Merge branch 'master' of https://git.overleaf.com/5cf56f861d72e9071d1a343c

3 files changed, 264 insertions, 4 deletions

diff --git a/main.tex b/main.tex
index ba35e1f..66459b6 100644
--- a/main.tex
+++ b/main.tex
@@ -609,6 +609,25 @@ is characterized by uniform $\Bog$ order. The staggered nematic of \ho\ is
 similar to the striped superconducting phase found in LBCO and other
 cuperates.\cite{Berg_2009b}
 
+{\color{blue}
+We can also connect our abstract order parameter to a physical picture of multipolar
+ordering.
+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.
+}
+
 The coincidence of our theory's orthorhombic high-pressure phase and \urusi's
 \afm\ is compelling, but our mean field theory does not make any explicit
 connection with the physics of \afm. Neglecting this physics could be
diff --git a/ref_response.tex b/ref_response.tex
new file mode 100644
index 0000000..9df15f7
--- /dev/null
+++ b/ref_response.tex
@@ -0,0 +1,228 @@
+\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 {\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
+
+\end{document}
\ No newline at end of file
diff --git a/referee_comments.txt b/referee_comments.txt
index 87f2116..18fd524 100644
--- a/referee_comments.txt
+++ b/referee_comments.txt
@@ -88,9 +88,22 @@ The behaviour of c/a is indeed interesting, but our model only considers the cou
 > simplest framework of localized f-electrons, what ionic moments would 
 > fit the present proposal?
 
-In a picture of localized U-5f electrons this
-This is a consistent with hexadecapolar order 
-[Mike, can you help with this?] [Yes I think he can! And it's a good point]
+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
+hexadecapolar order parameter 
+\[
+    H = \eta (J_x^2 - J_y^2)
+\]
+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.
 
 > ----------------------------------------------------------------------
 > Report of the Second Referee -- BN13654/Kent-Dobias
@@ -123,7 +136,7 @@ What our manuscript provides is a new way of interpreting a very clear experimen
 > 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 (c11-c12)/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 c66 and (c11-c12)/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 (c11-c12)/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 (c11-c12)/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 c66 and (c11-c12)/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 [this being thermally populated crystal field levels, right?] does not produce the sharp change in slope of (c11-c12)/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. 
 
 > 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.