pretty new figures

3 files changed, 137 insertions, 65 deletions

diff --git a/fig-stiffnesses.gplot b/fig-stiffnesses.gplot
index e132869..9fa686e 100644
--- a/fig-stiffnesses.gplot
+++ b/fig-stiffnesses.gplot
@@ -1,60 +1,85 @@
 
+# colors
 cc1 = "#5e81b5"
 cc2 = "#e19c24"
-cc3 = "#8fb032"
+cc3 = "#6fa012"
 cc4 = "#eb6235"
 
-set terminal epslatex size 8.68cm, 8.2cm standalone
+Tc = 17.26
+
+a1 = 71.1212
+b1 = 0.0104105
+c1 = 0.00378087
+d1 = 6.2662
+
+C10(T) = a1 - b1 * T
+C1(T) = C10(T) / (1 + d1 / (1 + c1 * abs(T - Tc)) / C10(T))
+
+a2 = 144.345
+b2 = 0.019492
+c2 = 120.462 
+
+C20(T) = a2 - b2 * T**2 / (c2 + T)
+
+set terminal epslatex size 8.68cm, 6.5cm standalone header \
+  '\usepackage{xcolor}\definecolor{mathc3}{HTML}{6fa012}\definecolor{mathc4}{HTML}{eb6235}'
 set output "fig-stiffnesses.tex"
 
-set multiplot  layout 3, 2  margins 0.13, 0.9, 0.1, 0.998  spacing 0.01, 0.01
+set multiplot  layout 2, 2  margins 0.1, 0.88, 0.1, 0.99  spacing 0.01, 0.01
 
 set nokey
-set xrange [0:315]
-set arrow 1 from 17.26,graph 0 to 17.26,graph 1 nohead lw 4 lc rgb cc2
 
-lam(T) = 71.1212 - 0.0104105 * T
 
-set format x ""
-set xtics 0, 100, 300 offset 0,0.5
 set mxtics 2
-set ylabel '\tiny $C / \mathrm{GPa}$' offset 4.5
-set format y '\tiny $%.1f$'
+set mx2tics 2
+set mytics 2
+set my2tics 2
+set format y '\tiny $%1.f$'
 set format y2 '\tiny $%.1f$'
+set format x '\tiny $%.0f$'
+#set format x2 '\tiny $%.0f$'
+set format x2 ''
 
+set arrow 1 from Tc,graph 0 to Tc,graph 1 nohead lw 5 lc rgb cc2
+unset xtics
+set x2tics 50, 50, 250 offset 0,-0.5 mirror
+#set x2label '\tiny $T / \mathrm K$' offset 0,-1.0
+set xrange [0:300]
+
+set ylabel '\scriptsize $C_{\mathrm{B_{2\mathrm g}}} / \mathrm{GPa}$' offset 3.5
 set yrange [140:145]
 set ytics 141,1,144 offset 0.5
-set title '\tiny (e) $C_{\mathrm{B_{2\mathrm g}}}$' offset -1,-6
 plot "data/c66.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
-     144.345 - 0.019492 * x**2 / (120.462 + x) dt 3 lw 4 lc rgb cc4
-
-set ylabel ''
+     C20(x) dt 3 lw 4 lc rgb cc4
 
+unset ylabel
+set y2label '\scriptsize $C_{\mathrm{B_{1\mathrm g}}} / \mathrm{GPa}$' offset -5 rotate by -90
 set yrange [65.05:65.7]
 set y2tics 62.1,0.1,65.6 offset -0.5 mirror
-set title '\tiny (c) $C_{\mathrm{B_{1\mathrm g}}}$' offset -1,-6
-plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3
+plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
+     C1(x) dt 3 lw 4 lc rgb cc4
 
-set format x '\tiny $%.0f$'
-set ylabel '\tiny $C / \mathrm{GPa}$'
-set xlabel '\tiny $T / \mathrm K$' offset 0,1
-unset y2tics
 
-set yrange [65:71.5]
+unset x2label
+unset x2tics
+
+set xlabel '\scriptsize $T / \mathrm K$' offset 0,1.25
+set xtics 50, 50, 250 offset 0,0.5 mirror
+unset y2tics
 set ytics 63,1,72
-set title '\tiny (c) $C_{\mathrm{B_{1\mathrm g}}}$' offset -1,-6
+unset y2label
+set ylabel '\scriptsize $\textcolor{mathc3}{C_{\mathrm{B_{1\mathrm g}}}},\textcolor{mathc4}{C_{\mathrm{B_{1\mathrm g}}}^0} / \mathrm{GPa}$' offset 3
+
+set yrange [64.5:71.5]
 plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
-     lam(x) dt 3 lw 4 lc rgb cc4
+     C10(x) dt 3 lw 4 lc rgb cc4
 
 set ylabel ''
-set y2label '\tiny $\log[(C^0 - C) / \mathrm{GPa}]$' offset -5 rotate by -90
-set xlabel '\tiny $\log(T / \mathrm K+D_\perp q_*^4/a)$' offset 0,1
-
-set yrange [1:1.85]
-set format x '\tiny $%0.1f$'
-set xrange [5.5:6.35]
-set y2tics 1,0.1,1.9 offset -0.7 mirror
-set xtics 5.6,0.2,6.4
-set title '\tiny (c) $C_{\mathrm{B_{1\mathrm g}}}$' offset -1,-6
-plot "data/c11mc12.dat" using (log($1 + 1 / 0.00378087)):(log(lam($1) - 100 * $2)) with lines lw 3 lc rgb cc3, \
-     7.3 - x
+set y2label '\scriptsize $[C^0(C^0/C - 1)]^{-1} \cdot \mathrm{GPa}$' offset -5.5 rotate by -90
+
+set format y2 '\tiny $%0.2f$'
+set format y '\tiny $%0.2f$'
+set yrange [0.12:0.345]
+set y2tics 0.15,0.05,0.39 offset -0.7 mirror
+plot "data/c11mc12.dat" using 1:(1 / (C10($1)*(C10($1) / (100 * $2) - 1))) with lines lw 3 lc rgb cc3, \
+     1/(C10(x) * (C10(x) / C1(x) - 1)) dt 3 lw 4 lc rgb cc4
diff --git a/fig-stiffnesses.pdf b/fig-stiffnesses.pdf
index 5962c50..de256aa 100644
--- a/fig-stiffnesses.pdf
+++ b/fig-stiffnesses.pdf
diff --git a/main.tex b/main.tex
index 84bb3f9..d79ef78 100644
--- a/main.tex
+++ b/main.tex
@@ -3,6 +3,41 @@
 \usepackage[utf8]{inputenc}
 \usepackage{amsmath,graphicx,upgreek,amssymb,xcolor}
 \usepackage[colorlinks=true,urlcolor=purple,citecolor=purple,filecolor=purple,linkcolor=purple]{hyperref}
+\usepackage[english]{babel}
+
+\definecolor{mathc1}{html}{5e81b5}
+\definecolor{mathc2}{html}{e19c24}
+\definecolor{mathc3}{html}{8fb032}
+\definecolor{mathc4}{html}{eb6235}
+
+\makeatletter
+% A change to a babel macro -- Don't ask!
+\def\bbl@set@language#1{%
+  \edef\languagename{%
+    \ifnum\escapechar=\expandafter`\string#1\@empty
+    \else\string#1\@empty\fi}%
+  %%%% ADDITION
+  \@ifundefined{babel@language@alias@\languagename}{}{%
+    \edef\languagename{\@nameuse{babel@language@alias@\languagename}}%
+  }%
+  %%%% END ADDITION
+  \select@language{\languagename}%
+  \expandafter\ifx\csname date\languagename\endcsname\relax\else
+    \if@filesw
+      \protected@write\@auxout{}{\string\select@language{\languagename}}%
+      \bbl@for\bbl@tempa\BabelContentsFiles{%
+        \addtocontents{\bbl@tempa}{\xstring\select@language{\languagename}}}%
+      \bbl@usehooks{write}{}%
+    \fi
+  \fi}
+% The user interface
+\newcommand{\DeclareLanguageAlias}[2]{%
+  \global\@namedef{babel@language@alias@#1}{#2}%
+}
+\makeatother
+
+\DeclareLanguageAlias{en}{english}
+
 
 \newcommand{\brad}[1]{{\color{red} #1}}
 
@@ -59,22 +94,41 @@
 \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 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.
 \end{abstract}
 
 \maketitle
 
 \emph{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. Our work here seeks to unify two experimental observations: one, the $\Bog$ ``nematic" elastic susceptibility $(c_{11}-c_{12})/2$ softens anomalously from room temperature down to T$_{\mathrm{HO}}=17.5~$ K \brad{find old citations for this data}; 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}.
+\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. Our work here seeks to unify two experimental
+observations: one, the $\Bog$ ``nematic" elastic susceptibility
+$(C_{11}-C_{12})/2$ softens anomalously from room temperature down to
+T$_{\mathrm{HO}}=17.5~$ K \brad{find old citations for this data}; 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 \brad{cite those ruled out} in a model-free way, but still leaves the microscopic nature of \ho~ undecided. 
 
@@ -357,10 +411,19 @@ $|\Delta\tilde r|^\gamma$ for $\gamma=1$.  \brad{I think this last sentence, whi
   \centering
   \includegraphics[width=\columnwidth]{fig-stiffnesses}
   \caption{
-   Resonant ultrasound spectroscopy measurements of the elastic moduli of
-   \urusi\ as a function of temperature for the six independent components of
-   strain. The vertical lines show the location of the \ho\ transition. 
-   \textbf{ONE FIGURE: Just B2g and B1g, vashni fit for one, our fit for the other, something else}
+   \Rus\ measurements of the elastic moduli of
+   \urusi\ as a function of temperature (green, solid) alongside fits to theory. The vertical yellow lines show the location of the \ho\ transition. (a) $\Btg$ modulus data and fit to standard form \cite{varshni_temperature_1970}. (b) $\Bog$ modulus data and fit
+    to \eqref{eq:elastic.susceptibility}. The fit gives
+    $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
+    $b^2/D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
+    q_*^4\simeq0.0038\,\K^{-1}$. Addition of an additional parameter to fit the standard bare modulus \cite{varshni_temperature_1970} led to sloppy fits. (c) $\Bog$ modulus data and fit of \emph{bare}
+    $\Bog$ modulus. (d) $\Bog$ modulus data and fit transformed using
+    $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is prediced from
+    \eqref{eq:susceptibility} and \eqref{eq:elastic.susceptibility} to be
+    linear. The failure of the Ginzburg--Landau prediction
+    below the transition is expected on the grounds that the \op\ is too large
+    for the free energy expansion to be valid by the time the Ginzburg
+    temperature is reached.
   }
   \label{fig:data}
 \end{figure}
@@ -385,22 +448,6 @@ Figure \ref{fig:fit}. The data and theory appear quantitatively consistent in
 the high temperature phase, suggesting that \ho\ can be described as a
 $\Bog$-nematic phase that is modulated at finite $q$ along the $c-$axis.
 
-\begin{figure}[htpb]
-  \includegraphics[width=\columnwidth]{fig-fit}
-  \caption{
-    Elastic modulus data for the $\Bog$ component of strain (solid) along with
-    a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$
-    (dashed). The fit gives
-    $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
-    $b^2/D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
-    q_*^4\simeq0.0038\,\K^{-1}$. 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}
-
 We have seen that the mean-field theory of a $\Bog$ \op\ recreates the topology
 of the \ho\ phase diagram and the temperature dependence of the $\Bog$ elastic modulus at zero pressure. This theory has several other physical implications. First,
 the association of a modulated $\Bog$ order with the \ho\ phase implies a