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\title{LETTER TO THE EDITOR/REFEREES} 

\pacs{} \maketitle

We thank the editor for organizing the review of our manuscript and are grateful to the referees for their valuable comments, which have strengthened our work and made it more impactful.
We respond in detail to the referees' comments below.
\\[12pt]
To summarize, we have:
\begin{enumerate}
  \item Added an additional interaction to our mean field theory free energy of
    the form $\epsilon^2 \eta^2$ and a corresponding appendix detailing its
    impact on the mean field modulus.
  \item Shown a new fit to the experimental data incorporating our new
    interaction, dramatically improving the fit below $T_c$.
  \item Incorporated a discussion of possible relationships between our
    abstract order parameter and physical order parameters in the conclusion
  \item Added a statement emphasizing the novelty of our incorporation of
    gradient terms to the mean-field analysis of ultrasound data
  \item Compared the relevant features of our experimental ultrasound data to
    previous ultrasound studies.
\end{enumerate}
We are confident that the revised version is much improved with the valuable new insights made possible by the referees suggestions. We hope the paper can now be published without delay.
\\[12pt]
Regards,
\\[12pt]
Jaron Kent-Dobias, Michael Matty, and Brad Ramshaw
\\[12pt]
\ruleline{Report of First Referee -- BN13654/Kent-Dobias}
\newline

{\color{blue}
    The work deals with a purely phenomenological model for the “hidden” 
    order parameter if URu2Si2, with particular emphasis on the expected 
    elastic properties. The work might eventually be suitable for Phys. 
    Rev. B, but some aspects are not clear to me. 
 
    The main result is Fig. 2, where the behavior around TN is difficult 
    to see. I suggest to add zooms on that crucial T-range, where it seems 
    to me that there is a qualitative difference between model and 
    experiments. The justification given by the Authors (“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”) does not 
    look plausible as the disagreement does not appear to develop slowly 
    as T decreases, but appears immediately below TN, where eta is small. 
}\\

We agree, it's not at all obvious that it is a failure of the small $\eta$ and $\epsilon$ expansion that leads to the gross disagreement between the data and theory below the transition. To investigate this, we added the  interaction in the mean-field free energy
of the form $\epsilon^2 \eta^2$. In a new appendix, we have worked through the mean field
modulus implied with this new interaction and a fit is now shown in Fig. 2 as a
dashed black line.

Though fine features of the low-temperature behavior are not reproduced, the
addition of further terms in the mean-field free energy supply finer
corrections. For instance, another correction at order $\eta^2$ (roughly linear in
$\Delta T$) is produced by a term of the form $\epsilon^4$, while myriad terms at yet higher
order provide corrections of order $\eta^4$ and up (quadratic and up in $\Delta T$).
Higher order corrections to the mean-field free energy produce arbitrary
analytic corrections to the low-temperature behavior, but do not influence
predictions above the transition. Those can only be further fit by more
complicated dependence of the present free-energy parameters on temperature and
pressure. We have also left the original fit (without the extra term) since the model is more simply described in the text.

As suggested, an inset with a zoom on the critical region has been added to the figure.\\

{\color{blue}
    Is it not clear how discriminatory is the agreement above TN in 2a, 2b 
    and 2c. Are calculation results robust over a wide range of fitting 
    parameters, or does the agreement result from a fine-tuning? (e.g., 
    the presence of a maximum at 120 K in 2b). 
}\\

In order to check the agreement in our fit, we performed the fit with a moving
temperature window that cuts off at $T_\text{max}$. The fit functions that
result from varying the cutoff are shown in Figure \ref{fig:parameter_curve}.
The result is fairly robust until $T_\text{max}$ is reduced to around
$90\,\mathrm K$.  As with any nonlinear least-squares fit, some linear
combinations of parameters are stiff and well-constrained while others are
sloppy and very poorly constrained. The parameters that vary most with the
temperature window are those that belong to the sloppiest linear combinations.
Notably, the parameters that make up the bare modulus $C_0$---and therefore
those responsible for transforming the experimental data in Figure 2(d)---are
robust as a function of the temperature cutoff and therefore so is the modified
Curie--Weiss behavior observed in that figure.\\


\begin{figure}
  \centering
  \includegraphics[width=0.7\textwidth]{referee_response_cutoff_curves.pdf}
  \caption{
    Fit to the data with the color given by value of $T_\text{max}$ (the maximum of temperature range of the data used in the fit) that produced the fit. Data shown as black line (mostly underneath the red curve).
  }
  \label{fig:parameter_curve}
\end{figure}

{\color{blue}
    Is it possible to say something about the c/a ratio, which displays a 
    non-trivial T-dependence? 
}\\

The behaviour of $c/a$ is indeed interesting, but our model only considers the
coupling to the two in-plane shear strains, since it is one of these that shows
the anomalous behaviour. To talk about the $c/a$ ratio we would have to
introduce coupling between the order parameter and the $\text A_\text{1g}$
strains ($\epsilon_{xx} + \epsilon_{yy}$, and $\epsilon_{zz}$). Because the
order parameter we consider breaks both translational and (locally) point-group
symmetries, this coupling would be quadratic-in-order-parameter,
linear-in-strain, and would thus be generic to any order parameter. Put more
simply - our model has special coupling to a particular shear strain, whereas
the c/a ratio is related to compressional strains, which couples to our order
parameter in the same way as it does to any other (non-A$_\text{1g}$) order
parameter. \\

{\color{blue}
    At last, I understand that the model is meant to be purely 
    phenomenological, but given the plethora of publications on URu2Si2 
    over 30 years, where any conceivable order parameter has been proposed 
    as candidate, the Authors should make a connection between their 
    abstract OP and possible physical realizations. For instance, in the 
    simplest framework of localized f-electrons, what ionic moments would 
    fit the present proposal?
}\\

We thank the referee for bringing up this point. We have added a statement
about possible physical realizations to the conclusion section of our
manuscript, which we believe broadens the appeal of our work by connecting it
to the large body of research concerning microscopic theories of hidden order.
As we now say in the manuscript, physically, our phenomenological order
parameter could correspond to B$_\text{1g}$ multipolar ordering originating
from the localized component of the U-5f electrons.  For the crystal field
states of URu$_2$Si$_2$, this could correspond either to electric quadropolar
or hexadecapolar order based on the available multipolar operators.\\

\ruleline{Report of the Second Referee -- BN13654/Kent-Dobias}\\

{\color{blue}
    In this paper, possible elastic properties of URu2Si2 are studied with 
    focusing on the long-standing hidden order (HO) problem. The authors 
    introduce a generic form of the free energy density for the elastic 
    energy, a modulated order parameter, and their mutual coupling, and 
    analyze the temperature dependences of the elastic constants by 
    minimizing the free energy. It is shown that the B1g component 
    exhibits a remarkable softening with decreasing temperature and a cusp 
    singularity at the HO transition point, and these results are compared 
    with recent ultrasound experiments. From the comparison, the authors 
    conclude that the HO phase of URu2Si2 originates from the modulated 
    B1g order parameter. 

    In the course of evaluation, the referee does not recommend the paper 
    to be published in PRB, mainly based on the following reason. 

    1) The scheme for the coupled strains in this paper is quite standard 
    within the mean-field treatment and does not provide a novel 
    theoretical advance. 
}\\

What our manuscript provides is a new way of interpreting a very clear
experimental signature -- that is, nearly perfect Curie--Weiss $1/(T-T_0)$ in
$(c_{11}-c_{12})/2$. We show that a staggered nematic order parameter explains
this behaviour. We agree that coupling strains and order parameters is not new,
but we do not believe that every scientific advance has to be accompanied by
new mathematical machinery for its own sake. Mean field theory happens to work
quite well here, and allows us to make clear symmetry-based statements. In
addition, the incorporation of gradient terms into the mean-field free energy
in the context of interpreting ultrasound data does appear to be somewhat novel.\\

{\color{blue}
  2) One can generically expect several sources for softening elastic 
  constants. For example, the authors in ref.25 also succeeded in the 
  quantitative fits in the framework of a 4f crystal field model for $T > 
  T_{HO}$. Thus, the fitting is not regarded as the decisive evidence on 
  the validity of the model. 
}\\

There are a couple of very important distinctions to be made between our work
and the work of ref. 25 (K.  Kuwahara et al.), which as the referee points out,
also identified softening in $(c_{11}-c_{12})/2$. First, the data in ref.\ 25
(figure 2c) appear to be contaminated by the $c_{66}$ mode, based on the fact that
the peak in $(c_{11}-c_{12})/2$ appears around 60 K. In the work of T. Yanagisawa et al
(Journal of the Physical Society of Japan 82 (2013) 013601), the peak is at 130
K, and the elastic constant softens back down to its room-temperature value by
$T_{HO}$. The data we show in figure 2b, obtained with resonant ultrasound,
also shows a maximum at around $130\, \mathrm K$, and also softens to its room-temperature
value by $T_{HO}$. The contamination in ref.\ 25 is likely an artifact of the
pulse-echo ultrasound technique, which can mix between $c_{66}$ and
$(c_{11}-c_{12})/2$ when the crystal is not perfectly aligned.

Second, and more
critically, the fit shown in figure 4 of ref. 25 does not show very good
agreement with the data at any temperature. The model used is one for
thermally-populated crystal field levels, and it does not directly relate to the phase 
transition at $T_{HO}$. This model does not produce the sharp change in slope
of $(c_{11}-c_{12})/2$ at $T_{HO}$, which is an essential singularity in the
thermodynamic free energy and which must appear in the elastic moduli at a second order phase transition. The crystal field model does attempt to reproduce the termination of the softening below $T_{HO}$, but it does so only in a smooth way, with no discontinuity in slope at $T_{HO}$.

To summarize, while ref. 25 does indeed propose a model to describe the softening seen in $(c_{11}-c_{12})/2$, the data in ref. 25 are contaminated by $c_{66}$, the model does not attribute the softening to the presence of an order parameter, it does not capture the singularity at the phase transition, and does not provide a good match to the Curie-Weiss behaviour of the elastic constant above $T_{HO}$. Whether or not the approach of ref. 25 is correct, it is entirely distinct from the approach we have taken here. \\

{\color{blue}
    3) The agreement of C[B1g] in the region $T<T_{HO}$ is poor, though only 
    the cusp at $T_{HO}$ seems qualitatively consistent with the experiment. 
    Moreover, the referee expects that even a cusp structure in the 
    elastic constants is not unique to this model; it can be obtained from 
    more general models beyond the linear coupling (4), within the 
    mean-field level. Therefore, the referee thinks that this analysis 
    does not lead uniquely to the authors' arguments on the realization of 
    the B1g order parameter. 
}\\

We agree that the poorness-of-fit below $T_{HO}$ is a concern. As we describe in our response above to referee 1, the addition of a higher power term in the free energy expansion goes a long way to resolving this, and further terms would improve the fit further. 

Importantly, While terms that modify the behavior below the transition provide cusp-like
features in the modulus for each strain symmetry, they cannot explain the $1/\Delta T$
softening seen in the high-temperature phase, since their contribution to the
response function is zero above $T_c$. Likewise, mean field theories with a
uniform order parameter cannot explain the finite value of the modulus at the
transition. Thus a B$_{\rm{1g}}$ order parameter is indeed unique in capturing both the behaviour above \textit{and} below $T_{HO}$.\\

{\color{blue}
    4) The most important point in the HO problem is the microscopic 
    identification of symmetry breaking and the order parameter. In spite 
    of the long history in research over almost 40 years, there is no 
    experimental evidence of the formation of any superlattice structure 
    at least at ambient pressure. So, the proposed modulated order is not 
    consistent with the absence or identification of symmetry breaking. 
    The authors do not provide any resolution on that point which is the 
    most relevant in this problem.
}\\

We agree that this is still a point under debate in the community, but there are many other experiments that give evidence for the formation of a superlattice structure along the c-axis at ambient pressure, e.g.:\\

C.\ Bareille, F.\ L.\ Boariu, H.\ Schwab, P.\ Lejay, F.\ Reinert, and A.\ F.
Santander-Syro, Nature Communications \textbf{5}, 4326 (2014).

R.\ Yoshida, Y.\ Nakamura, M.\ Fukui, Y.\ Haga, E.\ Yamamoto, Y.\ \=Onuki, M.\ Okawa,
S.\ Shin, M.\ Hirai, Y.\ Muraoka, and T.\ Yokoya, Physical Review B \textbf{82},
205108 (2010).

R.\ Yoshida, K.\  Tsubota, T.\  Ishiga, M.\  Sunagawa, J.\ Sonoyama, D.\ Aoki, J.
Flouquet, T.\ Wakita, Y.\ Muraoka, and T.\ Yokoya, Scientific Reports \textbf{3},
2750 (2013).

J.-Q.\ Meng, P.\ M.\ Oppeneer, J.\ A.\ Mydosh, P.\ S.\ Riseborough, K.\ Gofryk, J.\ J.
Joyce, E.\ D.\ Bauer, Y.\ Li, and T.\ Durakiewicz, Physical Review Letters
\textbf{111}, 127002 (2013).\\

While the ultrasound experiment cannot directly determine the wavevector, the most natural way to get Curie-Weiss susceptibility in the B$_{\rm{1g}}$ elastic modulus that is cut off (instead of falling all the way to zero) at $T_{\rm{HO}}$ is to have a B$_{\rm{1g}}$ order parameter modulated at a finite wavevector. This pushes the divergent susceptibility out to finite $q$ where it is unobserved (or not fully-observed) by the $q=0$ ultrasound. 

\end{document}
