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@@ -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} |