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| -rw-r--r-- | fig-stiffnesses.gplot | 28 | ||||
| -rw-r--r-- | fig-stiffnesses.pdf | bin | 332053 -> 333378 bytes | |||
| -rw-r--r-- | hidden_order.bib | 42 | ||||
| -rw-r--r-- | main.tex | 92 | ||||
| -rw-r--r-- | ref_response.tex | 190 | ||||
| -rw-r--r-- | referee_response_cutoff_curves.pdf | bin | 80978 -> 81698 bytes | |||
| -rw-r--r-- | referee_response_cutoff_eigenvectors.pdf | bin | 9073 -> 8958 bytes | |||
| -rw-r--r-- | referee_response_cutoff_parameters.pdf | bin | 9194 -> 9132 bytes |
8 files changed, 237 insertions, 115 deletions
diff --git a/fig-stiffnesses.gplot b/fig-stiffnesses.gplot index 52762dd..93b1f37 100644 --- a/fig-stiffnesses.gplot +++ b/fig-stiffnesses.gplot @@ -9,11 +9,11 @@ cc5 = "#000000" Tc = 17.26 -a1 = 72.9479505025133 -b1 = 0.01227234014115862 -c1 = 2675.3172792640935 -d1 = 8.418493827841177 -e1 = 19.456906054458745 +a1 = 71.13597475161484 +b1 = 0.010425748328804992 +c1 = 1665.064389018724 +d1 = 6.283065722894796 +e1 = 14.579664883988315 C10(T) = a1 - b1 * T C1(T) = C10(T) / (1 + 1 / (1 / d1 + abs(T - Tc) / c1) / C10(T)) @@ -63,16 +63,16 @@ set xrange [0:300] set title '(a)' offset 12,-2.7 set ylabel '$C_{\mathrm{B_{2\mathrm g}}} / \mathrm{GPa}$' offset 3 -set yrange [140:145] -set ytics 141,1,144 offset 0.5 mirror +set yrange [140.2:144.7] +set ytics 139,1,146 offset 0.5 mirror plot "data/c66.dat" using 1:(100 * $2) with lines lw 6 lc rgb cc3, \ C20(x) dt 3 lw 4 lc rgb cc4 set title '(b)' offset 12,-2.7 unset ylabel set y2label '$C_{\mathrm{B_{1\mathrm g}}} / \mathrm{GPa}$' offset -4 rotate by -90 -set yrange [65.05:65.7] -set y2tics 62.1,0.1,65.6 offset -0.5 mirror +set yrange [65.05:65.68] +set y2tics 62.1,0.1,66 offset -0.5 mirror set samples 1000 plot "data/c11mc12.dat" using 1:(100 * $2) with points pt 6 ps 0.4 lc rgb cc3, \ x > Tc ? C12(x) : 1 / 0 lw 3 lc rgb cc5, \ @@ -92,7 +92,7 @@ set ytics 63,1,74 unset y2label set ylabel '$\textcolor{mathc3}{C_{\mathrm{B_{1\mathrm g}}}},\textcolor{mathc4}{C_{\mathrm{B_{1\mathrm g}}}^0} / \mathrm{GPa}$' offset 3 -set yrange [64.5:73.5] +set yrange [64.5:71.5] plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 6 lc rgb cc3, \ C10(x) dt 3 lw 4 lc rgb cc4 @@ -102,8 +102,8 @@ set y2label '$\Big[C^0_{\mathrm{B_{1\mathrm g}}}(C^0_{\mathrm{B_{1\mathrm g}}}/C set title '(d)' offset 12,-2.7 set format y2 '\scriptsize$%0.2f$' set format y '\scriptsize$%0.2f$' -set yrange [0.11:0.25] -set y2tics 0.05,0.025,0.25 offset -0.7 mirror +set yrange [0.14:0.38] +set y2tics 0.05,0.05,0.5 offset -0.7 mirror plot "data/c11mc12.dat" using 1:(1 / (C10($1)*(C10($1) / (100 * $2) - 1))) with lines lw 6 lc rgb cc3, \ x > Tc ? 1/(C10(x) * (C10(x) / C12(x) - 1)) : 1 / 0 lw 3 lc cc5, \ x <= Tc ? 1/(C10(x) * (C10(x) / C12(x) - 1)) : 1 / 0 dt (2,1) lw 3 lc cc5, \ @@ -127,8 +127,8 @@ plot "data/c11mc12.dat" using 1:(100 * $2) with points pt 2 ps 0.15 lc rgb cc3, unset margins set origin 0.5,0.26 -set yrange [0.1155:0.123] -set y2tics 0.105,0.0025,0.131 +set yrange [0.154:0.166] +set y2tics 0.155,0.005,0.167 set format y2 '\scriptsize$%0.3f$' plot "data/c11mc12.dat" using 1:(1 / (C10($1)*(C10($1) / (100 * $2) - 1))) with points pt 2 ps 0.15 lc rgb cc3, \ diff --git a/fig-stiffnesses.pdf b/fig-stiffnesses.pdf Binary files differindex 2cc839f..56cf5b5 100644 --- a/fig-stiffnesses.pdf +++ b/fig-stiffnesses.pdf diff --git a/hidden_order.bib b/hidden_order.bib index df124d5..2e0de7c 100644 --- a/hidden_order.bib +++ b/hidden_order.bib @@ -12,6 +12,20 @@ doi = {10.1088/0034-4885/45/6/001} } +@article{Bareille_2014_Momentum-resolved, + author = {Bareille, C. and Boariu, F. L. and Schwab, H. and Lejay, P. and Reinert, F. and Santander-Syro, A. F.}, + title = {Momentum-resolved hidden-order gap reveals symmetry breaking and origin of entropy loss in {URu$_\text2$Si$_\text2$}}, + journal = {Nature Communications}, + publisher = {Springer Science and Business Media LLC}, + year = {2014}, + month = {7}, + number = {1}, + volume = {5}, + pages = {4326}, + url = {https://doi.org/10.1038%2Fncomms5326}, + doi = {10.1038/ncomms5326} +} + @article{Berg_2009b, author = {Berg, Erez and Fradkin, Eduardo and Kivelson, Steven A and Tranquada, John M}, title = {Striped superconductors: how spin, charge and superconducting orders intertwine in the cuprates}, @@ -636,4 +650,32 @@ doi = {10.1080/14786435.2013.878054} } +@article{Yoshida_2010_Signature, + author = {Yoshida, Rikiya and Nakamura, Yoshiaki and Fukui, Masaki and Haga, Yoshinori and Yamamoto, Etsuji and Ōnuki, Yoshichika and Okawa, Mario and Shin, Shik and Hirai, Masaaki and Muraoka, Yuji and Yokoya, Takayoshi}, + title = {Signature of hidden order and evidence for periodicity modification in {URu$_\text2$Si$_\text2$}}, + journal = {Physical Review B}, + publisher = {American Physical Society (APS)}, + year = {2010}, + month = {11}, + number = {20}, + volume = {82}, + pages = {205108}, + url = {https://doi.org/10.1103%2Fphysrevb.82.205108}, + doi = {10.1103/physrevb.82.205108} +} + +@article{Yoshida_2013_Translational, + author = {Yoshida, Rikiya and Tsubota, Koji and Ishiga, Toshihiko and Sunagawa, Masanori and Sonoyama, Jyunki and Aoki, Dai and Flouquet, Jacques and Wakita, Takanori and Muraoka, Yuji and Yokoya, Takayoshi}, + title = {Translational Symmetry Breaking and Gapping of Heavy-Quasiparticle Pocket in {URu$_\text2$Si$_\text2$}}, + journal = {Scientific Reports}, + publisher = {Springer Science and Business Media LLC}, + year = {2013}, + month = {10}, + number = {1}, + volume = {3}, + pages = {2750}, + url = {https://doi.org/10.1038%2Fsrep02750}, + doi = {10.1038/srep02750} +} + @@ -105,6 +105,7 @@ \maketitle \section{Introduction} + \urusi\ is a paradigmatic example of a material with an ordered state whose broken symmetry remains unknown. This state, known as \emph{hidden order} (\ho), sets the stage for unconventional superconductivity that emerges at even @@ -158,8 +159,8 @@ experiments under pressure approaching the Lifshitz point, which should find that the $(C_{11}-C_{12})/2$ modulus diverges as the uniform $\Bog$ strain of the high pressure phase is approached. - \section{Model \& Phase Diagram} + The point group of \urusi\ is \Dfh, and any theory must locally respect this symmetry in the high-temperature phase. Our phenomenological free energy density contains three parts: the elastic free energy, the \op, and the @@ -168,9 +169,9 @@ the strain $\epsilon$ is $f_\ee=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$. \footnote{Components of the elastic modulus tensor $C$ were given in the popular Voigt notation in the abstract and introduction. Here and henceforth the notation used is that natural for a rank-four tensor.} The form of the bare -moduli tensor $C^0$ is further restricted by symmetry. \cite{Landau_1986_Theory} Linear combinations of -the six independent components of strain form five irreducible components of -strain in \Dfh\ as +moduli tensor $C^0$ is further restricted by symmetry. +\cite{Landau_1986_Theory} Linear combinations of the six independent components +of strain form five irreducible components of strain in \Dfh\ as \begin{equation} \begin{aligned} & \epsilon_{\Aog,1}=\epsilon_{11}+\epsilon_{22} \hspace{0.15\columnwidth} && @@ -290,7 +291,7 @@ to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$. With the strain traced out, \eqref{eq:fo} describes the theory of a Lifshitz point at $\tilde r=c_\perp=0$.\cite{Lifshitz_1942_OnI, Lifshitz_1942_OnII} The properties discussed in the remainder of this section can all be found in a -standard text, e.g., in chapter 4 \S6.5 of Chaikin \& +standard text, e.g., in Chapter 4 \S6.5 of Chaikin \& Lubensky.\cite{Chaikin_1995} For a one-component \op\ ($\Bog$ or $\Btg$) and positive $c_\parallel$, it is traditional to make the field ansatz $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$, @@ -472,21 +473,22 @@ corresponding modulus. \centering \includegraphics{fig-stiffnesses} \caption{ - \Rus\ measurements of the elastic moduli of \urusi\ at ambient pressure as a - function of temperature from recent experiments\cite{Ghosh_2020_One-component} (blue, - solid) alongside fits to theory (magenta, dashed and black, solid). The solid yellow region - shows the location of the \ho\ phase. (a) $\Btg$ modulus data and a fit to - the standard form.\cite{Varshni_1970} (b) $\Bog$ modulus data and a fit to - \eqref{eq:static_modulus} (magenta, dashed) and a fit to \eqref{eq:C0} (black, solid). The fit gives - $C^0_\Bog\simeq\big[73-(0.012\,\K^{-1})T\big]\,\GPa$, $D_\perp - q_*^4/b^2\simeq0.12\,\GPa^{-1}$, and - $a/b^2\simeq3.7\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic - term in $C^0_\Bog$ was here not needed for the fit.\cite{Varshni_1970} (c) - $\Bog$ modulus data and the fit of the \emph{bare} $\Bog$ modulus. (d) - $\Bog$ modulus data and the fits transformed by - $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from - \eqref{eq:static_modulus} to equal $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, e.g., - an absolute value function. + \Rus\ measurements of the elastic moduli of \urusi\ at ambient pressure as + a function of temperature from recent + experiments\cite{Ghosh_2020_One-component} (blue, solid) alongside fits to + theory (magenta, dashed and black, dashed). The solid yellow region shows + the location of the \ho\ phase. (a) $\Btg$ modulus data and a fit to the + standard form.\cite{Varshni_1970} (b) $\Bog$ modulus data and a fit to + \eqref{eq:static_modulus} (magenta, dashed) and a fit to \eqref{eq:C0} + (black, dashed). The fit gives + $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $b^2/D_\perp + q_*^4\simeq6.28\,\GPa$, and $b^2/a\simeq1665\,\GPa\,\K^{-1}$. Addition of a + quadratic term in $C^0_\Bog$ was here not needed for the + fit.\cite{Varshni_1970} (c) $\Bog$ modulus data and the fit of the + \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and the fits + transformed by $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted + from \eqref{eq:static_modulus} to equal $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, + e.g., an absolute value function. } \label{fig:data} \end{figure*} @@ -494,7 +496,7 @@ corresponding modulus. \Rus\ experiments~\cite{Ghosh_2020_One-component} yield the individual elastic moduli broken into irreps; data for the $\Bog$ and $\Btg$ components defined in \eqref{eq:strain-components} are shown in Figures \ref{fig:data}(a--b). The -$\Btg$ in Fig.~\ref{fig:data}(a) modulus doesn't appear to have any response to +$\Btg$ modulus in Fig.~\ref{fig:data}(a) doesn't appear to have any response to the presence of the transition, exhibiting the expected linear stiffening upon cooling from room temperature, with a low-temperature cutoff at some fraction of the Debye temperature.\cite{Varshni_1970} The $\Bog$ modulus @@ -507,13 +509,13 @@ the result is shown in Figure \ref{fig:data}(b). The behavior of the modulus below the transition does not match \eqref{eq:static_modulus} well, but this is because of the truncation of the free energy expansion used above. Higher order terms like $\eta^2\epsilon^2$ -contribute to the modulus starting at order $\eta_*^2$, and therefore while -they do not affect the behavior above the transition, they change the behavior -below it. To demonstrate this, in Appendix~\ref{sec:higher-order} we compute -the modulus in a theory where the interaction free energy is truncated after -fourth order with new term $\frac12g\eta^2\epsilon^2$. The thin solid black -line in Fig.~\ref{fig:data} shows the fit of the \rus\ data to \eqref{eq:C0} -and shows that high-order corrections can account for the low-temperature +and $\epsilon^4$ contribute to the modulus starting at order $\eta_*^2$ and +therefore change the behavior below the transition but not above it. To +demonstrate this, in Appendix~\ref{sec:higher-order} we compute the modulus in +a theory where the interaction free energy is truncated after fourth order with +new term $\frac12g\eta^2\epsilon^2$. The dashed black line in +Fig.~\ref{fig:data} shows the fit of the \rus\ data to \eqref{eq:C0} and shows +that successive high-order corrections can account for the low-temperature behavior. The data and theory appear quantitatively consistent, suggesting that \ho\ can @@ -540,11 +542,13 @@ from the Lifshitz point we expect the wavevector to lock into values commensurate with the space group of the lattice, and moreover that at zero pressure, where the \rus\ data here was collected, the half-wavelength of the modulation should be commensurate with the lattice spacing $a_3\simeq9.68\,\A$, -or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$.\cite{Meng_2013_Imaging, Broholm_1991, Wiebe_2007, -Bourdarot_2010, Hassinger_2010} In between these two regimes, mean field theory -predicts that the ordering wavevector shrinks by jumping between ever-closer -commensurate values in the style of the devil's staircase.\cite{Bak_1982} In -reality the presence of fluctuations may wash out these transitions. +or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$.\cite{Bareille_2014_Momentum-resolved, +Yoshida_2010_Signature, Yoshida_2013_Translational, Meng_2013_Imaging, +Broholm_1991, Wiebe_2007, Bourdarot_2010, Hassinger_2010} In between these two +regimes, mean field theory predicts that the ordering wavevector shrinks by +jumping between ever-closer commensurate values in the style of the devil's +staircase.\cite{Bak_1982} In reality the presence of fluctuations may wash out +these transitions. This motivates future ultrasound experiments done under pressure, where the depth of the cusp in the $\Bog$ modulus should deepen (perhaps with these @@ -594,7 +598,9 @@ An ultrasound experiment with more precise temperature resolution near the critical point may be able to resolve a modified cusp exponent $\gamma\simeq1.31$,\cite{Guida_1998_Critical} since according to one analysis the universality class of a uniaxial modulated one-component \op\ is that of -the $\mathrm O(2)$, 3D XY transition.\cite{Garel_1976} +the $\mathrm O(2)$, 3D XY transition.\cite{Garel_1976} A crossover from mean +field theory may explain the small discrepancy in our fit very close to the +critical point. \section{Conclusion and Outlook.} We have developed a general phenomenological treatment of \ho\ \op s that have the potential for linear coupling to strain. @@ -608,8 +614,8 @@ similar to the striped superconducting phase found in LBCO and other cuperates.\cite{Berg_2009b} We can also connect our results to the large body of work concerning various -multipolar orders as candidate states for \ho\ (e.g. -refs.~\cite{Haule_2009,Ohkawa_1999,Santini_1994,Kiss_2005,Kung_2015,Kusunose_2011_On}). +multipolar orders as candidate states for \ho\ (e.g. refs.~\cite{Haule_2009, +Ohkawa_1999, Santini_1994, Kiss_2005, Kung_2015, Kusunose_2011_On}). Physically, our phenomenological order parameter could correspond to $\Bog$ multipolar ordering originating from the localized component of the U-5f electrons. For the crystal field states of \urusi, this could correspond either @@ -635,11 +641,11 @@ order, and preforming symmetry-sensitive thermodynamic experiments at pressure, such as ultrasound, that could further support or falsify this idea. \begin{acknowledgements} - Jaron Kent-Dobias is supported by NSF DMR-1719490, Michael Matty is supported by - NSF DMR-1719875, and Brad Ramshaw is supported by NSF DMR-1752784. We are - grateful for helpful discussions with Sri Raghu, Steve Kivelson, Danilo Liarte, and Jim - Sethna, and for permission to reproduce experimental data in our figure by - Elena Hassinger. We thank Sayak Ghosh for \rus\ data. + Jaron Kent-Dobias is supported by NSF DMR-1719490, Michael Matty is supported + by NSF DMR-1719875, and Brad Ramshaw is supported by NSF DMR-1752784. We are + grateful for helpful discussions with Sri Raghu, Steve Kivelson, Danilo + Liarte, and Jim Sethna, and for permission to reproduce experimental data in + our figure by Elena Hassinger. We thank Sayak Ghosh for \rus\ data. \end{acknowledgements} \appendix @@ -681,7 +687,7 @@ $\tilde r<c_\perp^2/4D=\tilde r_c$ with $q_*^2=-c_\perp/2D$ and \eta_*^2=\frac{c_\perp^2-4D\tilde r}{12D\tilde u} =\frac{|\Delta\tilde r|}{3\tilde u}. \end{equation} -We would like to calculate the $q$-dependant modulus +We would like to calculate the $q$-dependent modulus \begin{equation} C(q) =\frac1V\int dx\,dx'\,C(x,x')e^{-iq(x-x')}, @@ -837,7 +843,7 @@ $\eta_*^2$. With $r=a\Delta T+c^2/4D+b^2/C_0$, $u=\tilde u-b^2g/2C_0^2$, and -a\Delta T/3\tilde u & \Delta T \leq 0, \end{cases} \end{equation} -we can fit the ratios $b^2/a=1665\,\mathrm{GPa}\,\mathrm K$, $b^2/Dq_*^4=6.28\,\mathrm{GPa}$, and $b\sqrt{-g/\tilde u}=14.58\,\mathrm{GPa}$ with $C_0=(71.14-(0.010426\times T)/\mathrm K)\,\mathrm{GPa}$. The resulting fit the thin solid black line in Fig.~\ref{fig:data}. +we can fit the ratios $b^2/a=1665\,\mathrm{GPa}\,\mathrm K$, $b^2/Dq_*^4=6.28\,\mathrm{GPa}$, and $b\sqrt{-g/\tilde u}=14.58\,\mathrm{GPa}$ with $C_0=(71.14-(0.010426\,\mathrm K^{-1})T)\,\mathrm{GPa}$. The resulting fit is shown as a dashed black line in Fig.~\ref{fig:data}. \bibliographystyle{apsrev4-1} \bibliography{hidden_order} 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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