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diff --git a/poster_aps_mm_2020.tex b/poster_aps_mm_2020.tex index 658d988..a522399 100644 --- a/poster_aps_mm_2020.tex +++ b/poster_aps_mm_2020.tex @@ -3,7 +3,7 @@ \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[]{amsmath} -\usepackage{amssymb,latexsym,mathtools,multicol,lipsum,wrapfig} +\usepackage{amssymb,latexsym,mathtools,multicol,lipsum,wrapfig,floatrow} \usepackage[font=normalsize,labelfont=bf]{caption} \usepackage{tgheros} \usepackage[helvet]{sfmath} @@ -85,37 +85,97 @@ \vspace{1pc} \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} \Large - \begin{wrapfigure}{R}{.25\textwidth} + 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. + + \begin{figure} \centering - \includegraphics[width=0.25\textwidth]{rus_resonances.jpg} - \caption{Resonances } - \end{wrapfigure} + \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} - Strain measures the displacement of material from its equilibrium configuration. - \lipsum[1] + \section{Irreducible representations of strain} + \Large \begin{wrapfigure}{L}{.25\textwidth} \centering - \includegraphics[width=0.55\columnwidth]{urusi_modes.png} + \includegraphics[width=\columnwidth]{urusi_modes.png} \captionof{figure}{ - The crystal structure of \urusi\ and the influence of irreducible strains on it. + The crystal structure of \urusi\ and the influence of the irreducible + strains of \Dfh\ on it. } - \label{nets} + \label{fig:strains} \end{wrapfigure} - \lipsum[2-4] + The strain tensor $\epsilon$ has six independent components. For crystals + with a given point group linear combinations of those components 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 + 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 + 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 $\eta$ + \begin{equation} + f_\op=\frac12\big[r\eta^2+c_\parallel(\nabla_\parallel\eta)^2 + +c_\perp(\nabla_\perp\eta)^2 + +D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4, + \label{eq:fo} + \end{equation} + and the coupling between them $f_\i=-b\epsilon_\X\eta$. Tracing out the + strain gives an effective free energy with the form of \eqref{eq:fo} but with + $r\to\tilde r$. \begin{wrapfigure}{R}{0.6\columnwidth} \centering - \includegraphics[width=0.6\columnwidth]{paper/phase_diagram_experiments} + \includegraphics[width=\columnwidth]{paper/phase_diagram_experiments} \vspace{1em} - \includegraphics[width=0.3\columnwidth]{paper/phases_scalar}\hspace{-0.75em} - \includegraphics[width=0.3\columnwidth]{paper/phases_vector} + \includegraphics[width=0.51\columnwidth]{paper/phases_scalar}\hspace{-0.75em} + \includegraphics[width=0.51\columnwidth]{paper/phases_vector} \captionof{figure}{ Phase diagrams for (a) \urusi\ from Phys Rev B \textbf{77} 115117 (2008) (b) mean field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean @@ -125,12 +185,26 @@ \label{phase_diagram} \end{wrapfigure} - \lipsum[4-5] + 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. + + \section{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 + symmetry $\X$ of the order parameter is \begin{equation} - C_\X(0)=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. \begin{figure} \vspace{1pc} |