From 740356e2c23caa67f70e60ca4e86bc85a4e9ee3a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 17 Dec 2019 10:53:17 -0500
Subject: small edits from jim's suggestions

---
 main.tex | 43 +++++++++++++++++++++----------------------
 1 file changed, 21 insertions(+), 22 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index c02215a..8af828b 100644
--- a/main.tex
+++ b/main.tex
@@ -115,12 +115,14 @@ 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,
+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 proposes yet another, motivated by two experimental
-observations: first, the $\Bog$ ``nematic" elastic susceptibility
-$(C_{11}-C_{12})/2$ softens anomalously from room temperature down to
+with \ho. This work analyzes a phenomenological model of order parameters of
+general symmetry, linearly coupled to strain. We identify yet another mechanism
+that is best compatible with two experimental observations: first, the $\Bog$
+``nematic" elastic susceptibility $(C_{11}-C_{12})/2$ softens anomalously from
+room temperature down to
 $T_{\text{\ho}}=17.5\,\K$;\cite{de_visser_thermal_1986} and second, a $\Bog$
 nematic distortion is observed by x-ray scattering under sufficient pressure to
 destroy the \ho\ state.\cite{choi_pressure-induced_2018}
@@ -137,15 +139,14 @@ 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 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
-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. 
+(depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion,
+which might be related to the anomalous softening of the $\Bog$ elastic modulus
+$(C_{11}-C_{12})/2$ that occurs over a broad temperature range at zero
+pressure.\cite{wolf_elastic_1994, kuwahara_lattice_1997} Motivated by these
+results---which hint at a $\Bog$ strain susceptibility associated with the \ho\
+state---we construct a phenomenological mean field theory for an arbitrary \op\
+coupled to strain, and then determine the effect of its phase transitions on
+the elastic response in different symmetry channels. 
 
 We find that only one \op\ representation reproduces the anomalous $\Bog$
 elastic modulus, which softens in a Curie--Weiss-like manner from room
@@ -161,7 +162,7 @@ which should find that the $(C_{11}-C_{12})/2$ modulus diverges as the uniform
 $\Bog$ strain of the high pressure phase is approached.
 
 
-\section{Model}
+\section{Model \& Phase Diagram}
 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
@@ -292,7 +293,7 @@ to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$.
 With the strain traced out, \eqref{eq:fo} describes the theory of a Lifshitz
 point at $\tilde r=c_\perp=0$.\cite{lifshitz_theory_1942,
 lifshitz_theory_1942-1} The properties discussed in the remainder of this
-section can all be found in a standard text, e.g., Chaikin \&
+section can all be found in a standard text, e.g., in chapter 4 \S6.5  of Chaikin \&
 Lubensky.\cite{chaikin_principles_2000} For a one-component \op\ ($\Bog$ or
 $\Btg$) and positive $c_\parallel$, it is traditional to make the field ansatz
 $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$,
@@ -319,10 +320,10 @@ helical order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$.
 The uniform to modulated transition is now continuous. This does not reproduce
 the physics of \urusi, whose \ho\ phase is bounded by a line of first order transitions at high pressure,
 and so we will henceforth neglect the possibility of a multicomponent order
-parameter. The schematic phase diagrams for this model are shown in
+parameter. Schematic phase diagrams for both the one- and two-component models are shown in
 Figure~\ref{fig:phases}.
 
-\section{Results}
+\section{Susceptibility \& Elastic Moduli}
 We will now derive the effective elastic tensor $C$ that results from the
 coupling of strain to the \op. The ultimate result, found in
 \eqref{eq:elastic.susceptibility}, is that $C_\X$ differs from its bare value
@@ -388,7 +389,7 @@ on the configuration of the strain. Since $\eta_\star$ is a functional of $\epsi
 alone, only the modulus $C_\X$ will be modified from its bare value $C^0_\X$.
 
 Though the differential equation for $\eta_\star$ cannot be solved explicitly, we
-can use the inverse function theorem to make us \eqref{eq:implicit.eta} anyway.
+can use the inverse function theorem to make use of \eqref{eq:implicit.eta} anyway.
 First, denote by $\eta_\star^{-1}[\eta]$ the inverse functional of $\eta_\star$
 implied by \eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$
 corresponding to each solution of \eqref{eq:implicit.eta} it receives. This we
@@ -481,9 +482,7 @@ corresponding modulus.
    \eqref{eq:static_modulus}. The fit gives
    $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $D_\perp
    q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and
-   $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of an additional
-   parameter to fit the standard bare modulus\cite{varshni_temperature_1970}
-   led to poorly constrained fits. (c) $\Bog$ modulus data and the fit of the
+   $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic term in $C^0_\Bog$ was here not needed for the fit.\cite{varshni_temperature_1970} (c) $\Bog$ modulus data and the fit of the
    \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and the fit transformed
    by $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from
    \eqref{eq:static_modulus} to equal $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, e.g.,
-- 
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