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diff --git a/ref_response.tex b/ref_response.tex index 27b3468..69764dd 100644 --- a/ref_response.tex +++ b/ref_response.tex @@ -17,7 +17,7 @@ \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 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: @@ -60,8 +60,7 @@ Jaron Kent-Dobias, Michael Matty, and Brad Ramshaw 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 +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. @@ -75,9 +74,9 @@ 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. +pressure. We have also left the original fit (without the extra term) since the model is more simply described in the text. -An inset with a zoom on the critical region has been added to the figure.\\ +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 @@ -86,7 +85,7 @@ 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). }\\ -In order to check the agreement in our fit, we preformed the fit with a moving +In order to check the agreement in our fit, we performed the fit with a moving temperature window that cuts off at $T_\text{max}$. 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 @@ -110,7 +109,7 @@ down to cutoffs of $\sim90\,\mathrm K$. The fit functions that result from varyi 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 +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*} @@ -135,7 +134,7 @@ window means that transformation is likewise stable.\\ \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. + 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} @@ -209,7 +208,7 @@ 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.\\ +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 @@ -230,15 +229,17 @@ $T_{HO}$. 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 +$(c_{11}-c_{12})/2$ 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 +thermally-populated crystal field levels, and 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 must appear in the elastic moduli, and it does +thermodynamic free energy and must appear in the elastic moduli at a second order phase transition, and it does not produce $1/(T-T_0)$ strain susceptibility above $T_{HO}$, which is a -signature of strain and order parameter coupling. \\ +signature of strain and order parameter coupling. To summarize, while ref. 25 does indeed propose a model to describe the softening seen in $(c_{11}-c_{12})/2$, it does not attribute the softening to the presence of an order parameter, 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. \\ {\color{blue} 3) The agreement of C[B1g] in the region $T<T_{HO}$ is poor, though only @@ -251,29 +252,14 @@ signature of strain and order parameter coupling. \\ 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. +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. -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 +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.\\ +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 @@ -286,7 +272,7 @@ transition.\\ most relevant in this problem. }\\ -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.\\ +We agree that the presence of a super-lattice structure is still a debated point, 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). @@ -301,6 +287,8 @@ Flouquet, T.\ Wakita, Y.\ Muraoka, and T.\ Yokoya, Scientific Reports \textbf{3} 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). +\textbf{111}, 127002 (2013).\\ + +While the ultrasound experiment cannot determine the precise 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 zer) at $T_{\rm{HO}}$ is to have a B$_{\rm{1g}}$ order parameter modulated at a finite wavevector, pushing the divergent susceptibility out to finite $q$ where it is unobserved (or not fully-observed) by the $q=0$ ultrasound. \end{document} |