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

% Our mysterious boy
\def\urusi{URu$_{\text2}$Si$_{\text2}$}

\def\e{{\text e}} % "elastic"
\def\o{{\text o}} % "order parameter"
\def\i{{\text i}} % "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{\c A}}

% Other
\def\G{\text G} % Ginzburg

\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 order parameter, 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 pd paper], and at sufficiently large [hydrostatic?]
pressures, both give way to local moment antiferromagnetism.  Despite over
thirty years of effort, the symmetry of the hidden order 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].
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 order parameter. 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].  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 order parameter, and their interaction.
The most general quadratic free energy of the strain $\epsilon$ is
$f_\e=\lambda_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the $\lambda$
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}
\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\lambda_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},
\end{equation}
where the sum is over irreducible representations of the point group and the
stiffnesses $\lambda_\X^{(ij)}$ are
\begin{equation}
  \begin{aligned}
    &\lambda_{\Aog}^{(11)}=\tfrac12(\lambda_{1111}+\lambda_{1122}) &&
    \lambda_{\Aog}^{(22)}=\lambda_{3333} \\
    &\lambda_{\Aog}^{(12)}=\lambda_{1133} &&
    \lambda_{\Bog}^{(11)}=\tfrac12(\lambda_{1111}-\lambda_{1122}) \\
    &\lambda_{\Btg}^{(11)}=\lambda_{1212} &&
    \lambda_{\Eg}^{(11)}=\lambda_{1313}.
  \end{aligned}
\end{equation}
The interaction between strain and the order parameter $\eta$ 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 order parameter symmetries that produce linear couplings to
strain.

If the order parameter 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. 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}{\delta\epsilon_{\X i}(x)}=\lambda_\X\epsilon_{\X i}(x)
    +\frac12b\eta_i(x)
\end{equation}
gives $\epsilon_\X(x)=-(b/2\lambda_\X)\eta(x)$. Upon substitution into the free
energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$,
with $r\to\tilde r=r-b^2/4\lambda_\X$.

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 order parameter ($\Bog$ 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}
\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 order parameter ($\Eg$) 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 $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$ and
\begin{equation}
  \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{16D_\perp u}
    =\frac{\tilde r_c-\tilde r}{4u}
\end{equation}
The uniform--modulated transition is now continuous. 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.
  }
  \label{fig:phases}
\end{figure}

The susceptibility is given by
\begin{equation}
  \begin{aligned}
    &\chi_{ij}^{-1}(x,x')
    =\frac{\delta^2F}{\delta\eta_i(x)\delta\eta_j(x')} \\
    &\quad=\Big[\big(\tilde r-c_\parallel\nabla_\parallel^2
      -c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+4u\eta^2(x)\big)\delta_{ij} \\
    &\qquad\qquad +8u\eta_i(x)\eta_j(x)\Big]\delta(x-x'),
  \end{aligned}
\end{equation}
or in Fourier space,
\begin{equation}
  \begin{aligned}
    \chi_{ij}^{-1}(q)
      &=8u\sum_{q'}\tilde\eta_i(q')\eta_j(-q')+\bigg(\tilde r
        +c_\parallel q_\parallel^2-c_\perp q_\perp^2 \\
      &\qquad+D_\perp q_\perp^4+4u\sum_{q'}\tilde\eta_k(q')\tilde\eta_k(-q')\bigg)
        \delta_{ij}.
  \end{aligned}
\end{equation}
Near the unordered--modulated transition this yields
\begin{equation}
  \begin{aligned}
    \chi_{ij}(q)
    &=\frac{\delta_{ij}}{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
      +|\tilde r-\tilde r_c|} \\
    &=\frac{\delta_{ij}}{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 $\xi_\perp=(|\tilde r-\tilde r_c|/D_\perp)^{-1/4}$ and
$\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$.

The elastic susceptibility (inverse stiffness) is given in the same way: we
must trace over $\eta$ and take the second variation of the resulting free
energy. Extremizing over $\eta$ yields
\begin{equation}
  0=\frac{\delta F}{\delta\eta_i(x)}=\frac{\delta F_\o}{\delta\eta_i(x)}
    +\frac12b\epsilon_{\X i}(x),
  \label{eq:implicit.eta}
\end{equation}
which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this
cannot be solved explicitly, we can make use of the inverse function theorem to
write
\begin{equation}
  \begin{aligned}
    \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1}
    &=\frac{\delta\eta_j^{-1}[\eta](x)}{\delta\eta_i(x')}
    =-\frac2b\frac{\delta^2F_\o}{\delta\eta_i(x)\delta\eta_j(x')} \\
    &=-\frac2b\chi_{ij}^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta_{ij}\delta(x-x')
  \end{aligned}
  \label{eq:inv.func}
\end{equation}
It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the
susceptibility of the material to $\epsilon_\X$ strain is given by
\begin{widetext}
\begin{equation}
  \begin{aligned}
    \chi_{\X ij}^{-1}(x,x')
    &=\frac{\delta^2F}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\
    &=\lambda_\X\delta_{ij}\delta(x-x')+
    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
    +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_{\X i}(x')\delta\epsilon_{\X j}(x'')} \\
    &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o}{\delta\eta_k(x'')\delta\eta_\ell(x''')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')}
    +\int dx''\,\frac{\delta F_\o}{\delta\eta_k(x'')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ 
    &=\lambda_\X\delta_{ij}\delta(x-x')+
    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
    -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta_k(x'')}{\partial\epsilon_{\X\ell}(x''')}\bigg)^{-1}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} \\ 
    &=\lambda_\X\delta_{ij}\delta(x-x')+
    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
    -\frac12b\int dx''\,\delta_{i\ell}\delta(x-x'')\frac{\delta\eta_\ell(x'')}{\delta\epsilon_{\X j}(x')} 
    =\lambda_\X\delta_{ij}\delta(x-x')+
    \frac12b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')},
  \end{aligned}
\end{equation}
\end{widetext}
whose Fourier transform follows from \eqref{eq:inv.func} as
\begin{equation}
  \chi_{\X ij}(q)=\frac{\delta_{ij}}{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q).
  \label{eq:elastic.susceptibility}
\end{equation}
At $q=0$, which is where the stiffness measurements used here were taken, this
predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde
r_c|^\gamma$ 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 ultrasound. The vertical
    dashed lines show the location of the hidden order transition.
  }
  \label{fig:data}
\end{figure}

We have seen that mean field theory predicts that whatever component of strain
transforms like the order parameter will see a $t^{-1}$ softening in the
stiffness that ends in a cusp. Ultrasound experiments \textbf{[Elaborate???]}
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 \textbf{[What's this called?
Citation?]}. 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
$\lambda_\Bog^{(11)}$ 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
    $\lambda_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
    $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, 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, or since the correlation function is
$k_BT\chi(x,x')$,
\begin{equation}
  V_\xi^{-1}k_BT\int_{V_\xi}d^3x\,\chi(x,0)
  =\langle\delta\eta^2\rangle_{V_\xi}
  \lesssim\frac12\eta_*^2=\frac{|\Delta\tilde r|}{6u}
\end{equation}
with $V_\xi$ the correlation volume, which we will take to be a cylinder of
radius $\xi_\parallel/2$ and height $\xi_\perp$. Upon substitution of
\eqref{eq:susceptibility} and using the jump in the specific heat at the
transition from
\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}
this expression can be brought to the form
\begin{equation}
  \frac{2k_B}{\pi\Delta c_V\xi_{\perp0}\xi_{\parallel0}^2}
    \mathcal I(\xi_{\perp0} q_*|t|^{-1/4})
  \lesssim |t|^{13/4},
\end{equation}
where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I(x)\sim x^{-4}$ for large $x$, yielding
\begin{equation}
  t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4}
\end{equation}
Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$
\cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp
q_*^4/aT_c)^{1/4}\sim2$. We have reason to believe that at zero pressure, very
far from the Lifshitz point, $q_*$ is roughly the inverse lattice spacing
\textbf{[Why???]}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$,
we find $t_\G\sim0.04$, so that an experiment would need to be within
$\sim1\,\K$ to detect a deviation from mean field behavior. An ultrasound
experiment able to capture data over several decades within this vicinity of
$T_c$ may be able to measure a cusp with $|t|^\gamma$ for
$\gamma=\text{\textbf{???}}$, the empirical exponent \textbf{[Citation???]}.
Our analysis has looked at behavior for $T-T_c>1\,\K$, and so it remains
self-consistent.

There are two apparent discrepancies between the phase diagram presented in 
[cite] 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] 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. We 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. 
This is similar to the situation seen in [cite cuprate x-ray source]. 

\begin{acknowledgements}

\end{acknowledgements}

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

\end{document}