\documentclass[aps,prl,reprint]{revtex4-1}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,graphicx,upgreek,amssymb}

% 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}

% Units
\def\J{\mathrm J}
\def\m{\mathrm m}
\def\K{\mathrm K}
\def\GPa{\mathrm{GPa}}
\def\A{\mathrm{\c A}}

% Other
\def\G{\mathrm G} % Ginzburg

\begin{document}

\title{Elastic properties of \urusi are reproduced by modulated $\Bog$ order}
\author{Jaron Kent-Dobias}
\author{Michael Matty}
\author{Brad Ramshaw}
\affiliation{
  Laboratory of Atomic \& Solid State Physics, Cornell University,
  Ithaca, NY, USA
}

\date\today

\begin{abstract}
  We develop a phenomenological theory for the elastic response of materials
  with a \Dfh\ point group through phase transitions. The physics is
  generically that of Lifshitz points, with disordered, uniform ordered, and
  modulated ordered phases. Several experimental features of \urusi are
  reproduced when the order parameter has $\Bog$ symmetry: the topology of the
  temperature--pressure phase diagram, the response of the strain stiffness
  tensor above the hidden-order transition, and the strain response in the
  antiferromagnetic phase. In this scenario, the hidden order is a version of
  the high-pressure antiferromagnetic order modulated along the symmetry axis.
\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 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 study of phase transitions is a central theme of condensed matter physics. In many
cases, a phase transition between different states of matter is marked by a change in symmetry.
In this paradigm, the breaking of symmetry in an ordered phase corresponds to the condensation
of an order parameter (OP) that breaks the same symmetries. Near a second order phase
transition, the physics of the OP can often be described in the context of Landau-Ginzburg 
mean field theory. However, to construct such a theory, one must know the symmetries
of the order parameter, i.e. the symmetry of the ordered state.

One quintessential case where the symmetry of an ordered phase remains unknown is in \urusi.
\urusi is a heavy fermion superconductor in which superconductivity condenses out of a 
symmetry broken state referred to as hidden order (HO) [cite pd paper], and at sufficiently
large [hydrostatic?] pressures, both give way to local moment antiferromagnetism. 
Despite over thirty years of effort, the symmetry of the hidden order state remains unknown, and
modern theories [big citation chunk] propose a variety of possibilities. 
Many [all?] of these theories rely on the formulation of a microscopic model for the 
HO state, but without direct experimental observation of the broken symmetry, none 
have been confirmed. 

One case that does not rely on a microscopic model is recent work [cite RUS paper] 
that studies the HO transition using resonant ultrasound spectroscopy (RUS). 
RUS is an experimental technique that measures mechanical resonances of a sample. These
resonances contain information about the full elastic tensor of the material. Moreover, 
the frequency locations of the resonances are sensitive to symmetry breaking at an electronic
phase transition due to electron-phonon coupling [cite]. Ref. [RUS paper] uses this information
to place strict thermodynamic bounds on the symmetry of the HO OP, again, independent of 
any microscopic model. Motivated by these results, in this paper we consider a mean field theory
of an OP coupled to strain and the effect that the OP symmetry has on the elastic response
in different symmetry channels. Our study finds that a single possible OP symmetry
reproduces the experimental strain susceptibilities, and fits the experimental data well.

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)}=2\epsilon_{12}               \\
    \epsilon_\Eg^{(1)}=2\{\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
stiffnesses $\lambda_\X^{(ij)}$ 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)}=\lambda_{1212} &&
    \lambda_{\Eg}^{(11)}=\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 allowed 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}. 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}[htpb]
  \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_{ij}(q)
    &=\frac{\delta_{ij}}{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
      +|\tilde r-\tilde r_c|} \\
    &=\frac{\delta_{ij}}{D_\perp}\frac{\xi_\perp^4}
      {1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2},
  \end{aligned}
  \label{eq:susceptibility}
\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_{ij}^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta_{ij}\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_{ij}\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_{ij}\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_{ij}\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_{ij}\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)=\frac{\delta_{ij}}{\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}[htpb]
  \centering
  \includegraphics[width=\columnwidth]{fig-stiffnesses}
  \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.
  }
  \label{fig:data}
\end{figure}

We have seen that mean field theory predicts that whatever component of strain
transforms like the order parameter will see a $t^{-1}$ softening in the
stiffness that ends in a cusp. Ultrasound experiments \textbf{[Elaborate???]}
yield the strain stiffness for various components of the strain; this data is
shown in Figure \ref{fig:data}.  The $\Btg$ and $\Eg$ stiffnesses don't appear
to have any response to the presence of the transition, exhibiting the expected
linear stiffening with a low-temperature cutoff \textbf{[What's this called?
Citation?]}. The $\Bog$ stiffness has a dramatic response, softening over the
course of roughly $100\,\K$. There is a kink in the curve right at the
transition. While the low-temperature response is not as dramatic as the theory
predicts, mean field theory---which is based on a small-$\eta$ expansion---will
not work quantitatively far below the transition where $\eta$ has a large
nonzero value and higher powers in the free energy become important. The data
in the high-temperature phase can be fit to the theory
\eqref{eq:elastic.susceptibility}, with a linear background stiffness
$\lambda_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is
shown in Figure \ref{fig:fit}. The data and theory appear consistent.

\begin{figure}[htpb]
  \includegraphics[width=\columnwidth]{fig-fit}
  \caption{
    Strain stiffness 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
    $\lambda_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
    $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
    q_*^4\simeq0.0038\,\K^{-1}$.
  }
  \label{fig:fit}
\end{figure}

Mean field theory neglects the effect of fluctuations on critical behavior, yet
also predicts the magnitude of those fluctuations. This allows a mean field
theory to undergo an internal consistency check to ensure the predicted
fluctuations are indeed negligible. This is typically done by computing the
Ginzburg criterion \cite{ginzburg_remarks_1961}, which gives the proximity to
the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to
break down by comparing the magnitude of fluctuations in a correlation-length
sized box to the magnitude of the field, or since the correlation function is
$k_BT\chi(x,x')$,
\begin{equation}
  V_\xi^{-1}k_BT\int_{V_\xi}d^3x\,\chi(x,0)
  =\langle\delta\eta^2\rangle_{V_\xi}
  \lesssim\frac12\eta_*^2=\frac{|\Delta\tilde r|}{6u}
\end{equation}
with $V_\xi$ the correlation volume, which we will take to be a cylinder of
radius $\xi_\parallel/2$ and height $\xi_\perp$. Upon substitution of
\eqref{eq:susceptibility} and using the jump in the specific heat at the
transition from
\begin{equation}
  c_V=-T\frac{\partial^2f}{\partial T^2}
    =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c,\end{cases}
\end{equation}
this expression can be brought to the form
\begin{equation}
  \frac{2k_B}{\pi\Delta c_V\xi_{\perp0}\xi_{\parallel0}^2}
    \mathcal I(\xi_{\perp0} q_*|t|^{-1/4})
  \lesssim |t|^{13/4},
\end{equation}
where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I$
is defined by
\begin{equation}
  \mathcal I(x)=\frac1\pi\int_{-\infty}^\infty dy\,\frac{\sin\tfrac y2}y
    \bigg(\frac1{1+(y^2-x^2)^2}
      -\frac{K_1(\sqrt{1+(y^2-x^2)^2})}{\sqrt{1+(y^2-x^2)^2}}\bigg)
\end{equation}
For large argument, $\mathcal I(x)\sim x^{-4}$, yielding
\begin{equation}
  t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4}
\end{equation}
Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$
\cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp
q_*^4/aT_c)^{1/4}\sim2$. We have reason to believe that at zero pressure, very
far from the Lifshitz point, $q_*$ is roughly the inverse lattice spacing
\textbf{[Why???]}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$,
we find $t_\G\sim0.04$, so that an experiment would need to be within
$\sim1\,\K$ to detect a deviation from mean field behavior. An ultrasound
experiment able to capture data over several decades within this vicinity of
$T_c$ may be able to measure a cusp with $|t|^\gamma$ for
$\gamma=\text{\textbf{???}}$, the empirical exponent \textbf{[Citation???]}.
Our analysis has looked at behavior for $T-T_c>1\,\K$, and so it remains
self-consistent.

\begin{acknowledgements}

\end{acknowledgements}

\bibliography{hidden_order}

\end{document}