path: root/ref_response.tex

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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 the impact of our work.
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 {\color{red} 
        Added a statement emphasizing the novelty of our incorporation of gradient
        terms to the mean-field analysis of ultrasound data
    }
    \item {\color{red}
        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. 
}\\

The gross disagreement between the data and theory below the transition is
resolved by the addition of another 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.

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). 
}\\

{\color{red} Jaron please format this}\\

{\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 A1g 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$_{1g}$) 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 discussion of 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.
The U-5f electrons in URu$_2$Si$_2$ exhibit a moderate degree of localization [cite], which is
reflected in partial occupancy of many electronic states. Motivated by the results of refs [cite],
we assume that the dominant U state consists of $j = 5/2$ electrons in the U-5f2 configuration, which has 
total angular momentum $J = 4$. Within the $J=4$ multiplet, the precise energetic ordering
of the $D_{4h}$ crystal field states still remains a matter of debate [cite]. In a simple
framework of localized $j = 5/2$ electrons in the 5f2 configuration, our phenomenological theory
is consistent with the ground state being the B$_{1g}$ crystal field state with
order parameter 
\[
    H = \eta (J_x^2 - J_y^2)
\]
corresponding to hexadecapolar orbital order,
where here $\eta$ is taken to be modulated at $\vec{Q} = (0, 0, 1)$. 
The result of non-zero $\eta$ is a nematic distortion of the B1g orbitals, alternating along the c-axis.\\

\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 appears 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 c66 mode, based on the fact that the peak in c66 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 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. Perhaps more importantly, 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 has nothing to do with the phase transition at $T_{HO}$. This model 
{\color{red} (this being thermally populated crystal field levels, right?) }
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 must appear in the elastic moduli, and it does not produce $1/(T-T_0)$ strain susceptibility above $T_{HO}$, which is a signature of strain and order parameter coupling. \\

{\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. 
}\\

The gross disagreement between the data and theory below the transition is
resolved by the addition of another 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.

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.\\

{\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.
}\\

The articles below, also {\color{red}cited in our work}, all provide experimental evidence for the formation of a superlattice structure along the c-axis at ambient pressure.\\

Bareille C, Boariu F L, Schwab H, Lejay P, Reinert F and
Santander-Syro A 2014 Nat. Commun. 5 4326\\

Yoshida R et al 2010 Phys. Rev. B 82 205108\\

Yoshida R, Tsubota K, Ishiga T, Sunagawa M, Sonoyama J,
Aoki D, Flouquet J, Wakita T, Muraoka Y and Yokoya T
2013 Sci. Rep. 3 2750\\

Meng J-Q, Oppeneer P M, Mydosh J A, Riseborough P S,
Gofryk K, Joyce J, Bauer E D, Li Y and Durakiewicz T
2013 Phys. Rev. Lett. 110 127002

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