From 948efd8921bf228724e74e32b6ab074748698c07 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 5 Dec 2019 17:22:36 -0500
Subject: changed to prb style, reorganized citation locations, made some
 corrections for Brad

---
 main.tex | 163 ++++++++++++++++++++++++++++++++++-----------------------------
 1 file changed, 87 insertions(+), 76 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 9022bbc..1bdc030 100644
--- a/main.tex
+++ b/main.tex
@@ -1,5 +1,5 @@
 
-\documentclass[aps,prl,reprint,longbibliography,floatfix]{revtex4-1}
+\documentclass[aps,prb,reprint,longbibliography,floatfix]{revtex4-1}
 \usepackage[utf8]{inputenc}
 \usepackage{amsmath,graphicx,upgreek,amssymb,xcolor}
 \usepackage[colorlinks=true,urlcolor=purple,citecolor=purple,filecolor=purple,linkcolor=purple]{hyperref}
@@ -104,61 +104,71 @@
 
 \maketitle
 
-\emph{Introduction.}
+\section{Introduction}
 \urusi\ is a paradigmatic example of a material with an ordered state whose
 broken symmetry remains unknown. This state, known as \emph{hidden order}
 (\ho), sets the stage for unconventional superconductivity that emerges at even
 lower temperatures.  At sufficiently large hydrostatic pressures, both
-superconductivity and \ho\ give way to local moment antiferromagnetism (\afm)
-\cite{hassinger_temperature-pressure_2008}.  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. 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}.
+superconductivity and \ho\ give way to local moment antiferromagnetism
+(\afm).\cite{hassinger_temperature-pressure_2008}  Modern
+theories~\cite{kambe_odd-parity_2018, haule_arrested_2009,
+  kusunose_hidden_2011, kung_chirality_2015, cricchio_itinerant_2009,
+  ohkawa_quadrupole_1999, santini_crystal_1994, kiss_group_2005,
+  harima_why_2010, thalmeier_signatures_2011, tonegawa_cyclotron_2012,
+  rau_hidden_2012, riggs_evidence_2015, hoshino_resolution_2013,
+  ikeda_theory_1998, chandra_hastatic_2013, harrison_hidden_nodate,
+ikeda_emergent_2012} propose associating any of a variety of broken symmetries with \ho.
+This work 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}
 
 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
+$T_{\text{\ho}}$.\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,
+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 $\GPa$
+under pressure.\cite{choi_pressure-induced_2018} Above 0.13--0.5 $\GPa$
 (depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion.
 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
 transitions on the elastic response in different symmetry channels. 
 
-We find that only one \op\ symmetry \brad{we should be careful with our language. Can we talk about the "symmetry" of a state? Or is it the symmetries broken by the state? I don't think we can say \op\ symmetry, maybe the \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 $\Bog$ order, and the triple point between them with a Lifshitz point. In addition to the agreement with the ultrasound data across a broad temperature range, the theory predicts uniform $\Bog$ strain at high pressure---the same distortion that was recently seen in x-ray scattering experiments \cite{choi_pressure-induced_2018}. This theory strongly motivates future ultrasound experiments under pressure approaching the Lifshitz point, which should find that the $(C_{11}-C_{12})/2$ modulus diverges as the uniform $\Bog$ strain of the \afm\ phase is approached. \brad{Do we want to make the bold statement here that maybe the AFM is just a parasitically-induced by product of the high-field phase?}
-
-
-\emph{Model.}
-The point group of \urusi\ is \Dfh, and any coarse-grained \brad{like mustard? Is there a reason that we say coarse grained, though? I think microscopic hamiltonians have to obey the symmetry of the lattice as well, no?} theory must locally
+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
+$\Bog$ order, and the triple point between them with a Lifshitz point. In
+addition to the agreement with the ultrasound data across a broad temperature
+range, the theory predicts uniform $\Bog$ strain at high pressure---the same
+distortion that was recently seen in x-ray scattering
+experiments.\cite{choi_pressure-induced_2018} This theory strongly motivates
+future ultrasound experiments under pressure approaching the Lifshitz point,
+which should find that the $(C_{11}-C_{12})/2$ modulus diverges as the uniform
+$\Bog$ strain of the \afm\ 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
 interaction between strain and \op. The most general quadratic free energy of
-the strain $\epsilon$ is $f_\e=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$
+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
+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 in \Dfh\ as
 \begin{equation}
@@ -194,20 +204,25 @@ coupling to linear order is
 \begin{equation}
   f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta.
 \end{equation}
-If there exists no component of strain that transforms like the
-representation $\X$ then there can be no linear coupling. The next-order coupling is linear in strain, quadratic in order parameter, and the effect of this coupling at a continuous phase transition is to produce a jump in the $\Aog$
-elastic moduli if $\eta$ is single-component \cite{luthi_sound_1970,
-ramshaw_avoided_2015, shekhter_bounding_2013}, and jumps in other elastic
-moduli if multicomponent \cite{ghosh_single-component_nodate}. Because we are
-interested in physics that anticipates the phase transition \brad{"anticipates the phase transition" I think is too vague on its own - i added something after it as an example}---i.e. that the growing order parameter susceptibility is reflected directly in the elastic susceptibility--- we will focus our
-attention on \op s that can produce linear couplings to strain.  Looking at the
-components present in \eqref{eq:strain-components}, this rules out all of the
-u-reps (which are odd under inversion), the $\Atg$ irrep, and all half-integer (spinor) representations \brad{added spinors because these actually form some of the theories}.
+If there exists no component of strain that transforms like the representation
+$\X$ then there can be no linear coupling. The next-order coupling is linear in
+strain, quadratic in order parameter, and the effect of this coupling at a
+continuous phase transition is to produce a jump in the $\Aog$ elastic moduli
+if $\eta$ is single-component, \cite{luthi_sound_1970, ramshaw_avoided_2015,
+shekhter_bounding_2013} and jumps in other elastic moduli if
+multicomponent.\cite{ghosh_single-component_nodate} Because we are interested
+in physics that anticipates the phase transition---for instance, that the
+growing \op\ susceptibility is reflected directly in the elastic
+susceptibility---we will focus our attention on \op s that can produce linear
+couplings to strain.  Looking at the components present in
+\eqref{eq:strain-components}, this rules out all of the u-reps (which are odd
+under inversion), the $\Atg$ irrep, and all half-integer (spinor)
+representations.
 
 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 $\Aog$
+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
@@ -259,9 +274,9 @@ to $f_\op$ with $r\to\tilde r=r-b^2/2C^0_\X$.
 
   \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em}
   \includegraphics[width=0.51\columnwidth]{phases_vector}
-  \caption{\brad{where are we submitting this? we should prooooobbably get permission from the journal to reproduce this data. It's usually easy, i think?}
+  \caption{
     Phase diagrams for (a) \urusi\ from experiments (neglecting the
-    superconducting phase) \cite{hassinger_temperature-pressure_2008} (b) mean
+    superconducting phase)~\cite{hassinger_temperature-pressure_2008} (b) mean
     field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean
     field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote
     continuous transitions, while dashed lines denote first order transitions.
@@ -273,16 +288,16 @@ to $f_\op$ with $r\to\tilde r=r-b^2/2C^0_\X$.
 \end{figure}
 
 With the strain traced out, \eqref{eq:fo} describes the theory of a Lifshitz
-point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942,
-lifshitz_theory_1942-1}. The properties discused below can all be found in a
-standard text, e.g.,~\cite{chaikin_principles_2000}. For a one-component \op\
+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
 $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. 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 with the \op\ symmetry, e.g., $\Bog$ or $\Btg$. For $c_\perp<0$
+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}
@@ -300,12 +315,12 @@ between the two components of the \op. In this case the uniform ordered phase
 is only stable for $c_\perp>0$, and the modulated phase is now characterized by
 helical order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$.
 The uniform to modulated transition is now continuous. This does not reproduce
-the physics of \ho, which has a first order transition between \ho\ and \afm \brad{This is where we need to be clearer: we say that the transition to AFM is first order, but our theory doesn't transition to AFM. So maybe we need to introduce the idea earlier of a "high-pressure" phase, of which AFM is one characteristic (and maybe that's an induced or parasitic aspect), but its defining characteristic is broken rotational symmetry},
+the physics of \urusi, whose \ho\ phase is bounded by a line of first order transitions at high pressure,
 and so we will henceforth neglect the possibility of a multicomponent order
 parameter. The schematic phase diagrams for this model are shown in
 Figure~\ref{fig:phases}.
 
-\emph{Results.}
+\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
 \eqref{eq:elastic.susceptibility}, is that $C_\X$ differs from its bare value
@@ -394,18 +409,18 @@ the second variation
 \begin{widetext}
 \begin{equation}
   \begin{aligned}
-    \frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} 
-    &=C^0_\X\delta(x-x')-
+    &\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\qquad+\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')-
+    &\qquad=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\qquad\qquad+\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')-
+    &\qquad=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')} 
     =C^0_\X\delta(x-x')-b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}.
@@ -455,15 +470,15 @@ The remaining clue at $q=0$ is a particular kink in the corresponding modulus.
   \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)
+   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
+   form.\cite{varshni_temperature_1970} (b) $\Bog$ modulus data and 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}
+   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
@@ -476,14 +491,14 @@ The remaining clue at $q=0$ is a particular kink in the corresponding modulus.
   \label{fig:data}
 \end{figure}
 
-\emph{Comparison to experiment.}
-\Rus\ experiments \cite{ghosh_single-component_nodate} yield the individual
+\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 $\Bog$
+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
@@ -508,7 +523,7 @@ 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}. 
+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
@@ -516,11 +531,10 @@ from the Lifshitz point we expect the wavevector to lock into values
 commensurate with the space group of the lattice, and moreover that at zero
 pressure, where the \rus\ data here was collected, the half-wavelength of the
 modulation should be commensurate with the lattice spacing $a_3\simeq9.68\,\A$,
-or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$ \cite{meng_imaging_2013,
-broholm_magnetic_1991, wiebe_gapped_2007, bourdarot_precise_2010}. In between
+or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$.\cite{meng_imaging_2013,
+broholm_magnetic_1991, wiebe_gapped_2007, bourdarot_precise_2010} 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}. 
+ever-closer commensurate values in the style of the devil's staircase.\cite{bak_commensurate_1982} 
 
 This motivates future ultrasound experiments done under
 pressure, where the depth of the cusp in the $\Bog$ modulus should deepen
@@ -537,18 +551,16 @@ 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
 behavior at a one-component disordered to modulated transition, and therefore
-is not expected to modify the thermodynamic behavior otherwise
-\cite{garel_commensurability_1976}. 
+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
+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
+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
@@ -558,8 +570,8 @@ 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
 should break down sufficiently close to the critical point due to fluctuations,
-at the Ginzburg temperature \cite{hornreich_lifshitz_1980,
-ginzburg_remarks_1961}. Magnetic phase transitions tend to have Ginzburg
+at the Ginzburg temperature. \cite{hornreich_lifshitz_1980,
+ginzburg_remarks_1961} Magnetic phase transitions tend to have Ginzburg
 temperature of order one.  Our fit above gives $\xi_{\perp0}q_*=(D_\perp
 q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of
 $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ at about
@@ -569,14 +581,13 @@ 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
+$\gamma\simeq1.31$,\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.
 
-\emph{Conclusion and Outlook.} We have developed a general phenomenological
+\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
@@ -588,8 +599,8 @@ 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
-\cite{choi_pressure-induced_2018}, except for the apparent earlier onset in
+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
-- 
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