diff options
| -rw-r--r-- | fig-stiffnesses.gplot | 93 | ||||
| -rw-r--r-- | fig-stiffnesses.pdf | bin | 70347 -> 83184 bytes | |||
| -rw-r--r-- | main.tex | 109 |
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 Binary files differindex 5962c50..de256aa 100644 --- a/fig-stiffnesses.pdf +++ b/fig-stiffnesses.pdf @@ -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 |