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diff --git a/ref_response.tex b/ref_response.tex index f02cbc3..586521f 100644 --- a/ref_response.tex +++ b/ref_response.tex @@ -22,21 +22,17 @@ 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. - } + \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 + 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] @@ -90,14 +86,77 @@ An inset with a zoom on the critical region has been added to the figure.\\ the presence of a maximum at 120 K in 2b). }\\ -{\color{red} Jaron please format this}\\ +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 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. \\ +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 @@ -109,27 +168,15 @@ The behaviour of c/a is indeed interesting, but our model only considers the cou 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$_{1g}$ 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. -% 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.\\ +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}\\ @@ -154,19 +201,44 @@ hexadecapolar order based on the available multipolar operators. 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.\\ +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. + 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. \\ +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 $T<T_{HO}$ is poor, though only @@ -214,19 +286,21 @@ transition.\\ 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.\\ +The articles below, also 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\\ +C.\ Bareille, F.\ L.\ Boariu, H.\ Schwab, P.\ Lejay, F.\ Reinert, and A.\ F. +Santander-Syro, Nature Communications \textbf{5}, 4326 (2014). -Yoshida R et al 2010 Phys. Rev. B 82 205108\\ +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). -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\\ +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). -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 +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). -\end{document}
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