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diff --git a/poster_aps_mm_2020.tex b/poster_aps_mm_2020.tex index 84d0496..d63be44 100644 --- a/poster_aps_mm_2020.tex +++ b/poster_aps_mm_2020.tex @@ -11,8 +11,6 @@ \usepackage[export]{adjustbox} \renewcommand*\familydefault{\sfdefault} -\mathtoolsset{showonlyrefs=true} - \setlength\textwidth{194pc} \begin{document} @@ -80,33 +78,43 @@ \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}\hspace{37.5pc}\textbf{\LARGE arXiv:1910.01669 [cond-mat.str-el]} +\Huge \textbf{Elastic properties of hidden order in URu$_{\text2}$Si$_{\text2}$ are reproduced by\\ staggered nematic order} +\hspace{37.5pc}\textbf{\LARGE arXiv:1910.01669 [cond-mat.str-el]} \bigskip\\ -\huge \textbf{Jaron~Kent-Dobias, Michael Matty \& B J Ramshaw}\hspace{7pc}\ding{81}\hspace{7pc}\textbf{Cornell University} +\huge \textbf{Jaron~Kent-Dobias, Michael Matty \& B J Ramshaw} +\hspace{7pc}\ding{81}\hspace{7pc}\textbf{Cornell University} \begin{multicols}{2} \Large - \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. + \hspace{1em} + \textbf{ \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). It also has a + superconducting phase within \ho\ that is destroyed by \afm. + + \hspace{1em} + The modulus tensor of \urusi\ anomalously softens over several hundred + Kelvin and is cut off by the \ho\ transition. + \textbf{ + We show that only one order parameter symmetry is consistent with 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 - \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. + \hspace{1em} + Resonant ultrasound spectroscopy drives a sample with sound and measures its + resonances by looking for spikes in the response. Using the sample geometry + and the location of sufficiently many resonances, the modulus tensor $C$ + can be calculated. + \textbf{ + The structure of the modulus tensor and its temperature dependence yield + a lot of information about symmetry breaking. + } \begin{figure} \centering @@ -114,50 +122,66 @@ \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 + \textbf{Left:} + Response 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. + strains of \Dfh. } \label{fig:rus} \vspace{1em} \end{figure} - The strain tensor $\epsilon$ has six independent components that can be - divided into tuples upon which symmetry transformations act with irreducible - 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). + \hspace{1em} + The strain tensor $\epsilon$ has six independent components. These components can be + divided into tuples that symmetry transformations of the crystal act on with irreducible + representations (irreps) of its point group. + \textbf{ + \urusi's point group \Dfh\ yields five `irreducible' strains, shown in + Fig.~\ref{fig:rus}. + } + Four are one-tuples (single-component). These are uniform compression in and + out of the plane ($\Aog$) and in-plane shear along the faces and diagonals of + the unit cell ($\Bog$ and $\Btg$). One is a two-tuple (multi-component), and + corresponds to out of plane shears ($\Eg$). + \section{Landau--Ginzburg theory} \Large - 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 $\eta$ + \hspace{1em} + The free energy must be invariant under symmetry transformations. This + constrains the way fields can couple together. Two fields can linearly couple + only if they correspond to the same irrep. Higher order couplings lead to + thermodynamic discontinuities but not diverging responses. + \textbf{ + The anomalous softening of \urusi\ suggests the hidden order parameter + couples linearly to strain. + } + + \hspace{1em} + An $\Aog$ order parameter results in first-order transitions and can't + describe hidden order. If the order parameter $\eta$ corresponds to any of + the remaining irreps $\X$ present in strain, the free energy is + $f=f_\e+f_\i+f_\op$, where \begin{equation} - f_\op=\frac12\big[r\eta^2+c_\parallel(\nabla_\parallel\eta)^2 + \begin{aligned} + & \hspace{2em}f_\e=\sum_\X C^0_\X\epsilon_\X\epsilon_\X + \hspace{4em} f_\i=-b\epsilon_\X\eta \\ + & 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, + +D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4 + \end{aligned} \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 - $r\to\tilde r$. + The strain $\epsilon$ can be traced out exactly, which results in a free + energy density with the form of $f_\op$ but with $r\to\tilde + r=r-b^2/2C^0_\X$. - \vfill\null + \vfill \begin{wrapfigure}{R}{0.6\columnwidth} \centering @@ -168,55 +192,74 @@ \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 - field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote - continuous transitions, while dashed lines denote first order transitions. + Phase diagrams for (a) \urusi\ from Phys Rev B \textbf{77} 115117 (2008) + (b) theory for a one-component order ($\Bog$ or $\Btg$) (c) theory for a + two-component order ($\Eg$). Solid lines denote continuous transitions, + while dashed denote first order. } \label{phase_diagram} \end{wrapfigure} - 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 $\Btg$. + \hspace{1em} + \textbf{ + This theory has a \emph{Lifshitz triple point} at $\tilde r=c_\perp=0$. + } + The three phases that meet are + \begin{itemize} + \item\hspace{0.25em}unordered ($\eta=0$) + \item\hspace{0.25em}uniform ($\eta\neq0$) + \item\hspace{0.25em}modulated ($\eta\propto\cos q_*x_3$) + \end{itemize} + Phase diagrams are in Fig.~\ref{phase_diagram}(b--c). The order of the + transition between the uniform and modulated phases depends on the number of + order parameter components: one-component theories have a first order + transition while two-component theories are continuous. + \textbf{ + The only order parameter irreps consistent with the first order transition + between \ho\ and \afm\ in \urusi\ are $\Bog$ and $\Btg$. + } - \section{Anomalous stiffness response} + \section{Anomalous modulus response} \Large - 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 + \hspace{1em} + The modulus $C$ can be calculated like any other response function. + Approaching the modulated phase, the modulus with the symmetry irrep + $\X$ 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} \label{eq:static_modulus} \end{equation} - 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. + \hspace{1em} + \textbf{ + At the critical point the modulus with the symmetry of the order parameter + has an inverted cusp. + } + Compared with experiments, this doesn't resemble the $\Btg$ modulus in + Fig.~\ref{fig:plots}(a) but fits the $\Bog$ modulus in + Fig.~\ref{fig:plots}(b). \textbf{This suggests hidden order is a $\Bog$ + nematic modulated along the \textit{c}-axis.} + + \hspace{1em} + 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 $\Bog$ modulus at the Lifshitz point. \begin{figure} - \vspace{1pc} + \vspace{1em} \centering \includegraphics[width=\columnwidth]{paper/fig-stiffnesses.pdf} \captionof{figure}{ - \Rus\ measurements of the elastic moduli of \urusi\ at ambient pressure - as a function of temperature from \texttt{arXiv:1903.00552 - [cond-mat.str-el]} (blue, solid) alongside fits to theory (magenta, - dashed). The solid yellow region shows the location of the \ho\ phase. - (a) $\Btg$ modulus data and a fit to the standard form. (b) $\Bog$ + \Rus\ measurements of \urusi's moduli as a function of temperature from + \texttt{arXiv:1903.00552 [cond-mat.str-el]} (blue, solid) alongside fits + to this theory (magenta, dashed). The solid yellow region shows the \ho\ + phase. (a) $\Btg$ modulus data and a fit to standard form. (b) $\Bog$ modulus data and a fit to \eqref{eq:static_modulus}. (c) $\Bog$ modulus data and the fit of the \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus - data and the fit 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. + data and fit transformed in way predicted from \eqref{eq:static_modulus} + to be an absolute value function. } \label{fig:plots} \vspace{1pc} |