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@@ -110,56 +110,51 @@ broken symmetry remains unknown. This state, known as \emph{hidden order} (\ho), sets the stage for unconventional superconductivity that emerges at even lower temperatures. At sufficiently large hydrostatic pressures, both superconductivity and \ho\ give way to local moment antiferromagnetism -(\afm).\cite{hassinger_temperature-pressure_2008} Modern -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, -ikeda_emergent_2012} propose associating any of a variety of broken symmetries -with \ho. 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 $(C_{11}-C_{12})/2$ 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} +(\afm).\cite{Hassinger_2008} Modern theories~\cite{Kambe_2018, Haule_2009, + Kusunose_2011, Kung_2015, Cricchio_2009, Ohkawa_1999, Santini_1994, +Kiss_2005, Harima_2010, Thalmeier_2011, Tonegawa_2012, Rau_2012, Riggs_2015, +Hoshino_2013, Ikeda_1998, Chandra_2013a, 1902.06588v2, Ikeda_2012} propose +associating any of a variety of broken symmetries with \ho. 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 +$(C_{11}-C_{12})/2$ softens anomalously from room temperature down to +$T_{\text{\ho}}=17.5\,\K$;\cite{deVisser_1986} and second, a $\Bog$ nematic +distortion is observed by x-ray scattering under sufficient pressure to destroy +the \ho\ state.\cite{Choi_2018} Recent resonant ultrasound spectroscopy (\rus) measurements were used to examine the thermodynamic discontinuities in the elastic moduli at -$T_{\text{\ho}}$.\cite{ghosh_single-component_nodate} The observation of -discontinues only in compressional, or $\Aog$, elastic moduli requires that the -point-group representation of \ho\ be one-dimensional. This rules out many -order parameter candidates~\cite{thalmeier_signatures_2011, -tonegawa_cyclotron_2012, rau_hidden_2012, riggs_evidence_2015, -hoshino_resolution_2013, ikeda_emergent_2012, chandra_origin_2013} in a -model-independent way, but doesn't differentiate between those that remain. +$T_{\text{\ho}}$.\cite{1903.00552v1} The observation of discontinues only in +compressional, or $\Aog$, elastic moduli requires that the point-group +representation of \ho\ be one-dimensional. This rules out many order parameter +candidates~\cite{Thalmeier_2011, Tonegawa_2012, Rau_2012, Riggs_2015, +Hoshino_2013, Ikeda_2012, Chandra_2013b} in a 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, -which might be related to the anomalous softening of the $\Bog$ elastic modulus +under pressure.\cite{Choi_2018} Above 0.13--0.5 $\GPa$ (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. +pressure.\cite{Wolf_1994, Kuwahara_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 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 pressure---the same -distortion that 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 $(C_{11}-C_{12})/2$ modulus diverges as the uniform -$\Bog$ strain of the high pressure phase is approached. +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 +pressure---the same distortion that was recently seen in x-ray scattering +experiments.\cite{Choi_2018} This theory 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. \section{Model \& Phase Diagram} @@ -211,13 +206,12 @@ If there exists no component of strain that transforms like the representation $\X$ then there can be no linear coupling. The next-order coupling is linear in strain, quadratic in order parameter, and the effect of this coupling at a continuous phase transition is to produce a jump in the $\Aog$ elastic moduli -if $\eta$ is single-component, \cite{luthi_sound_1970, ramshaw_avoided_2015, -shekhter_bounding_2013} and jumps in other elastic moduli if -multicomponent.\cite{ghosh_single-component_nodate} Because we are interested -in physics that anticipates the phase transition---for instance, that the -growing \op\ susceptibility is reflected directly in the elastic -susceptibility---we will focus our attention on \op s that can produce linear -couplings to strain. Looking at the components present in +if $\eta$ is single-component, \cite{Luthi_1970, Ramshaw_2015, Shekhter_2013} +and jumps in other elastic moduli if multicomponent.\cite{1903.00552v1} Because +we are interested in physics that anticipates the phase transition---for +instance, that the growing \op\ susceptibility is reflected directly in the +elastic susceptibility---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 u-reps (which are odd under inversion), the $\Atg$ irrep, and all half-integer (spinor) representations. @@ -225,10 +219,9 @@ representations. If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd terms are allowed in its free energy and without fine-tuning any transition will be first order and not continuous. Since the \ho\ phase transition is -second-order,\cite{de_visser_thermal_1986} we will henceforth rule out $\Aog$ -\op s as well. For the \op\ representation $\X$ as any of those -remaining---$\Bog$, $\Btg$, or $\Eg$---the most general quadratic free energy -density is +second-order,\cite{deVisser_1986} we will henceforth rule out $\Aog$ \op s as +well. For the \op\ representation $\X$ as any of those remaining---$\Bog$, +$\Btg$, 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 @@ -279,23 +272,23 @@ to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$. \includegraphics[width=0.51\columnwidth]{phases_vector} \caption{ Phase diagrams for (a) \urusi\ from experiments (neglecting the - superconducting phase)~\cite{hassinger_temperature-pressure_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 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 $\Delta\tilde r=\tilde - r-\tilde r_c=a(T-T_c)$. + 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 + 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 $\Delta\tilde r=\tilde r-\tilde + r_c=a(T-T_c)$. } \label{fig:phases} \end{figure} 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., 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 +point at $\tilde r=c_\perp=0$.\cite{Lifshitz_1942a, Lifshitz_1942b} The +properties discussed in the remainder of this section can all be found in a +standard text, e.g., in chapter 4 \S6.5 of Chaikin \& +Lubensky.\cite{Chaikin_1995} 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 $c_\perp>0$, 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 @@ -313,15 +306,16 @@ transition between the uniform and modulated orderings is first order for a one-component \op\ and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. -For a two-component \op\ ($\Eg$) we must also allow a relative phase -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 \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- and two-component models are shown in -Figure~\ref{fig:phases}. +For a two-component \op\ ($\Eg$) we must also allow a relative phase 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 \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- +and two-component models are shown in Figure~\ref{fig:phases}. + \section{Susceptibility \& Elastic Moduli} We will now derive the effective elastic tensor $C$ that results from the @@ -406,9 +400,8 @@ derivative of $\eta^{-1}_\star[\eta]$ with respect to $\eta$, yielding \end{aligned} \label{eq:inv.func} \end{equation} -Next, \eqref{eq:implicit.eta} and \eqref{eq:inv.func} -can be used in concert with the ordinary rules of functional calculus to yield -the second variation +Next, \eqref{eq:implicit.eta} and \eqref{eq:inv.func} can be used in concert +with the ordinary rules of functional calculus to yield the second variation \begin{widetext} \begin{equation} \begin{aligned} @@ -452,9 +445,9 @@ the result, we finally arrive at \end{equation} Though not relevant here, this result generalizes to multicomponent \op s. -What does \eqref{eq:elastic.susceptibility} predict in the vicinity of the -\ho\ transition? Near the disordered to modulated transition---the zero-pressure transition to the HO state---the -static modulus is given by +What does \eqref{eq:elastic.susceptibility} predict in the vicinity of the \ho\ +transition? Near the disordered to modulated transition---the zero-pressure +transition to the HO state---the static modulus is given by \begin{equation} C_\X(0)=C_\X^0\bigg[1+\frac{b^2}{C_\X^0}\big(D_\perp q_*^4+|\Delta\tilde r|\big)^{-1}\bigg]^{-1}. \label{eq:static_modulus} @@ -474,17 +467,18 @@ corresponding modulus. \includegraphics[width=\columnwidth]{fig-stiffnesses} \caption{ \Rus\ measurements of the elastic moduli of \urusi\ at ambient pressure as a - function of temperature from recent - experiments\cite{ghosh_single-component_nodate} (blue, solid) alongside fits - to theory (magenta, dashed). The solid yellow region shows the location of - the \ho\ phase. (a) $\Btg$ modulus data and a fit to the standard - form.\cite{varshni_temperature_1970} (b) $\Bog$ modulus data and a fit to + function of temperature from recent experiments\cite{1903.00552v1} (blue, + solid) alongside fits to theory (magenta, dashed). The solid yellow region + shows the location of the \ho\ phase. (a) $\Btg$ modulus data and a fit to + the standard form.\cite{Varshni_1970} (b) $\Bog$ modulus data and a fit to \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 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 + $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic + term in $C^0_\Bog$ was here not needed for the fit.\cite{Varshni_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., an absolute value function. The failure of the Ginzburg--Landau prediction below the transition is expected on the grounds that the \op\ is too large @@ -495,14 +489,13 @@ corresponding modulus. \end{figure} \section{Comparison to experiment} -\Rus\ experiments~\cite{ghosh_single-component_nodate} yield the individual -elastic moduli broken into irreps; data for the $\Bog$ and $\Btg$ -components defined in \eqref{eq:strain-components} are shown in Figures -\ref{fig:data}(a--b). The $\Btg$ in Fig.~\ref{fig:data}(a) modulus doesn't -appear to have any response to the presence of the transition, exhibiting the -expected linear stiffening upon cooling from room temperature, with a -low-temperature cutoff at some fraction of the Debye -temperature.\cite{varshni_temperature_1970} The $\Bog$ modulus +\Rus\ experiments~\cite{1903.00552v1} yield the individual elastic moduli +broken into irreps; data for the $\Bog$ and $\Btg$ components defined in +\eqref{eq:strain-components} are shown in Figures \ref{fig:data}(a--b). The +$\Btg$ in Fig.~\ref{fig:data}(a) modulus doesn't appear to have any response to +the presence of the transition, exhibiting the expected linear stiffening upon +cooling from room temperature, with a low-temperature cutoff at some fraction +of the Debye temperature.\cite{Varshni_1970} The $\Bog$ modulus Fig.~\ref{fig:data}(b) has a dramatic response, softening over the course of roughly $100\,\K$ and then cusping at the \ho\ transition. While the low-temperature response is not as dramatic as the theory predicts, mean field @@ -528,7 +521,7 @@ $\langle\epsilon_\Bog\rangle^2=b^2\tilde r/4u(C^0_\Bog)^2$, which corresponds to an orthorhombic structural phase. The onset of orthorhombic symmetry breaking was recently detected at high pressure in \urusi\ using x-ray diffraction, a further consistency of this theory with the phenomenology of -\urusi.\cite{choi_pressure-induced_2018} +\urusi.\cite{Choi_2018} Second, as the Lifshitz point is approached from low pressure, this theory predicts that the modulation wavevector $q_*$ should vanish continuously. Far @@ -536,12 +529,11 @@ from the Lifshitz point we expect the wavevector to lock into values commensurate with the space group of the lattice, and moreover that at zero pressure, where the \rus\ data here was collected, the half-wavelength of the modulation should be commensurate with the lattice spacing $a_3\simeq9.68\,\A$, -or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$.\cite{meng_imaging_2013, -broholm_magnetic_1991, wiebe_gapped_2007, bourdarot_precise_2010, hassinger_similarity_2010} In between -these two regimes, mean field theory predicts that the ordering wavevector -shrinks by jumping between ever-closer commensurate values in the style of the -devil's staircase.\cite{bak_commensurate_1982} In reality the presence of -fluctuations may wash out these transitions. +or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$.\cite{Meng_2013, Broholm_1991, Wiebe_2007, +Bourdarot_2010, Hassinger_2010} In between these two regimes, mean field theory +predicts that the ordering wavevector shrinks by jumping between ever-closer +commensurate values in the style of the devil's staircase.\cite{Bak_1982} In +reality the presence of fluctuations may wash out these transitions. This motivates future ultrasound experiments done under pressure, where the depth of the cusp in the $\Bog$ modulus should deepen (perhaps with these @@ -549,47 +541,51 @@ 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_ultrasonic_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 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 one-component disordered to modulated transition, and therefore -is not expected to modify the thermodynamic behavior -otherwise.\cite{garel_commensurability_1976} +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 +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 +one-component disordered to modulated transition, and therefore is not expected +to modify the thermodynamic behavior otherwise.\cite{Garel_1976} There are two apparent discrepancies between the orthorhombic strain in the -phase diagram presented by recent x-ray data\cite{choi_pressure-induced_2018}, -and that predicted by our mean field theory if its uniform $\Bog$ phase is -taken to be coincident with \urusi's \afm. The first is the apparent onset of -the orthorhombic phase in the \ho\ state at slightly lower pressures than the onset of \afm. As the -recent x-ray research\cite{choi_pressure-induced_2018} notes, this misalignment of the two transitions as function of doping could be due -to the lack of an ambient pressure calibration for the lattice constant. The -second discrepancy is the onset of orthorhombicity at higher temperatures than -the onset of \afm. We note that magnetic susceptibility data sees no trace of another phase -transition at these higher temperatures. \cite{inoue_high-field_2001} It is therefore possible that the high-temperature orthorhombic signature in x-ray scattering is not the result of a bulk thermodynamic phase, but instead marks the onset of short-range correlations, as it does in the high-T$_{\mathrm{c}}$ cuprates \cite{ghiringhelli2012long} (where the onset of CDW correlations also lacks a thermodynamic phase transition). +phase diagram presented by recent x-ray data\cite{Choi_2018}, and that +predicted by our mean field theory if its uniform $\Bog$ phase is taken to be +coincident with \urusi's \afm. The first is the apparent onset of the +orthorhombic phase in the \ho\ state at slightly lower pressures than the onset +of \afm. As the recent x-ray research\cite{Choi_2018} notes, this misalignment +of the two transitions as function of doping could be due to the lack of an +ambient pressure calibration for the lattice constant. The second discrepancy +is the onset of orthorhombicity at higher temperatures than the onset of \afm. +We note that magnetic susceptibility data sees no trace of another phase +transition at these higher temperatures. \cite{Inoue_2001} It is therefore +possible that the high-temperature orthorhombic signature in x-ray scattering +is not the result of a bulk thermodynamic phase, but instead marks the onset of +short-range correlations, as it does in the high-T$_{\mathrm{c}}$ cuprates +\cite{Ghiringhelli_2012} (where the onset of CDW correlations also lacks a +thermodynamic phase transition). Three dimensions is below the upper critical dimension $4\frac12$ of a one-component disordered-to-modulated transition, and so mean field theory should break down sufficiently close to the critical point due to fluctuations, -at the Ginzburg temperature. \cite{hornreich_lifshitz_1980, -ginzburg_remarks_1961} Magnetic phase transitions tend to have a Ginzburg -temperature of order one. Our fit 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 $\xi_{\perp0}$ at 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 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 critical point may be able to resolve a modified cusp -exponent $\gamma\simeq1.31$,\cite{guida_critical_1998} since the universality -class of a uniaxial modulated one-component \op\ is $\mathrm -O(2)$.\cite{garel_commensurability_1976} We should not expect any quantitative -agreement between mean field theory and experiment in the low temperature phase -since, by the point the Ginzburg criterion is satisfied, $\eta$ is order one -and the Landau--Ginzburg free energy expansion is no longer valid. +at the Ginzburg temperature. \cite{Hornreich_1980, Ginzburg_1961} Magnetic +phase transitions tend to have a Ginzburg temperature of order one. Our fit +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 +$\xi_{\perp0}$ at 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 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 critical point may be able to +resolve a modified cusp exponent $\gamma\simeq1.31$,\cite{Guida_1998} since the +universality class of a uniaxial modulated one-component \op\ is $\mathrm +O(2)$.\cite{Garel_1976} We should not expect any quantitative agreement between +mean field theory and experiment in the low temperature phase since, by the +point the Ginzburg criterion is satisfied, $\eta$ is order one and the +Landau--Ginzburg free energy expansion is no longer valid. \section{Conclusion and Outlook.} We have developed a general phenomenological treatment of \ho\ \op s that have the potential for linear coupling to strain. @@ -598,24 +594,27 @@ the phase diagram of \urusi\ are $\Bog$ and $\Btg$. Of these, only a staggered $\Bog$ \op\ is consistent with zero-pressure \rus\ data, with a cusp appearing in the associated elastic modulus. In this picture, the \ho\ phase is characterized by uniaxial modulated $\Bog$ order, while the high pressure phase -is characterized by uniform $\Bog$ order. The staggered nematic of \ho\ is similar to the striped superconducting phase found in LBCO and other cuperates.\cite{berg_striped_2009} +is characterized by uniform $\Bog$ order. The staggered nematic of \ho\ is +similar to the striped superconducting phase found in LBCO and other +cuperates.\cite{Berg_2009b} 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 -connection with the physics of \afm. Neglecting this physics could be reasonable since correlations -often lead to \afm\ as a secondary effect, like what occurs in many Mott insulators. An -electronic theory of this phase diagram may find that the \afm\ observed in -\urusi\ indeed follows along with an independent high-pressure orthorhombic phase associated with -uniform $\Bog$ electronic order. +connection with the physics of \afm. Neglecting this physics could be +reasonable since correlations often lead to \afm\ as a secondary effect, like +what occurs in many Mott insulators. An electronic theory of this phase diagram +may find that the \afm\ observed in \urusi\ indeed follows along with an +independent high-pressure orthorhombic phase associated with uniform $\Bog$ +electronic order. The corresponding prediction of uniform $\Bog$ symmetry breaking in the high pressure phase is consistent with recent diffraction experiments, -\cite{choi_pressure-induced_2018} except for the apparent earlier onset in -temperature of the $\Bog$ symmetry breaking, which we believe may be due to -fluctuating order at temperatures above the actual transition temperature. This work motivates both +\cite{Choi_2018} except for the apparent earlier onset in temperature of the +$\Bog$ 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. +order, and preforming symmetry-sensitive thermodynamic experiments at pressure, +such as ultrasound, that could further support or falsify this idea. \begin{acknowledgements} Jaron Kent-Dobias is supported by NSF DMR-1719490, Michael Matty is supported by |