7 +238,7 @@ $\Btg$, or $\Eg$---the most general quadratic free energy density is where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$, and $\nabla_\perp=\partial_3$ transforms like $\Atu$. Other quartic terms are allowed---especially many for an $\Eg$ \op---but we have included only those -terms necessary for stability when either $r$ or $c_\perp$ become negative. The +terms necessary for stability when either $r$ or $c_\perp$ become negative as a function of temperature. The full free energy functional of $\eta$ and $\epsilon$ is \begin{equation} \begin{aligned} @@ -279,7 +279,7 @@ to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$. Phase diagrams for (a) \urusi\ from experiments (neglecting the superconducting phase)~\cite{Hassinger_2008} (b) mean field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean field theory of a - two-component ($\Eg$) Lifshitz point. Solid lines denote continuous + two-component ($\Eg$) Lifshitz point. Solid lines denote second-order transitions, while dashed lines denote first order transitions. Later, when we fit the elastic moduli predictions for a $\Bog$ \op\ to data along the ambient pressure line, we will take $\Delta\tilde r=\tilde r-\tilde @@ -318,7 +318,7 @@ 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. Schematic phase diagrams for both the one- +of a multicomponent order parameter---consistent with earlier ultrasound measurements \cite{Ghosh_2020_One-component}. Schematic phase diagrams for both the one- and two-component models are shown in Figure~\ref{fig:phases}. @@ -372,7 +372,7 @@ perpendicular and parallel to the plane, respectively. 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 the lower point $a(T-T_c)=\Delta\tilde -r=-D_\perp q_*^4<0$, this is cut off with a cusp at $\Delta\tilde r=0$. +r=-D_\perp q_*^4<0$, this is cut off with a cusp at the phase transition at $\Delta\tilde r=0$. The elastic susceptibility, which is the reciprocal of the effective elastic modulus, is found in a similar way to the \op\ susceptibility: we must trace @@ -510,7 +510,7 @@ The behavior of the modulus below the transition does not match \eqref{eq:static_modulus} well, but this is because of the truncation of the free energy expansion used above. Higher order terms like $\eta^2\epsilon^2$ and $\epsilon^4$ contribute to the modulus starting at order $\eta_*^2$ and -therefore change the behavior below the transition but not above it. To +therefore change the behavior below the transition, where the expectation value of $\eta$ is finite, but not above it, where the expectation value of $\eta$ is zero. To demonstrate this, in Appendix~\ref{sec:higher-order} we compute the modulus in a theory where the interaction free energy is truncated after fourth order with new term $\frac12g\eta^2\epsilon^2$. The dashed black line in @@ -556,9 +556,7 @@ commensurability jumps) at low pressure and approach zero as $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. Alternatively, \rus\ done at ambient pressure might examine the heavy Fermi liquid to \afm\ transition by doping. Though previous \rus\ studies have doped \urusi\ with -rhodium,\cite{Yanagisawa_2014} the magnetic rhodium dopants likely promote -magnetic phases. A non-magnetic dopant such as phosphorous may more faithfully -explore the transition out of the HO phase. Our work also motivates experiments +rhodium,\cite{Yanagisawa_2014} rhodium changes the carrier concentration as well as the lattice spacing, and may favour the promotion of the magnetic phase. An iso-electronic (as well as iso-magnetic) dopant such as iron may more faithfully explore the transition out of the HO phase. Our work also motivates experiments that can probe the entire correlation function---like x-ray and neutron scattering---and directly resolve its finite-$q$ divergence. The presence of spatial commensurability is known to be irrelevant to critical behavior at a @@ -591,10 +589,10 @@ above gives $\xi_{\perp0}q_*=(D_\perp q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ on the order of lattice constant, which is about what one would expect for a generic magnetic transition. The agreement of this data in the -$t\sim0.1$--10 range with the mean field exponent suggests that this region is +$(T-T_{\rm{HO}})/T_{\rm{HO}}\sim0.1$--10 range with the mean field exponent suggests that this region is outside the Ginzburg region, but an experiment may begin to see deviations from mean field behavior within approximately several Kelvin of the critical point. -An ultrasound experiment with more precise temperature resolution near the +An ultrasound experiment with finer temperature resolution near the critical point may be able to resolve a modified cusp exponent $\gamma\simeq1.31$,\cite{Guida_1998_Critical} since according to one analysis the universality class of a uniaxial modulated one-component \op\ is that of @@ -620,7 +618,7 @@ Physically, our phenomenological order parameter could correspond to $\Bog$ multipolar ordering originating from the localized component of the U-5f electrons. For the crystal field states of \urusi, this could correspond either to electric quadropolar or hexadecapolar order based on the available -multipolar operators. \cite{Kusunose_2011_On} +multipolar operators. \cite{Kusunose_2011_On} The coincidence of our theory's orthorhombic high-pressure phase and \urusi's \afm\ is compelling, but our mean field theory does not make any explicit @@ -638,7 +636,7 @@ $\Bog$ symmetry breaking, which we believe may be due to fluctuating order at temperatures above the actual transition temperature. This work motivates both further theoretical work regarding a microscopic theory with modulated $\Bog$ order, and preforming symmetry-sensitive thermodynamic experiments at pressure, -such as ultrasound, that could further support or falsify this idea. +such as pulse-echo ultrasound, that could further support or falsify this idea. \begin{acknowledgements} Jaron Kent-Dobias is supported by NSF DMR-1719490, Michael Matty is supported |