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@@ -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 |