From b3ceea417882c84e681728bc5d5517caf05367cf Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Sun, 10 Nov 2019 17:17:43 -0500
Subject: lots of small edits

---
 main.tex | 258 ++++++++++++++++++++++++++++++++-------------------------------
 1 file changed, 132 insertions(+), 126 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 1d6ba0f..589d0e3 100644
--- a/main.tex
+++ b/main.tex
@@ -77,7 +77,7 @@
 
 \begin{document}
 
-\title{Elastic properties of hidden order in \urusi\ are reproduced by staggered nematic order}
+\title{Elastic properties of hidden order in \urusi\ are reproduced by a staggered nematic}
 \author{Jaron Kent-Dobias}
 \author{Michael Matty}
 \author{Brad Ramshaw}
@@ -89,17 +89,17 @@
 \date\today
 
 \begin{abstract}
-  We develop a phenomenological mean field theory of the hidden order phase in
-  \urusi\ as a ``staggered nematic" order. Several experimental features are
-  reproduced when the order parameter is a nematic of the $\Bog$
-  representation, staggered along the c-axis: the topology of the
-  temperature--pressure phase diagram, the response of the elastic modulus
-  $(C_{11}-C_{12})/2$ above the hidden-order transition at zero pressure, and
-  orthorhombic symmetry breaking in the high-pressure antiferromagnetic phase.
-  In this scenario, hidden order is characterized by broken rotational symmetry
-  that is modulated along the $c$-axis, the primary order of the high-pressure
-  phase is an unmodulated nematic state, and the triple point joining those two
-  phases with the high-temperature paramagnetic phase is a Lifshitz point.
+  We develop a phenomenological mean field theory describing the hidden order
+  phase in \urusi\ as a nematic of the $\Bog$ representation staggered along
+  the $c$-axis. Several experimental features are reproduced by this theory:
+  the topology of the temperature--pressure phase diagram, the response of the
+  elastic modulus $(C_{11}-C_{12})/2$ above the transition at ambient pressure,
+  and orthorhombic symmetry breaking in the high-pressure antiferromagnetic
+  phase.  In this scenario, hidden order is characterized by broken rotational
+  symmetry that is modulated along the $c$-axis, the primary order of the
+  high-pressure phase is an unmodulated nematic, and the triple point
+  joining those two phases with the high-temperature paramagnetic phase is a
+  Lifshitz point.
 \end{abstract}
 
 \maketitle
@@ -113,52 +113,51 @@ superconductivity and \ho\ give way to local moment antiferromagnetism (\afm)
 \cite{hassinger_temperature-pressure_2008}.  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,
+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. Our work here seeks to unify two experimental
-observations: one, the $\Bog$ ``nematic" elastic susceptibility
+variety of possibilities. 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$_{\mathrm{HO}}=17.5~$ K \cite{de_visser_thermal_1986}; and two, a
-$\Bog$ nematic distortion is observed by x-ray scattering under sufficient
-pressure to destroy the \ho\ state \cite{choi_pressure-induced_2018}.
-
-Recent \emph{resonant ultrasound spectroscopy} (\rus) measurements examined the
-thermodynamic discontinuities in the elastic moduli at T$_{\mathrm{HO}}$
-\cite{ghosh_single-component_nodate}. The observation of discontinues only in
-compressional, or $\Aog$, elastic moduli requires that the point-group
-representation of \ho\ is one-dimensional. This rules out a large number of
-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-free way, but
-still leaves the microscopic nature of \ho~ undecided. 
-
-Recent X-ray experiments discovered rotational symmetry breaking in \urusi\
+$T_{\text{\ho}}=17.5~$ 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
+$T_{\text{\ho}}$ \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. 
+
+Recent x-ray experiments discovered rotational symmetry breaking in \urusi\
 under pressure \cite{choi_pressure-induced_2018}. Above 0.13--0.5 $\GPa$
 (depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion.
-While it is still unclear as to whether this is a true thermodynamic phase
+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---$(C_{11}-C_{12})/2$ in Voigt notation---that occurs over a broad
-temperature range at zero-pressure \cite{wolf_elastic_1994, kuwahara_lattice_1997}. Motivated by
-these results, hinting 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 the determine the effect of its phase transitions
-on the elastic response in different symmetry channels. 
+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 the determine the effect of its phase
+transitions on the elastic response in different symmetry channels. 
 
 We find that only one \op\ symmetry reproduces the anomalous $\Bog$ elastic
-modulus, which softens in a Curie-Weiss like manner from room temperature, but
-which cusps at T$_{\mathrm{HO}}$. That theory associates \ho\ with a $\Bog$
-\op\ \emph{modulated along the $c$- axis}, the \afm\ state with uniform $\Bog$
-order, and the triple point between them with a Lifshitz point. Besides the
-agreement with ultrasound data across a broad temperature range, the theory
-predicts uniform $\Bog$ strain at high pressure---the same distortion which 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 $(C_{11}-C_{12})/2$ diverges once the uniform $\Bog$
-strain sets in.
+modulus, which softens in a Curie--Weiss-like manner from room temperature but
+cusps at $T_{\text{\ho}}$. That theory associates \ho\ with a $\Bog$ \op\
+modulated along the $c$-axis, the \afm\ state with uniform $\Bog$ order, and
+the triple point between them with a Lifshitz point. Besides the agreement with
+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 \afm\ phase is approached.
 
 
 \emph{Model.}
@@ -166,9 +165,12 @@ The point group of \urusi\ is \Dfh, and any coarse-grained 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 $f_\e=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$. Linear
-combinations of the six independent components of strain form five irreducible
-components of strain as
+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
+strain as
 \begin{equation}
   \begin{aligned}
     & \epsilon_{\Aog,1}=\epsilon_{11}+\epsilon_{22} \hspace{0.15\columnwidth} &&
@@ -205,20 +207,21 @@ coupling to linear order is
 If there doesn't exist a component of strain that transforms like the
 representation $\X$ there can be no linear coupling, and the effect of the \op\
 condensing at a continuous phase transition is to produce a jump in the $\Aog$
-elastic modului if $\eta$ is single-component \cite{luthi_sound_1970,
+elastic moduli if $\eta$ is single-component \cite{luthi_sound_1970,
 ramshaw_avoided_2015, shekhter_bounding_2013}, and jumps in other elastic
-moduli if multicompenent \cite{ghosh_single-component_nodate}. Because we are
+moduli if multicomponent \cite{ghosh_single-component_nodate}. Because we are
 interested in physics that anticipates the phase transition, 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
-\emph{u}-reps (which are odd under inversion) and the $\Atg$ irrep.
+u-reps (which are odd under inversion) and the $\Atg$ irrep.
 
 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
-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 $\Bog$, $\Btg$, or $\Eg$, the
-most general quadratic free energy density 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
 \begin{equation}
   \begin{aligned}
     f_\op=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
@@ -242,12 +245,12 @@ full free energy functional of $\eta$ and $\epsilon$ is
 \end{equation}
 
 Rather than analyze this two-argument functional directly, we begin by tracing
-out the strain and studying the behavior of \op\ alone, assuming the
-strain is equilibrated. Later we will invert this procedure and trace out the
-\op when we compute the effective elastic moduli. The only strain relevant to
-the \op\ at linear coupling is $\epsilon_\X$, which can be traced out of the
-problem exactly in mean field theory. Extremizing the functional
-\eqref{eq:free_energy} with respect to $\epsilon_\X$ gives
+out the strain and studying the behavior of \op\ alone. Later we will invert
+this procedure and trace out the \op\ when we compute the effective elastic
+moduli. The only strain relevant to the \op\ at linear coupling is
+$\epsilon_\X$, which can be traced out of the problem exactly in mean field
+theory. Extremizing the functional \eqref{eq:free_energy} with respect to
+$\epsilon_\X$ gives
 \begin{equation}
   0
     =\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_\X(x)}\bigg|_{\epsilon=\epsilon_\star}
@@ -256,9 +259,9 @@ problem exactly in mean field theory. Extremizing the functional
 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 free energy, the resulting single-argument free energy
-functional $F[\eta,\epsilon_\star[\eta]]$ has a density identical to $f_\op$
-with $r\to\tilde r=r-b^2/2C^0_\X$. 
+substitution into the \eqref{eq:free_energy}, the resulting single-argument
+free energy functional $F[\eta,\epsilon_\star[\eta]]$ has a density identical
+to $f_\op$ with $r\to\tilde r=r-b^2/2C^0_\X$. 
 
 \begin{figure}[htpb]
   \includegraphics[width=\columnwidth]{phase_diagram_experiments}
@@ -274,7 +277,7 @@ with $r\to\tilde r=r-b^2/2C^0_\X$.
     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 zero (atmospheric) pressure line, we will take $\Delta\tilde r=\tilde
+    data along the ambient pressure line, we will take $\Delta\tilde r=\tilde
     r-\tilde r_c=a(T-T_c)$.
   }
   \label{fig:phases}
@@ -301,7 +304,9 @@ $q_*^2=-c_\perp/2D_\perp$ and
 with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The
 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
+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 $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$.
@@ -312,32 +317,30 @@ parameter. The schematic phase diagrams for this model are shown in
 Figure~\ref{fig:phases}.
 
 \emph{Results.}
-We will now derive the \emph{effective elastic tensor} $C$ that results from
-the coupling of strain to the \op. The ultimate result, found in
+We will now derive the effective elastic tensor $C$ that results from 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 symmetry $\X$ of the \op. Moreover, the effective elastic
-moduli does not vanish at the unordered to modulated transition---as it would
-if the transition were a $q=0$ structural phase transition---but instead
-exhibits a \emph{cusp}. To show this, we will first compute the susceptibility
-of the \op, which will both be demonstrative of how the modulus is calculated
-and prove useful in expressing the functional form of the modulus. Then we will
-compute the elastic modulus using techniques from functional calculus.
-
-The generalized susceptibility of a single component ($\Bog$ or $\Btg$) \op\ is
+$C^0_\X$ only for the symmetry $\X$ of the \op. Moreover, this modulus does not
+vanish at the unordered to modulated transition---as it would if the transition
+were a $q=0$ structural phase transition---but 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}
   \begin{aligned}
     &\chi^\recip(x,x')
       =\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'),
+        &\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
+where $\recip$ indicates a functional reciprocal defined as
 \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
+Taking the Fourier transform and integrating out $q'$ gives
 \begin{equation}
   \chi(q)
     =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4
@@ -347,8 +350,8 @@ Near the unordered to modulated transition this yields
 \begin{equation}
   \begin{aligned}
     \chi(q)
-      &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
-        +|\Delta\tilde r|} \\
+      &=\big[c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
+    +|\Delta\tilde r|\big]^{-1} \\
       &=\frac1{D_\perp}\frac{\xi_\perp^4}
         {1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2},
   \end{aligned}
@@ -359,26 +362,27 @@ $\xi_\parallel=(|\Delta\tilde
 r|/c_\parallel)^{-1/2}=\xi_{\parallel0}|t|^{-1/2}$, where $t=(T-T_c)/T_c$ is
 the reduced temperature and $\xi_{\perp0}=(D_\perp/aT_c)^{1/4}$ and
 $\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths
-parallel and perpendicular to the axis of symmetry, respectively.  Notice that
-the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde r|)^{-1}$ does
-not diverge at the unordered to modulated transition. Though it anticipates a
-transition with Curie--Weiss-like divergence at $a(T-T_c)=\Delta\tilde
-r=-D_\perp q_*^4$, this is cut off with a cusp at $\Delta\tilde r=0$. 
-
-The elastic susceptibility, which corresponds with the reciprocal of the elastic modulus,
-is given in a similar way to the \op\ susceptibility: we must trace over $\eta$
-and take the second variation of the resulting effective free energy functional
-of $\epsilon$. Extremizing over $\eta$ yields
+perpendicular and parallel to the plane, respectively.  The static
+susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde r|)^{-1}$ does not diverge
+at the unordered to modulated transition. Though it anticipates a transition
+with Curie--Weiss-like divergence at the lower point $a(T-T_c)=\Delta\tilde
+r=-D_\perp q_*^4<0$, this is cut off with a cusp at $\Delta\tilde r=0$. 
+
+The elastic susceptibility, which is the reciprocal of the effective elastic
+modulus, is found in a similar way to the \op\ susceptibility: we must trace
+over $\eta$ and take the second variation of the resulting effective free
+energy functional of $\epsilon$ alone. Extremizing over $\eta$ yields
 \begin{equation}
-  0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}=
-    \frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}-b\epsilon_\X(x),
+  0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}
+    =\frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}-b\epsilon_\X(x),
   \label{eq:implicit.eta}
 \end{equation}
-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 modulus $C_\X$ can be modified from its bare value $C^0_\X$.
-Though this differential equation for $\eta_*$ cannot be solved explicitly, we
-can make use of the inverse function theorem.  First, denote by
+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
@@ -390,8 +394,8 @@ 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^*[\epsilon]} \\
+    &=b^{-1}\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}.
   \end{aligned}
   \label{eq:inv.func}
 \end{equation}
@@ -404,12 +408,14 @@ the second variation
     \frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} 
     &=C^0_\X\delta(x-x')-
     2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
-    -b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') +\int dx''\,\frac{\delta^2\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]}\\
-    &\qquad\qquad+\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]} \\ 
+    -b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') \\
+    &+\int dx''\,\frac{\delta^2\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]}
+    +\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]} \\ 
     &=C^0_\X\delta(x-x')-
     2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
-    -b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') +\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}(b\epsilon_\X(x''))\\
-    &\qquad\qquad+b\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\\ 
+    -b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') \\
+    &+\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}(b\epsilon_\X(x''))
+    +b\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^0_\X\delta(x-x')-
     2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
     +b\int dx''\,\delta(x-x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x')} 
@@ -452,7 +458,7 @@ 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 finite $q$.
+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]
@@ -483,13 +489,13 @@ The remaining clue at $q=0$ is a particular kink in the corresponding modulus.
 
 \emph{Comparison to experiment.}
 \Rus\ experiments \cite{ghosh_single-component_nodate} yield the individual
-elastic moduli broken into irrep symmetry; 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, on the other
-hand, has a dramatic response, softening over the course of roughly $100\,\K$,
+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
@@ -540,9 +546,9 @@ 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 the critical
+The presence of spatial commensurability known to be irrelevant to critical
 behavior at a one-component disordered to modulated transition, and therefore
-is not expected to modify the critical behavior otherwise
+is not expected to modify the thermodynamic behavior otherwise
 \cite{garel_commensurability_1976}. 
 
 There are two apparent discrepancies between the orthorhombic strain in the
@@ -553,12 +559,12 @@ phase in the \ho\ state prior to the onset of \afm.  As
 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 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. \brad{I think
-this paragraph should probably be tigtened up a bit, we need to be more
-specific about "don't expect there to be one" and "fluctuations"}. 
+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
-- 
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