From 57151805ef46c88b532f94826930ea366862be14 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sat, 24 Aug 2019 15:51:05 -0400
Subject: many changes

---
 main.tex | 526 +++++++++++++++++++++++++++++++++++++--------------------------
 1 file changed, 307 insertions(+), 219 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 80b1c3c..8070a95 100644
--- a/main.tex
+++ b/main.tex
@@ -6,9 +6,8 @@
 % Our mysterious boy
 \def\urusi{URu$_{\text2}$Si$_{\text2}$}
 
-\def\e{{\text{\textsc{Elastic}}}} % "elastic"
-\def\o{{\text{\textsc{Op}}}} % "order parameter"
-\def\i{{\text{\textsc{Int}}}} % "interaction"
+\def\e{{\text{\textsc{elastic}}}} % "elastic"
+\def\i{{\text{\textsc{int}}}} % "interaction"
 
 \def\Dfh{D$_{\text{4h}}$}
 
@@ -40,12 +39,13 @@
 \def\op{\textsc{op}} % order parameter
 \def\ho{\textsc{ho}} % hidden order
 \def\rus{\textsc{rus}} % Resonant ultrasound spectroscopy 
+\def\afm{\textsc{afm}} % Antiferromagnetism 
 \def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy 
 \def\recip{{\{-1\}}} % functional reciprocal
 
 \begin{document}
 
-\title{Elastic properties of \urusi\ are reproduced by modulated $\Bog$ order}
+\title{Elastic properties of hidden order in \urusi\ reproduced by modulated $\Bog$ order}
 \author{Jaron Kent-Dobias}
 \author{Michael Matty}
 \author{Brad Ramshaw}
@@ -57,101 +57,86 @@
 \date\today
 
 \begin{abstract}
-  We develop a phenomenological theory for the elastic response of materials
-  with a \Dfh\ point group through phase transitions. The physics is
-  generically that of Lifshitz points, with disordered, uniform ordered, and
-  modulated ordered phases. Several experimental features of \urusi\ are
-  reproduced when the order parameter has $\Bog$ symmetry: the topology of the
-  temperature--pressure phase diagram, the response of the strain stiffness
-  tensor above the hidden-order transition, and the strain response in the
-  antiferromagnetic phase. In this scenario, the hidden order is a version of
-  the high-pressure antiferromagnetic order modulated along the symmetry axis.
+  We develop a phenomenological mean field theory for the elastic response of
+  \urusi\ through its hidden order transition. Several experimental features
+  are reproduced when the order parameter has $\Bog$ symmetry: the topology of
+  the temperature--pressure phase diagram, the response of the strain stiffness
+  tensor above the hidden-order transition at zero pressure, and orthorhombic
+  symmetry breaking in the high-pressure antiferromagnetic phase. In this
+  scenario, the hidden order is a version of the high-pressure
+  antiferromagnetic order modulated along the symmetry axis, and the triple
+  point joining those two phases with the paramagnetic phase is a Lifshitz point.
 \end{abstract}
 
 \maketitle
 
-% \begin{enumerate}
-%   \item Introduction
-%     \begin{enumerate}
-%       \item \urusi\ hidden order intro paragraph, discuss the phase diagram
-%       \item Strain/OP coupling discussion/RUS
-%       \item Discussion of experimental data
-%       \item We look at MFT's for OP's of various symmetries
-%     \end{enumerate}
-      
-%   \item Theory
-%     \begin{enumerate}
-%       \item Introduce various pieces of free energy
-      
-%       \item Summary of MFT results
-%     \end{enumerate}
-    
-%     \item Data piece
-    
-%     \item Talk about more cool stuff like AFM C4 breaking etc
-% \end{enumerate}
-
 The study of phase transitions is a central theme of condensed matter physics.
 In many cases, a phase transition between different states of matter is marked
 by a change in symmetry.  In this paradigm, the breaking of symmetry in an
 ordered phase corresponds to the condensation of an order parameter (\op) that
 breaks the same symmetries. Near a second order phase transition, the physics
-of the \op\ can often be described in the context of Landau-Ginzburg mean field
+of the \op\ can often be described in the context of Landau--Ginzburg mean field
 theory. However, to construct such a theory, one must know the symmetries of
 the \op, i.e. the symmetry of the ordered state.
 
 A paradigmatic example where the symmetry of an ordered phase remains unknown
 is in \urusi.  \urusi\ is a heavy fermion superconductor in which
 superconductivity condenses out of a symmetry broken state referred to as
-hidden order (\ho) \cite{hassinger_temperature-pressure_2008}, and at sufficiently large [hydrostatic?]
-pressures, both give way to local moment antiferromagnetism.  Despite over
-thirty years of effort, the symmetry of the \ho\ state remains unknown,
-and 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_2019, ikeda_emergent_2012} propose a
-variety of possibilities.  Many [all?] of these theories rely on the
-formulation of a microscopic model for the \ho\ state, but without direct
-experimental observation of the broken symmetry, none have been confirmed. 
+hidden order (\ho) \cite{hassinger_temperature-pressure_2008}, and at
+sufficiently large hydrostatic pressures, both give way to local moment
+antiferromagnetism (\afm).  Despite over thirty years of effort, the symmetry of the
+\ho\ state remains unknown, and 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 a variety of
+possibilities.  Many of these theories rely on the formulation of a microscopic
+model for the \ho\ state, but without direct experimental observation of the
+broken symmetry, none have been confirmed. 
 
 One case that does not rely on a microscopic model is recent work
-\cite{ghosh_single-component_2019} that studies the \ho\ transition using
-resonant ultrasound spectroscopy (\rus).  \Rus\ is an experimental technique that
-measures mechanical resonances of a sample. These resonances contain
-information about the full elastic tensor of the material. Moreover, the
-frequency locations of the resonances are sensitive to symmetry breaking at an
-electronic phase transition due to electron-phonon coupling \cite{shekhter_bounding_2013}.
-Ref.~\cite{ghosh_single-component_2019} uses this information to place strict
-thermodynamic bounds on the symmetry of the \ho\ \op, again, independent of any
-microscopic model. Motivated by these results, in this paper we consider a mean
-field theory of an \op\ coupled to strain and the effect that the \op\ symmetry has
-on the elastic response in different symmetry channels. Our study finds that a
-single possible \op\ symmetry reproduces the experimental strain
-susceptibilities, and fits the experimental data well.
-
-We first present a phenomenological Landau-Ginzburg mean field theory of strain
-coupled to an \op. We examine the phase diagram predicted by this
-theory and compare it to the experimentally obtained phase diagram of \urusi.
-Then we compute the elastic response to strain, and examine the response
-function dependence on the symmetry of the \op.  We proceed to compare the
-results from mean field theory with data from \rus\ experiments.  We further
-examine the consequences of our theory at non-zero applied pressure in
-comparison with recent x-ray scattering experiments \cite{choi_pressure-induced_2018}.  Finally, we
-discuss our conclusions and the future experimental and theoretical work motivated
-by our results.
+\cite{ghosh_single-component_nodate} that studies the \ho\ transition using
+resonant ultrasound spectroscopy (\rus).  \Rus\ is an experimental technique
+that measures mechanical resonances of a sample. These resonances contain
+information about the full strain stiffness tensor of the material. Moreover,
+the frequency locations of the resonances are sensitive to symmetry breaking at
+an electronic phase transition due to electron-phonon coupling
+\cite{shekhter_bounding_2013}.  Ref.~\cite{ghosh_single-component_nodate} uses
+this information to place strict thermodynamic bounds on the symmetry of the
+\ho\ \op, again, independent of any microscopic model. Motivated by these
+results, in this paper we consider a mean field theory of an \op\ coupled to
+strain and the effect that the \op\ symmetry has on the elastic response in
+different symmetry channels. Our study finds that a single possible \op\
+symmetry reproduces the experimental strain susceptibilities and fits the
+experimental data well. The resulting theory associates \ho\ with $\Bog$ order
+\emph{modulated along the rotation axis}, \afm\ with uniform $\Bog$ order, and
+a Lifshitz point with the triple point between them.
+
+We first present a phenomenological Landau--Ginzburg mean field theory of
+strain coupled to an \op. We examine the phase diagrams predicted by this
+theory for various \op\ symmetries and compare them to the experimentally
+obtained phase diagram of \urusi.  Then we compute the elastic response to
+strain, and examine the response function dependence on the symmetry of the
+\op.  We compare the results from mean field theory with data from \rus\
+experiments.  We further examine the consequences of our theory at non-zero
+applied pressure in comparison with recent x-ray scattering experiments
+\cite{choi_pressure-induced_2018}. We preform a consistency check for the
+applicability of our mean field theory for the \rus\ data.  Finally, we discuss
+our conclusions and the future experimental and theoretical work motivated by
+our results.
 
 The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
 respect this symmetry. We will introduce a phenomenological free energy density
 in three parts: that of the strain, the \op, and their interaction.
 The most general quadratic free energy of the strain $\epsilon$ is
-$f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare stiffness tensor $C$
-tensor is constrained by both that $\epsilon$ is a symmetric tensor and by the
-point group symmetry \cite{landau_theory_1995}. The latter can be seen in a
-systematic way. First, the six independent components of strain are written as
-linear combinations that behave like irreducible representations under the
-action of the point group, or
+$f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare strain
+stiffness tensor $C$ tensor is constrained by both that $\epsilon$ is a
+symmetric tensor and by the point group symmetry \cite{landau_theory_1995}. The
+latter can be seen in a systematic way. First, the six independent components
+of strain are written as linear combinations that behave like irreducible
+representations under the action of the point group, or
 \begin{equation}
   \begin{aligned}
     \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth}
@@ -187,68 +172,49 @@ representation is $\X$, then the most general coupling to linear order is
 \end{equation}
 If $\X$ is a representation not present in the strain there can be no linear
 coupling, and the effect of $\eta$ going through a continuous phase transition
-is to produce a jump in the $\Aog$ strain stiffness. We will therefore focus
-our attention on \op\ symmetries that produce linear couplings to
-strain. Looking at the components present in \eqref{eq:strain-components}, this rules out all of the u-reps (odd under inversion) and the $\Atg$ irrep as having any anticipatory response in the strain stiffness.
+is to produce a jump in the $\Aog$ strain stiffness \cite{luthi_sound_1970,
+ramshaw_avoided_2015, shekhter_bounding_2013}. We will therefore focus our
+attention on \op\ symmetries that produce linear couplings to strain. Looking
+at the components present in \eqref{eq:strain-components}, this rules out all
+of the u-reps (odd under inversion) and the $\Atg$ irrep as having any
+anticipatory response in the strain stiffness.
 
 If the \op\ transforms like $\Aog$, odd terms are allowed in its
 free energy and any transition will be abrupt and not continuous without
-tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic
+fine-tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quadratic
 free energy density is
 \begin{equation}
   \begin{aligned}
-    f_\o=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
+    f_\op=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
       +c_\perp(\nabla_\perp\eta)^2 \\
-      &\quad+D_\parallel(\nabla_\parallel^2\eta)^2
-      +D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4
+      &\qquad\qquad\qquad\quad+D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4
   \end{aligned}
   \label{eq:fo}
 \end{equation}
 where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and
-$\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$
-since this does not affect the physics at hand. The full free energy functional of $\eta$ and $\epsilon$ is then
+$\nabla_\perp=\partial_3$ 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
+full free energy functional of $\eta$ and $\epsilon$ is then
 \begin{equation}
   \begin{aligned}
     F[\eta,\epsilon]
-      &=F_\o[\eta]+F_\e[\epsilon]+F_\i[\eta,\epsilon] \\
-      &=\int dx\,(f_\o+f_\e+f_\i)
+      &=F_\op[\eta]+F_\e[\epsilon]+F_\i[\eta,\epsilon] \\
+      &=\int dx\,(f_\op+f_\e+f_\i)
   \end{aligned}
 \end{equation}
-Neglecting interaction terms
-higher than quadratic order, the only strain relevant to the problem is
-$\epsilon_\X$, and this can be traced out of the problem exactly, since
+The only strain relevant to the \op\ is $\epsilon_\X$, which can be traced out
+of the problem exactly in mean field theory. Extremizing with respect to
+$\epsilon_\X$,
 \begin{equation}
-  0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_{\X i}(x)}=C_\X\epsilon_{\X i}(x)
+  0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_{\X i}(x)}\bigg|_{\epsilon=\epsilon_\star}=C_\X\epsilon^\star_{\X i}(x)
     +\frac12b\eta_i(x)
 \end{equation}
-gives $\epsilon_\X[\eta]=-(b/2C_\X)\eta$. Upon substitution into the free
-energy, the resulting effective free energy $F[\eta,\epsilon[\eta]]$ has a density identical to $f_\o$
-with $r\to\tilde r=r-b^2/4C_\X$.
-
-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}. For a one-component \op\ ($\Bog$ or $\Btg$) it is
-traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. 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. 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}
-    =\frac{\tilde r_c-\tilde r}{3u}
-    =\frac{\Delta\tilde r}{3u}
-\end{equation}
-with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The
-transition between the uniform and modulated orderings is abrupt for a one-component
-field 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 field. 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 $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$.
-The uniform--modulated transition is now continuous. This already does not reproduce the physics of \ho, and so we will henceforth neglect this possibility. The schematic phase
-diagrams for this model are shown in Figure \ref{fig:phases}.
+gives the optimized strain conditional on the \op\ as
+$\epsilon_\X^\star[\eta](x)=-(b/2C_\X)\eta(x)$ and $\epsilon_\Y^\star[\eta]=0$
+for all other $\Y$. Upon substitution into the free energy, the resulting
+effective free energy $F[\eta,\epsilon_\star[\eta]]$ has a density identical to
+$f_\op$ with $r\to\tilde r=r-b^2/4C_\X$.
 
 \begin{figure}[htpb]
   \includegraphics[width=\columnwidth]{phase_diagram_experiments}
@@ -270,28 +236,62 @@ diagrams for this model are shown in Figure \ref{fig:phases}.
   \label{fig:phases}
 \end{figure}
 
-We will now proceed to derive the effective strain stiffness tensor $\lambda$ that results from the coupling of strain to the \op. The ultimate result, in \eqref{eq:elastic.susceptibility}, is that $\lambda_\X$ only differs from its bare value for the symmetry $\X$ of the order parameter. To show this, we will first compute the susceptibility of the \op, which will both be demonstrative of how the stiffness is calculated and prove useful in expressing the functional form of the stiffness. Then, we will compute the strain stiffness using some tricks from functional calculus.
-
-The susceptibility of the order parameter to a field linearly coupled to it is given by
+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}. For a one-component \op\ ($\Bog$ or $\Btg$) 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. 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}
+    =\frac{\tilde r_c-\tilde r}{3u}
+    =\frac{|\Delta\tilde r|}{3u}
+\end{equation}
+with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The
+transition between the uniform and modulated orderings is abrupt for a
+one-component field 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 field. 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--modulated transition is now continuous. This does not
+reproduce the physics of \ho, which has an abrupt 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}.
+
+We will now proceed to derive the \emph{effective strain stiffness tensor}
+$\lambda$ that results from the coupling of strain to the \op. The ultimate
+result, found in \eqref{eq:elastic.susceptibility}, is that $\lambda_\X$
+differs from its bare value $C_\X$ only for the symmetry $\X$ of the \op. To
+show this, we will first compute the susceptibility of the \op, which will both
+be demonstrative of how the stiffness is calculated and prove useful in
+expressing the functional form of the stiffness. Then we will compute the
+strain stiffness using some tricks from functional calculus.
+
+The susceptibility of a single component ($\Bog$ or $\Btg$) \op\ to a field
+linearly coupled to it is given by
 \begin{equation}
   \begin{aligned}
     &\chi^\recip(x,x')
-    =\frac{\delta^2F[\eta,\epsilon[\eta]]}{\delta\eta(x)\delta\eta(x')}
-      =\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2 \\
-    &\qquad\qquad+D_\perp\nabla_\perp^4+12u\eta^2(x)\big)
+    =\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'),
   \end{aligned}
   \label{eq:sus_def}
 \end{equation}
 where $\recip$ indicates a \emph{functional reciprocal} in the sense that
-\[
+\begin{equation}
   \int dx''\,\chi^\recip(x,x'')\chi(x'',x')=\delta(x-x').
-\]
+\end{equation}
 Taking the Fourier transform and integrating over $q'$ we have
 \begin{equation}
     \chi(q)
     =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4
-    +12u\sum_{q'}\tilde\eta_{q'}\tilde\eta_{-q'}\big)^{-1}.
+    +12u\sum_{q'}\langle\tilde\eta_{q'}\rangle\langle\tilde\eta_{-q'}\rangle\big)^{-1}.
 \end{equation}
 Near the unordered--modulated transition this yields
 \begin{equation}
@@ -305,89 +305,111 @@ Near the unordered--modulated transition this yields
   \label{eq:susceptibility}
 \end{equation}
 with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}$ and
-$\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}$. We must emphasize that this is \emph{not} the magnetic susceptibility because a $\Bog$ or $\Btg$ \op\ cannot couple linearly to a uniform magnetic field. The object defined in \eqref{eq:sus_def} is most readily interpreted as proportional to the two-point connected correlation function $\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$.
+$\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}$. We must emphasize that
+this is \emph{not} the magnetic susceptibility because a $\Bog$ or $\Btg$ \op\
+cannot couple linearly to a uniform magnetic field. The object defined in
+\eqref{eq:sus_def} is most readily interpreted as proportional to the two-point
+connected correlation function
+$\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$.
 
 The strain stiffness is given in a similar way to the inverse susceptibility: we
 must trace over $\eta$ and take the second variation of the resulting free
 energy functional of $\epsilon$. Extremizing over $\eta$ yields
 \begin{equation}
-  0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}=\frac{\delta F_\o[\eta]}{\delta\eta(x)}
-    +\frac12b\epsilon_\X(x),
+  0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}=
+    \frac12b\epsilon_\X(x)+\frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star},
   \label{eq:implicit.eta}
 \end{equation}
-which implicitly gives $\eta[\epsilon]$. Since $\eta$ is a functional of $\epsilon_\X$ alone, only the stiffness $\lambda_\X$ is modified from its bare value $C_\X$. Though this
-cannot be solved explicitly, we can make use of the inverse function theorem.
-First, denote by $\eta^{-1}[\eta]$ the inverse functional of $\eta$ implied by
+which implicitly gives $\eta_\star[\epsilon]$, the optimized \op\ conditioned on the strain. Since $\eta_\star$ is a functional of $\epsilon_\X$
+alone, only the stiffness $\lambda_\X$ is modified from its bare value $C_\X$.
+Though this differential equation for $\eta_*$ cannot be solved explicitly, we
+can make use of the inverse function theorem.  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 $\eta^{-1}[\eta](x)=-2/b(\delta F_\o[\eta]/\delta\eta(x))$.
-Now, we use the inverse function
-theorem to relate the functional reciprocal of the derivative of $\eta[\epsilon]$ with respect to $\epsilon_\X$ to the derivative of $\eta^{-1}[\eta]$ with respect to $\eta$, yielding
+to each solution of \eqref{eq:implicit.eta} it receives. This we can
+immediately identify from \eqref{eq:implicit.eta} as
+$\eta^{-1}_\star[\eta](x)=-2/b(\delta F_\op[\eta]/\delta\eta(x))$.  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(x)}{\delta\epsilon_\X(x')}\bigg)^\recip
-    &=\frac{\delta\eta^{-1}[\eta](x)}{\delta\eta(x')}
-    =-\frac2b\frac{\delta^2F_\o[\eta]}{\delta\eta(x)\delta\eta(x')} \\
-    &=-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x'),
+    \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]} 
+    =-\frac2b\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}.
   \end{aligned}
   \label{eq:inv.func}
 \end{equation}
-where we have used what we already know about the variation of $F_\o[\eta]$ with respect to $\eta$.
-Finally, \eqref{eq:implicit.eta} and \eqref{eq:inv.func} can be used in concert with the ordinary rules of functional calculus to yield the strain stiffness
+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}
-    \lambda_\X(x,x')
-    &=\frac{\delta^2F[\eta[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\
+    \frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} 
     &=C_\X\delta(x-x')+
-    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
-    +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')} \\
-    &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o[\eta]}{\delta\eta(x'')\delta\eta(x''')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')}
-    +\int dx''\,\frac{\delta F_\o[\eta]}{\delta\eta(x'')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\ 
+    b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
+    +\frac12b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') \\
+    &\quad+\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\frac{\delta^2F_\op[\eta]}{\delta\eta(x'')\delta\eta(x''')}\bigg|_{\eta=\eta_\star[\epsilon]}
+    +\int dx''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}\frac{\delta F_\op[\eta]}{\delta\eta(x'')}\bigg|_{\eta=\eta_\star[\epsilon]} \\ 
     &=C_\X\delta(x-x')+
-    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
-    -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta(x'')}{\partial\epsilon_\X(x''')}\bigg)^{-1}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')} \\ 
+    b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
+    -\frac12b\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\bigg(\frac{\partial\eta_\star[\epsilon](x'')}{\partial\epsilon_\X(x''')}\bigg)^\recip \\ 
     &=C_\X\delta(x-x')+
-    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
-    -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta(x'')}{\delta\epsilon_\X(x')} 
+    b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
+    -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x')} 
     =C_\X\delta(x-x')+
-    \frac12b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')},
+    \frac12b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}.
   \end{aligned}
+  \label{eq:big.boy}
 \end{equation}
 \end{widetext}
-whose Fourier transform follows from \eqref{eq:inv.func} as
+The strain stiffness is given by the second variation evaluated at the
+extremized solution $\langle\epsilon\rangle$. 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}
+  \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip\bigg|_{\epsilon=\langle\epsilon\rangle}
+    =-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x').
+  \label{eq:recip.deriv.op}
+\end{equation}
+Upon substitution into \eqref{eq:big.boy} and taking the Fourier transform of
+the result, we finally arrive at
 \begin{equation}
-  \lambda_\X(q)=C_\X\bigg(1+\frac{b^2}{4C_\X}\chi(q)\bigg)^{-1}.
+  \lambda_\X(q)
+  =C_\X-\frac b2\bigg(\frac2{b\chi(q)}+\frac b{2C_\X}\bigg)^{-1}
+  =C_\X\bigg(1+\frac{b^2}{4C_\X}\chi(q)\bigg)^{-1}.
   \label{eq:elastic.susceptibility}
 \end{equation}
-Though not relevant here, this result generalizes to multicomponent order parameters.
-At $q=0$, which is where the stiffness measurements used here were taken, this
-predicts a cusp in the elastic susceptibility of the form $|\Delta\tilde r|^\gamma$ for $\gamma=1$. 
+Though not relevant here, this result generalizes to multicomponent \op s.  At
+$q=0$, which is where the stiffness measurements used here were taken, this
+predicts a cusp in the strain stiffness of the form $|\Delta\tilde
+r|^\gamma$ for $\gamma=1$. 
 \begin{figure}[htpb]
   \centering
   \includegraphics[width=\columnwidth]{fig-stiffnesses}
   \caption{
     Measurements of the effective strain stiffness as a function of temperature
-    for the six independent components of strain from \rus. The vertical
-    lines show the location of the \ho\ transition.
+    for the six independent components of strain from \rus. The vertical lines
+    show the location of the \ho\ transition.
   }
   \label{fig:data}
 \end{figure}
 
-We have seen that mean field theory predicts that whatever component of strain
-transforms like the \op\ will see a $t^{-1}$ softening in the
-stiffness that ends in a cusp. \Rus\ experiments \cite{ghosh_single-component_2019}
-yield the strain stiffness for various components of the strain; this data is
-shown in Figure \ref{fig:data}.  The $\Btg$ and $\Eg$ stiffnesses don't appear
-to have any response to the presence of the transition, exhibiting the expected
-linear stiffening with a low-temperature cutoff \cite{varshni_temperature_1970}. The $\Bog$ stiffness has a dramatic response, softening over the
-course of roughly $100\,\K$. There is a kink in the curve right at the
-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:elastic.susceptibility}, with a linear background stiffness
-$C_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is
+\Rus\ experiments \cite{ghosh_single-component_nodate} yield the strain
+stiffness for various components of the strain; this data is shown in Figure
+\ref{fig:data}.  The $\Btg$ stiffness doesn't appear to have any response to
+the presence of the transition, exhibiting the expected linear stiffening with
+a low-temperature cutoff \cite{varshni_temperature_1970}. The $\Bog$ stiffness
+has a dramatic response, softening over the course of roughly $100\,\K$. There
+is a kink in the curve right at the 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:elastic.susceptibility}, with a linear background
+stiffness $C_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is
 shown in Figure \ref{fig:fit}. The data and theory appear consistent.
 
 \begin{figure}[htpb]
@@ -398,29 +420,75 @@ shown in Figure \ref{fig:fit}. The data and theory appear consistent.
     (dashed). The fit gives
     $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
     $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
-    q_*^4\simeq0.0038\,\K^{-1}$.
+    q_*^4\simeq0.0038\,\K^{-1}$. 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:fit}
 \end{figure}
 
-Mean field theory neglects the effect of fluctuations on critical behavior, yet
-also predicts the magnitude of those fluctuations. This allows a mean field
-theory to undergo an internal consistency check to ensure the predicted
-fluctuations are indeed negligible. This is typically done by computing the
-Ginzburg criterion \cite{ginzburg_remarks_1961}, which gives the proximity to
-the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to
-break down by comparing the magnitude of fluctuations in a correlation-length
-sized box to the magnitude of the field. In the modulated phase the spatially
-averaged magnitude is zero, and so we will instead compare fluctuations in the
+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 $\Bog$ strain
+stiffness at zero pressure. There are several implications of this theory. 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 $\langle\epsilon_\Bog\rangle^2=b^2\tilde
+r/16uC_\Bog^2$, which corresponds to an orthorhombic 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}. Second, as the Lifshitz point is
+approached from low pressure this theory predicts the modulation wavevector
+$q_*$ should continuously vanish. Far 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, or $q_*\simeq0.328\,\A^{-1}$ \cite{meng_imaging_2013}. In between
+these two regimes, the ordering wavevector should shrink by jumping between
+ever-closer commensurate values in the style of the devil's staircase
+\cite{bak_commensurate_1982}. This motivates future \rus\ experiments done at
+pressure, where the depth of the cusp in the $\Bog$ stiffness should deepen
+(perhaps with these commensurability jumps) at low pressure and approach zero
+like $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. The presence of
+spatial commensurability is not expected to modify the critical 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}, and therefore we don't
+in fact expect there to be one. We do expect that this could be due to the high
+energy nature of x-rays as an experimental probe: orthorhombic fluctuations
+could appear at higher temperatures than the true onset of an orthorhombic
+phase. 
+
+Three dimensions is below the upper critical dimension $4\frac12$, and so mean
+field theory should break down sufficiently close to the critical point due to
+fluctuations \cite{hornreich_lifshitz_1980}. Mean field theory neglects the
+effect of fluctuations on critical behavior, yet also predicts the magnitude of
+those fluctuations. This allows a mean field theory to undergo an internal
+consistency check to ensure the predicted fluctuations are indeed negligible in the region under consideration.
+This is typically done by computing the Ginzburg temperature
+\cite{ginzburg_remarks_1961}, which gives the proximity to the critical point
+$t=(T-T_c)/T_c$ at which mean field theory is expected to break down by
+comparing the magnitude of fluctuations in a correlation-length sized box to
+the magnitude of the field. In the modulated phase the spatially averaged
+magnitude is zero, and so we will instead compare fluctuations in the
 \emph{amplitude} at $q_*$ to the magnitude of that amplitude. Defining the
 field $\alpha$ by $\eta(x)=\alpha(x)e^{-iq_*x_3}$, it follows that in the
-modulated phase $\alpha(x)=\alpha_0$ for $\alpha_0^2=|\delta \tilde r|/4u$. In
-the modulated phase, the $q$-dependant fluctuations in $\alpha$ are given by
-\[
+modulated phase $\langle\alpha(x)\rangle=\alpha_0$ for $\alpha_0^2=|\delta
+\tilde r|/4u$. In the modulated phase, the $q$-dependant fluctuations in
+$\alpha$ are given by
+\begin{equation}
   G_\alpha(q)=k_BT\chi_\alpha(q)=\frac1{c_\parallel q_\parallel^2+D_\perp(4q_*^2q_\perp^2+q_\perp^4)+2|\delta r|},
-\]
+\end{equation}
 An estimate of the Ginzburg criterion is then given by the temperature at which
-$V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle\simeq\langle\alpha\rangle^2=\alpha_0^2$,
+$V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle_{V_\xi}\simeq\langle\alpha\rangle^2=\alpha_0^2$,
 where $V_\xi=\xi_\perp\xi_\parallel^2$ is a correlation volume. The parameter $u$
 can be replaced in favor of the jump in the specific heat at the transition
 using
@@ -428,37 +496,57 @@ using
   c_V=-T\frac{\partial^2f}{\partial T^2}
     =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c.\end{cases}
 \end{equation}
-The integral over the correlation function $G_\alpha$ can be preformed up to one integral analytically using a Gaussian-bounded correlation volume, yielding $\langle\Delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}\delta r^{-1}\mathcal I(\xi_\perp q_*)$ for
-\[
-  \mathcal I(x)=-2^{3/2}\pi^{5/2}\int dy\,e^{[2+(4x^2-1)y^2+y^4]/2}
-\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big)
-\]
+The integral over the correlation function $G_\alpha$ can be preformed up to
+one integral analytically using a Gaussian-bounded correlation volume, yielding
+$\langle\Delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}|\Delta \tilde r|^{-1}\mathcal
+I(\xi_\perp q_*)$ for
+\begin{equation}
+  \begin{aligned}
+    \mathcal I(x)=-2^{3/2}\pi^{5/2}\int& dy\,e^{[2+(4x^2-1)y^2+y^4]/2} \\
+      &\times\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big).
+  \end{aligned}
+\end{equation}
 This gives a transcendental equation
-\[
-  \mathcal I(\xi_{\perp0}q_*\delta t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}\delta t_\G^{3/4}.
-\]
+\begin{equation}
+  \mathcal I(\xi_{\perp0}q_*t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}t_\G^{3/4},
+\end{equation}
+with $\xi=\xi_0|t|^{-\nu}$ defining the bare correlation lengths.
 Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$
-\cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp
-q_*^4/aT_c)^{1/4}\simeq2$. We have reason to believe that at zero pressure, very
-far from the Lifshitz point, the half-wavelength of the modulation should be commensurate with the lattice, giving $q_*\simeq0.328\,\A^{-1}$ 
-\cite{meng_imaging_2013}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$,
-we find $\delta t_\G\sim0.4$, though this estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in $\delta t_\G$ over nearly four orders of magnitude.
-The estimate here predicts that an experiment may begin to see deviations from
-mean field behavior within around $5\,\K$ of the critical point. A \rus\ 
-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 scalar \op\ is $\mathrm O(2)$ \cite{garel_commensurability_1976}. Our work here appears self--consistent, given that our fit is mostly concerned with temperatures farther than this from the critical point. This analysis also indicates that 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.
-
-There are two apparent discrepancies between the phase diagram presented in 
-\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 ref.\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}, and therefore we don't in fact expect there to be one. We do expect that this could be due to the
-high energy nature of x-rays as an experimental probe: orthorhombic fluctuations
-could appear at higher temperatures than the true onset of an orthorhombic phase. 
+\cite{fisher_specific_1990}. Our fit above gives $\xi_{\perp0}q_*=(D_\perp
+q_*^4/aT_c)^{1/4}\simeq2$. Further supposing that
+$\xi_{\parallel0}\simeq\xi_{\perp0}$, we find $t_\G\sim0.4$, though this
+estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our
+estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in
+$t_\G$ over nearly four orders of magnitude.  The estimate here predicts
+that an experiment may begin to see deviations from mean field behavior within
+around several degrees Kelvin of the critical point. A \rus\ 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 scalar \op\ is $\mathrm O(2)$
+\cite{garel_commensurability_1976}. Our work here appears self--consistent,
+given that our fit is mostly concerned with temperatures ten to hundreds of
+Kelvin from the critical point. This analysis also indicates that 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.
+
+We have preformed a general treatment of phenomenological \ho\ \op s with the
+potential for linear coupling to strain. The possibilities with consistent mean
+field phase diagrams are $\Bog$ and $\Btg$, and the only of these consistent
+with zero-pressure \rus\ data is $\Bog$, with a cusp appearing in the
+associated stiffness. In this picture, the \ho\ phase is characterized by
+uniaxial modulated $\Bog$ order, while the \afm\ phase is characterized by
+uniform $\Bog$ order. The corresponding prediction of uniform $\Bog$ symmetry
+breaking in the \afm\ phase is consistent with recent diffraction experiments
+\cite{choi_pressure-induced_2018}. 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}
-
+  This research was supported by NSF DMR-1719490, [Mike's grant], [Brad's
+  grants????]. The authors would like to thank [ask Brad] for helpful
+  correspondence.
 \end{acknowledgements}
 
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