
\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}

% 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 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 mean field theory for the strain in \urusi\
  through its hidden order transition. Several experimental features 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 at zero pressure, and orthorhombic
  symmetry breaking in the high-pressure antiferromagnetic phase. In this
  scenario, the hidden order is characterized by the order parameter in the
  high-pressure antiferromagnetic phase modulated along the symmetry axis, and
  the triple point joining those two phases with the 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.

A paradigmatic example of a material with an ordered state whose broken
symmetry remains unknown is in \urusi.  \urusi\ is a heavy fermion
superconductor in which superconductivity condenses out of a symmetry broken
state referred to as \emph{hidden order} (\ho)
\cite{hassinger_temperature-pressure_2008}, and at sufficiently large
hydrostatic pressures, both give way to local moment antiferromagnetism (\afm).
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.  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. 

Recent work that studied the \ho\ transition using \emph{resonant ultrasound
spectroscopy} (\rus) was able to shed light on the symmetry of the ordered
state without the formulation of any microscopic model
\cite{ghosh_single-component_nodate}.  \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.

Motivated by these results, 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 features of the
experimental strain stiffness. That theory associates the \ho\ state with a
$\Bog$ \op\ \emph{modulated along the rotation axis}, the \afm\ state with
uniform $\Bog$ order, and the triple point between them with a Lifshitz point.
Besides the agreement with \rus\ data in the \ho\ state, the theory predicts
uniform $\Bog$ strain in the \afm\ state, which was recently seen in x-ray
scattering experiments \cite{choi_pressure-induced_2018}. The theory's
implications for the dependence of the strain stiffness on pressure and doping
strongly motivates future \rus\ experiments that could either further support
or falsify it.

The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
respect this symmetry. Our phenomenological free energy density contains three
parts: the free energy for the strain, the \op, and their interaction.  The
most general quadratic free energy of the strain $\epsilon$ is
$f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, 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}. The
six independent components of strain can written as linear combinations that
each 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}
  \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 stiffnesses $C_\X^{(ij)}$ are
\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 representation
of the point group that $\eta$ transforms as. If this representation is $\X$,
the most general coupling to linear order is
\begin{equation}
  f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta.
\end{equation}
If the representation $\X$ is not present in the strain 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 $\Aog$ strain stiffness if $\eta$ is
single-component \cite{luthi_sound_1970, ramshaw_avoided_2015,
shekhter_bounding_2013}, and jumps in other strain stiffnesses 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\ symmetries that can produce linear couplings to strain.  Looking at the
components present in \eqref{eq:strain-components}, this rules out all of the
u-reps (which are odd under inversion) and the $\Atg$ irrep.

If the \op\ transforms like $\Aog$, odd terms are allowed in its free energy
and any transition will be abrupt and not continuous without fine-tuning. Since
this is not a feature of \urusi\ \ho\ physics, we will henceforth rule it out
as well. For $\X$ 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\ 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 $r\to\tilde r=r-b^2/2C_\X$.

\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}

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
$\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. 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}
    =\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 field 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. 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. This does not
reproduce the physics of \ho, which has an abrupt 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 proceed to derive the \emph{effective strain stiffness 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. Moreover, the effective strain stiffness does not vanish at the unordered--modulated transition, but 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.

The susceptibility of a single component ($\Bog$ or $\Btg$) \op\ to a
thermodynamically conjugate field 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 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.
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 $\Delta\tilde r=0$. 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. The object
defined in \eqref{eq:sus_def} is most readily interpreted as proportional to
the two-point connected correlation function
$\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$.

The strain stiffness 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 $\gamma=1$. 
\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 \rus. The vertical lines
    show the location of the \ho\ transition.
  }
  \label{fig:data}
\end{figure}

\Rus\ experiments \cite{ghosh_single-component_nodate} yield the strain
stiffness tensor; the data 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 with a low-temperature
cutoff \cite{varshni_temperature_1970}. 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
$C_\Bog^{(11)}$ 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.

\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 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. Alternatively,
\rus\ done at ambient pressure might examine the heavy fermi liquid to \afm\
transition by doping. The presence of spatial commensurability is not
expected to modify the critical behavior otherwise
\cite{garel_commensurability_1976}.

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. 

Three dimensions is below the upper critical dimension $4\frac12$, 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.

We have preformed a general treatment of phenomenological \ho\ \op s with the
potential for linear coupling to strain. The possibilities with consistent mean
field phase diagrams are $\Bog$ and $\Btg$, and the only of these consistent
with zero-pressure \rus\ data is $\Bog$, with a cusp appearing in the
associated stiffness. In this picture, the \ho\ phase is characterized by
uniaxial modulated $\Bog$ order, while the \afm\ phase is characterized by
uniform $\Bog$ order. The corresponding prediction of uniform $\Bog$ symmetry
breaking in the \afm\ phase is consistent with recent diffraction experiments
\cite{choi_pressure-induced_2018}. 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}

\bibliographystyle{apsrev4-1}
\bibliography{hidden_order}

\end{document}
