path: root/main.tex

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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 a staggered nematic}
\author{Jaron Kent-Dobias}
\author{Michael Matty}
\author{B.~J. Ramshaw}
\affiliation{
  Laboratory of Atomic \& Solid State Physics, Cornell University,
  Ithaca, NY, USA
}

\date\today

\begin{abstract}
  We develop a phenomenological mean field theory describing the hidden order
  phase in \urusi\ as a nematic of the $\Bog$ representation staggered along
  the $c$-axis. Several experimental features are reproduced by this theory:
  the topology of the temperature--pressure phase diagram, the response of the
  elastic modulus $(C_{11}-C_{12})/2$ above the transition at ambient 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, and the triple point
  joining those two phases with the high-temperature paramagnetic phase is a
  Lifshitz point.
\end{abstract}

\maketitle

\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}  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 associating any of a variety of broken symmetries
with \ho.  This work proposes yet another, motivated by two experimental
observations: first, the $\Bog$ ``nematic" elastic susceptibility
$(C_{11}-C_{12})/2$ softens anomalously from room temperature down to
$T_{\text{\ho}}=17.5\,\K$;\cite{de_visser_thermal_1986} and second, a $\Bog$
nematic distortion is observed by x-ray scattering under sufficient pressure to
destroy the \ho\ state.\cite{choi_pressure-induced_2018}

Recent resonant ultrasound spectroscopy (\rus) measurements were used to
examine the thermodynamic discontinuities in the elastic moduli at
$T_{\text{\ho}}$.\cite{ghosh_single-component_nodate} The observation of
discontinues only in compressional, or $\Aog$, elastic moduli requires that the
point-group representation of \ho\ be one-dimensional. This rules out many
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-independent way, but doesn't differentiate between those that remain. 

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 remains 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.\cite{wolf_elastic_1994, kuwahara_lattice_1997} Motivated by
these results---which hint 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 then determine the effect of its phase
transitions on the elastic response in different symmetry channels. 

We find that only one \op\ representation reproduces the anomalous $\Bog$
elastic modulus, which softens in a Curie--Weiss-like manner from room
temperature and then cusps at $T_{\text{\ho}}$. That theory associates \ho\
with a $\Bog$ \op\ modulated along the $c$-axis, the high pressure state with uniform
$\Bog$ order, and the triple point between them with a Lifshitz point. In
addition to the agreement with the ultrasound data across a broad temperature
range, the theory predicts uniform $\Bog$ strain at high pressure---the same
distortion that 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 $(C_{11}-C_{12})/2$ modulus diverges as the uniform
$\Bog$ strain of the high pressure phase is approached.


\section{Model}
The point group of \urusi\ is \Dfh, and any 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 $f_\e=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$.
\footnote{Components of the elastic modulus tensor $C$ were given in the
  popular Voigt notation in the abstract and introduction. Here and henceforth
the notation used is that natural for a rank-four tensor.} The form of the bare
moduli tensor $C^0$ is further restricted by symmetry. Linear combinations of
the six independent components of strain form five irreducible components of
strain in \Dfh\ 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 exists no component of strain that transforms like the representation
$\X$ then there can be no linear coupling. The next-order coupling is linear in
strain, quadratic in order parameter, and the effect of this coupling at a
continuous phase transition is to produce a jump in the $\Aog$ elastic moduli
if $\eta$ is single-component, \cite{luthi_sound_1970, ramshaw_avoided_2015,
shekhter_bounding_2013} and jumps in other elastic moduli if
multicomponent.\cite{ghosh_single-component_nodate} Because we are interested
in physics that anticipates the phase transition---for instance, that the
growing \op\ susceptibility is reflected directly in the elastic
susceptibility---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 u-reps (which are odd
under inversion), the $\Atg$ irrep, and all half-integer (spinor)
representations.

If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd
terms are allowed in its free energy and without fine-tuning any transition
will be first order and not continuous. Since the \ho\ phase transition is
second-order,\cite{de_visser_thermal_1986} we will henceforth rule out $\Aog$
\op s as well.  For the \op\ representation $\X$ as any of those
remaining---$\Bog$, $\Btg$, 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 the \op\ alone. Later we will invert
this procedure and trace out the \op\ when we compute the effective elastic
moduli. The only strain relevant to an \op\ of representation $\X$ 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 \eqref{eq:free_energy}, the resulting single-argument
free energy functional $F[\eta,\epsilon_\star[\eta]]$ has a density identical
to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$. 

\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 ambient 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 discussed in the remainder of this
section can all be found in a standard text, e.g., Chaikin \&
Lubensky.\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 of the \op\ representation, 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 \urusi, whose \ho\ phase is bounded by a line of first order transitions at high pressure,
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}.

\section{Results}
We will now derive the 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 representation $\X$ of the \op. Moreover, this modulus
does not vanish at the unordered to modulated transition---as it would if the
transition were a $q=0$ phase transition---but instead ends in a cusp. In this
section we start by computing the susceptibility of the \op\ at the unordered
to modulated transition, and then compute the elastic modulus for the same. 

The 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-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 functional reciprocal defined as
\begin{equation}
  \int dx''\,\chi^\recip(x,x'')\chi(x'',x')=\delta(x-x').
\end{equation}
Taking the Fourier transform and integrating out $q'$ gives
\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)
      &=\big[c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
    +|\Delta\tilde r|\big]^{-1} \\
      &=\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
perpendicular and parallel to the plane, respectively.  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 the lower point $a(T-T_c)=\Delta\tilde
r=-D_\perp q_*^4<0$, this is cut off with a cusp at $\Delta\tilde r=0$. 

The elastic susceptibility, which is the reciprocal of the effective elastic
modulus, is found 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 $\epsilon$ alone. 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 \op\ conditioned
on the configuration of the strain. Since $\eta_\star$ is a functional of $\epsilon_\X$
alone, only the modulus $C_\X$ will be modified from its bare value $C^0_\X$.

Though the differential equation for $\eta_\star$ cannot be solved explicitly, we
can use the inverse function theorem to make us \eqref{eq:implicit.eta} anyway.
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_\star[\epsilon]} \\
    &=b^{-1}\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta_\star[\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'') \\
    &\qquad\qquad\qquad+\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]}
    +\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]} \\ 
    &\qquad=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'') \\
    &\qquad\qquad\qquad\qquad+\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}(b\epsilon_\X(x''))
    +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\\ 
    &\qquad=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 \eqref{eq:big.boy}
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 zero-pressure transition to the HO state---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 approaching 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 nonzero $q=q_*$.  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\ at ambient pressure as a
   function of temperature from recent
   experiments\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 a fit to the standard
   form.\cite{varshni_temperature_1970} (b) $\Bog$ modulus data and a 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 the fit of the
   \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and the 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}

\section{Comparison to experiment}
\Rus\ experiments~\cite{ghosh_single-component_nodate} yield the individual
elastic moduli broken into irrep symmetries; data for the $\Bog$ and $\Btg$
components defined in \eqref{eq:strain-components} are shown in Figures
\ref{fig:data}(a--b).  The $\Btg$ in Fig.~\ref{fig:data}(a) 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
Fig.~\ref{fig:data}(b) 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 high pressure 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.  The onset of orthorhombic symmetry
breaking was recently detected at high pressure in \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 ultrasound experiments done under
pressure, where the depth of the cusp in the $\Bog$ modulus should deepen
(perhaps with these commensurability jumps) at low pressure and approach zero
as $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. 
Alternatively, \rus\ done at ambient pressure might examine the heavy fermi
liquid to \afm\ transition by doping. Though previous \rus\ studies have doped
\urusi\ with Rhodium,\cite{yanagisawa_ultrasonic_2014} the magnetic nature of
Rhodium ions likely artificially promotes magnetic phases. A dopant like
phosphorous that only exerts chemical pressure might more faithfully explore
the pressure axis of the phase diagram. Our work also motivates experiments
that can probe the entire correlation function---like x-ray and neutron
scattering---and directly resolve its finite-$q$ divergence.  The presence of
spatial commensurability is known to be irrelevant to critical behavior at a
one-component disordered to modulated transition, and therefore is not
expected to modify the thermodynamic behavior
otherwise.\cite{garel_commensurability_1976} 

There are two apparent discrepancies between the orthorhombic strain in the
phase diagram presented by recent x-ray data\cite{choi_pressure-induced_2018}
and that predicted by our mean field theory when its uniform ordered phase is
taken to be coincident with \urusi's \afm.  The first is the apparent onset of
the orthorhombic phase in the \ho\ state prior to the onset of \afm.  As the
recent x-ray research\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} We suspect
that the high-temperature orthorhombic signature is not the result of a bulk
phase, and could be due to the high energy (small-wavelength) nature of x-rays
as an experimental probe: \op\ fluctuations should lead to the formation of
orthorhombic regions on the order of the correlation length that become larger
and more persistent as the transition is approached. 

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. An ultrasound 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.

\section{Conclusion and Outlook.} We have developed a general phenomenological
treatment of  \ho\ \op s that have 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 high pressure phase
is characterized by uniform $\Bog$ order. 

The coinciding of our theory's orthorhombic high-pressure phase and \urusi's
\afm\ is compelling, but our mean field theory does not make any explicit
connection with the physics of \afm.  This may be reasonable since correlations
often lead to \afm\ as a secondary effect, like in many Mott insulators. An
electronic theory of this phase diagram may find that the \afm\ observed in
\urusi\ indeed follows along with a high-pressure orthorhombic phase associated with
uniform $\Bog$ electronic order. 

The corresponding prediction of uniform $\Bog$ symmetry breaking in the high
pressure phase is consistent with recent diffraction experiments,
\cite{choi_pressure-induced_2018} except for the apparent earlier onset in
temperature of the $\Bog$ symmetry breaking, which we believe to be due to
fluctuating order above the actual phase transition.  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}
  Jaron Kent-Dobias is supported by NSF DMR-1719490, Michael Matty is supported by
  NSF DMR-1719875, and Brad Ramshaw is supported by NSF DMR-1752784. We are
  grateful for helpful discussions with Sri Raghu, Danilo Liarte, and Jim
  Sethna, and for permission to reproduce experimental data in our figure by
  Elena Hassinger. We thank Sayak Ghosh for \rus\ data.
\end{acknowledgements}

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