\documentclass[aps,prl,reprint,longbibliography,floatfix]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,graphicx,upgreek,amssymb} % Our mysterious boy \def\urusi{URu$_{\text2}$Si$_{\text2}$} \def\e{{\text{\textsc{Elastic}}}} % "elastic" \def\o{{\text{\textsc{Op}}}} % "order parameter" \def\i{{\text{\textsc{Int}}}} % "interaction" \def\Dfh{D$_{\text{4h}}$} % Irreducible representations (use in math mode) \def\Aog{{\text A_{\text{1g}}}} \def\Atg{{\text A_{\text{2g}}}} \def\Bog{{\text B_{\text{1g}}}} \def\Btg{{\text B_{\text{2g}}}} \def\Eg {{\text E_{\text g}}} \def\Aou{{\text A_{\text{1u}}}} \def\Atu{{\text A_{\text{2u}}}} \def\Bou{{\text B_{\text{1u}}}} \def\Btu{{\text B_{\text{2u}}}} \def\Eu {{\text E_{\text u}}} % Variables to represent some representation \def\X{\text X} \def\Y{\text Y} % Units \def\J{\text J} \def\m{\text m} \def\K{\text K} \def\GPa{\text{GPa}} \def\A{\text{\r A}} % Other \def\G{\text G} % Ginzburg \def\op{\textsc{op}} % order parameter \def\ho{\textsc{ho}} % hidden order \def\rus{\textsc{rus}} % Resonant ultrasound spectroscopy \def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy \def\recip{{\{-1\}}} % functional reciprocal \begin{document} \title{Elastic properties of \urusi\ are reproduced by modulated $\Bog$ order} \author{Jaron Kent-Dobias} \author{Michael Matty} \author{Brad Ramshaw} \affiliation{ Laboratory of Atomic \& Solid State Physics, Cornell University, Ithaca, NY, USA } \date\today \begin{abstract} We develop a phenomenological theory for the elastic response of materials with a \Dfh\ point group through phase transitions. The physics is generically that of Lifshitz points, with disordered, uniform ordered, and modulated ordered phases. Several experimental features of \urusi\ are reproduced when the order parameter has $\Bog$ symmetry: the topology of the temperature--pressure phase diagram, the response of the strain stiffness tensor above the hidden-order transition, and the strain response in the antiferromagnetic phase. In this scenario, the hidden order is a version of the high-pressure antiferromagnetic order modulated along the symmetry axis. \end{abstract} \maketitle % \begin{enumerate} % \item Introduction % \begin{enumerate} % \item \urusi\ hidden order intro paragraph, discuss the phase diagram % \item Strain/OP coupling discussion/RUS % \item Discussion of experimental data % \item We look at MFT's for OP's of various symmetries % \end{enumerate} % \item Theory % \begin{enumerate} % \item Introduce various pieces of free energy % \item Summary of MFT results % \end{enumerate} % \item Data piece % \item Talk about more cool stuff like AFM C4 breaking etc % \end{enumerate} The study of phase transitions is a central theme of condensed matter physics. In many cases, a phase transition between different states of matter is marked by a change in symmetry. In this paradigm, the breaking of symmetry in an ordered phase corresponds to the condensation of an order parameter (\op) that breaks the same symmetries. Near a second order phase transition, the physics of the \op\ can often be described in the context of Landau-Ginzburg mean field theory. However, to construct such a theory, one must know the symmetries of the \op, i.e. the symmetry of the ordered state. A paradigmatic example where the symmetry of an ordered phase remains unknown is in \urusi. \urusi\ is a heavy fermion superconductor in which superconductivity condenses out of a symmetry broken state referred to as hidden order (\ho) \cite{hassinger_temperature-pressure_2008}, and at sufficiently large [hydrostatic?] pressures, both give way to local moment antiferromagnetism. Despite over thirty years of effort, the symmetry of the \ho\ state remains unknown, and modern theories \cite{kambe_odd-parity_2018, haule_arrested_2009, kusunose_hidden_2011, kung_chirality_2015, cricchio_itinerant_2009, ohkawa_quadrupole_1999, santini_crystal_1994, kiss_group_2005, harima_why_2010, thalmeier_signatures_2011, tonegawa_cyclotron_2012, rau_hidden_2012, riggs_evidence_2015, hoshino_resolution_2013, ikeda_theory_1998, chandra_hastatic_2013, harrison_hidden_2019, ikeda_emergent_2012} propose a variety of possibilities. Many [all?] of these theories rely on the formulation of a microscopic model for the \ho\ state, but without direct experimental observation of the broken symmetry, none have been confirmed. One case that does not rely on a microscopic model is recent work \cite{ghosh_single-component_2019} that studies the \ho\ transition using resonant ultrasound spectroscopy (\rus). \Rus\ is an experimental technique that measures mechanical resonances of a sample. These resonances contain information about the full elastic tensor of the material. Moreover, the frequency locations of the resonances are sensitive to symmetry breaking at an electronic phase transition due to electron-phonon coupling \cite{shekhter_bounding_2013}. Ref.~\cite{ghosh_single-component_2019} uses this information to place strict thermodynamic bounds on the symmetry of the \ho\ \op, again, independent of any microscopic model. Motivated by these results, in this paper we consider a mean field theory of an \op\ coupled to strain and the effect that the \op\ symmetry has on the elastic response in different symmetry channels. Our study finds that a single possible \op\ symmetry reproduces the experimental strain susceptibilities, and fits the experimental data well. We first present a phenomenological Landau-Ginzburg mean field theory of strain coupled to an \op. We examine the phase diagram predicted by this theory and compare it to the experimentally obtained phase diagram of \urusi. Then we compute the elastic response to strain, and examine the response function dependence on the symmetry of the \op. We proceed to compare the results from mean field theory with data from \rus\ experiments. We further examine the consequences of our theory at non-zero applied pressure in comparison with recent x-ray scattering experiments \cite{choi_pressure-induced_2018}. Finally, we discuss our conclusions and the future experimental and theoretical work motivated by our results. The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally respect this symmetry. We will introduce a phenomenological free energy density in three parts: that of the strain, the \op, and their interaction. The most general quadratic free energy of the strain $\epsilon$ is $f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare stiffness tensor $C$ tensor is constrained by both that $\epsilon$ is a symmetric tensor and by the point group symmetry \cite{landau_theory_1995}. The latter can be seen in a systematic way. First, the six independent components of strain are written as linear combinations that behave like irreducible representations under the action of the point group, or \begin{equation} \begin{aligned} \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth} \epsilon_\Aog^{(2)}=\epsilon_{33} \\ \epsilon_\Bog^{(1)}=\epsilon_{11}-\epsilon_{22} && \epsilon_\Btg^{(1)}=2\epsilon_{12} \\ \epsilon_\Eg^{(1)}=2\{\epsilon_{11},\epsilon_{22}\}. \end{aligned} \label{eq:strain-components} \end{equation} Next, all quadratic combinations of these irreducible strains that transform like $\Aog$ are included in the free energy as \begin{equation} f_\e=\frac12\sum_\X C_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)}, \end{equation} where the sum is over irreducible representations of the point group and the stiffnesses $C_\X^{(ij)}$ are \begin{equation} \begin{aligned} &C_{\Aog}^{(11)}=\tfrac12(C_{1111}+C_{1122}) && C_{\Aog}^{(22)}=C_{3333} \\ &C_{\Aog}^{(12)}=C_{1133} && C_{\Bog}^{(11)}=\tfrac12(C_{1111}-C_{1122}) \\ &C_{\Btg}^{(11)}=C_{1212} && C_{\Eg}^{(11)}=C_{1313}. \end{aligned} \end{equation} The interaction between strain and the \op\ $\eta$ depends on the representation of the point group that $\eta$ transforms as. If this representation is $\X$, then the most general coupling to linear order is \begin{equation} f_\i=b^{(i)}\epsilon_\X^{(i)}\eta \end{equation} If $\X$ is a representation not present in the strain there can be no linear coupling, and the effect of $\eta$ going through a continuous phase transition is to produce a jump in the $\Aog$ strain stiffness. We will therefore focus our attention on \op\ symmetries that produce linear couplings to strain. Looking at the components present in \eqref{eq:strain-components}, this rules out all of the u-reps (odd under inversion) and the $\Atg$ irrep as having any anticipatory response in the strain stiffness. If the \op\ transforms like $\Aog$, odd terms are allowed in its free energy and any transition will be abrupt and not continuous without tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic free energy density is \begin{equation} \begin{aligned} f_\o=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2 +c_\perp(\nabla_\perp\eta)^2 \\ &\quad+D_\parallel(\nabla_\parallel^2\eta)^2 +D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4 \end{aligned} \label{eq:fo} \end{equation} where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and $\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$ since this does not affect the physics at hand. The full free energy functional of $\eta$ and $\epsilon$ is then \begin{equation} F[\eta,\epsilon]=\int dx\,(f_\o+f_\e+f_\i) \end{equation} Neglecting interaction terms higher than quadratic order, the only strain relevant to the problem is $\epsilon_\X$, and this can be traced out of the problem exactly, since \begin{equation} 0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_{\X i}(x)}=C_\X\epsilon_{\X i}(x) +\frac12b\eta_i(x) \end{equation} gives $\epsilon_\X[\eta]=-(b/2C_\X)\eta$. Upon substitution into the free energy, the resulting effective free energy $F_\o[\eta]=F[\eta,\epsilon[\eta]]$ has a density identical to $f_\o$ with $r\to\tilde r=r-b^2/4C_\X$. With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942, lifshitz_theory_1942-1}. For a one-component \op\ ($\Bog$ or $\Btg$) it is traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$, or $\tilde r0$, and the modulated phase is now characterized by helical order with $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. The uniform--modulated transition is now continuous. This already does not reproduce the physics of \ho, and so we will henceforth neglect this possibility. The schematic phase diagrams for this model are shown in Figure \ref{fig:phases}. \begin{figure}[htpb] \includegraphics[width=\columnwidth]{phase_diagram_experiments} \vspace{1em} \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em} \includegraphics[width=0.51\columnwidth]{phases_vector} \caption{ Phase diagrams for (a) \urusi\ from experiments (neglecting the superconducting phase) \cite{hassinger_temperature-pressure_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 transitions, while dashed lines denote abrupt transitions. Later, when we fit the elastic stiffness predictions for a $\Bog$ \op\ to data along the zero (atmospheric) pressure line, we will take $\Delta\tilde r=\tilde r-\tilde r_c=a(T-T_c)$. } \label{fig:phases} \end{figure} The susceptibility of the order parameter to a field linearly coupled to it is given by \begin{equation} \begin{aligned} &\chi^\recip(x,x') =\frac{\delta^2F_\o[\eta]}{\delta\eta(x)\delta\eta(x')} =\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2 \\ &\qquad\qquad+D_\perp\nabla_\perp^4+12u\eta^2(x)\big) \delta(x-x'), \end{aligned} \label{eq:sus_def} \end{equation} where $\recip$ indicates a \emph{functional reciprocal} in the sense that \[ \int dx''\,\chi^\recip(x,x'')\chi(x'',x')=\delta(x-x'). \] Taking the Fourier transform and integrating over $q'$ we have \begin{equation} \chi(q) =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4 +12u\sum_{q'}\tilde\eta_{q'}\tilde\eta_{-q'}\big)^{-1}. \end{equation} Near the unordered--modulated transition this yields \begin{equation} \begin{aligned} \chi(q) &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2 +|\Delta\tilde r|} \\ &=\frac1{D_\perp}\frac{\xi_\perp^4} {1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2}, \end{aligned} \label{eq:susceptibility} \end{equation} with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}$ and $\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}$. We must emphasize that this is \emph{not} the magnetic susceptibility because a $\Bog$ or $\Btg$ \op\ cannot couple linearly to a uniform magnetic field. The object defined in \eqref{eq:sus_def} is most readily interpreted as proportional to the two-point connected correlation function $\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$. The strain stiffness is given in a similar way to the inverse susceptibility: we must trace over $\eta$ and take the second variation of the resulting free energy functional of $\epsilon$. Extremizing over $\eta$ yields \begin{equation} 0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}=\frac{\delta F_\o[\eta]}{\delta\eta(x)} +\frac12b\epsilon_\X(x), \label{eq:implicit.eta} \end{equation} which implicitly gives $\eta[\epsilon]$ and $F_\e[\epsilon]=F[\eta[\epsilon],\epsilon]$. Since $\eta$ is a functional of $\epsilon_\X$ alone, only the stiffness $\lambda_\X$ is modified from its bare value $C_\X$. Though this cannot be solved explicitly, we can make use of the inverse function theorem. First, denote by $\eta^{-1}[\eta]$ the inverse functional of $\eta$ implied by \eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$ corresponding to each solution of \eqref{eq:implicit.eta} it receives. Now, we use the inverse function theorem to relate the functional reciprocal of the derivative of $\eta[\epsilon]$ with respect to $\epsilon_\X$ to the derivative of $\eta^{-1}[\eta]$ with respect to $\eta$, yielding \begin{equation} \begin{aligned} \bigg(\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}\bigg)^\recip &=\frac{\delta\eta^{-1}[\eta](x)}{\delta\eta(x')} =-\frac2b\frac{\delta^2F_\o[\eta]}{\delta\eta(x)\delta\eta(x')} \\ &=-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x'), \end{aligned} \label{eq:inv.func} \end{equation} where we have used what we already know about the variation of $F_\o[\eta]$ with respect to $\eta$. Finally, \eqref{eq:implicit.eta} and \eqref{eq:inv.func} can be used in concert with the ordinary rules of functional calculus to yield the strain stiffness \begin{widetext} \begin{equation} \begin{aligned} \lambda_\X(x,x') &=\frac{\delta^2F_\e[\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\ &=C_\X\delta(x-x')+ b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')} +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')} \\ &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o[\eta]}{\delta\eta(x'')\delta\eta(x''')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')} +\int dx''\,\frac{\delta F_\o[\eta]}{\delta\eta(x'')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\ &=C_\X\delta(x-x')+ b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')} -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta(x'')}{\partial\epsilon_\X(x''')}\bigg)^{-1}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')} \\ &=C_\X\delta(x-x')+ b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')} -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta(x'')}{\delta\epsilon_\X(x')} =C_\X\delta(x-x')+ \frac12b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}, \end{aligned} \end{equation} \end{widetext} whose Fourier transform follows from \eqref{eq:inv.func} as \begin{equation} \lambda_\X(q)=C_\X\bigg(1+\frac{b^2}{4C_\X}\chi(q)\bigg)^{-1}. \label{eq:elastic.susceptibility} \end{equation} Though not relevant here, this result generalizes to multicomponent order parameters. At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the elastic susceptibility of the form $|\Delta\tilde r|^\gamma$ for $\gamma=1$. \begin{figure}[htpb] \centering \includegraphics[width=\columnwidth]{fig-stiffnesses} \caption{ Measurements of the effective strain stiffness as a function of temperature for the six independent components of strain from \rus. The vertical lines show the location of the \ho\ transition. } \label{fig:data} \end{figure} We have seen that mean field theory predicts that whatever component of strain transforms like the \op\ will see a $t^{-1}$ softening in the stiffness that ends in a cusp. \Rus\ experiments \cite{ghosh_single-component_2019} yield the strain stiffness for various components of the strain; this data is shown in Figure \ref{fig:data}. The $\Btg$ and $\Eg$ stiffnesses don't appear to have any response to the presence of the transition, exhibiting the expected linear stiffening with a low-temperature cutoff \cite{varshni_temperature_1970}. The $\Bog$ stiffness has a dramatic response, softening over the course of roughly $100\,\K$. There is a kink in the curve right at the transition. While the low-temperature response is not as dramatic as the theory predicts, 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. The data in the high-temperature phase can be fit to the theory \eqref{eq:elastic.susceptibility}, with a linear background stiffness $C_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure \ref{fig:fit}. The data and theory appear consistent. \begin{figure}[htpb] \includegraphics[width=\columnwidth]{fig-fit} \caption{ Strain stiffness data for the $\Bog$ component of strain (solid) along with a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$ (dashed). The fit gives $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp q_*^4\simeq0.0038\,\K^{-1}$. } \label{fig:fit} \end{figure} Mean field theory neglects the effect of fluctuations on critical behavior, yet also predicts the magnitude of those fluctuations. This allows a mean field theory to undergo an internal consistency check to ensure the predicted fluctuations are indeed negligible. This is typically done by computing the Ginzburg criterion \cite{ginzburg_remarks_1961}, which gives the proximity to the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to break down by comparing the magnitude of fluctuations in a correlation-length sized box to the magnitude of the field. In the modulated phase the spatially averaged magnitude is zero, and so we will instead compare fluctuations in the \emph{amplitude} at $q_*$ to the magnitude of that amplitude. Defining the field $\alpha$ by $\eta(x)=\alpha(x)e^{-iq_*x_3}$, it follows that in the modulated phase $\alpha(x)=\alpha_0$ for $\alpha_0^2=|\delta \tilde r|/4u$. In the modulated phase, the $q$-dependant fluctuations in $\alpha$ are given by \[ G_\alpha(q)=k_BT\chi_\alpha(q)=\frac1{c_\parallel q_\parallel^2+D_\perp(4q_*^2q_\perp^2+q_\perp^4)+2|\delta r|}, \] An estimate of the Ginzburg criterion is then given by the temperature at which $V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle\simeq\langle\alpha\rangle^2=\alpha_0^2$, where $V_\xi=\xi_\perp\xi_\parallel^2$ is a correlation volume. The parameter $u$ can be replaced in favor of the jump in the specific heat at the transition using \begin{equation} c_V=-T\frac{\partial^2f}{\partial T^2} =\begin{cases}0&T>T_c\\Ta^2/12 u&T