From b8871420207aef84b24cab0f2aa880244cc78425 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 23 Aug 2019 11:43:06 -0400
Subject: changed stiffness notation and more our acronyms more consistent

---
 main.tex | 105 ++++++++++++++++++++++++++++++++++-----------------------------
 1 file changed, 56 insertions(+), 49 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 3f9c01f..706b74f 100644
--- a/main.tex
+++ b/main.tex
@@ -1,5 +1,5 @@
 
-\documentclass[aps,prl,reprint,longbibliography]{revtex4-1}
+\documentclass[aps,prl,reprint,longbibliography,floatfix]{revtex4-1}
 \usepackage[utf8]{inputenc}
 \usepackage{amsmath,graphicx,upgreek,amssymb}
 
@@ -37,6 +37,10 @@
 
 % Other
 \def\G{\text G} % Ginzburg
+\def\op{\textsc{op}} % order parameter
+\def\ho{\textsc{ho}} % hidden order
+\def\rus{\textsc{rus}} % Resonant ultrasound spectroscopy 
+\def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy 
 
 \begin{document}
 
@@ -89,18 +93,18 @@
 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
+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
+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.
+the \op, 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{hassinger_temperature-pressure_2008}, and at sufficiently large [hydrostatic?]
+hidden order (\ho) \cite{hassinger_temperature-pressure_2008}, 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,
+thirty years of effort, the symmetry of the \ho\ 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,
@@ -108,30 +112,30 @@ 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
+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
+\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{shekhter_bounding_2013}.
 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
+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
+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
+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
+coupled to an \op. 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
+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{choi_pressure-induced_2018}.  Finally, we
 discuss our conclusions and the future experimental and theoretical work motivated
@@ -139,9 +143,9 @@ by our results.
 
 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.
+in three parts: that of the strain, the \op, and their interaction.
 The most general quadratic free energy of the strain $\epsilon$ is
-$f_\e=\lambda_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the $\lambda$
+$f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare stiffness tensor $C$
 tensor is constrained by both that $\epsilon$ is a symmetric tensor and by the
 point group symmetry \cite{landau_theory_1995}. The latter can be seen in a
 systematic way. First, the six independent components of strain are written as
@@ -160,21 +164,21 @@ action of the point group, or
 Next, all quadratic combinations of these irreducible strains that transform
 like $\Aog$ are included in the free energy as
 \begin{equation}
-  f_\e=\frac12\sum_\X\lambda_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},
+  f_\e=\frac12\sum_\X C_\X^{(ij)}\epsilon_\X^{(i)}\epsilon_\X^{(j)},
 \end{equation}
 where the sum is over irreducible representations of the point group and the
-stiffnesses $\lambda_\X^{(ij)}$ are
+stiffnesses $C_\X^{(ij)}$ are
 \begin{equation}
   \begin{aligned}
-    &\lambda_{\Aog}^{(11)}=\tfrac12(\lambda_{1111}+\lambda_{1122}) &&
-    \lambda_{\Aog}^{(22)}=\lambda_{3333} \\
-    &\lambda_{\Aog}^{(12)}=\lambda_{1133} &&
-    \lambda_{\Bog}^{(11)}=\tfrac12(\lambda_{1111}-\lambda_{1122}) \\
-    &\lambda_{\Btg}^{(11)}=\lambda_{1212} &&
-    \lambda_{\Eg}^{(11)}=\lambda_{1313}.
+    &C_{\Aog}^{(11)}=\tfrac12(C_{1111}+C_{1122}) &&
+    C_{\Aog}^{(22)}=C_{3333} \\
+    &C_{\Aog}^{(12)}=C_{1133} &&
+    C_{\Bog}^{(11)}=\tfrac12(C_{1111}-C_{1122}) \\
+    &C_{\Btg}^{(11)}=C_{1212} &&
+    C_{\Eg}^{(11)}=C_{1313}.
   \end{aligned}
 \end{equation}
-The interaction between strain and the order parameter $\eta$ depends on the
+The interaction between strain and the \op\ $\eta$ depends on the
 representation of the point group that $\eta$ transforms as. If this
 representation is $\X$, then the most general coupling to linear order is
 \begin{equation}
@@ -183,10 +187,10 @@ representation is $\X$, then the most general coupling to linear order is
 If $\X$ is a representation not present in the strain there can be no linear
 coupling, and the effect of $\eta$ going through a continuous phase transition
 is to produce a jump in the $\Aog$ strain stiffness. We will therefore focus
-our attention on order parameter symmetries that produce linear couplings to
+our attention on \op\ symmetries that produce linear couplings to
 strain. Looking at the components present in \eqref{eq:strain-components}, this rules out all of the u-reps (odd under inversion) and the $\Atg$ irrep as having any anticipatory response in the strain stiffness.
 
-If the order parameter transforms like $\Aog$, odd terms are allowed in its
+If the \op\ transforms like $\Aog$, odd terms are allowed in its
 free energy and any transition will be abrupt and not continuous without
 tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic
 free energy density is
@@ -205,16 +209,16 @@ since this does not affect the physics at hand. Neglecting interaction terms
 higher than quadratic order, the only strain relevant to the problem is
 $\epsilon_\X$, and this can be traced out of the problem exactly, since
 \begin{equation}
-  0=\frac{\delta F}{\delta\epsilon_{\X i}(x)}=\lambda_\X\epsilon_{\X i}(x)
+  0=\frac{\delta F}{\delta\epsilon_{\X i}(x)}=C_\X\epsilon_{\X i}(x)
     +\frac12b\eta_i(x)
 \end{equation}
-gives $\epsilon_\X(x)=-(b/2\lambda_\X)\eta(x)$. Upon substitution into the free
+gives $\epsilon_\X(x)=-(b/2C_\X)\eta(x)$. Upon substitution into the free
 energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$,
-with $r\to\tilde r=r-b^2/4\lambda_\X$.
+with $r\to\tilde r=r-b^2/4C_\X$.
 
 With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz
 point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942,
-lifshitz_theory_1942-1}. For a one-component order parameter ($\Bog$ or $\Btg$) it is
+lifshitz_theory_1942-1}. For a one-component \op\ ($\Bog$ or $\Btg$) it is
 traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. For $\tilde
 r>0$ and $c_\perp>0$, or $\tilde r<c_\perp^2/4D_\perp$ and $c_\perp<0$, the
 only stable solution is $\eta_*=q_*=0$ and the system is unordered. For $\tilde
@@ -229,7 +233,7 @@ $q_*^2=-c_\perp/2D_\perp$ and
 with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The
 transition between the uniform and modulated orderings is abrupt for a one-component
 field and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. For a
-two-component order parameter ($\Eg$) we must also allow a relative phase between the
+two-component \op\ ($\Eg$) we must also allow a relative phase between the
 two components of the field. In this case the uniform ordered phase is only
 stable for $c_\perp>0$, and the modulated phase is now characterized by helical
 order with $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$ and
@@ -253,6 +257,9 @@ diagrams for this model are shown in Figure \ref{fig:phases}.
     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 abrupt transitions.
+    Later, when we fit the elastic stiffness predictions for a $\Bog$ \op\ to
+    data along the zero (atmospheric) pressure line, we will take $\tilde
+    r-\tilde r_c=a(T-T_c)$.
   }
   \label{fig:phases}
 \end{figure}
@@ -307,7 +314,7 @@ write
     \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1}
     &=\frac{\delta\eta_j^{-1}[\eta](x)}{\delta\eta_i(x')}
     =-\frac2b\frac{\delta^2F_\o}{\delta\eta_i(x)\delta\eta_j(x')} \\
-    &=-\frac2b\chi_{ij}^{-1}(x,x')-\frac{b}{2\lambda_\X}\delta_{ij}\delta(x-x')
+    &=-\frac2b\chi_{ij}^{-1}(x,x')-\frac{b}{2C_\X}\delta_{ij}\delta(x-x')
   \end{aligned}
   \label{eq:inv.func}
 \end{equation}
@@ -316,27 +323,27 @@ susceptibility of the material to $\epsilon_\X$ strain is given by
 \begin{widetext}
 \begin{equation}
   \begin{aligned}
-    \chi_{\X ij}^{-1}(x,x')
+    \lambda_{\X ij}(x,x')
     &=\frac{\delta^2F}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\
-    &=\lambda_\X\delta_{ij}\delta(x-x')+
+    &=C_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_{\X i}(x')\delta\epsilon_{\X j}(x'')} \\
     &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o}{\delta\eta_k(x'')\delta\eta_\ell(x''')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')}
     +\int dx''\,\frac{\delta F_\o}{\delta\eta_k(x'')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ 
-    &=\lambda_\X\delta_{ij}\delta(x-x')+
+    &=C_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta_k(x'')}{\partial\epsilon_{\X\ell}(x''')}\bigg)^{-1}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} \\ 
-    &=\lambda_\X\delta_{ij}\delta(x-x')+
+    &=C_\X\delta_{ij}\delta(x-x')+
     b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
     -\frac12b\int dx''\,\delta_{i\ell}\delta(x-x'')\frac{\delta\eta_\ell(x'')}{\delta\epsilon_{\X j}(x')} 
-    =\lambda_\X\delta_{ij}\delta(x-x')+
+    =C_\X\delta_{ij}\delta(x-x')+
     \frac12b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')},
   \end{aligned}
 \end{equation}
 \end{widetext}
 whose Fourier transform follows from \eqref{eq:inv.func} as
 \begin{equation}
-  \chi_{\X ij}(q)=\frac{\delta_{ij}}{\lambda_\X}+\frac{b^2}{4\lambda_\X^2}\chi_{ij}(q).
+  \lambda_{\X ij}^{-1}(q)=\frac{\delta_{ij}}{C_\X}+\frac{b^2}{4C_\X^2}\chi_{ij}(q).
   \label{eq:elastic.susceptibility}
 \end{equation}
 At $q=0$, which is where the stiffness measurements used here were taken, this
@@ -348,15 +355,15 @@ r_c|^\gamma$ for $\gamma=1$.
   \includegraphics[width=\columnwidth]{fig-stiffnesses}
   \caption{
     Measurements of the effective strain stiffness as a function of temperature
-    for the six independent components of strain from ultrasound. The vertical
-    lines show the location of the hidden order transition.
+    for the six independent components of strain from \rus. The vertical
+    lines show the location of the \ho\ transition.
   }
   \label{fig:data}
 \end{figure}
 
 We have seen that mean field theory predicts that whatever component of strain
-transforms like the order parameter will see a $t^{-1}$ softening in the
-stiffness that ends in a cusp. Ultrasound experiments \cite{ghosh_single-component_2019}
+transforms like the \op\ will see a $t^{-1}$ softening in the
+stiffness that ends in a cusp. \Rus\ experiments \cite{ghosh_single-component_2019}
 yield the strain stiffness for various components of the strain; this data is
 shown in Figure \ref{fig:data}.  The $\Btg$ and $\Eg$ stiffnesses don't appear
 to have any response to the presence of the transition, exhibiting the expected
@@ -368,7 +375,7 @@ not work quantitatively far below the transition where $\eta$ has a large
 nonzero value and higher powers in the free energy become important. The data
 in the high-temperature phase can be fit to the theory
 \eqref{eq:elastic.susceptibility}, with a linear background stiffness
-$\lambda_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is
+$C_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is
 shown in Figure \ref{fig:fit}. The data and theory appear consistent.
 
 \begin{figure}[htpb]
@@ -377,7 +384,7 @@ shown in Figure \ref{fig:fit}. The data and theory appear consistent.
     Strain stiffness data for the $\Bog$ component of strain (solid) along with
     a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$
     (dashed). The fit gives
-    $\lambda_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
+    $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$,
     $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp
     q_*^4\simeq0.0038\,\K^{-1}$.
   }
@@ -425,13 +432,13 @@ far from the Lifshitz point, the half-wavelength of the modulation should be com
 \cite{meng_imaging_2013}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$,
 we find $\delta t_\G\sim0.4$, though this estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in $\delta t_\G$ over nearly four orders of magnitude.
 The estimate here predicts that an experiment may begin to see deviations from
-mean field behavior within around $5\,\K$ of the critical point. An ultrasound
+mean field behavior within around $5\,\K$ of the critical point. A \rus\ 
 experiment with more precise temperature resolution near the critical point may
-be able to resolve a modified cusp exponent $\gamma\simeq1.31$ \cite{guida_critical_1998}, since the universality class of a uniaxial modulated scalar order parameter is $\mathrm O(2)$ \cite{garel_commensurability_1976}. Our work here appears self--consistent, given that our fit is mostly concerned with temperatures farther than this from the critical point. This analysis also indicates that we should not expect any quantitative agreement between mean field theory and experiment in the low temperature phase since, by the point the Ginzburg criterion is satisfied, $\eta$ is order one and the Landau--Ginzburg free energy expansion is no longer valid.
+be able to resolve a modified cusp exponent $\gamma\simeq1.31$ \cite{guida_critical_1998}, since the universality class of a uniaxial modulated scalar \op\ is $\mathrm O(2)$ \cite{garel_commensurability_1976}. Our work here appears self--consistent, given that our fit is mostly concerned with temperatures farther than this from the critical point. This analysis also indicates that we should not expect any quantitative agreement between mean field theory and experiment in the low temperature phase since, by the point the Ginzburg criterion is satisfied, $\eta$ is order one and the Landau--Ginzburg free energy expansion is no longer valid.
 
 There are two apparent discrepancies between the phase diagram presented in 
 \cite{choi_pressure-induced_2018} and that predicted by our mean field theory. The first is the apparent
-onset of the orthorhombic phase in the HO state prior to the onset of AFM. 
+onset of the orthorhombic phase in the \ho\ state prior to the onset of AFM. 
 As ref.\cite{choi_pressure-induced_2018} notes, this could be due to the lack of an ambient pressure calibration
 for the lattice constant. The second discrepancy is the onset of orthorhombicity 
 at higher temperatures than the onset of AFM. Susceptibility data sees no trace of another phase transition at these higher temperatures \cite{inoue_high-field_2001}, and therefore we don't in fact expect there to be one. We do expect that this could be due to the
-- 
cgit v1.2.3-70-g09d2

