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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{177pc}\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}
\bigskip\\
\huge \textbf{Jaron~Kent-Dobias, Michael Matty \& Brad J Ramshaw}
\vspace{1pc}

\begin{multicols}{2}
  \Large
  \urusi\ is a heavy-fermi material with a novel continuous phase transition whose broken symmetry is not known, earning it the name ``hidden order.'' Under sufficient pressure it instead transitions into an antiferromagnet, and the transitions between these phases is first-order.

  Recent resonant ultrasound spectroscopy experiments measured a novel behavior of the stiffness tensor over several hundred Kelvin approaching the transition. Inspired by this, we develop the most general mean field theory consistent with that behavior and the topology of the phase diagram and find that the only consistent symmetry reproduces the experimental data. 

  \section{Resonant ultrasound spectroscopy}
  \Large

  When periodically strained by a driving sound, materials respond with
  oscillations the response can be measured. Resonances are produced by driving
  at the natural frequency of a normal mode of the material and are visible as
  a spike in the response amplitude. The stiffness tensor $C$ is uniquely
  determined by the natural frequencies of sufficiently many normal modes.
  Preforming this decomposition at each temperature in a sweep gives a lot of
  information about thermodynamic functions.

  \begin{figure}
    \centering
    \vspace{1em}
    \floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=0.4\textwidth}}]{figure}[\FBwidth]
    {
      \caption{
        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.
      }
      \label{fig:rus}
    }
    {\includegraphics[width=0.5\textwidth]{rus_resonances.jpg}}
    \vspace{1em}
  \end{figure}

  \section{Irreducible representations of strain}
  \Large

  \begin{wrapfigure}{L}{.25\textwidth}
    \centering
    \includegraphics[width=\columnwidth]{urusi_modes.png}
    \captionof{figure}{
      The crystal structure of \urusi\ and the influence of the irreducible
      strains of \Dfh\ on it.
    }
    \label{fig:strains}
  \end{wrapfigure}

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

  Dividing the strain this way simplifies expression its coupling to an order
  parameter, which Landau tells us must be described by an irreducible action
  by the symmetry group. If our order parameter shares its representation with
  a strain, they can couple linearly. Otherwise their coupling must be
  higher-order, which leads to thermodynamic discontinuities but not diverging
  responses.

  \section{Landau--Ginzburg theory}
  \Large

  We want to write down every theory of an order parameter $\eta$ that
  \begin{itemize}
    \item couples linearly to strain
    \item has continuous transitions into hidden order and \afm
    \item has a first order transition between hidden order and \afm
  \end{itemize}
  Of the irreps corresponding to strains, $\Aog$ yields first-order transitions
  and shouldn't be considered.  For $\eta$ of the 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
  $r\to\tilde r$.

  \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 system with a \emph{Lifshitz point} at $\tilde
   r=c_\perp=0$. This triple point lies at the confluence of three phases: an
   unordered phase, a phase with uniform nonzero $\eta$, and a phase with
   modulated nonzero $\eta$, as seen in Fig.~\ref{phase_diagram}. The nature of
   the boundaries between these phases depends on how many components $\eta$
   has---since the two-component theory has transitions inconsistent with
   \urusi\ we no longer consider it. This leaves $\Bog$ and $\Btg$ as
   candidates.

  \section{Stiffness response}
  \Large

  Within this theory the stiffness can be calculated like any other response
  function. As the modulated phase is approached, the strain stiffness with the
  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. 

  \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{homo}
    \vspace{1pc}
  \end{figure}
\end{multicols}

\end{document}
