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\begin{document}

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\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
  \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). \urusi\ also has a lower-temperature
  superconducting phase within \ho\ that is destroyed by the \afm.

  Resonant ultrasound spectroscopy measures anomalous softening in its stiffness
  tensor over several hundred Kelvin approaching the \ho\ transition. Starting
  with a general Ginzburg--Landau theory of \urusi, we show that only one
  order parameter symmetry is consistent with both 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

  \Rus\ drives material with sound and measures the response. Resonances
  produced by driving at the natural frequency of a normal mode are visible as
  spikes in the response amplitude. The stiffness tensor $C$ is uniquely
  determined by the frequencies of sufficiently many normal modes and the
  sample geometry. The structure of the stiffness tensor and its behavior as a
  function of temperature 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 amplitude 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\ on it.
    }
    \label{fig:rus}
    \vspace{1em}
  \end{figure}

  The strain tensor $\epsilon$ has six independent components that can be
  divided into tuples upon which symmetry transformations act with irreducible
  representations of the crystallographic point group. \urusi's point group is \Dfh\ and has
  five `irreducible' strains, depicted in Fig.~\ref{fig:rus}. Of these, four
  are one-tuples (single-component) and one is a two-tuple (multi-component). 


  \section{Landau--Ginzburg theory}
  \Large

  Symmetry transformations must act trivially on the free energy, so all its
  terms must correspond to $\Aog$ irreps. An order parameter can only linearly
  couple to a strain that shares its irrep. Any other coupling must be
  higher-order, which leads to thermodynamic discontinuities but not diverging
  responses. The anomalous softening seen in \urusi\ suggests the \ho\ order
  parameter $\eta$ couple linearly to a strain.

  Of the irreps found in strains, $\Aog$ yields a first-order transition and
  therefore can't describe \ho.  For any of those remaining, the effective free
  energy has three components: the bare elastic energy of the strain
  $f_\e=\sum_\X C^0_\X\epsilon_\X\epsilon_\X$, the bare energy of the order
  parameter $\eta$
  \begin{equation}
      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,
    \label{eq:fo}
  \end{equation}
  and the coupling between them $f_\i=-b\epsilon_\X\eta$. Tracing out the
  strain gives an effective free energy with the form of \eqref{eq:fo} but with
  $r\to\tilde r$.

  \vfill\null

  \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) mean
      field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean
      field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote
      continuous transitions, while dashed lines denote first order transitions.
    }
    \label{phase_diagram}
  \end{wrapfigure}

   This is the theory of a \emph{Lifshitz point} at $\tilde r=c_\perp=0$. This
   triple point lies where three phases meet: an unordered phase with $\eta=0$,
   a uniform ordered phase with $\eta\neq0$, and a modulated ordered phase with
   $\eta\propto\cos q_*x_3$. Phase diagrams are shown in Fig.~\ref{phase_diagram}(b--c). The type of transition between the ordered phases
   depends on how many components $\eta$ has: a one-component theory is first
   order while a two-component one is continuous. The irreps
   consistent with the first order transition between \ho\ and \afm\ in \urusi\
   are $\Bog$ and $\Btg$.

  \section{Anomalous stiffness response}
  \Large

  The stiffness $C$ can be calculated like any other response
  function. Approaching the modulated phase, the strain stiffness with the same
  symmetry $\X$ 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}
  which has the form of an inverted cusp at the critical point. Compared with
  experimental \rus\ measurements, this doesn't resemble the $\Btg$ stiffness
  in Fig.~\ref{fig:plots}(a), but fits the $\Bog$ stiffness in
  Fig.~\ref{fig:plots}(b). This suggests \ho\ is a $\Bog$ nematic modulated along the $c$-axis.

  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 $\Bog$ stiffness at the Lifshitz point.

  \begin{figure}
    \vspace{1pc}
    \centering
    \includegraphics[width=\columnwidth]{paper/fig-stiffnesses.pdf}
    \captionof{figure}{
     \Rus\ measurements of the elastic moduli of \urusi\ at ambient pressure
     as a function of temperature from \texttt{arXiv:1903.00552
     [cond-mat.str-el]} (blue, solid) alongside fits to theory (magenta,
     dashed). The solid yellow region shows the location of the \ho\ phase.
     (a) $\Btg$ modulus data and a fit to the 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 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., an absolute value function. 
    }
    \label{fig:plots}
    \vspace{1pc}
  \end{figure}
\end{multicols}

\end{document}