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| -rw-r--r-- | main.tex | 28 | ||||
| -rw-r--r-- | ref_response.tex | 54 |
2 files changed, 34 insertions, 48 deletions
@@ -116,7 +116,7 @@ Kusunose_2011_On, Kung_2015, Cricchio_2009, Ohkawa_1999, Santini_1994, Kiss_2005, Harima_2010, Thalmeier_2011, Tonegawa_2012_Cyclotron, Rau_2012_Hidden, Riggs_2015_Evidence, Hoshino_2013_Resolution, Ikeda_1998_Theory, Chandra_2013_Hastatic, Harrison_2019_Hidden, Ikeda_2012} propose -associating any of a variety of broken symmetries with \ho. This work analyzes +associating any of a variety of broken symmetries with \ho. Motivated by the anomalous temperature dependence of one of the elastic moduli , this work analyzes a family of phenomenological models with order parameters of general symmetry that couple linearly to strain. Of these, only one is compatible with two experimental observations: first, the $\Bog$ ``nematic" elastic susceptibility @@ -152,9 +152,9 @@ temperature and then cusps at $T_{\text{\ho}}$. That theory associates \ho\ with a $\Bog$ \op\ modulated along the $c$-axis, the high pressure state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. In addition to the agreement with the ultrasound data across a broad -temperature range, the theory predicts uniform $\Bog$ strain at high +temperature range, our model predicts uniform $\Bog$ strain at high pressure---the same distortion that was recently seen in x-ray scattering -experiments.\cite{Choi_2018} This theory strongly motivates future ultrasound +experiments.\cite{Choi_2018} This work strongly motivates future ultrasound 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. @@ -238,7 +238,7 @@ $\Btg$, or $\Eg$---the most general quadratic free energy density is where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$, and $\nabla_\perp=\partial_3$ transforms like $\Atu$. Other quartic terms are allowed---especially many for an $\Eg$ \op---but we have included only those -terms necessary for stability when either $r$ or $c_\perp$ become negative. The +terms necessary for stability when either $r$ or $c_\perp$ become negative as a function of temperature. The full free energy functional of $\eta$ and $\epsilon$ is \begin{equation} \begin{aligned} @@ -279,7 +279,7 @@ to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$. Phase diagrams for (a) \urusi\ from experiments (neglecting the superconducting phase)~\cite{Hassinger_2008} (b) mean field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean field theory of a - two-component ($\Eg$) Lifshitz point. Solid lines denote continuous + two-component ($\Eg$) Lifshitz point. Solid lines denote second-order transitions, while dashed lines denote first order transitions. Later, when we fit the elastic moduli predictions for a $\Bog$ \op\ to data along the ambient pressure line, we will take $\Delta\tilde r=\tilde r-\tilde @@ -318,7 +318,7 @@ order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. The uniform to modulated transition is now continuous. This does not reproduce the physics of \urusi, whose \ho\ phase is bounded by a line of first order transitions at high pressure, and so we will henceforth neglect the possibility -of a multicomponent order parameter. Schematic phase diagrams for both the one- +of a multicomponent order parameter---consistent with earlier ultrasound measurements \cite{Ghosh_2020_One-component}. Schematic phase diagrams for both the one- and two-component models are shown in Figure~\ref{fig:phases}. @@ -372,7 +372,7 @@ perpendicular and parallel to the plane, respectively. The static susceptibility $\chi(0)=(D_\perp q_*^4+|\Delta\tilde r|)^{-1}$ does not diverge at the unordered to modulated transition. Though it anticipates a transition with Curie--Weiss-like divergence at the lower point $a(T-T_c)=\Delta\tilde -r=-D_\perp q_*^4<0$, this is cut off with a cusp at $\Delta\tilde r=0$. +r=-D_\perp q_*^4<0$, this is cut off with a cusp at the phase transition at $\Delta\tilde r=0$. The elastic susceptibility, which is the reciprocal of the effective elastic modulus, is found in a similar way to the \op\ susceptibility: we must trace @@ -510,7 +510,7 @@ 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 $\eta^2\epsilon^2$ 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 +therefore change the behavior below the transition, where the expectation value of $\eta$ is finite, but not above it, where the expectation value of $\eta$ is zero. 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 @@ -556,9 +556,7 @@ commensurability jumps) at low pressure and approach zero as $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. Alternatively, \rus\ done at ambient pressure might examine the heavy Fermi liquid to \afm\ transition by doping. Though previous \rus\ studies have doped \urusi\ with -rhodium,\cite{Yanagisawa_2014} the magnetic rhodium dopants likely promote -magnetic phases. A non-magnetic dopant such as phosphorous may more faithfully -explore the transition out of the HO phase. Our work also motivates experiments +rhodium,\cite{Yanagisawa_2014} rhodium changes the carrier concentration as well as the lattice spacing, and may favour the promotion of the magnetic phase. An iso-electronic (as well as iso-magnetic) dopant such as iron may more faithfully explore the transition out of the HO phase. Our work also motivates experiments that can probe the entire correlation function---like x-ray and neutron scattering---and directly resolve its finite-$q$ divergence. The presence of spatial commensurability is known to be irrelevant to critical behavior at a @@ -591,10 +589,10 @@ above gives $\xi_{\perp0}q_*=(D_\perp q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ on the order of lattice constant, which is about what one would expect for a generic magnetic transition. The agreement of this data in the -$t\sim0.1$--10 range with the mean field exponent suggests that this region is +$(T-T_{\rm{HO}})/T_{\rm{HO}}\sim0.1$--10 range with the mean field exponent suggests that this region is outside the Ginzburg region, but an experiment may begin to see deviations from mean field behavior within approximately several Kelvin of the critical point. -An ultrasound experiment with more precise temperature resolution near the +An ultrasound experiment with finer 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 @@ -620,7 +618,7 @@ 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 to electric quadropolar or hexadecapolar order based on the available -multipolar operators. \cite{Kusunose_2011_On} +multipolar operators. \cite{Kusunose_2011_On} 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 @@ -638,7 +636,7 @@ $\Bog$ symmetry breaking, which we believe may be due to fluctuating order at temperatures above the actual transition temperature. This work motivates both further theoretical work regarding a microscopic theory with modulated $\Bog$ order, and preforming symmetry-sensitive thermodynamic experiments at pressure, -such as ultrasound, that could further support or falsify this idea. +such as pulse-echo ultrasound, that could further support or falsify this idea. \begin{acknowledgements} Jaron Kent-Dobias is supported by NSF DMR-1719490, Michael Matty is supported 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} |