path: root/main.tex

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

% 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{An exact relation between response functions of linearly coupled fields:}
\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 elastic response of
  \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 a version of the high-pressure
  antiferromagnetic order 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 a central theme of condensed matter physics.
In many cases, a phase transition between different states of matter is marked
by a change in symmetry.  In this paradigm, the breaking of symmetry in an
ordered phase corresponds to the condensation of an order parameter (\op) that
breaks the same symmetries. Near a second order phase transition, the physics
of the \op\ can often be described in the context of Landau--Ginzburg mean field
theory. However, to construct such a theory, one must know the symmetries of
the \op, i.e. the symmetry of the ordered state.

A paradigmatic example where the symmetry of an ordered phase remains unknown
is in \urusi.  \urusi\ is a heavy fermion superconductor in which
superconductivity condenses out of a symmetry broken state referred to as
hidden order (\ho) \cite{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 without direct experimental observation of the
broken symmetry, none have been confirmed. 

One case that does not rely on a microscopic model is recent work
\cite{ghosh_single-component_nodate} that studies the \ho\ transition using
resonant ultrasound spectroscopy (\rus).  \Rus\ is an experimental technique
that measures mechanical resonances of a sample. These resonances contain
information about the full strain stiffness tensor of the material. Moreover,
the frequency locations of the resonances are sensitive to symmetry breaking at
an electronic phase transition due to electron-phonon coupling
\cite{shekhter_bounding_2013}.  Ref.~\cite{ghosh_single-component_nodate} uses
this information to place strict thermodynamic bounds on the symmetry of the
\ho\ \op, again, independent of any microscopic model. Motivated by these
results, in this paper we consider a mean field theory of an \op\ coupled to
strain and the effect that the \op\ symmetry has on the elastic response in
different symmetry channels. Our study finds that a single possible \op\
symmetry reproduces the experimental strain susceptibilities and fits the
experimental data well. The resulting theory associates \ho\ with $\Bog$ order
\emph{modulated along the rotation axis}, \afm\ with uniform $\Bog$ order, and
a Lifshitz point with the triple point between them.

We first present a phenomenological Landau--Ginzburg mean field theory of
strain coupled to an \op. We examine the phase diagrams predicted by this
theory for various \op\ symmetries and compare them to the experimentally
obtained phase diagram of \urusi.  Then we compute the elastic response to
strain, and examine the response function dependence on the symmetry of the
\op.  We compare the results from mean field theory with data from \rus\
experiments.  We further examine the consequences of our theory at non-zero
applied pressure in comparison with recent x-ray scattering experiments
\cite{choi_pressure-induced_2018}. Finally, we discuss our conclusions and the
future experimental and theoretical work motivated by our results.

\begin{equation}
  F[\varphi,\vartheta]=F_1[\varphi]+F_2[\vartheta]-\int dx\,b\varphi(x)\vartheta(x)
\end{equation}
\begin{equation}
  \begin{aligned}
    0&=\frac{\delta F[\varphi,\vartheta]}{\delta\vartheta(x)}\bigg|_{\vartheta=\vartheta_\star}
    =\frac{\delta F_2[\vartheta]}{\delta\vartheta(x)}\bigg|_{\vartheta=\vartheta_\star}-b\varphi(x)
  \end{aligned}
\end{equation}
\begin{equation}
  \vartheta_\star^{-1}[\vartheta](x)=\frac1b\frac{\delta F_2[\vartheta]}{\delta\vartheta(x)}
\end{equation}
\begin{equation}
  \bigg(\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}\bigg)^\recip
  =\frac{\delta\vartheta_\star^{-1}[\vartheta](x)}{\delta\vartheta(x')}\bigg|_{\vartheta=\vartheta_\star[\varphi]}
  =\frac1b\frac{\delta^2F_2[\vartheta]}{\delta\vartheta(x)\delta\vartheta(x')}\bigg|_{\vartheta=\vartheta_\star[\varphi]}
\end{equation}
\begin{widetext}
\begin{equation}
  \begin{aligned}
    \frac{\delta^2F[\varphi,\vartheta_\star(\varphi)]}{\delta\varphi(x)\delta\varphi(x')} 
    &=\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}-
    2b\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}
    -b\int dx''\,\frac{\delta^2\vartheta_\star[\varphi](x)}{\delta\varphi(x')\delta\varphi(x'')}\varphi(x'') +\int dx''\,\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x)\delta\varphi(x')}\frac{\delta F_2[\vartheta]}{\delta\vartheta(x'')}\bigg|_{\vartheta=\vartheta_\star[\varphi]} \\
    &\qquad\qquad+\int dx''\,dx'''\,\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x)}\frac{\delta\vartheta_\star[\varphi](x''')}{\delta\varphi(x')}\frac{\delta^2F_2[\vartheta]}{\delta\vartheta(x'')\delta\vartheta(x''')}\bigg|_{\vartheta=\vartheta_\star[\varphi]}
    \\ 
    &=\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}
    -2b\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}-b\int dx''\,\frac{\delta^2\vartheta_\star[\varphi](x)}{\delta\varphi(x')\delta\varphi(x'')}\varphi(x'') +\int dx''\,\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x)\delta\varphi(x')}(b\varphi(x'')) \\
    &\qquad\qquad+b\int dx''\,dx'''\,\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x)}\frac{\delta\vartheta_\star[\varphi](x''')}{\delta\varphi(x')}\bigg(\frac{\partial\vartheta_\star[\varphi](x'')}{\partial\varphi(x''')}\bigg)^\recip \\ 
    &=\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}-
    2b\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}
    +b\int dx''\,\delta(x-x'')\frac{\delta\vartheta_\star[\varphi](x'')}{\delta\varphi(x')} 
    =\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}-
    b\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}.
  \end{aligned}
\end{equation}
since $\vartheta_\star[\langle\varphi\rangle]=\langle\vartheta\rangle$,
\begin{equation}
  \bigg(\frac{\delta\vartheta_\star[\varphi](x)}{\delta\varphi(x')}\bigg)^\recip\bigg|_{\varphi=\langle\varphi\rangle}
  =\frac1b\frac{\delta^2F_2[\vartheta]}{\delta\vartheta(x)\delta\vartheta(x')}\bigg|_{\vartheta=\langle\vartheta\rangle}
  =\frac1b\chi_\vartheta^\recip(x,x')+\frac{\delta\varphi_\star[\vartheta](x)}{\delta\vartheta(x')}\bigg|_{\vartheta=\langle\vartheta\rangle}
  =\frac1b\chi_\vartheta^\recip(x,x')+\frac1b\frac{\delta^2F_1[\varphi]}{\delta\varphi(x)\delta\varphi(x')}\bigg|_{\varphi=\langle\varphi\rangle}
\end{equation}
\begin{equation}
  \begin{aligned}
    \chi_\varphi^\recip(x,x')=
    \frac{\delta^2F[\varphi,\vartheta_\star(\varphi)]}{\delta\varphi(x)\delta\varphi(x')} \bigg|_{\varphi=\langle\varphi\rangle}
    =
      \end{aligned}
\end{equation}
\end{widetext}

The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
respect this symmetry. We will introduce a phenomenological free energy density
in three parts: that of the strain, the \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 that $\epsilon$ is a
symmetric tensor and by the point group symmetry \cite{landau_theory_1995}. The
latter can be seen in a systematic way. First, the six independent components
of strain are written as linear combinations that behave like irreducible
representations under the action of the point group, or
\begin{equation}
  \begin{aligned}
    \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth}
    \epsilon_\Aog^{(2)}=\epsilon_{33}               \\
    \epsilon_\Bog^{(1)}=\epsilon_{11}-\epsilon_{22} &&
    \epsilon_\Btg^{(1)}=2\epsilon_{12}               \\
    \epsilon_\Eg^{(1)}=2\{\epsilon_{11},\epsilon_{22}\}.
  \end{aligned}
  \label{eq:strain-components}
\end{equation}
Next, all quadratic combinations of these irreducible strains that transform
like $\Aog$ are included in the free energy as
\begin{equation}
  f_\e=\frac12\sum_\X C_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},
\end{equation}
where the sum is over irreducible representations of the point group and the
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 the \op\ $\eta$ depends on the
representation of the point group that $\eta$ transforms as. If this
representation is $\X$, then the most general coupling to linear order is
\begin{equation}
  f_\i=b^{(i)}\epsilon_\X^{(i)}\eta
\end{equation}
If $\X$ is a representation not present in the strain there can be no linear
coupling, and the effect of $\eta$ going through a continuous phase transition
is to produce a jump in the $\Aog$ strain stiffness \cite{luthi_sound_1970,
ramshaw_avoided_2015, shekhter_bounding_2013}. We will therefore focus our
attention on \op\ symmetries that produce linear couplings to strain. Looking
at the components present in \eqref{eq:strain-components}, this rules out all
of the u-reps (odd under inversion) and the $\Atg$ irrep as having any
anticipatory response in the strain stiffness.

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. 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 then
\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 i}(x)}\bigg|_{\epsilon=\epsilon_\star}=C_\X\epsilon^\star_{\X i}(x)
    +\frac12b\eta_i(x)
\end{equation}
gives the optimized strain conditional on the \op\ as
$\epsilon_\X^\star[\eta](x)=-(b/2C_\X)\eta(x)$ 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 $r\to\tilde r=r-b^2/4C_\X$.

\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. 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 field
linearly coupled to it 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. 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 free
energy functional of $\epsilon$. Extremizing over $\eta$ yields
\begin{equation}
  0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}=
    \frac12b\epsilon_\X(x)+\frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star},
  \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 $\lambda_\X$ is 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
$\eta^{-1}_\star[\eta](x)=-2/b(\delta F_\op[\eta]/\delta\eta(x))$.  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]} 
    =-\frac2b\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')+
    b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
    +\frac12b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') \\
    &\quad+\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]}
    +\int dx''\,\frac{\delta\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]} \\ 
    &=C_\X\delta(x-x')+
    b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
    -\frac12b\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')+
    b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}
    -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x')} 
    =C_\X\delta(x-x')+
    \frac12b\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 solution $\langle\epsilon\rangle$. 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}
    =-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x'),
  \label{eq:recip.deriv.op}
\end{equation}
where $\chi^\recip$ is the \op\ susceptibilty 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-\frac b2\bigg(\frac2{b\chi(q)}+\frac b{2C_\X}\bigg)^{-1}
  =C_\X\bigg(1+\frac{b^2}{4C_\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 strain stiffness 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 for various components of the strain; this data 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 consistent.

\begin{figure}[htpb]
  \includegraphics[width=\columnwidth]{fig-fit}
  \caption{
    Strain stiffness data for the $\Bog$ component of strain (solid) along with
    a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$
    (dashed). The fit gives
    $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
    $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
    q_*^4\simeq0.0038\,\K^{-1}$. 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. There are several implications of this theory. 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 $\langle\epsilon_\Bog\rangle^2=b^2\tilde
r/16uC_\Bog^2$, 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 the modulation wavevector
$q_*$ should continuously vanish. 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. 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
in fact 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 argeement 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, [Mike's grant], [Brad's
  grants????]. The authors would like to thank [ask Brad] for helpful
  correspondence.
\end{acknowledgements}

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

\end{document}