From 64ebc5efdb0498c459cc1d280a9acac62b68f151 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 5 Aug 2019 21:47:14 -0400 Subject: redo of the phase diagrams --- main.tex | 163 +++++++++++++++++++++++++++++++++------------------------------ 1 file changed, 85 insertions(+), 78 deletions(-) (limited to 'main.tex') diff --git a/main.tex b/main.tex index aac94e5..2017904 100644 --- a/main.tex +++ b/main.tex @@ -1,46 +1,46 @@ -\documentclass[aps,prl,reprint]{revtex4-1} +\documentclass[aps,prl,reprint,longbibliography]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,graphicx,upgreek,amssymb} % Our mysterious boy -\def\urusi{URu$_2$Si$_2\ $} +\def\urusi{URu$_{\text2}$Si$_{\text2}$} -\def\e{{\mathrm e}} % "elastic" -\def\o{{\mathrm o}} % "order parameter" -\def\i{{\mathrm i}} % "interaction" +\def\e{{\text e}} % "elastic" +\def\o{{\text o}} % "order parameter" +\def\i{{\text i}} % "interaction" -\def\Dfh{D$_{4\mathrm h}$} +\def\Dfh{D$_{\text{4h}}$} % Irreducible representations (use in math mode) -\def\Aog{{\mathrm A_{1\mathrm g}}} -\def\Atg{{\mathrm A_{2\mathrm g}}} -\def\Bog{{\mathrm B_{1\mathrm g}}} -\def\Btg{{\mathrm B_{2\mathrm g}}} -\def\Eg {{\mathrm E_{ \mathrm g}}} -\def\Aou{{\mathrm A_{1\mathrm u}}} -\def\Atu{{\mathrm A_{2\mathrm u}}} -\def\Bou{{\mathrm B_{1\mathrm u}}} -\def\Btu{{\mathrm B_{2\mathrm u}}} -\def\Eu {{\mathrm E_{ \mathrm u}}} +\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{\mathrm X} -\def\Y{\mathrm Y} +\def\X{\text X} +\def\Y{\text Y} % Units -\def\J{\mathrm J} -\def\m{\mathrm m} -\def\K{\mathrm K} -\def\GPa{\mathrm{GPa}} -\def\A{\mathrm{\c A}} +\def\J{\text J} +\def\m{\text m} +\def\K{\text K} +\def\GPa{\text{GPa}} +\def\A{\text{\c A}} % Other -\def\G{\mathrm G} % Ginzburg +\def\G{\text G} % Ginzburg \begin{document} -\title{Elastic properties of \urusi are reproduced by modulated $\Bog$ order} +\title{Elastic properties of \urusi\ are reproduced by modulated $\Bog$ order} \author{Jaron Kent-Dobias} \author{Michael Matty} \author{Brad Ramshaw} @@ -55,7 +55,7 @@ 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 + 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 @@ -68,7 +68,7 @@ \begin{enumerate} \item Introduction \begin{enumerate} - \item \urusi hidden order intro paragraph, discuss the phase diagram + \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 @@ -86,48 +86,58 @@ \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 order parameter, 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 pd paper], and at sufficiently -large [hydrostatic?] pressures, both give way to local moment antiferromagnetism. -Despite over thirty years of effort, the symmetry of the hidden order state remains unknown, and modern theories -\cite{kambe:pr2018a, haule:np2009a, kusunose:jpsj2011a, kung:s2015a,cricchio:prl2009a,ohkawa:jpcm1999a,santini:prl1994a,kiss:ap2004a,harima:jpsj2010a,thalmeier:pr2011a,tonegawa:prl2012a,rau:pr2012a,riggs:nc2015a,hoshino:jpsj2013a,ikeda:prl1998a,chandra:n2013a,harrison:apa2019a,ikeda:np2012a} -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:apa2019a} -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]. Ref.~\cite{ghosh:apa2019a} 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 -order parameter. 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]. -Finally, we discuss our conclusions and future experimental and theoretical work that our results motivate. - -The point group of \urusi is \Dfh, and any coarse-grained theory must locally +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 order parameter, 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 pd paper], and at sufficiently large [hydrostatic?] +pressures, both give way to local moment antiferromagnetism. Despite over +thirty years of effort, the symmetry of the hidden order 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]. +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 order parameter. 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]. Finally, we +discuss our conclusions and future experimental and theoretical work that our +results motivate. + +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 order parameter, and their interaction. The most general quadratic free energy of the strain $\epsilon$ is @@ -230,6 +240,10 @@ The uniform--modulated transition is now continuous. 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{ @@ -397,14 +411,7 @@ this expression can be brought to the form \mathcal I(\xi_{\perp0} q_*|t|^{-1/4}) \lesssim |t|^{13/4}, \end{equation} -where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I$ -is defined by -\begin{equation} - \mathcal I(x)=\frac1\pi\int_{-\infty}^\infty dy\,\frac{\sin\tfrac y2}y - \bigg(\frac1{1+(y^2-x^2)^2} - -\frac{K_1(\sqrt{1+(y^2-x^2)^2})}{\sqrt{1+(y^2-x^2)^2}}\bigg) -\end{equation} -For large argument, $\mathcal I(x)\sim x^{-4}$, yielding +where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I(x)\sim x^{-4}$ for large $x$, yielding \begin{equation} t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4} \end{equation} @@ -426,6 +433,6 @@ self-consistent. \end{acknowledgements} \bibliographystyle{apsrev4-1} -\bibliography{hidden_order,library} +\bibliography{hidden_order, library} \end{document} -- cgit v1.2.3-70-g09d2