From 0d2d0cde33d298d3ac4d2357a1b3008292a96773 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 5 Dec 2019 21:23:05 -0500
Subject: lots of small spot changes, and a big purge of afm talk

---
 main.tex | 261 +++++++++++++++++++++++++++++++++------------------------------
 1 file changed, 135 insertions(+), 126 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 1bdc030..385ddcf 100644
--- a/main.tex
+++ b/main.tex
@@ -117,12 +117,13 @@ theories~\cite{kambe_odd-parity_2018, haule_arrested_2009,
   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 seeks to unify 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}
+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
+$T_{\text{\ho}}=17.5\,\K$;\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}
 
 Recent resonant ultrasound spectroscopy (\rus) measurements were used to
 examine the thermodynamic discontinuities in the elastic moduli at
@@ -140,7 +141,7 @@ under pressure.\cite{choi_pressure-induced_2018} Above 0.13--0.5 $\GPa$
 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
+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
@@ -149,7 +150,7 @@ 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 \afm\ state with uniform
+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
@@ -157,19 +158,20 @@ 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 \afm\ phase is approached.
+$\Bog$ strain of the high pressure phase is approached.
 
 
 \section{Model}
-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
+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
 interaction between strain and \op. The most general quadratic free energy of
 the strain $\epsilon$ is $f_\e=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$.
 \footnote{Components of the elastic modulus tensor $C$ were given in the
-popular Voigt notation in the abstract and introduction. Here and henceforth
-the notation used is that natural for a rank-four tensor.} Linear combinations
-of the six independent components of strain form five irreducible components of
+  popular Voigt notation in the abstract and introduction. Here and henceforth
+the notation used is that natural for a rank-four tensor.} The form of the bare
+moduli tensor $C^0$ is further restricted by symmetry. Linear combinations of
+the six independent components of strain form five irreducible components of
 strain in \Dfh\ as
 \begin{equation}
   \begin{aligned}
@@ -220,8 +222,8 @@ under inversion), the $\Atg$ irrep, and all half-integer (spinor)
 representations.
 
 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
+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
@@ -263,9 +265,9 @@ $\epsilon_\X$ gives
 which in turn gives the strain field conditioned on the state of the \op\ field
 as $\epsilon_\X^\star[\eta](x)=(b/C^0_\X)\eta(x)$ at all spatial coordinates
 $x$, and $\epsilon_\Y^\star[\eta]=0$ for all other irreps $\Y\neq\X$. Upon
-substitution into the \eqref{eq:free_energy}, the resulting single-argument
+substitution into \eqref{eq:free_energy}, the resulting single-argument
 free energy functional $F[\eta,\epsilon_\star[\eta]]$ has a density identical
-to $f_\op$ with $r\to\tilde r=r-b^2/2C^0_\X$. 
+to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$. 
 
 \begin{figure}[htpb]
   \includegraphics[width=\columnwidth]{phase_diagram_experiments}
@@ -289,16 +291,16 @@ to $f_\op$ with $r\to\tilde r=r-b^2/2C^0_\X$.
 
 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 discused below can all be found in a
-standard text, e.g., 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
+lifshitz_theory_1942-1} The properties discussed in the remainder of this
+section can all be found in a standard text, e.g., 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 $c_\perp>0$,
 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 of the \op\ representation, e.g., $\Bog$ or $\Btg$. For $c_\perp<0$
-and $\tilde r<c_\perp^2/4D_\perp$ there are free energy minima for
+uniform order of the \op\ representation, e.g., $\Bog$ or $\Btg$. 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}
@@ -321,14 +323,14 @@ parameter. The schematic phase diagrams for this model are shown in
 Figure~\ref{fig:phases}.
 
 \section{Results}
-We will now derive the effective elastic tensor $C$ that results from coupling
-of strain to the \op. The ultimate result, found in
+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
-$C^0_\X$ only for the representation $\X$ of the \op. Moreover, this modulus does not
-vanish at the unordered to modulated transition---as it would if the transition
-were a $q=0$ phase transition---but instead ends in a cusp. In this section
-we start by computing the susceptibility of the \op\ at the unordered to
-modulated transition, and then compute the elastic modulus for the same. 
+$C^0_\X$ only for the representation $\X$ of the \op. Moreover, this modulus
+does not vanish at the unordered to modulated transition---as it would if the
+transition were a $q=0$ phase transition---but instead ends in a cusp. In this
+section we start by computing the susceptibility of the \op\ at the unordered
+to modulated transition, and then compute the elastic modulus for the same. 
 
 The susceptibility of a single-component ($\Bog$ or $\Btg$) \op\ is
 \begin{equation}
@@ -385,21 +387,21 @@ which implicitly gives $\eta_\star[\epsilon]$, the \op\ conditioned
 on the configuration of the strain. Since $\eta_\star$ is a functional of $\epsilon_\X$
 alone, only the modulus $C_\X$ will be modified from its bare value $C^0_\X$.
 
-Though the differential equation for $\eta_*$ cannot be solved explicitly, we
-can use the inverse function theorem to make use of it 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 can
-immediately identify from \eqref{eq:implicit.eta} as
-$\eta^{-1}_\star[\eta](x)=b^{-1}(\delta F_\op[\eta]/\delta\eta(x))$.  Now, we use
-the inverse function theorem to relate the functional reciprocal of the
+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.
+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
+can immediately identify from \eqref{eq:implicit.eta} as
+$\eta^{-1}_\star[\eta](x)=b^{-1}(\delta F_\op[\eta]/\delta\eta(x))$.  Now, we
+use the inverse function theorem to relate the functional reciprocal of the
 derivative of $\eta_\star[\epsilon]$ with respect to $\epsilon_\X$ to the
 derivative of $\eta^{-1}_\star[\eta]$ with respect to $\eta$, yielding
 \begin{equation}
   \begin{aligned}
     \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip
-    &=\frac{\delta\eta_\star^{-1}[\eta](x)}{\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]} \\
-    &=b^{-1}\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}.
+    &=\frac{\delta\eta_\star^{-1}[\eta](x)}{\delta\eta(x')}\bigg|_{\eta=\eta_\star[\epsilon]} \\
+    &=b^{-1}\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta_\star[\epsilon]}.
   \end{aligned}
   \label{eq:inv.func}
 \end{equation}
@@ -428,9 +430,9 @@ the second variation
   \label{eq:big.boy}
 \end{equation}
 \end{widetext}
-The elastic modulus is given by the second variation evaluated at the
-extremized strain $\langle\epsilon\rangle$. To calculate it, note that
-evaluating the second variation of $F_\op$ in \eqref{eq:inv.func} at
+The elastic modulus is given by the second variation \eqref{eq:big.boy}
+evaluated at the extremized strain $\langle\epsilon\rangle$. To calculate it,
+note that evaluating the second variation of $F_\op$ in \eqref{eq:inv.func} at
 $\langle\epsilon\rangle$ (or
 $\eta_\star(\langle\epsilon\rangle)=\langle\eta\rangle$) yields
 \begin{equation}
@@ -456,74 +458,78 @@ static modulus is given by
   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}
 \end{equation}
-This corresponds to a softening in the $\X$-modulus at the transition that is
-cut off with a cusp of the form $|\Delta\tilde r|^\gamma\propto|T-T_c|^\gamma$
-with $\gamma=1$. This is our main result. The only \op\ irreps that couple
-linearly with strain and reproduce the topology of the \urusi\ phase diagram
-are $\Bog$ and $\Btg$. For either of these irreps, the transition into a
-modulated rather than uniform phase masks traditional signatures of a
-continuous transition by locating thermodynamic singularities at nonzero $q=q_*$.
-The remaining clue at $q=0$ is a particular kink in the corresponding modulus.
+This corresponds to a softening in the $\X$-modulus approaching the transition
+that is cut off with a cusp of the form $|\Delta\tilde
+r|^\gamma\propto|T-T_c|^\gamma$ with $\gamma=1$. This is our main result. The
+only \op\ irreps that couple linearly with strain and reproduce the topology of
+the \urusi\ phase diagram are $\Bog$ and $\Btg$. For either of these irreps,
+the transition into a modulated rather than uniform phase masks traditional
+signatures of a continuous transition by locating thermodynamic singularities
+at nonzero $q=q_*$.  The remaining clue at $q=0$ is a particular kink in the
+corresponding modulus.
 
 \begin{figure}[htpb]
   \centering
   \includegraphics[width=\columnwidth]{fig-stiffnesses}
   \caption{
-   \Rus\ measurements of the elastic moduli of \urusi\ as a function of
-   temperature from~\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 fit to standard
-   form.\cite{varshni_temperature_1970} (b) $\Bog$ modulus data and fit to
+   \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
    \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 fit of
-   \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and 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 for the free energy expansion to
-   be valid by the time the Ginzburg temperature is reached.
+   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
+   \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
+   for the free energy expansion to be valid by the time the Ginzburg
+   temperature is reached.
   }
   \label{fig:data}
 \end{figure}
 
 \section{Comparison to experiment}
 \Rus\ experiments~\cite{ghosh_single-component_nodate} yield the individual
-elastic moduli broken into irrep symmetries; the $\Bog$ and $\Btg$ components
-defined in \eqref{eq:strain-components} are shown in Figures
-\ref{fig:data}(a--b).  The $\Btg$ 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 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 theory---which is based on a
-small-$\eta$ expansion---will not work quantitatively far below the transition
-where $\eta$ has a large nonzero value and higher powers in the free energy
-become important. The data in the high-temperature phase can be fit to the
-theory \eqref{eq:static_modulus}, with a linear background modulus $C^0_\Bog$
-and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure
-\ref{fig:data}(b). The data and theory appear quantitatively consistent in the
-high temperature phase, suggesting that \ho\ can be described as a
-$\Bog$-nematic phase that is modulated at finite $q$ along the $c-$axis. The
-predicted softening appears over hundreds of Kelvin; Figures
-\ref{fig:data}(c--d) show the background modulus $C_\Bog^0$ and the
+elastic moduli broken into irrep symmetries; 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
+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
+theory---which is based on a small-$\eta$ expansion---will not work
+quantitatively far below the transition where $\eta$ has a large nonzero value
+and higher powers in the free energy become important. The data in the
+high-temperature phase can be fit to the theory \eqref{eq:static_modulus}, with
+a linear background modulus $C^0_\Bog$ and $\tilde r-\tilde r_c=a(T-T_c)$, and
+the result is shown in Figure \ref{fig:data}(b). The data and theory appear
+quantitatively consistent in the high temperature phase, suggesting that \ho\
+can be described as a $\Bog$-nematic phase that is modulated at finite $q$
+along the $c-$axis. The predicted softening appears over hundreds of Kelvin;
+Figures \ref{fig:data}(c--d) show the background modulus $C_\Bog^0$ and the
 \op--induced response isolated from each other. 
 
 We have seen that the mean-field theory of a $\Bog$ \op\ recreates the topology
 of the \ho\ phase diagram and the temperature dependence of the $\Bog$ elastic
 modulus at zero pressure. This theory has several other physical implications.
 First, the association of a modulated $\Bog$ order with the \ho\ phase implies
-a \emph{uniform} $\Bog$ order associated with the \afm\ phase, and moreover a
-uniform $\Bog$ strain of magnitude $\langle\epsilon_\Bog\rangle^2=b^2\tilde
-r/4u(C^0_\Bog)^2$, which corresponds to an orthorhombic structural phase.
-Orthorhombic symmetry breaking was recently detected in the \afm\ phase of
-\urusi\ using x-ray diffraction, a further consistency of this theory with the
-phenomenology of \urusi.\cite{choi_pressure-induced_2018} 
+a \emph{uniform} $\Bog$ order associated with the high pressure phase, and
+moreover a uniform $\Bog$ strain of magnitude
+$\langle\epsilon_\Bog\rangle^2=b^2\tilde r/4u(C^0_\Bog)^2$, 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} 
 
 Second, as the Lifshitz point is approached from low pressure, this theory
 predicts that the modulation wavevector $q_*$ should vanish continuously. Far
@@ -548,24 +554,26 @@ motivate x-ray and neutron-diffraction experiments to look for new q's -
 mentioning this is important if we want to get others interested, no one else
 does RUS...} Alternatively, \rus\ done at ambient pressure might examine the
 heavy fermi liquid to \afm\ transition by doping. \brad{We have to be careful,
-someone did do some doping studies and it's not clear exctly what's going on}.
-The presence of spatial commensurability known to be irrelevant to critical
+someone did do some doping studies and it's not clear exactly what's going on}.
+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} 
 
 There are two apparent discrepancies between the orthorhombic strain in the
-phase diagram presented by \cite{choi_pressure-induced_2018} and that predicted
-by our mean field theory.  The first is the apparent onset of the orthorhombic
-phase in the \ho\ state prior to the onset of \afm.  As~\cite{choi_pressure-induced_2018} notes, this 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.
-Susceptibility data sees no trace of another phase transition at these higher
-temperatures.\cite{inoue_high-field_2001} We suspect that the high-temperature
-orthorhombic signature is not the result of a bulk phase, and could be due to
-the high energy (small-wavelength) nature of x-rays as an experimental probe:
-\op\ fluctuations should lead to the formation of orthorhombic regions on the
-order of the correlation length that become larger and more persistent as the
-transition is approached. 
+phase diagram presented by recent x-ray data\cite{choi_pressure-induced_2018}
+and that predicted by our mean field theory when its uniform ordered phase is
+taken to be coincident with \urusi's \afm.  The first is the apparent onset of
+the orthorhombic phase in the \ho\ state prior to the onset of \afm.  As the
+recent x-ray research\cite{choi_pressure-induced_2018} notes, this 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.  Susceptibility data sees no trace of another phase
+transition at these higher temperatures.\cite{inoue_high-field_2001} We suspect
+that the high-temperature orthorhombic signature is not the result of a bulk
+phase, and could be due to the high energy (small-wavelength) nature of x-rays
+as an experimental probe: \op\ fluctuations should lead to the formation of
+orthorhombic regions on the order of the correlation length that become larger
+and more persistent as the transition is approached. 
 
 Three dimensions is below the upper critical dimension $4\frac12$ of a
 one-component disordered to modulated transition, and so mean field theory
@@ -579,33 +587,34 @@ 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 around several degrees 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
+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.
 
 \section{Conclusion and Outlook.} We have developed a general phenomenological
-treatment of  \ho\ \op s with the potential for linear coupling to strain. The
-two representations with mean field phase diagrams that are consistent with 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
+treatment of  \ho\ \op s that have the potential for linear coupling to strain.
+The two representations with mean field phase diagrams that are consistent with
+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 \afm\ phase is
-characterized by uniform $\Bog$ order. \brad{We need to be a bit more explicit
-about what we think is going on with \afm - is it just a parasitic phase? Is
-our modulated phase somehow "moduluated \afm" (can you modualte AFM in such
-as way as to make it disappear? Some combination of orbitals?)} The
-corresponding prediction of uniform $\Bog$ symmetry breaking in the \afm\ phase
-is consistent with recent diffraction experiments,
+characterized by uniaxial modulated $\Bog$ order, while the high pressure phase
+is characterized by uniform $\Bog$ order. \brad{We need to be a bit more
+explicit about what we think is going on with \afm - is it just a parasitic
+phase? Is our modulated phase somehow "moduluated \afm" (can you modualte AFM
+in such as way as to make it disappear? Some combination of orbitals?)} 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 than AFM, which we believe to be
-due to fluctuating order above the actual phase transition.  This work
-motivates both further theoretical work regarding a microscopic theory with
-modulated $\Bog$ order, and preforming \rus\ experiments at pressure that could
-further support or falsify this idea.
+temperature of the $\Bog$ symmetry breaking, which we believe to be due to
+fluctuating order above the actual phase transition.  This work motivates both
+further theoretical work regarding a microscopic theory with modulated $\Bog$
+order, and preforming \rus\ experiments at pressure that could further support
+or falsify this idea.
 
 \begin{acknowledgements}
   Jaron Kent-Dobias had support from NSF DMR-1719490 and Michael Matty had
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