From 1fe05733b0635a16bc8f4c21b72320e3633c2b58 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 22 Oct 2019 15:56:33 -0400
Subject: more fixes

---
 main.tex | 43 ++++++++++++++++++++++++++-----------------
 1 file changed, 26 insertions(+), 17 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 3a43f91..4f8ab62 100644
--- a/main.tex
+++ b/main.tex
@@ -102,7 +102,7 @@ ikeda_emergent_2012} propose a variety of possibilities. Our work here seeks to
 
 Recent \emph{resonant ultrasound spectroscopy} (\rus) measurements examined the thermodynamic discontinuities in the elastic moduli at T$_{\mathrm{HO}}$ \cite{ghosh_single-component_nodate}. The observation of discontinues only in compressional, or $\Aog$, elastic moduli requires that the point-group representation of \ho\ is one-dimensional. This rules out a large number of order parameter candidates \brad{cite those ruled out} in a model-free way, but still leaves the microscopic nature of \ho~ undecided. 
 
-Recent X-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure \cite{choi_pressure-induced_2018}. Above \brad{whatever pressure they find it at...}, \urusi\ undergoes a $\Bog$ nematic distortion. While it is still unclear as to whether this is a true thermodynamic phase transition, it may be related to the anomalous softening of the $\Bog$ elastic modulus---$(c_{11}-c_{12})/2$---that occurs over a broad temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by these results, hinting at a $\Bog$ strain susceptibility associated with the \ho\ state, we construct a phenomenological mean field theory for an arbitrary \op\ coupled to strain, and the determine the effect of its phase transitions on the elastic response in different symmetry channels. 
+Recent X-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure \cite{choi_pressure-induced_2018}. Above 0.13--0.5 $\GPa$ (depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion. While it is still unclear as to whether this is a true thermodynamic phase transition, it may be related to the anomalous softening of the $\Bog$ elastic modulus---$(c_{11}-c_{12})/2$---that occurs over a broad temperature range at zero-pressure \brad{cite old ultrasound}. Motivated by these results, hinting at a $\Bog$ strain susceptibility associated with the \ho\ state, we construct a phenomenological mean field theory for an arbitrary \op\ coupled to strain, and the determine the effect of its phase transitions on the elastic response in different symmetry channels. 
 
 We find that only one \op\ symmetry reproduces the anomalous $(c_{11}-c_{12})/2$ elastic modulus, which softens in a Curie-Weiss like manner from room temperature, but which cusps at T$_{\mathrm{HO}}$. That theory associates \ho\ with a $\Bog$ \op\ \emph{modulated along the $c$- axis}, the \afm\ state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. Besides the agreement with ultrasound data across a broad temperature range, the theory predicts uniform $\Bog$ strain at high pressure---the same distortion which was recently seen in x-ray scattering experiments \cite{choi_pressure-induced_2018}. This theory
 strongly motivates future ultrasound experiments under pressure approaching the Lifshitz point, which should find that the $(c_{11}-c_{12})/2$ diverges once the uniform $\Bog$ strain sets in.
@@ -197,7 +197,7 @@ gives the optimized strain conditional on the \op\ as
 $\epsilon_\X^\star[\eta](x)=(b/C_\X)\eta(x)$ and $\epsilon_\Y^\star[\eta]=0$
 for all other $\Y$. Upon substitution into the free energy, the resulting
 effective free energy $F[\eta,\epsilon_\star[\eta]]$ has a density identical to
-$f_\op$ with $r\to\tilde r=r-b^2/2C_\X$. \brad{need a sentence along the lines of "As $r$ is typically associated with $T-T_c$, this substitution has the effect of shifting the bare $T_c$ due to linear coupling between strain and order parameter", or something like that. Actually I'm a bit confused, shouldn't the new Tc be proportional to strain? Or is this just the correct even in the absence of any applied strain}.
+$f_\op$ with $r\to\tilde r=r-b^2/2C_\X$. 
 
 \begin{figure}[htpb]
   \includegraphics[width=\columnwidth]{phase_diagram_experiments}
@@ -219,16 +219,19 @@ $f_\op$ with $r\to\tilde r=r-b^2/2C_\X$. \brad{need a sentence along the lines o
   \label{fig:phases}
 \end{figure}
 
-\section{Results}
 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 \op\ ($\Bog$ or $\Btg$) it is
+lifshitz_theory_1942-1}. The properties discused below can all be found in a
+standard text, e.g.,~\cite{chaikin_principles_2000}. For a one-component \op\
+($\Bog$ or $\Btg$) and positive $c_\parallel$, it is
 traditional to make the field ansatz
-$\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$ \brad{Why is it traditional to ignore any in-plane modulation (x1, x2)?}. For $\tilde r>0$ and $c_\perp>0$,
+$\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. 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 $\tilde r<0$ there are free
-energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has uniform order of whatever . For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free
-energy minima for $q_*^2=-c_\perp/2D_\perp$ and
+energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has
+uniform order with the \op\ symmetry, e.g., $\Bog$ or $\Btg$. For $c_\perp<0$
+and $\tilde r<c_\perp^2/4D_\perp$ there are free energy minima for
+$q_*^2=-c_\perp/2D_\perp$ and
 \begin{equation}
   \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u}
     =\frac{\tilde r_c-\tilde r}{3u}
@@ -238,11 +241,17 @@ 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 \op\ and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde
 r/5}$. For a two-component \op\ ($\Eg$) we must also allow a relative phase
-between the two components of the field \brad{Unless there is a specific reason, we should probably stick to \op\ instead of "field"}. In this case the uniform ordered phase
+between the two components of the \op. 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 $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. The uniform--modulated transition is now continuous \brad{Is this at all obvious? Do we have to "show" it in some way? Or cite something?}. This does not reproduce the physics of \ho, which has a first-order \brad{I think we should keep the language "First order" rather than "abrupt", which doesn't really mean anything specific} transition between \ho\ and \afm, and so we will henceforth neglect the possibility of a multicomponent order parameter. The schematic phase diagrams for this model are shown in Figure
-\ref{fig:phases}.
+helical order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$.
+The uniform--modulated transition is now continuous. This does not reproduce
+the physics of \ho, which has a first-order \brad{I think we should keep the
+language "First order" rather than "abrupt", which doesn't really mean anything
+specific} transition between \ho\ and \afm, and so we will henceforth neglect
+the possibility of a multicomponent order parameter. The schematic phase
+diagrams for this model are shown in Figure~\ref{fig:phases}.
 
+\section{Results}
 We will now derive the \emph{effective elastic tensor}
 $\lambda$ that results from the coupling of strain to the \op. The ultimate
 result, found in \eqref{eq:elastic.susceptibility}, is that $\lambda_\X$
@@ -379,9 +388,9 @@ $|\Delta\tilde r|^\gamma$ for $\gamma=1$.  \brad{I think this last sentence, whi
   \centering
   \includegraphics[width=\columnwidth]{fig-stiffnesses}
   \caption{
-   Resonant ultrasound spectroscopy measurements of the elastic moduli of \urusi\ as a function of temperature
-    for the six independent components of strain. The vertical lines
-    show the location of the \ho\ transition. \brad{Can you move the labels on the right-hand panels over to the right-hand axis? Also, can you make the labels smaller and the actual panels bigger}
+   Resonant ultrasound spectroscopy measurements of the elastic moduli of
+   \urusi\ as a function of temperature for the six independent components of
+   strain. The vertical lines show the location of the \ho\ transition. 
   }
   \label{fig:data}
 \end{figure}
@@ -442,9 +451,9 @@ pressure, where the depth of the cusp in the $\Bog$ stiffness should deepen
 (perhaps with these commensurability jumps) at low pressure and approach zero
 like $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. \brad{Should also motivate x-ray and neutron-diffraction experiments to look for new q's - mentioning this is important if we want to get others interested, no one else does RUS...} Alternatively,
 \rus\ done at ambient pressure might examine the heavy fermi liquid to \afm\
-transition by doping. \brad{We have to be careful, someone did do some doping studies and it's not clear exctly what's going on}. The presence of spatial commensurability is not
+transition by doping. \brad{We have to be careful, someone did do some doping studies and it's not clear exctly what's going on}. The presence of spatial commensurability known to be irrelevant to the critical behavior at a one-component disordered to modulated transition, and therefore is not
 expected to modify the critical behavior otherwise
-\cite{garel_commensurability_1976}. \brad{this thought feels half-finished, where was it going?}
+\cite{garel_commensurability_1976}. 
 
 There are two apparent discrepancies between the orthorhombic strain in the
 phase diagram presented by \cite{choi_pressure-induced_2018} and that predicted
@@ -460,7 +469,7 @@ energy nature of x-rays as an experimental probe: orthorhombic fluctuations
 could appear at higher temperatures than the true onset of an orthorhombic
 phase. \brad{I think this paragraph should probably be tigtened up a bit, we need to be more specific about "don't expect there to be one" and "fluctuations"}. 
 
-Three dimensions is below the upper critical dimension $4\frac12$ \brad{upper critical dimension of what, all Landau mean field theories?}, and so mean
+Three dimensions is below the upper critical dimension $4\frac12$ of a one-component disordered to modulated transition, and so mean
 field theory should break down sufficiently close to the critical point due to
 fluctuations, at the Ginzburg temperature \cite{hornreich_lifshitz_1980, ginzburg_remarks_1961}. Magnetic phase transitions tend to have Ginzburg temperature of order one.
 Our fit above gives $\xi_{\perp0}q_*=(D_\perp
@@ -469,7 +478,7 @@ The agreement of this data in the $t\sim0.1$--10 range with the mean field expon
 around several degrees Kelvin 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 \op\ is $\mathrm O(2)$
+universality class of a uniaxial modulated one-component \op\ is $\mathrm O(2)$
 \cite{garel_commensurability_1976}. 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
-- 
cgit v1.2.3-70-g09d2

