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\begin{document}

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\font\cpp=cmr24
\def\max{\mathrm{max}}

% Our mysterious boy
\def\urusi{URu$_{\text2}$Si$_{\text2}$}

\def\e{{\text{\textsc{elastic}}}} % "elastic"
\def\i{{\text{\textsc{int}}}} % "interaction"

\def\Dfh{D$_{\text{4h}}$}

% Irreducible representations (use in math mode)
\def\Aog{{\text A_{\text{1g}}}}
\def\Atg{{\text A_{\text{2g}}}}
\def\Bog{{\text B_{\text{1g}}}}
\def\Btg{{\text B_{\text{2g}}}}
\def\Eg {{\text E_{\text  g}}}
\def\Aou{{\text A_{\text{1u}}}}
\def\Atu{{\text A_{\text{2u}}}}
\def\Bou{{\text B_{\text{1u}}}}
\def\Btu{{\text B_{\text{2u}}}}
\def\Eu {{\text E_{\text  u}}}

% Variables to represent some representation
\def\X{\text X}
\def\Y{\text Y}

% Units
\def\J{\text J}
\def\m{\text m}
\def\K{\text K}
\def\GPa{\text{GPa}}
\def\A{\text{\r A}}

% Other
\def\op{\textsc{op}} % order parameter
\def\ho{\textsc{ho}} % hidden order
\def\rus{\textsc{rus}} % resonant ultrasound spectroscopy 
\def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy 
\def\afm{\textsc{afm}} % antiferromagnetism 
\def\recip{{\{-1\}}} % functional reciprocal

\noindent\hspace{176pc}\includegraphics[width=18pc]{CULogo-red120px.eps}
\vspace{-24.5pc}\\
\Huge \textbf{Elastic properties of hidden order in URu$_{\text2}$Si$_{\text2}$ are reproduced by\\ staggered nematic order}
\hspace{37.5pc}\textbf{\LARGE arXiv:1910.01669 [cond-mat.str-el]}
\bigskip\\
\huge \textbf{Jaron~Kent-Dobias, Michael Matty \& B J Ramshaw}
\hspace{7pc}\ding{81}\hspace{7pc}\textbf{Cornell University}

\begin{multicols}{2}
  \Large
  \hspace{1em}
  \textbf{ \urusi\ is a heavy fermion material with a continuous
    transition into a `hidden order' (\ho) phase whose broken symmetry is
    unknown.
  } Under sufficient pressure it instead transitions into an antiferromagnet
  (\afm), with phase diagram in Fig.~\ref{phase_diagram}(a). It also has a
  superconducting phase within \ho\ that is destroyed by \afm.

  \hspace{1em}
  The modulus tensor of \urusi\ anomalously softens over several hundred
  Kelvin and is cut off by the \ho\ transition.
  \textbf{
    We show that only one order parameter symmetry is consistent with this
    softening and the topology of the phase diagram.
  }
  This choice reproduces other \urusi\ phenomena and motivates new experiments.

  \section{Resonant ultrasound spectroscopy \&\\ irreducible strains}
  \Large

  \hspace{1em}
  Resonant ultrasound spectroscopy drives a sample with sound and measures its
  resonances by looking for spikes in the response. Using the sample geometry
  and the location of sufficiently many resonances, the modulus tensor $C$
  can be calculated.
  \textbf{
    The structure of the modulus tensor and its temperature dependence yield
    a lot of information about symmetry breaking.
   }

  \begin{figure}
    \centering
    \vspace{1em}
    \includegraphics[width=0.5\textwidth]{rus_resonances.jpg}
    \hfill\raisebox{2em}{\includegraphics[width=0.48\textwidth]{urusi_modes.png}}
    \captionof{figure}{
      \textbf{Left:}
      Response versus driving frequency for a sample at some
      temperature. The spikes correspond to resonances, and the strains
      corresponding to a few dominant modes are depicted in the cartoons.
      \textbf{Right:}
      The crystal structure of \urusi\ and the influence of the irreducible
      strains of \Dfh.
    }
    \label{fig:rus}
    \vspace{1em}
  \end{figure}

  \hspace{1em}
  The strain tensor $\epsilon$ has six independent components. These components can be
  divided into tuples that symmetry transformations of the crystal act on with irreducible
  representations (irreps) of its point group.
  \textbf{
    \urusi's point group \Dfh\ yields five `irreducible' strains, shown in
    Fig.~\ref{fig:rus}.
  }
  Four are one-tuples (single-component). These are uniform compression in and
  out of the plane ($\Aog$) and in-plane shear along the faces and diagonals of
  the unit cell ($\Bog$ and $\Btg$). One is a two-tuple (multi-component), and
  corresponds to out of plane shears ($\Eg$).



  \section{Landau--Ginzburg theory}
  \Large

  \hspace{1em}
  The free energy must be invariant under symmetry transformations. This
  constrains the way fields can couple together. Two fields can linearly couple
  only if they correspond to the same irrep. Higher order couplings lead to
  thermodynamic discontinuities but not diverging responses.
  \textbf{
    The anomalous softening of \urusi\ suggests the hidden order parameter
    couples linearly to strain.
  }

  \hspace{1em}
  An $\Aog$ order parameter results in first-order transitions and can't
  describe hidden order. If the order parameter $\eta$ corresponds to any of
  the remaining irreps $\X$ present in strain, the free energy is
  $f=f_\e+f_\i+f_\op$, where
  \begin{equation}
    \begin{aligned}
      & \hspace{2em}f_\e=\sum_\X C^0_\X\epsilon_\X\epsilon_\X
        \hspace{4em} f_\i=-b\epsilon_\X\eta \\ 
      & f_\op=\frac12\big[r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
        +c_\perp(\nabla_\perp\eta)^2
        +D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4
    \end{aligned}
    \label{eq:fo}
  \end{equation}
  The strain $\epsilon$ can be traced out exactly, which results in a free
  energy density with the form of $f_\op$ but with $r\to\tilde
  r=r-b^2/2C^0_\X$.

  \vfill

  \begin{wrapfigure}{R}{0.6\columnwidth}
    \centering
    \includegraphics[width=\columnwidth]{paper/phase_diagram_experiments}

    \vspace{1em}

    \includegraphics[width=0.51\columnwidth]{paper/phases_scalar}\hspace{-0.75em}
    \includegraphics[width=0.51\columnwidth]{paper/phases_vector}
    \captionof{figure}{
      Phase diagrams for (a) \urusi\ from Phys Rev B \textbf{77} 115117 (2008)
      (b) theory for a one-component order ($\Bog$ or $\Btg$) (c) theory for a
      two-component order ($\Eg$). Solid lines denote continuous transitions,
      while dashed denote first order.
    }
    \label{phase_diagram}
  \end{wrapfigure}

  \hspace{1em}
  \textbf{
    This theory has a \emph{Lifshitz triple point} at $\tilde r=c_\perp=0$.
  }
  The three phases that meet are
  \begin{itemize}
    \item\hspace{0.25em}unordered ($\eta=0$)
    \item\hspace{0.25em}uniform ($\eta\neq0$)
    \item\hspace{0.25em}modulated ($\eta\propto\cos q_*x_3$)
  \end{itemize}
  Phase diagrams are in Fig.~\ref{phase_diagram}(b--c). The order of the
  transition between the uniform and modulated phases depends on the number of
  order parameter components: one-component theories have a first order
  transition while two-component theories are continuous.
  \textbf{
    The only order parameter irreps consistent with the first order transition
    between \ho\ and \afm\ in \urusi\ are $\Bog$ and $\Btg$.
  }

  \section{Anomalous modulus response}
  \Large

  \hspace{1em}
  The modulus $C$ can be calculated like any other response function.
  Approaching the modulated phase, the modulus with the symmetry irrep
  $\X$ of the order parameter is
  \begin{equation}
    C_\X=C_\X^0\bigg[1+\frac{b^2}{C_\X^0}\big(D_\perp q_*^4+|\Delta\tilde r|\big)^{-1}\bigg]^{-1}
    \label{eq:static_modulus}
  \end{equation}
  \hspace{1em}
  \textbf{
    At the critical point the modulus with the symmetry of the order parameter
    has an inverted cusp.
  }
  Compared with experiments, this doesn't resemble the $\Btg$ modulus in
  Fig.~\ref{fig:plots}(a) but fits the $\Bog$ modulus in
  Fig.~\ref{fig:plots}(b). \textbf{This suggests hidden order is a $\Bog$
  nematic modulated along the \textit{c}-axis.}

  \hspace{1em}
  This theory explains the presence of the 0.4--0.5 inverse lattice constant
  scattering peak seen in some experiments and is consistent with $\Bog$
  symmetry breaking observed in the \afm\ phase. \Rus\ experiments at pressure
  might resolve the divergence of the modulation wavevector $q_*$ and the
  vanishing of the $\Bog$ modulus at the Lifshitz point.

  \begin{figure}
    \vspace{1em}
    \centering
    \includegraphics[width=\columnwidth]{paper/fig-stiffnesses.pdf}
    \captionof{figure}{
     \Rus\ measurements of \urusi's moduli as a function of temperature from
     \texttt{arXiv:1903.00552 [cond-mat.str-el]} (blue, solid) alongside fits
     to this theory (magenta, dashed). The solid yellow region shows the \ho\
     phase.  (a) $\Btg$ modulus data and a fit to standard form. (b) $\Bog$
     modulus data and a fit to \eqref{eq:static_modulus}. (c) $\Bog$ modulus
     data and the fit of the \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus
     data and fit transformed in way predicted from \eqref{eq:static_modulus}
     to be an absolute value function. 
    }
    \label{fig:plots}
    \vspace{1pc}
  \end{figure}
\end{multicols}

\end{document}
