\documentclass[aps,prl,reprint,longbibliography,floatfix]{revtex4-1}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,graphicx,upgreek,amssymb,xcolor}
\usepackage[colorlinks=true,urlcolor=purple,citecolor=purple,filecolor=purple,linkcolor=purple]{hyperref}

\newcommand{\brad}[1]{{\color{red} #1}}

% 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\G{\text G} % Ginzburg
\def\op{\textsc{op}} % order parameter
\def\ho{\textsc{ho}} % hidden order
\def\rus{\textsc{rus}} % Resonant ultrasound spectroscopy 
\def\afm{\textsc{afm}} % Antiferromagnetism 
\def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy 
\def\recip{{\{-1\}}} % functional reciprocal

\begin{document}

\title{Elastic properties of hidden order in \urusi\ are reproduced by staggered nematic 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 mean field theory of the hidden order phase in \urusi\ as a ``staggered nematic" order. Several experimental features are reproduced when the order parameter is a nematic of the $\Bog$ representation, staggered along the c-axis: the topology of the temperature--pressure phase diagram, the response of the elastic modulus $(c_{11}-c_{12})/2$ above the hidden-order transition at zero pressure, and orthorhombic symmetry breaking in the high-pressure antiferromagnetic phase. In this scenario, hidden order is characterized by broken rotational symmetry that is modulated along the $c$-axis, the primary order of the high-pressure phase is an unmodulated nematic state, and the triple point joining those two phases with the high-temperature paramagnetic phase is a Lifshitz point.
\end{abstract}

\maketitle

%The study of phase transitions is central to condensed matter physics.  Phase
%transitions are often accompanied by a change in symmetry whose emergence can
%be described by the condensation of an order parameter (\op) that breaks the
%same symmetries. Near a continuous phase transition, the physics of the \op\
%can often be qualitatively and sometimes quantitatively described by
%Landau--Ginzburg mean field theories. These depend on little more than the
%symmetries of the \op, and coincidence of their predictions with experimental
%signatures of the \op\ is evidence of the symmetry of the corresponding ordered
%state.

% Many of these
%theories rely on the formulation of a microscopic model for the \ho\ state, but
%since there has not been direct experimental observation of the broken
%symmetry, none can been confirmed. 

%\Rus\ is an experimental technique that
%measures mechanical resonances of a sample. These resonances contain
%information about the sample's full strain stiffness tensor. Moreover, the
%frequency locations of the resonances are sensitive to symmetry breaking at an
%electronic phase transition due to electron-phonon coupling
%\cite{shekhter_bounding_2013}.  Ref.~\cite{ghosh_single-component_nodate} uses
%this information to place strict thermodynamic bounds on the dimension of the
%\ho\ \op\ independent of any microscopic model.
\section{Introduction}
\urusi\ is a paradigmatic example of a material with an ordered state whose broken symmetry remains unknown. This state, known as \emph{hidden order} (\ho), sets the stage for unconventional superconductivity that emerges at even lower temperatures. 
At sufficiently large hydrostatic pressures, both superconductivity and \ho\ give way to local moment antiferromagnetism (\afm) \cite{hassinger_temperature-pressure_2008}.
Despite over thirty years of effort, the symmetry of the \ho\ state remains
unknown, and modern theories \cite{kambe_odd-parity_2018, haule_arrested_2009,
  kusunose_hidden_2011, kung_chirality_2015, cricchio_itinerant_2009,
  ohkawa_quadrupole_1999, santini_crystal_1994, kiss_group_2005,
  harima_why_2010, thalmeier_signatures_2011, tonegawa_cyclotron_2012,
rau_hidden_2012, riggs_evidence_2015, hoshino_resolution_2013,
ikeda_theory_1998, chandra_hastatic_2013, harrison_hidden_nodate,
ikeda_emergent_2012} propose a variety of possibilities. Our work here seeks to unify two experimental observations: one, the $\Bog$ ``nematic" elastic susceptibility $(c_{11}-c_{12})/2$ softens anomalously from room temperature down to T$_{\mathrm{HO}}=17.5~$ K \brad{find old citations for this data}; and two, a $\Bog$ nematic distortion is observed by x-ray scattering under sufficient pressure to destroy the \ho\ state \cite{choi_pressure-induced_2018}.

Recent \emph{resonant ultrasound spectroscopy} (\rus) measurements examined the thermodynamic discontinuities in the elastic moduli at T$_{\mathrm{HO}}$ \cite{ghosh_single-component_nodate}. The observation of discontinues only in compressional, or $\Aog$, elastic moduli requires that the point-group representation of \ho\ is one-dimensional. This rules out a large number of order parameter candidates \brad{cite those ruled out} in a model-free way, but still leaves the microscopic nature of \ho~ undecided. 

Recent X-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure \cite{choi_pressure-induced_2018}. Above \brad{whatever pressure they find it at...}, \urusi\ undergoes a $\Bog$ nematic distortion. While it is still unclear as to whether this is a true thermodynamic phase transition, it may be related to the anomalous softening of the $\Bog$ elastic modulus---$(c_{11}-c_{12})/2$---that occurs over a broad temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by these results, hinting at a $\Bog$ strain susceptibility associated with the \ho\ state, we construct a phenomenological mean field theory for an arbitrary \op\ coupled to strain, and the determine the effect of its phase transitions on the elastic response in different symmetry channels. 

We find that only one \op\ symmetry reproduces the anomalous $(c_{11}-c_{12})/2$ elastic modulus, which softens in a Curie-Weiss like manner from room temperature, but which cusps at T$_{\mathrm{HO}}$. That theory associates \ho\ with a $\Bog$ \op\ \emph{modulated along the $c$- axis}, the \afm\ state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. Besides the agreement with ultrasound data across a broad temperature range, the theory predicts uniform $\Bog$ strain at high pressure---the same distortion which was recently seen in x-ray scattering experiments \cite{choi_pressure-induced_2018}. This theory
strongly motivates future ultrasound experiments under pressure approaching the Lifshitz point, which should find that the $(c_{11}-c_{12})/2$ diverges once the uniform $\Bog$ strain sets in.


%but the form of the bare strain
%stiffness tensor $C$ tensor is constrained by both the index symmetry of the
%strain tensor and by the point group symmetry \cite{landau_theory_1995} \brad{why %is there a Landau paper from 1995?!}. 


\section{Model}
The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
respect this symmetry in the high-temperature phase. Our phenomenological free energy density contains three parts: the elastic free energy, the \op, and the interaction between strain and \op. The most general quadratic free energy of the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, where the six irreducible components of strain are
\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}
  \label{eq:strain-components}
\end{equation}
All quadratic combinations of these irreducible strains that transform like
$\Aog$ are included in the free energy,
\begin{equation}
  f_\e=\frac12\sum_\X C_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},
\end{equation}
where the sum is over irreducible representations of the point group and the
bare elastic moduli $C_\X^{(ij)}$ are \brad{I would write these in Voigt notation, so c1111 is just c11, c1122 is c12, etc.}
\begin{equation}
  \begin{aligned}
    &C_{\Aog}^{(11)}=\tfrac12(C_{1111}+C_{1122}) &&
    C_{\Aog}^{(22)}=C_{3333} \\
    &C_{\Aog}^{(12)}=C_{1133} &&
    C_{\Bog}^{(11)}=\tfrac12(C_{1111}-C_{1122}) \\
    &C_{\Btg}^{(11)}=C_{1212} &&
    C_{\Eg}^{(11)}=C_{1313}.
  \end{aligned}
\end{equation}
The interaction between strain and an \op\ $\eta$ depends on the point group representation of $\eta$. If this representation is $\X$, the most general coupling to linear order is \brad{why the negative sign?}
\begin{equation}
  f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta.
\end{equation}
If the representation $\X$ is not present in the strain \brad{what does "present in the strain" mean?} there can be no linear
coupling, and the effect of the \op\ condensing at a continuous phase
transition is to produce a jump in the $\Aog$ elastic modului if $\eta$ is
single-component \cite{luthi_sound_1970, ramshaw_avoided_2015,
shekhter_bounding_2013}, and jumps in other elastic moduli if multicompenent \cite{ghosh_single-component_nodate}. Because we are interested
in physics that anticipates the phase transition, we will focus our attention
on \op s that can produce linear couplings to strain.  Looking at the
components present in \eqref{eq:strain-components}, this rules out all of the
\emph{u}-reps (which are odd under inversion) and the $\Atg$ irrep.

If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd terms are allowed in its free energy and any transition will be first-order and not continuous without fine-tuning. Since the \ho\ phase transition is second-order \brad{cite something}, we will henceforth rule out $\Aog$ \op s as well. 

For the \op\ representation $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general
quadratic free energy density is
\begin{equation}
  \begin{aligned}
    f_\op=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
      +c_\perp(\nabla_\perp\eta)^2 \\
      &\qquad\qquad\qquad\quad+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 $\Atu$. Other quartic terms are
allowed---especially many for an $\Eg$ \op---but we have included only those
terms necessary for stability when either $r$ or $c_\perp$ become negative. The
full free energy functional of $\eta$ and $\epsilon$ is
\begin{equation}
  \begin{aligned}
    F[\eta,\epsilon]
      &=F_\op[\eta]+F_\e[\epsilon]+F_\i[\eta,\epsilon] \\
      &=\int dx\,(f_\op+f_\e+f_\i).
  \end{aligned}
\end{equation}
The only strain relevant to the \op\ at linear coupling is $\epsilon_\X$, which can be traced out
of the problem exactly in mean field theory. Extremizing with respect to
$\epsilon_\X$,
\begin{equation}
  0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_\X(x)}\bigg|_{\epsilon=\epsilon_\star}=C_\X\epsilon^\star_\X(x)
    -b\eta(x)
\end{equation}
gives the optimized strain conditional on the \op\ as
$\epsilon_\X^\star[\eta](x)=(b/C_\X)\eta(x)$ and $\epsilon_\Y^\star[\eta]=0$
for all other $\Y$. Upon substitution into the free energy, the resulting
effective free energy $F[\eta,\epsilon_\star[\eta]]$ has a density identical to
$f_\op$ with $r\to\tilde r=r-b^2/2C_\X$. \brad{need a sentence along the lines of "As $r$ is typically associated with $T-T_c$, this substitution has the effect of shifting the bare $T_c$ due to linear coupling between strain and order parameter", or something like that. Actually I'm a bit confused, shouldn't the new Tc be proportional to strain? Or is this just the correct even in the absence of any applied strain}.

\begin{figure}[htpb]
  \includegraphics[width=\columnwidth]{phase_diagram_experiments}

  \vspace{1em}

  \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em}
  \includegraphics[width=0.51\columnwidth]{phases_vector}
  \caption{
    Phase diagrams for (a) \urusi\ from experiments (neglecting the
    superconducting phase) \cite{hassinger_temperature-pressure_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 abrupt transitions.
    Later, when we fit the elastic stiffness predictions for a $\Bog$ \op\ to
    data along the zero (atmospheric) pressure line, we will take $\Delta\tilde r=\tilde
    r-\tilde r_c=a(T-T_c)$.
  }
  \label{fig:phases}
\end{figure}

\section{Results}
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 one-component \op\ ($\Bog$ or $\Btg$) it is
traditional to make the field ansatz
$\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$ \brad{Why is it traditional to ignore any in-plane modulation (x1, x2)?}. 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 \brad{probably helpful to specify what kind of order here - uniform $\Bog$ order, correct?}. 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}
    =\frac{|\Delta\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
one-component \op\ and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde
r/5}$. For a two-component \op\ ($\Eg$) we must also allow a relative phase
between the two components of the field \brad{Unless there is a specific reason, we should probably stick to \op\ instead of "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 $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. The uniform--modulated transition is now continuous \brad{Is this at all obvious? Do we have to "show" it in some way? Or cite something?}. This does not reproduce the physics of \ho, which has a first-order \brad{I think we should keep the language "First order" rather than "abrupt", which doesn't really mean anything specific} transition between \ho\ and \afm, and so we will henceforth neglect the possibility of a multicomponent order parameter. The schematic phase diagrams for this model are shown in Figure
\ref{fig:phases}.

We will now derive the \emph{effective elastic tensor}
$\lambda$ that results from the coupling of strain to the \op. The ultimate
result, found in \eqref{eq:elastic.susceptibility}, is that $\lambda_\X$
differs from its bare value $C_\X$ only for the symmetry $\X$ of the \op\ \brad{Why the mixed $\lambda$ and C notation? Why not C and C dagger or tilde or hat?}. Moreover, the effective strain stiffness \brad{I think "elastic moduli" is a lot more familiar to people than "Strain stiffness"} does not vanish at the unordered--modulated transition \brad{"unordered--modulated transition" is confusing language}---as it would if the transition were a $q=0$ structural phase transition---but instead exhibits a \emph{cusp}. To
show this, we will first compute the susceptibility of the \op, which will both
be demonstrative of how the stiffness is calculated and prove useful in
expressing the functional form of the stiffness. Then we will compute the
strain stiffness using some tricks from functional calculus \brad{"tricks" will be a bit too colloquial for most referees}.

The susceptibility of a single component ($\Bog$ or $\Btg$) \op\ $\eta$ to a
thermodynamically conjugate field (such as strain) \brad{it it clear why they are conjugate fields? Because they are linearlly coupled?} is given by
\begin{equation}
  \begin{aligned}
    &\chi^\recip(x,x')
    =\frac{\delta^2F[\eta,\epsilon_\star[\eta]]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\langle\eta\rangle}
      =\big[\tilde r-c_\parallel\nabla_\parallel^2 \\
    &\qquad\qquad-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+12u\langle\eta(x)\rangle^2\big]
    \delta(x-x'),
  \end{aligned}
  \label{eq:sus_def}
\end{equation}
where $\recip$ indicates a \emph{functional reciprocal} in the sense that
\begin{equation}
  \int dx''\,\chi^\recip(x,x'')\chi(x'',x')=\delta(x-x').
\end{equation}
Taking the Fourier transform and integrating over $q'$ we have
\begin{equation}
    \chi(q)
    =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4
    +12u\sum_{q'}\langle\tilde\eta_{q'}\rangle\langle\tilde\eta_{-q'}\rangle\big)^{-1}.
\end{equation}
Near the unordered--modulated transition \brad{again, this needs clearer language} this yields
\begin{equation}
  \begin{aligned}
    \chi(q)
    &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
      +|\Delta\tilde r|} \\
    &=\frac1{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 $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}=\xi_{\perp0}|t|^{-1/4}$ and
$\xi_\parallel=(|\Delta\tilde
r|/c_\parallel)^{-1/2}=\xi_{\parallel0}|t|^{-1/2}$, where $t=(T-T_c)/T_c$ is
the reduced temperature and $\xi_{\perp0}=(D_\perp/aT_c)^{1/4}$ and
$\xi_{\parallel0}=(c_\parallel/aT_c)^{1/2}$ are the bare correlation lengths \brad{needs a descriptor like "in and perpendicular to the x-y plane" or something like that}.
Notice that the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde
r|)^{-1}$ does not diverge at the unordered--modulated transition. Though it
anticipates a transition with Curie--Weiss-like divergence at $\Delta\tilde
r=-D_\perp q_*^4$, this is cut off with a cusp at $\Delta\tilde r=0$ \brad{this will all be clearer if you remind the reader that this is Tc, or the new renormalized Tc, or whatever it is}. We must
emphasize that this is \emph{not} the magnetic susceptibility because a $\Bog$
or $\Btg$ \op\ cannot couple linearly to a uniform magnetic field \brad{not sure that this reminder is important, I don't think anyone things we are dealing with magnetic fields.}. The object
defined in \eqref{eq:sus_def} is most readily interpreted as proportional to
the two-point connected correlation function
$\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$ \brad{is this important?}.

The strain stiffness \brad{elastic modulus? Or is this elastic compliance?} is given in a similar way to the inverse susceptibility:
we must trace over $\eta$ and take the second variation of the resulting
effective free energy functional of $\epsilon$. Extremizing over $\eta$ yields
\begin{equation}
  0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}=
    \frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}-b\epsilon_\X(x),
  \label{eq:implicit.eta}
\end{equation}
which implicitly gives $\eta_\star[\epsilon]$, the optimized \op\ conditioned
on the strain. Since $\eta_\star$ is a functional of $\epsilon_\X$
alone, only the stiffness $\lambda_\X$ can be modified from its bare value $C_\X$.
Though this differential equation for $\eta_*$ cannot be solved explicitly, we
can make use of the inverse function theorem.  First, denote by
$\eta_\star^{-1}[\eta]$ the inverse functional of $\eta_\star$ implied by
\eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$ corresponding
to each solution of \eqref{eq:implicit.eta} it receives. This we can
immediately identify from \eqref{eq:implicit.eta} as
$\eta^{-1}_\star[\eta](x)=b^{-1}(\delta F_\op[\eta]/\delta\eta(x))$.  Now, we use
the inverse function theorem to relate the functional reciprocal of the
derivative of $\eta_\star[\epsilon]$ with respect to $\epsilon_\X$ to the
derivative of $\eta^{-1}_\star[\eta]$ with respect to $\eta$, yielding
\begin{equation}
  \begin{aligned}
    \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip
    &=\frac{\delta\eta_\star^{-1}[\eta](x)}{\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]} 
    =b^{-1}\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}.
  \end{aligned}
  \label{eq:inv.func}
\end{equation}
Next, \eqref{eq:implicit.eta} and \eqref{eq:inv.func}
can be used in concert with the ordinary rules of functional calculus to yield
the second variation
\begin{widetext}
\begin{equation}
  \begin{aligned}
    \frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} 
    &=C_\X\delta(x-x')-
    2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
    -b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') +\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}\frac{\delta F_\op[\eta]}{\delta\eta(x'')}\bigg|_{\eta=\eta_\star[\epsilon]}\\
    &\qquad\qquad+\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\frac{\delta^2F_\op[\eta]}{\delta\eta(x'')\delta\eta(x''')}\bigg|_{\eta=\eta_\star[\epsilon]} \\ 
    &=C_\X\delta(x-x')-
    2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
    -b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') +\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}(b\epsilon_\X(x''))\\
    &\qquad\qquad+b\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')} \bigg(\frac{\partial\eta_\star[\epsilon](x'')}{\partial\epsilon_\X(x''')}\bigg)^\recip\\ 
    &=C_\X\delta(x-x')-
    2b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
    +b\int dx''\,\delta(x-x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x')} 
    =C_\X\delta(x-x')-b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}.
  \end{aligned}
  \label{eq:big.boy}
\end{equation}
\end{widetext}
The strain stiffness is given by the second variation evaluated at the
extremized strain $\langle\epsilon\rangle$. To calculate it, note that
evaluating the second variation of $F_\op$ in \eqref{eq:inv.func} at
$\langle\epsilon\rangle$ (or
$\eta_\star(\langle\epsilon\rangle)=\langle\eta\rangle$) yields
\begin{equation}
  \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip\bigg|_{\epsilon=\langle\epsilon\rangle}
    =b^{-1}\chi^\recip(x,x')+\frac{b}{C_\X}\delta(x-x'),
  \label{eq:recip.deriv.op}
\end{equation}
where $\chi^\recip$ is the \op\ susceptibility given by \eqref{eq:sus_def}.
Upon substitution into \eqref{eq:big.boy} and taking the Fourier transform of
the result, we finally arrive at
\begin{equation}
  \lambda_\X(q)
  =C_\X-b\bigg(\frac1{b\chi(q)}+\frac b{C_\X}\bigg)^{-1}
  =C_\X\bigg(1+\frac{b^2}{C_\X}\chi(q)\bigg)^{-1}.
  \label{eq:elastic.susceptibility}
\end{equation}
Though not relevant here, this result generalizes to multicomponent \op s.  At
$q=0$, which is where the stiffness measurements used here were taken, this
predicts a cusp in the static strain stiffness $\lambda_\X(0)$ of the form
$|\Delta\tilde r|^\gamma$ for $\gamma=1$.  \brad{I think this last sentence, which is the point of the whole paper, needs to be expanded upon and emphasized. It needs to be clear that what we have done is consider a general OP of B1g or B2g type modulated along the c-axis. For a general Landau free energy, it will develop order at some finite q, but if you measure at q=0, which is what ultraound typically does, you still see "remnant" behaviour that cusps at the transition}
\begin{figure}[htpb]
  \centering
  \includegraphics[width=\columnwidth]{fig-stiffnesses}
  \caption{
   Resonant ultrasound spectroscopy measurements of the elastic moduli of \urusi\ as a function of temperature
    for the six independent components of strain. The vertical lines
    show the location of the \ho\ transition. \brad{Can you move the labels on the right-hand panels over to the right-hand axis? Also, can you make the labels smaller and the actual panels bigger}
  }
  \label{fig:data}
\end{figure}

\section{Comparison to experiment}
\Rus\ experiments \cite{ghosh_single-component_nodate} yield the full elasticity tensor; the moduli broken into the irrep components defined in
\eqref{eq:strain-components} is shown in Figure \ref{fig:data}.  The $\Btg$
stiffness doesn't appear to have any response to the presence of the
transition, exhibiting the expected linear stiffening upon cooling from room temperature, with a low-temperature
cutoff at some fraction of the Debye temperature\cite{varshni_temperature_1970}. The $\Bog$ stiffness, on the other hand, has a dramatic
response, softening over the course of roughly $100\,\K$, and then cusping at the \ho\ 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
$C_\Bog^{(11)}$ 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 quantitatively consistent in
the high temperature phase, suggesting that \ho\ can be described as a $\Bog$-nematic phase that is modulated at finite $q$ along the $c-$axis.

\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
    $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
    $b^2/D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
    q_*^4\simeq0.0038\,\K^{-1}$. The failure of the Ginzburg--Landau prediction
    below the transition is expected on the grounds that the \op\ is too large
    for the free energy expansion to be valid by the time the Ginzburg
    temperature is reached.
  }
  \label{fig:fit}
\end{figure}

We have seen that the mean-field theory of a $\Bog$ \op\ recreates the topology
of the \ho\ phase diagram and the temperature dependence of the $\Bog$ strain
stiffness at zero pressure. This theory has several other physical implications. First,
the association of a modulated $\Bog$ order with the \ho\ phase implies a
\emph{uniform} $\Bog$ order associated with the \afm\ phase, and moreover a
uniform $\Bog$ strain of magnitude $\langle\epsilon_\Bog\rangle^2=b^2\tilde
r/4uC_\Bog^2$, which corresponds to an orthorhombic structural phase. Orthorhombic
symmetry breaking was recently detected in the \afm\ phase of \urusi\ using
x-ray diffraction, a further consistency of this theory with the phenomenology
of \urusi\ \cite{choi_pressure-induced_2018}. Second, as the Lifshitz point is
approached from low pressure, this theory predicts that the modulation wavevector
$q_*$ should vanish continuously. Far from the Lifshitz point we expect the
wavevector to lock into values commensurate with the space group of the
lattice, and moreover that at zero pressure, where the \rus\ data here was
collected, the half-wavelength of the modulation should be commensurate with
the lattice spacing $a_3\simeq9.68\,\A$, or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$ \cite{meng_imaging_2013,
broholm_magnetic_1991, wiebe_gapped_2007, bourdarot_precise_2010}. In between
these two regimes, the ordering wavevector should shrink by jumping between
ever-closer commensurate values in the style of the devil's staircase
\cite{bak_commensurate_1982}. This motivates future \rus\ experiments done at
pressure, where the depth of the cusp in the $\Bog$ stiffness should deepen
(perhaps with these commensurability jumps) at low pressure and approach zero
like $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. \brad{Should also motivate x-ray and neutron-diffraction experiments to look for new q's - mentioning this is important if we want to get others interested, no one else does RUS...} Alternatively,
\rus\ done at ambient pressure might examine the heavy fermi liquid to \afm\
transition by doping. \brad{We have to be careful, someone did do some doping studies and it's not clear exctly what's going on}. The presence of spatial commensurability is not
expected to modify the critical behavior otherwise
\cite{garel_commensurability_1976}. \brad{this thought feels half-finished, where was it going?}

There are two apparent discrepancies between the orthorhombic strain in the
phase diagram presented by \cite{choi_pressure-induced_2018} and that predicted
by our mean field theory.  The first is the apparent onset of the orthorhombic
phase in the \ho\ state prior to the onset of \afm.  As
\cite{choi_pressure-induced_2018} notes, this could be due to the lack of
an ambient pressure calibration for the lattice constant. The second
discrepancy is the onset of orthorhombicity at higher temperatures than the
onset of \afm. Susceptibility data sees no trace of another phase transition at
these higher temperatures \cite{inoue_high-field_2001}, and therefore we don't
expect there to be one. We do expect that this could be due to the high
energy nature of x-rays as an experimental probe: orthorhombic fluctuations
could appear at higher temperatures than the true onset of an orthorhombic
phase. \brad{I think this paragraph should probably be tigtened up a bit, we need to be more specific about "don't expect there to be one" and "fluctuations"}. 

Three dimensions is below the upper critical dimension $4\frac12$ \brad{upper critical dimension of what, all Landau mean field theories?}, and so mean
field theory should break down sufficiently close to the critical point due to
fluctuations, at the Ginzburg temperature \cite{hornreich_lifshitz_1980, ginzburg_remarks_1961}. Magnetic phase transitions tend to have Ginzburg temperature of order one.
Our fit above gives $\xi_{\perp0}q_*=(D_\perp
q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ at about what one would expect for a generic magnetic transition.
The agreement of this data in the $t\sim0.1$--10 range with the mean field exponent suggests that this region is outside the Ginzburg region, but an experiment may begin to see deviations from mean field behavior within
around several degrees Kelvin of the critical point. A \rus\ experiment with more precise
temperature resolution near the critical point may be able to resolve a
modified cusp exponent $\gamma\simeq1.31$ \cite{guida_critical_1998}, since the
universality class of a uniaxial modulated scalar \op\ is $\mathrm O(2)$
\cite{garel_commensurability_1976}. We should not expect any quantitative
agreement between mean field theory and experiment in the low temperature phase
since, by the point the Ginzburg criterion is satisfied, $\eta$ is order one
and the Landau--Ginzburg free energy expansion is no longer valid.

\section{Conclusion and Outlook}
We have developed a general phenomenological treatment of  \ho\ \op s with the potential for linear coupling to strain. The two representations with mean
field phase diagrams that are consistent with the phase diagram of \urusi\ are $\Bog$ and $\Btg$. Of these, only a staggered $\Bog$ \op is consistent with zero-pressure \rus\ data, with a cusp appearing in the
associated elastic stiffness. In this picture, the \ho\ phase is characterized by
uniaxial modulated $\Bog$ order, while the \afm\ phase is characterized by
uniform $\Bog$ order. \brad{We need to be a bit more explicit about what we think is going on with \afm - is it just a parasitic phase? Is our modulated phase somehow "moduluated \afm" (can you modualte AFM in such as way as to make it disappear? Some combination of orbitals?)} The corresponding prediction of uniform $\Bog$ symmetry
breaking in the \afm\ phase is consistent with recent diffraction experiments
\cite{choi_pressure-induced_2018} \brad{needs a caveat about temperature, so that we're being transparent}. This work motivates both further theoretical
work regarding a microscopic theory with modulated $\Bog$ order, and preforming
\rus\ experiments at pressure that could further support or falsify this idea.

\begin{acknowledgements}
  This research was supported by NSF DMR-1719490 and DMR-1719875.
\end{acknowledgements}

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