\documentclass[prb,amsmath,amssymb,floatfix,superscriptaddress]{revtex4} \usepackage{bm} \usepackage{url} \usepackage{graphicx} \usepackage{physics} \usepackage{epsfig} \usepackage{subfigure} \usepackage[usenames]{color} \usepackage{hyperref} \usepackage{enumerate} \newcommand*\ruleline[1]{\par\noindent\raisebox{.8ex}{\makebox[\linewidth]{\hrulefill\hspace{1ex}\raisebox{-.8ex}{#1}\hspace{1ex}\hrulefill}}} \begin{document} \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 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. }\\ 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). }\\ In order to check the agreement in our fit, we preformed the fit with a moving temperature window that cuts off at $T_\text{max}$. Our fits' parameters are $x_i$ for $i = 1, \ldots, 5$ for $C_0 = x_1 - x_2 (T / \mathrm K)$, $x_3 = b^2/a$, $x_4 = b^2/Dq_*^4$, and $x_5 = b \sqrt{-g/u}$. The variation of these parameters as a function of $T_\text{max}$ are shown on the top of Figure \ref{fig:parameter_cutoff}. The parameter $x_1$ is fairly stable at all temperature cutoffs, while the rest vary by 1.5--$2.5\times$ their $275\,\mathrm K$ value down to cutoffs of $\sim90\,\mathrm K$. The fit functions that result from varying the cutoff are shown in Figure \ref{fig:parameter_curve}. \begin{figure} \centering \includegraphics[width=0.7\textwidth]{referee_response_cutoff_parameters.pdf} \includegraphics[width=0.7\textwidth]{referee_response_cutoff_eigenvectors.pdf} \caption{ Fit parameters as a function of the cutoff temperature $T_\text{max}$. (Top) Bare fit parameters corresponding to ratios of Landau coefficients. (Bottom) Linear combinations of bare fit parameters corresponding to eigendirections of the covariance matrix at $T_\text{max}=275\,\mathrm K$. } \label{fig:parameter_cutoff} \end{figure} More insight into the consistency of the fit comes from examining the linear combinations of parameters that form eigenvectors of the fit covariance matrix, since—unlike the natural parameters of the mean field theory—these have uncorrelated uncertainties. For the fit including all temperatures (up to $275\,\mathrm K$), these are (in order of fit uncertainty): \begin{align*} y_1 &= -0.0020 x_1 + 2.2 \times 10^{-6} x_2 - 1.0 x_3 - 0.0023 x_4 - 0.0056 x_5 \\ y_2 &= -0.015 x_1 + 0.000042 x_2 - 0.0055 x_3 - 0.021 x_4 + 1.0 x_5 \\ y_3 &= -0.64 x_1 + 0.0020 x_2 + 0.0032 x_3 - 0.77 x_4 - 0.025 x_5 \\ y_4 &= -0.77 x_1 + 0.0066 x_2 + 0.000075 x_3 + 0.64 x_4 + 0.0014 x_5 \\ y_5 &= 0.0064 x_1 + 1.0 x_2 - 4.3 \times 10^{-6} x_3 - 0.0027 x_4 - 4.9 \times 10^{-7} x_5 \end{align*} The variation of these parameter combinations as a function of $T_\text{max}$ are shown on the bottom of Figure \ref{fig:parameter_cutoff}. The parameter $y_1$, which is principally $x_3 = a/b^2$, varies the most with the cutoff, at most around $2\times$ its value until $\sim90\,\mathrm K$. The parameter $y_2$, which is principally $x_5 = b \sqrt{-g/u}$, varies at most around $1.25\times$ its value until $\sim90\,\mathrm K$. The other three parameters are stable at any cutoff, and are mixed combinations of $x_1$, $x_2$, and $x_4$. Notably, $x_1$ and $x_2$ are the only parameters involved in transforming the experimental data in Figure 2(d), and their stability as a function of the data window means that transformation is likewise stable.\\ \begin{figure} \centering \includegraphics[width=0.7\textwidth]{referee_response_cutoff_curves.pdf} \caption{ Fit function with color given by value of $T_\text{max}$ that produced it. Data shown as black line. } \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 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 $130\, \mathrm K$, 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. 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 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