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\o{{\text{\textsc{Op}}}} % "order parameter"
\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\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy 
\def\recip{{\{-1\}}} % functional reciprocal

\begin{document}

\title{Elastic properties of \urusi\ are reproduced by modulated $\Bog$ order}
\author{Jaron Kent-Dobias}
\author{Michael Matty}
\author{Brad Ramshaw}
\affiliation{
  Laboratory of Atomic \& Solid State Physics, Cornell University,
  Ithaca, NY, USA
}

\date\today

\begin{abstract}
  We develop a phenomenological theory for the elastic response of materials
  with a \Dfh\ point group through phase transitions. The physics is
  generically that of Lifshitz points, with disordered, uniform ordered, and
  modulated ordered phases. Several experimental features of \urusi\ 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, and the strain response in the
  antiferromagnetic phase. In this scenario, the hidden order is a version of
  the high-pressure antiferromagnetic order modulated along the symmetry axis.
\end{abstract}

\maketitle

% \begin{enumerate}
%   \item Introduction
%     \begin{enumerate}
%       \item \urusi\ hidden order intro paragraph, discuss the phase diagram
%       \item Strain/OP coupling discussion/RUS
%       \item Discussion of experimental data
%       \item We look at MFT's for OP's of various symmetries
%     \end{enumerate}
      
%   \item Theory
%     \begin{enumerate}
%       \item Introduce various pieces of free energy
      
%       \item Summary of MFT results
%     \end{enumerate}
    
%     \item Data piece
    
%     \item Talk about more cool stuff like AFM C4 breaking etc
% \end{enumerate}

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.  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_2019, ikeda_emergent_2012} propose a
variety of possibilities.  Many [all?] 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_2019} 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 elastic 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_2019} 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.

We first present a phenomenological Landau-Ginzburg mean field theory of strain
coupled to an \op. We examine the phase diagram predicted by this
theory and compare it 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 proceed to 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.

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 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. 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
tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic
free energy density is
\begin{equation}
  \begin{aligned}
    f_\o=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2
      +c_\perp(\nabla_\perp\eta)^2 \\
      &\quad+D_\parallel(\nabla_\parallel^2\eta)^2
      +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 $\Atu$. We'll take $D_\parallel=0$
since this does not affect the physics at hand. The full free energy functional of $\eta$ and $\epsilon$ is then
\begin{equation}
  F[\eta,\epsilon]=\int dx\,(f_\o+f_\e+f_\i)
\end{equation}
Neglecting interaction terms
higher than quadratic order, the only strain relevant to the problem is
$\epsilon_\X$, and this can be traced out of the problem exactly, since
\begin{equation}
  0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_{\X i}(x)}=C_\X\epsilon_{\X i}(x)
    +\frac12b\eta_i(x)
\end{equation}
gives $\epsilon_\X[\eta]=-(b/2C_\X)\eta$. Upon substitution into the free
energy, the resulting effective free energy $F_\o[\eta]=F[\eta,\epsilon[\eta]]$ has a density identical to $f_\o$
with $r\to\tilde r=r-b^2/4C_\X$.

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 $\eta(x)=\eta_*\cos(q_*x_3)$. 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 $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$.
The uniform--modulated transition is now continuous. This already does not reproduce the physics of \ho, and so we will henceforth neglect this possibility. The schematic phase
diagrams for this model are shown in Figure \ref{fig:phases}.

\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}

The susceptibility of the order parameter to a field linearly coupled to it is given by
\begin{equation}
  \begin{aligned}
    &\chi^\recip(x,x')
      =\frac{\delta^2F_\o[\eta]}{\delta\eta(x)\delta\eta(x')}
      =\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2 \\
    &\qquad\qquad+D_\perp\nabla_\perp^4+12u\eta^2(x)\big)
    \delta(x-x'),
  \end{aligned}
  \label{eq:sus_def}
\end{equation}
where $\recip$ indicates a \emph{functional reciprocal} in the sense that
\[
  \int dx''\,\chi^\recip(x,x'')\chi(x'',x')=\delta(x-x').
\]
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'}\tilde\eta_{q'}\tilde\eta_{-q'}\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 $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}$ and
$\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}$. 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)}=\frac{\delta F_\o[\eta]}{\delta\eta(x)}
    +\frac12b\epsilon_\X(x),
  \label{eq:implicit.eta}
\end{equation}
which implicitly gives $\eta[\epsilon]$ and $F_\e[\epsilon]=F[\eta[\epsilon],\epsilon]$. Since $\eta$ is a functional of $\epsilon_\X$ alone, only the stiffness $\lambda_\X$ is modified from its bare value $C_\X$. Though this
cannot be solved explicitly, we can make use of the inverse function theorem.
First, denote by $\eta^{-1}[\eta]$ the inverse functional of $\eta$ implied by
\eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$ corresponding
to each solution of \eqref{eq:implicit.eta} it receives. Now, we use the inverse function
theorem to relate the functional reciprocal of the derivative of $\eta[\epsilon]$ with respect to $\epsilon_\X$ to the derivative of $\eta^{-1}[\eta]$ with respect to $\eta$, yielding
\begin{equation}
  \begin{aligned}
    \bigg(\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}\bigg)^\recip
    &=\frac{\delta\eta^{-1}[\eta](x)}{\delta\eta(x')}
    =-\frac2b\frac{\delta^2F_\o[\eta]}{\delta\eta(x)\delta\eta(x')} \\
    &=-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x'),
  \end{aligned}
  \label{eq:inv.func}
\end{equation}
where we have used what we already know about the variation of $F_\o[\eta]$ with respect to $\eta$.
Finally, \eqref{eq:implicit.eta} and \eqref{eq:inv.func} can be used in concert with the ordinary rules of functional calculus to yield the strain stiffness
\begin{widetext}
\begin{equation}
  \begin{aligned}
    \lambda_\X(x,x')
    &=\frac{\delta^2F_\e[\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\
    &=C_\X\delta(x-x')+
    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
    +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')} \\
    &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o[\eta]}{\delta\eta(x'')\delta\eta(x''')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')}
    +\int dx''\,\frac{\delta F_\o[\eta]}{\delta\eta(x'')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\ 
    &=C_\X\delta(x-x')+
    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
    -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta(x'')}{\partial\epsilon_\X(x''')}\bigg)^{-1}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')} \\ 
    &=C_\X\delta(x-x')+
    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
    -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta(x'')}{\delta\epsilon_\X(x')} 
    =C_\X\delta(x-x')+
    \frac12b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')},
  \end{aligned}
\end{equation}
\end{widetext}
whose Fourier transform follows from \eqref{eq:inv.func} as
\begin{equation}
  \lambda_\X(q)=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 order parameters.
At $q=0$, which is where the stiffness measurements used here were taken, this
predicts a cusp in the elastic susceptibility 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}

We have seen that mean field theory predicts that whatever component of strain
transforms like the \op\ will see a $t^{-1}$ softening in the
stiffness that ends in a cusp. \Rus\ experiments \cite{ghosh_single-component_2019}
yield the strain stiffness for various components of the strain; this data is
shown in Figure \ref{fig:data}.  The $\Btg$ and $\Eg$ stiffnesses don'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 $a/D_\perp
    q_*^4\simeq0.0038\,\K^{-1}$.
  }
  \label{fig:fit}
\end{figure}

Mean field theory neglects the effect of fluctuations on critical behavior, yet
also predicts the magnitude of those fluctuations. This allows a mean field
theory to undergo an internal consistency check to ensure the predicted
fluctuations are indeed negligible. This is typically done by computing the
Ginzburg criterion \cite{ginzburg_remarks_1961}, which gives the proximity to
the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to
break down by comparing the magnitude of fluctuations in a correlation-length
sized box to the magnitude of the field. In the modulated phase the spatially
averaged magnitude is zero, and so we will instead compare fluctuations in the
\emph{amplitude} at $q_*$ to the magnitude of that amplitude. Defining the
field $\alpha$ by $\eta(x)=\alpha(x)e^{-iq_*x_3}$, it follows that in the
modulated phase $\alpha(x)=\alpha_0$ for $\alpha_0^2=|\delta \tilde r|/4u$. In
the modulated phase, the $q$-dependant fluctuations in $\alpha$ are given by
\[
  G_\alpha(q)=k_BT\chi_\alpha(q)=\frac1{c_\parallel q_\parallel^2+D_\perp(4q_*^2q_\perp^2+q_\perp^4)+2|\delta r|},
\]
An estimate of the Ginzburg criterion is then given by the temperature at which
$V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle\simeq\langle\alpha\rangle^2=\alpha_0^2$,
where $V_\xi=\xi_\perp\xi_\parallel^2$ is a correlation volume. The parameter $u$
can be replaced in favor of the jump in the specific heat at the transition
using
\begin{equation}
  c_V=-T\frac{\partial^2f}{\partial T^2}
    =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c.\end{cases}
\end{equation}
The integral over the correlation function $G_\alpha$ can be preformed up to one integral analytically using a Gaussian-bounded correlation volume, yielding $\langle\Delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}\delta r^{-1}\mathcal I(\xi_\perp q_*)$ for
\[
  \mathcal I(x)=-2^{3/2}\pi^{5/2}\int dy\,e^{[2+(4x^2-1)y^2+y^4]/2}
\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big)
\]
This gives a transcendental equation
\[
  \mathcal I(\xi_{\perp0}q_*\delta t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}\delta t_\G^{3/4}.
\]
Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$
\cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp
q_*^4/aT_c)^{1/4}\simeq2$. We have reason to believe that at zero pressure, very
far from the Lifshitz point, the half-wavelength of the modulation should be commensurate with the lattice, giving $q_*\simeq0.328\,\A^{-1}$ 
\cite{meng_imaging_2013}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$,
we find $\delta t_\G\sim0.4$, though this estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in $\delta t_\G$ over nearly four orders of magnitude.
The estimate here predicts that an experiment may begin to see deviations from
mean field behavior within around $5\,\K$ 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}. Our work here appears self--consistent, given that our fit is mostly concerned with temperatures farther than this from the critical point. This analysis also indicates that 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.

There are two apparent discrepancies between the phase diagram presented in 
\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 ref.\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. 

\begin{acknowledgements}

\end{acknowledgements}

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

\end{document}