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\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\op{\textsc{op}} % order parameter
\def\ho{\textsc{ho}} % hidden order
\def\rus{\textsc{rus}} % resonant ultrasound spectroscopy 
\def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy 
\def\afm{\textsc{afm}} % antiferromagnetism 
\def\recip{{\{-1\}}} % functional reciprocal

\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

\emph{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
\cite{thalmeier_signatures_2011,tonegawa_cyclotron_2012,rau_hidden_2012,riggs_evidence_2015,hoshino_resolution_2013,ikeda_emergent_2012,chandra_origin_2013}
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 0.13--0.5 $\GPa$
(depending on temperature), \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---$(c_{11}-c_{12})/2$ in Voigt notation---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 $\Bog$ 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.


\emph{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 $f_\e=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$. Linear
combinations of the six independent components of strain form five irreducible
components of strain as
\begin{equation}
  \begin{aligned}
    & \epsilon_{\Aog,1}=\epsilon_{11}+\epsilon_{22} \hspace{0.15\columnwidth} &&
      \epsilon_\Bog=\epsilon_{11}-\epsilon_{22} \\
    & \epsilon_{\Aog,2}=\epsilon_{33} &&
      \epsilon_\Btg=2\epsilon_{12} \\
    & \epsilon_\Eg=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^0_{\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 $C^0_\X$ are 
\begin{equation}
  \begin{aligned}
    & C^0_{\Aog,11}=\tfrac12(C^0_{1111}+C^0_{1122}) &&
      C^0_{\Bog}=\tfrac12(C^0_{1111}-C^0_{1122}) \\
    & C^0_{\Aog,22}=C^0_{3333} &&
      C^0_{\Btg}=C^0_{1212} \\
    & C^0_{\Aog,12}=C^0_{1133} &&
      C^0_{\Eg}=C^0_{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
\begin{equation}
  f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta.
\end{equation}
If there doesn't exist a component of strain that transforms like the
representation $\X$ 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}
  \label{eq:free_energy}
\end{equation}

Rather than analyze this two-argument functional directly, we begin by tracing
out the strain and studying the behavior of \op\ alone, assuming the
strain is equilibrated. Later we will invert this procedure and trace out the
\op when we compute the effective elastic moduli. 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 the functional
\eqref{eq:free_energy} with respect to $\epsilon_\X$ gives
\begin{equation}
  0
    =\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_\X(x)}\bigg|_{\epsilon=\epsilon_\star}
    =C^0_\X\epsilon^\star_\X(x)-b\eta(x),
\end{equation}
which in turn gives the strain field conditioned on the state of the \op\ field
as $\epsilon_\X^\star[\eta](x)=(b/C^0_\X)\eta(x)$ at all spatial coordinates
$x$, and $\epsilon_\Y^\star[\eta]=0$ for all other irreps $\Y\neq\X$. Upon
substitution into the free energy, the resulting single-argument free energy
functional $F[\eta,\epsilon_\star[\eta]]$ has a density identical to $f_\op$
with $r\to\tilde r=r-b^2/2C^0_\X$. 

\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 first order transitions.
    Later, when we fit the elastic moduli 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}. The properties discused below can all be found in a
standard text, e.g.,~\cite{chaikin_principles_2000}. For a one-component \op\
($\Bog$ or $\Btg$) and positive $c_\parallel$, 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 with the \op\ symmetry, e.g., $\Bog$ or $\Btg$. 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 first order 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 \op. 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 to modulated transition is now continuous. This does not reproduce
the physics of \ho, which has a first order 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}.

\emph{Results.}
We will now derive the \emph{effective elastic tensor} $C$ that results from
the coupling of strain to the \op. The ultimate result, found in
\eqref{eq:elastic.susceptibility}, is that $C_\X$ differs from its bare value
$C^0_\X$ only for the symmetry $\X$ of the \op. Moreover, the effective elastic
moduli does not vanish at the unordered to modulated transition---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 modulus is calculated
and prove useful in expressing the functional form of the modulus. Then we will
compute the elastic modulus using techniques from functional calculus.

The generalized susceptibility of a single component ($\Bog$ or $\Btg$) \op\ is
\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 to 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
parallel and perpendicular to the axis of symmetry, respectively.  Notice that
the static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde r|)^{-1}$ does
not diverge at the unordered to modulated transition. Though it anticipates a
transition with Curie--Weiss-like divergence at $a(T-T_c)=\Delta\tilde
r=-D_\perp q_*^4$, this is cut off with a cusp at $\Delta\tilde r=0$. 

The elastic susceptibility, which corresponds with the reciprocal of the elastic modulus,
is given in a similar way to the \op\ 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 modulus $C_\X$ can be modified from its bare value $C^0_\X$.
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^0_\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^0_\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^0_\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^0_\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 elastic modulus 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^0_\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}
  C_\X(q)
  =C^0_\X-b\bigg(\frac1{b\chi(q)}+\frac b{C^0_\X}\bigg)^{-1}
  =C^0_\X\bigg(1+\frac{b^2}{C^0_\X}\chi(q)\bigg)^{-1}.
  \label{eq:elastic.susceptibility}
\end{equation}
Though not relevant here, this result generalizes to multicomponent \op s.

What does \eqref{eq:elastic.susceptibility} predict in the vicinity of the
\ho\ transition? Near the disordered to modulated transition, the
static modulus is given by
\begin{equation}
  C_\X(0)=C_\X^0\bigg[1+\frac{b^2}{C_\X^0}\big(D_\perp q_*^4+|\Delta\tilde r|\big)^{-1}\bigg]^{-1}.
  \label{eq:static_modulus}
\end{equation}
This corresponds to a softening in the $\X$-modulus at the transition that is
cut off with a cusp of the form $|\Delta\tilde r|^\gamma\propto|T-T_c|^\gamma$
with $\gamma=1$. This is our main result. The only \op\ irreps that couple
linearly with strain and reproduce the topology of the \urusi\ phase diagram
are $\Bog$ and $\Btg$. For either of these irreps, the transition into a
modulated rather than uniform phase masks traditional signatures of a
continuous transition by locating thermodynamic singularities at finite $q$.
The remaining clue at $q=0$ is a particular kink in the corresponding modulus.

\begin{figure}[htpb]
  \centering
  \includegraphics[width=\columnwidth]{fig-stiffnesses}
  \caption{
   \Rus\ measurements of the elastic moduli of \urusi\ as a function of
   temperature from \cite{ghosh_single-component_nodate} (blue, solid)
   alongside fits to theory (magenta, dashed). The solid yellow region shows
   the location of the \ho\ phase. (a) $\Btg$ modulus data and fit to standard
   form \cite{varshni_temperature_1970}. (b) $\Bog$ modulus data and fit to
   \eqref{eq:static_modulus}. The fit gives
   $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $D_\perp
   q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and
   $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of an additional
   parameter to fit the standard bare modulus \cite{varshni_temperature_1970}
   led to poorly constrained fits. (c) $\Bog$ modulus data and fit of
   \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and fit transformed by
   $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from
   \eqref{eq:static_modulus} to equal
   $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, e.g., an absolute value function. 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:data}
\end{figure}

\emph{Comparison to experiment.}
\Rus\ experiments \cite{ghosh_single-component_nodate} yield the individual
elastic moduli broken into irrep symmetry; the $\Bog$ and $\Btg$ components
defined in \eqref{eq:strain-components} are shown in Figures \ref{fig:data}(a--b).
The $\Btg$ modulus 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 $\Bog$ modulus, 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:static_modulus}, with a linear background modulus $C^0_\Bog$
and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure
\ref{fig:data}(b). 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. The
predicted softening appears over hundreds of Kelvin; Figures
\ref{fig:data}(c--d) show the background modulus $C_\Bog^0$ and the
\op--induced response isolated from each other.

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 $\Bog$ elastic
modulus 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 $\langle\epsilon_\Bog\rangle^2=b^2\tilde
r/4u(C^0_\Bog)^2$, 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 $\Bog$ modulus 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 known to be irrelevant to the critical
behavior at a one-component disordered to modulated transition, and therefore
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. \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 $4\frac12$ of a
one-component disordered to modulated transition, 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 one-component \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.

\emph{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 modulus. 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}
up to 
\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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