\documentclass[aps,prl,reprint]{revtex4-2}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,graphicx}

% Our mysterious boy
\def\urusi{URu$_2$Si$_2\ $}

\def\e{{\mathrm e}} % "elastic"
\def\o{{\mathrm o}} % "order parameter"
\def\i{{\mathrm i}} % "interaction"

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

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

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

\begin{document}

\title{\urusi mft}
\author{Jaron Kent-Dobias}
\author{Mike Matty}
\author{Brad Ramshaw}
\affiliation{Laboratory of Atomic \& Solid State Physics, Cornell University, Ithaca, NY, USA}

\date\today

\begin{abstract}
  blah blah blah its-a abstract
\end{abstract}

\maketitle

\begin{enumerate}
  \item Introduction
    \begin{enumerate}
      \item \urusi hidden order intro paragraph, discuss the phase diagram
      \item Strain/OP coupling discussion/RUS
      \item Discussion of experimental data
      \item Analogy of lack of divergence/AFM w/ FM $\chi$
      \item We look at MFT's for OP's of various symmetries
    \end{enumerate}
      
  \item Theory
    \begin{enumerate}
      \item Introduce various pieces of free energy
      
      \item Summary of MFT results
    \end{enumerate}
    
    \item Data piece
    
    \item Talk about more cool stuff like AFM C4 breaking etc
\end{enumerate}

The point group of \urusi is \Dfh, and any coarse-grained theory must locally respect this symmetry. We will introduce a phenomenological free energy density in three parts: that of the strain, the order parameter, and their interaction. The most general quadratic free energy of the strain $\epsilon$ is $f_\e=\lambda_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the $\lambda$ tensor is constrained by both that $\epsilon$ is a symmetric tensor and by the point group symmetry \cite{landau_theory_1995}. The latter can be seen in a systematic way. First, the six independent components of strain are written as linear combinations that behave like irreducible representations under the action of the point group, or
\begin{equation}
  \begin{aligned}
    \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth}
    \epsilon_\Aog^{(2)}=\epsilon_{33}               \\
    \epsilon_\Bog^{(1)}=\epsilon_{11}-\epsilon_{22} &&
    \epsilon_\Btg^{(1)}=\epsilon_{12}               \\
    \epsilon_\Eg^{(1)} =\{\epsilon_{11},\epsilon_{22}\}.
  \end{aligned}
\end{equation}
Next, all quadratic combinations of these irreducible strains that transform like $\Aog$ are included in the free energy as
\begin{equation}
  f_\e=\frac12\sum_\X\lambda_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},
\end{equation}
where the sum is over irreducible representations of the point group and the $\lambda_\X^{(ij)}$ are
\begin{equation}
  \begin{aligned}
    &\lambda_{\Aog}^{(11)}=\tfrac12(\lambda_{1111}+\lambda_{1122}) &&
    \lambda_{\Aog}^{(22)}=\lambda_{3333} \\
    &\lambda_{\Aog}^{(12)}=\lambda_{1133} &&
    \lambda_{\Bog}^{(11)}=\tfrac12(\lambda_{1111}-\lambda_{1122}) \\
    &\lambda_{\Btg}^{(11)}=4\lambda_{1212} &&
    \lambda_{\Eg}^{(11)}=4\lambda_{1313}.
  \end{aligned}
\end{equation}
The interaction between strain and the order parameter $\eta$ depends on the representation of the point group that $\eta$ transforms as. If this representation is $\X$, then the most general coupling to linear order is
\begin{equation}
  f_\i=b^{(i)}\epsilon_\X^{(i)}\eta
\end{equation}
If $\X$ is a representation not present in the strain there can be no linear coupling, and the effect of $\eta$ going through a continuous phase transition is to produce a jump in the $\Aog$ strain stiffness. We will therefore focus our attention on order parameter symmetries that produce linear couplings to strain.

If the order parameter transforms like $\Aog$, odd terms are allow in its free energy and any transition will be abrupt and not continuous without tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic free energy density is
\begin{equation}
  \begin{aligned}
    f_\o=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
      +c_\perp(\nabla_\perp\eta)^2 \\
      &\quad+D_\parallel(\nabla_\parallel^2\eta)^2
      +D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4
  \end{aligned}
  \label{eq:fo}
\end{equation}
where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and $\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$ since this does not affect the physics at hand. Neglecting interaction terms higher than quadratic order, the only strain relevant to the problem is $\epsilon_\X$, and this can be traced out of the problem exactly, since
\begin{equation}
  0=\frac{\delta F}{\delta\epsilon_{\X i}(x)}=\lambda_\X\epsilon_{\X i}(x)+\frac12b\eta_i(x)
\end{equation}
gives $\epsilon_\X(x)=-(b/2\lambda_\X)\eta(x)$. Upon substitution into the free energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$, with $r\to\tilde r=r-b^2/4\lambda_\X$.

With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942, lifshitz_theory_1942-1, hornreich_lifshitz_1980}. For a scalar order parameter ($\Bog$ or $\Btg$) it is traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$, or $\tilde r<c_\perp^2/4D_\perp$ and $c_\perp<0$, the only stable solution is $\eta_*=q_*=0$ and the system is unordered. For $\tilde r<0$ there are free energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has uniform order. For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free energy minima for $q_*^2=-c_\perp/2D_\perp$ and
\begin{equation}
  \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u}=\frac{\tilde r_c-\tilde r}{3u}
\end{equation}
with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The transition between the uniform and modulated orderings is abrupt for a scalar field and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. For a vector order parameter ($\Eg$) we must also allow a relative phase between the two components of the field. In this case the uniform ordered phase is only stable for $c_\perp>0$, and the modulated phase is now characterized by helical order with $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$ and
\begin{equation}
  \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{16D_\perp u}=\frac{\tilde r_c-\tilde r}{4u}
\end{equation}
The uniform--modulated transition is now continuous. The schematic phase diagrams for this model are shown in Figure \ref{fig:phases}.

\begin{figure}
  \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em}
  \includegraphics[width=0.51\columnwidth]{phases_vector}
  \caption{Schematic phase diagrams for this model. Solid lines denote
    continuous transitions, while dashed lines indicated abrupt transitions. (a)
    The phases for a scalar ($\Bog$ or $\Btg$). (b) The phases for a vector
    ($\Eg$).}
  \label{fig:phases}
\end{figure}

The susceptibility is given by
\begin{equation}
  \begin{aligned}
    &\chi_{ij}^{-1}(x,x')
    =\frac{\delta^2F}{\delta\eta_i(x)\delta\eta_j(x')} \\
    &\quad=\Big[\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+4u\eta^2(x)\big)\delta_{ij} \\
    &\qquad\qquad +8u\eta_i(x)\eta_j(x)\Big]\delta(x-x'),
  \end{aligned}
\end{equation}
or in Fourier space,
\begin{equation}
  \begin{aligned}
    \chi_{ij}^{-1}(q)
      &=8u\sum_{q'}\tilde\eta_i(q')\eta_j(-q')+\bigg(\tilde r+c_\parallel q_\parallel^2-c_\perp q_\perp^2 \\
      &\qquad+D_\perp q_\perp^4+4u\sum_{q'}\tilde\eta_k(q')\tilde\eta_k(-q')\bigg)\delta_{ij}.
  \end{aligned}
\end{equation}
Near the unordered--modulated transition this yields
\begin{equation}
  \begin{aligned}
  \chi(q)
  &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2+|\tilde r-\tilde r_c|} \\
  &=\frac1{D_\perp}\frac{\xi_\perp^4}{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2},
  \end{aligned}
\end{equation}
with $\xi_\perp=(|\tilde r-\tilde r_c|/D_\perp)^{-1/4}$ and $\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$.

The elastic susceptibility (inverse stiffness) is given in the same way: we must trace over $\eta$ and take the second variation of the resulting free energy. Extremizing over $\eta$ yields
\begin{equation}
  0=\frac{\delta F}{\delta\eta_i(x)}=\frac{\delta F_\o}{\delta\eta_i(x)}+\frac12b\epsilon_{\X i}(x),
  \label{eq:implicit.eta}
\end{equation}
which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this cannot be solved explicitly, we can make use of the inverse function theorem to write
\begin{equation}
  \begin{aligned}
    \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1}
    &=\frac{\delta\eta_j^{-1}[\eta](x)}{\delta\eta_i(x')}
    =-\frac2b\frac{\delta^2F_\o}{\delta\eta_i(x)\delta\eta_j(x')} \\
    &=-\frac2b\chi^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta(x-x')
  \end{aligned}
  \label{eq:inv.func}
\end{equation}
It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the susceptibility of the material to $\epsilon_\X$ strain is given by
\begin{widetext}
\begin{equation}
  \begin{aligned}
    \chi_{\X ij}^{-1}(x,x')
    &=\frac{\delta^2F}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\
    &=\lambda_\X\delta(x-x')+
    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
    +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_{\X i}(x')\delta\epsilon_{\X j}(x'')} \\
    &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o}{\delta\eta_k(x'')\delta\eta_\ell(x''')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')}
    +\int dx''\,\frac{\delta F_\o}{\delta\eta_k(x'')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ 
    &=\lambda_\X\delta(x-x')+
    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
    -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta_k(x'')}{\partial\epsilon_{\X\ell}(x''')}\bigg)^{-1}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} \\ 
    &=\lambda_\X\delta(x-x')+
    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
    -\frac12b\int dx''\,\delta_{i\ell}\delta(x-x'')\frac{\delta\eta_\ell(x'')}{\delta\epsilon_{\X j}(x')} 
    =\lambda_\X\delta(x-x')+
    \frac12b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')},
  \end{aligned}
\end{equation}
\end{widetext}
whose Fourier transform follows from \eqref{eq:inv.func} as
\begin{equation}
  \chi_{\X ij}(q)=\frac1{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q).
  \label{eq:elastic.susceptibility}
\end{equation}
At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde r_c|^\gamma$ for $\gamma=1$.

\begin{figure}
  \centering
  \includegraphics[width=0.49\columnwidth]{stiff_a11.pdf}
  \includegraphics[width=0.49\columnwidth]{stiff_a22.pdf}
  \includegraphics[width=0.49\columnwidth]{stiff_a12.pdf}
  \includegraphics[width=0.49\columnwidth]{stiff_b1.pdf}
  \includegraphics[width=0.49\columnwidth]{stiff_b2.pdf}
  \includegraphics[width=0.49\columnwidth]{stiff_e.pdf}
  \caption{
    Measurements of the effective strain stiffness as a function of temperature
    for the six independent components of strain from ultrasound. The vertical
    dashed lines show the location of the hidden order transition.
  }
\end{figure}

\begin{figure}
  \includegraphics[width=\columnwidth]{cusp}
  \caption{
    Strain stiffness data for the $\Bog$ component of strain (solid) along with
    a fit of \eqref{eq:elastic.susceptibility} (dashed).
  }
  }
\end{figure}

\begin{acknowledgements}

\end{acknowledgements}

\bibliography{hidden_order}

\end{document}