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diff --git a/poster_aps_mm_2020.tex b/poster_aps_mm_2020.tex index a522399..3246d92 100644 --- a/poster_aps_mm_2020.tex +++ b/poster_aps_mm_2020.tex @@ -1,5 +1,6 @@ \documentclass[portrait]{a0poster} +\usepackage{pifont} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[]{amsmath} @@ -77,84 +78,72 @@ \def\afm{\textsc{afm}} % antiferromagnetism \def\recip{{\{-1\}}} % functional reciprocal -\noindent\hspace{177pc}\includegraphics[width=18pc]{CULogo-red120px.eps} +\noindent\hspace{176pc}\includegraphics[width=18pc]{CULogo-red120px.eps} \vspace{-24.5pc}\\ -\Huge \textbf{Elastic properties of hidden order in URu$_{\text2}$Si$_{\text2}$ are reproduced by\\ staggered nematic order} +\Huge \textbf{Elastic properties of hidden order in URu$_{\text2}$Si$_{\text2}$ are reproduced by\\ staggered nematic order}\hspace{37pc}\textbf{\LARGE arXiv:1910.01669 [cond-mat.str-el]} \bigskip\\ -\huge \textbf{Jaron~Kent-Dobias, Michael Matty \& Brad J Ramshaw} -\vspace{1pc} +\huge \textbf{Jaron~Kent-Dobias, Michael Matty \& Brad J Ramshaw}\hspace{3.5pc}\ding{81}\hspace{3.5pc}\textbf{Cornell University} \begin{multicols}{2} \Large - \urusi\ is a heavy-fermi material with a novel continuous phase transition whose broken symmetry is not known, earning it the name ``hidden order.'' Under sufficient pressure it instead transitions into an antiferromagnet, and the transitions between these phases is first-order. - - Recent resonant ultrasound spectroscopy experiments measured a novel behavior of the stiffness tensor over several hundred Kelvin approaching the transition. Inspired by this, we develop the most general mean field theory consistent with that behavior and the topology of the phase diagram and find that the only consistent symmetry reproduces the experimental data. - - \section{Resonant ultrasound spectroscopy} + \urusi\ is a heavy fermion material with a continuous transition into a `hidden + order' (\ho) phase whose broken symmetry is unknown. Under sufficient + pressure it instead transitions into an antiferromagnet (\afm), with phase + diagram in Fig.~\ref{phase_diagram}(a). \urusi\ also has a lower-temperature + superconducting phase within \ho\ that is destroyed by the \afm. + + Resonant ultrasound spectroscopy measures anomalous softening in its stiffness + tensor over several hundred Kelvin approaching the \ho\ transition. Starting + with a general Ginzburg--Landau theory of \urusi, we show that only one + order parameter symmetry is consistent with both this softening and the + topology of the phase diagram. This choice reproduces other \urusi\ phenomena and motivates new experiments. + + \section{Resonant ultrasound spectroscopy \&\\ irreducible strains} \Large - When periodically strained by a driving sound, materials respond with - oscillations the response can be measured. Resonances are produced by driving - at the natural frequency of a normal mode of the material and are visible as - a spike in the response amplitude. The stiffness tensor $C$ is uniquely - determined by the natural frequencies of sufficiently many normal modes. - Preforming this decomposition at each temperature in a sweep gives a lot of - information about thermodynamic functions. + \Rus\ drives material with sound and measures the response. Resonances + produced by driving at the natural frequency of a normal mode are visible as + spikes in the response amplitude. The stiffness tensor $C$ is uniquely + determined by the frequencies of sufficiently many normal modes and the + sample geometry. The structure of the stiffness tensor and its behavior as a + function of temperature yield a lot of information about symmetry breaking. \begin{figure} \centering \vspace{1em} - \floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,center},capbesidewidth=0.4\textwidth}}]{figure}[\FBwidth] - { - \caption{ - Response amplitude versus driving frequency for a sample at some - temperature. The spikes correspond to resonances, and the strains - corresponding to a few dominant modes are depicted in the cartoons. - } - \label{fig:rus} - } - {\includegraphics[width=0.5\textwidth]{rus_resonances.jpg}} - \vspace{1em} - \end{figure} - - \section{Irreducible representations of strain} - \Large - - \begin{wrapfigure}{L}{.25\textwidth} - \centering - \includegraphics[width=\columnwidth]{urusi_modes.png} + \includegraphics[width=0.5\textwidth]{rus_resonances.jpg} + \hfill\raisebox{2em}{\includegraphics[width=0.48\textwidth]{urusi_modes.png}} \captionof{figure}{ + \textbf{Left:} Response amplitude versus driving frequency for a sample at some + temperature. The spikes correspond to resonances, and the strains + corresponding to a few dominant modes are depicted in the cartoons. + \textbf{Right:} The crystal structure of \urusi\ and the influence of the irreducible strains of \Dfh\ on it. } - \label{fig:strains} - \end{wrapfigure} + \label{fig:rus} + \vspace{1em} + \end{figure} - The strain tensor $\epsilon$ has six independent components. For crystals - with a given point group linear combinations of those components can be + The strain tensor $\epsilon$ has six independent components that can be divided into tuples upon which symmetry transformations act with irreducible - representations of the point group. \urusi's point group is \Dfh\ and has - five `irreducible' strains depicted in Fig.~\ref{fig:strains}. Of these, four + representations of the crystallographic point group. \urusi's point group is \Dfh\ and has + five `irreducible' strains, depicted in Fig.~\ref{fig:rus}. Of these, four are one-tuples (single-component) and one is a two-tuple (multi-component). - Dividing the strain this way simplifies expression its coupling to an order - parameter, which Landau tells us must be described by an irreducible action - by the symmetry group. If our order parameter shares its representation with - a strain, they can couple linearly. Otherwise their coupling must be - higher-order, which leads to thermodynamic discontinuities but not diverging - responses. \section{Landau--Ginzburg theory} \Large - We want to write down every theory of an order parameter $\eta$ that - \begin{itemize} - \item couples linearly to strain - \item has continuous transitions into hidden order and \afm - \item has a first order transition between hidden order and \afm - \end{itemize} - Of the irreps corresponding to strains, $\Aog$ yields first-order transitions - and shouldn't be considered. For $\eta$ of the remaining, the effective free + Symmetry transformations must act trivially on the free energy, so all its + terms must correspond to $\Aog$ irreps. An order parameter can only linearly + couple to a strain that shares its irrep. Any other coupling must be + higher-order, which leads to thermodynamic discontinuities but not diverging + responses. The anomalous softening seen in \urusi\ suggests the \ho\ order + parameter $\eta$ couple linearly to a strain. + + Of the irreps found in strains, $\Aog$ yields a first-order transition and + therefore can't describe \ho. For any of those remaining, the effective free energy has three components: the bare elastic energy of the strain $f_\e=\sum_\X C^0_\X\epsilon_\X\epsilon_\X$, the bare energy of the order parameter $\eta$ @@ -168,6 +157,8 @@ strain gives an effective free energy with the form of \eqref{eq:fo} but with $r\to\tilde r$. + \vfill\null + \begin{wrapfigure}{R}{0.6\columnwidth} \centering \includegraphics[width=\columnwidth]{paper/phase_diagram_experiments} @@ -185,26 +176,31 @@ \label{phase_diagram} \end{wrapfigure} - This is the theory of a system with a \emph{Lifshitz point} at $\tilde - r=c_\perp=0$. This triple point lies at the confluence of three phases: an - unordered phase, a phase with uniform nonzero $\eta$, and a phase with - modulated nonzero $\eta$, as seen in Fig.~\ref{phase_diagram}. The nature of - the boundaries between these phases depends on how many components $\eta$ - has---since the two-component theory has transitions inconsistent with - \urusi\ we no longer consider it. This leaves $\Bog$ and $\Btg$ as - candidates. + This is the theory of a \emph{Lifshitz point} at $\tilde r=c_\perp=0$. This + triple point lies where three phases meet: an unordered phase with $\eta=0$, + a uniform ordered phase with $\eta\neq0$, and a modulated ordered phase with + $\eta\propto\cos q_*x_3$. Phase diagrams are shown in Fig.~\ref{phase_diagram}(b--c). The type of transition between the ordered phases + depends on how many components $\eta$ has: a one-component theory is first + order while a two-component one is continuous. The irreps + consistent with the first order transition between \ho\ and \afm\ in \urusi\ + are $\Bog$ and $\Btg$. - \section{Stiffness response} + \section{Anomalous stiffness response} \Large - Within this theory the stiffness can be calculated like any other response - function. As the modulated phase is approached, the strain stiffness with the + The stiffness $C$ can be calculated like any other response + function. Approaching the modulated phase, the strain stiffness with the same symmetry $\X$ of the order parameter is \begin{equation} - C_\X=C_\X^0\bigg[1+\frac{b^2}{C_\X^0}\big(D_\perp q_*^4+|\Delta\tilde r|\big)^{-1}\bigg]^{-1}. + C_\X=C_\X^0\bigg[1+\frac{b^2}{C_\X^0}\big(D_\perp q_*^4+|\Delta\tilde r|\big)^{-1}\bigg]^{-1} \label{eq:static_modulus} \end{equation} - which has the form of an inverted cusp at the critical point. + which has the form of an inverted cusp at the critical point. Compared with + experimental \rus\ measurements, this doesn't resemble the $\Btg$ stiffness + in Fig.~\ref{fig:plots}(a), but fits the $\Bog$ stiffness in + Fig.~\ref{fig:plots}(b). This suggests \ho\ is a $\Bog$ nematic modulated along the $c$-axis. + + This theory explains the presence of the 0.4--0.5 inverse lattice constant scattering peak seen in some experiments and is consistent with $\Bog$ symmetry breaking observed in the \afm\ phase. \Rus\ experiments at pressure might resolve the divergence of the modulation wavevector $q_*$ and the vanishing of the $\Bog$ stiffness at the Lifshitz point. \begin{figure} \vspace{1pc} @@ -222,7 +218,7 @@ 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{homo} + \label{fig:plots} \vspace{1pc} \end{figure} \end{multicols} |