redo of the phase diagrams

1 files changed, 85 insertions, 78 deletions

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}