From 4f302eacb4373b7aa1a82810ed8e58ef201d64dc Mon Sep 17 00:00:00 2001
From: bradramshaw undefined <bradramshaw@cornell.edu>
Date: Wed, 4 Dec 2019 03:12:54 +0000
Subject: Update on Overleaf.

---
 main.tex | 51 ++++++++++++++++++++-------------------------------
 1 file changed, 20 insertions(+), 31 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index af82865..9022bbc 100644
--- a/main.tex
+++ b/main.tex
@@ -80,7 +80,7 @@
 \title{Elastic properties of hidden order in \urusi\ are reproduced by a staggered nematic}
 \author{Jaron Kent-Dobias}
 \author{Michael Matty}
-\author{Brad Ramshaw}
+\author{B.~J. Ramshaw}
 \affiliation{
   Laboratory of Atomic \& Solid State Physics, Cornell University,
   Ithaca, NY, USA
@@ -144,24 +144,14 @@ modulus $(C_{11}-C_{12})/2$ that occurs over a broad temperature range at
 zero-pressure \cite{wolf_elastic_1994, kuwahara_lattice_1997}. Motivated by
 these results---which hint 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
+arbitrary \op\ coupled to strain, and then 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 $\Bog$ elastic
-modulus, which softens in a Curie--Weiss-like manner from room temperature but
-cusps at $T_{\text{\ho}}$. That theory associates \ho\ with a $\Bog$ \op\
-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 that 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 $(C_{11}-C_{12})/2$ modulus diverges
-as the uniform $\Bog$ strain of the \afm\ phase is approached.
+We find that only one \op\ symmetry \brad{we should be careful with our language. Can we talk about the "symmetry" of a state? Or is it the symmetries broken by the state? I don't think we can say \op\ symmetry, maybe the \op\ representation?} reproduces the anomalous $\Bog$ elastic modulus, which softens in a Curie--Weiss-like manner from room temperature and then cusps at $T_{\text{\ho}}$. That theory associates \ho\ with a $\Bog$ \op\ modulated along the $c$-axis, the \afm\ state with uniform $\Bog$ order, and the triple point between them with a Lifshitz point. In addition to the agreement with the ultrasound data across a broad temperature range, the theory predicts uniform $\Bog$ strain at high pressure---the same distortion that 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 $(C_{11}-C_{12})/2$ modulus diverges as the uniform $\Bog$ strain of the \afm\ phase is approached. \brad{Do we want to make the bold statement here that maybe the AFM is just a parasitically-induced by product of the high-field phase?}
 
 
 \emph{Model.}
-The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally
+The point group of \urusi\ is \Dfh, and any coarse-grained \brad{like mustard? Is there a reason that we say coarse grained, though? I think microscopic hamiltonians have to obey the symmetry of the lattice as well, no?} theory must locally
 respect this symmetry in the high-temperature phase. Our phenomenological free
 energy density contains three parts: the elastic free energy, the \op, and the
 interaction between strain and \op. The most general quadratic free energy of
@@ -170,7 +160,7 @@ the strain $\epsilon$ is $f_\e=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$
 popular Voigt notation in the abstract and introduction. Here and henceforth
 the notation used is that natural for a rank-four tensor.}. Linear combinations
 of the six independent components of strain form five irreducible components of
-strain as
+strain in \Dfh\ as
 \begin{equation}
   \begin{aligned}
     & \epsilon_{\Aog,1}=\epsilon_{11}+\epsilon_{22} \hspace{0.15\columnwidth} &&
@@ -204,16 +194,15 @@ coupling to linear order is
 \begin{equation}
   f_\i=-b^{(i)}\epsilon_\X^{(i)}\eta.
 \end{equation}
-If there doesn't exist a component of strain that transforms like the
-representation $\X$ there can be no linear coupling, and the effect of the \op\
-condensing at a continuous phase transition is to produce a jump in the $\Aog$
+If there exists no component of strain that transforms like the
+representation $\X$ then there can be no linear coupling. The next-order coupling is linear in strain, quadratic in order parameter, and the effect of this coupling at a continuous phase transition is to produce a jump in the $\Aog$
 elastic moduli if $\eta$ is single-component \cite{luthi_sound_1970,
 ramshaw_avoided_2015, shekhter_bounding_2013}, and jumps in other elastic
 moduli if multicomponent \cite{ghosh_single-component_nodate}. Because we are
-interested in physics that anticipates the phase transition, we will focus our
+interested in physics that anticipates the phase transition \brad{"anticipates the phase transition" I think is too vague on its own - i added something after it as an example}---i.e. that the growing order parameter susceptibility is reflected directly in the elastic susceptibility--- we will focus our
 attention on \op s that can produce linear couplings to strain.  Looking at the
 components present in \eqref{eq:strain-components}, this rules out all of the
-u-reps (which are odd under inversion) and the $\Atg$ irrep.
+u-reps (which are odd under inversion), the $\Atg$ irrep, and all half-integer (spinor) representations \brad{added spinors because these actually form some of the theories}.
 
 If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd
 terms are allowed in its free energy and any transition will be first order and
@@ -245,9 +234,9 @@ full free energy functional of $\eta$ and $\epsilon$ is
 \end{equation}
 
 Rather than analyze this two-argument functional directly, we begin by tracing
-out the strain and studying the behavior of \op\ alone. Later we will invert
+out the strain and studying the behavior of the \op\ alone. Later we will invert
 this procedure and trace out the \op\ when we compute the effective elastic
-moduli. The only strain relevant to the \op\ at linear coupling is
+moduli. The only strain relevant to an \op\ of representation $\X$ at linear coupling is
 $\epsilon_\X$, which can be traced out of the problem exactly in mean field
 theory. Extremizing the functional \eqref{eq:free_energy} with respect to
 $\epsilon_\X$ gives
@@ -270,7 +259,7 @@ to $f_\op$ with $r\to\tilde r=r-b^2/2C^0_\X$.
 
   \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em}
   \includegraphics[width=0.51\columnwidth]{phases_vector}
-  \caption{
+  \caption{\brad{where are we submitting this? we should prooooobbably get permission from the journal to reproduce this data. It's usually easy, i think?}
     Phase diagrams for (a) \urusi\ from experiments (neglecting the
     superconducting phase) \cite{hassinger_temperature-pressure_2008} (b) mean
     field theory of a one-component ($\Bog$ or $\Btg$) Lifshitz point (c) mean
@@ -311,7 +300,7 @@ 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 to modulated transition is now continuous. This does not reproduce
-the physics of \ho, which has a first order transition between \ho\ and \afm,
+the physics of \ho, which has a first order transition between \ho\ and \afm \brad{This is where we need to be clearer: we say that the transition to AFM is first order, but our theory doesn't transition to AFM. So maybe we need to introduce the idea earlier of a "high-pressure" phase, of which AFM is one characteristic (and maybe that's an induced or parasitic aspect), but its defining characteristic is broken rotational symmetry},
 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}.
@@ -320,9 +309,9 @@ Figure~\ref{fig:phases}.
 We will now derive the effective elastic tensor $C$ that results from coupling
 of strain to the \op. The ultimate result, found in
 \eqref{eq:elastic.susceptibility}, is that $C_\X$ differs from its bare value
-$C^0_\X$ only for the symmetry $\X$ of the \op. Moreover, this modulus does not
+$C^0_\X$ only for the representation $\X$ of the \op. Moreover, this modulus does not
 vanish at the unordered to modulated transition---as it would if the transition
-were a $q=0$ structural phase transition---but ends in a cusp. In this section
+were a $q=0$ phase transition---but instead ends in a cusp. In this section
 we start by computing the susceptibility of the \op\ at the unordered to
 modulated transition, and then compute the elastic modulus for the same. 
 
@@ -446,7 +435,7 @@ the result, we finally arrive at
 Though not relevant here, this result generalizes to multicomponent \op s.
 
 What does \eqref{eq:elastic.susceptibility} predict in the vicinity of the
-\ho\ transition? Near the disordered to modulated transition, the
+\ho\ transition? Near the disordered to modulated transition---the zero-pressure transition to the HO state---the
 static modulus is given by
 \begin{equation}
   C_\X(0)=C_\X^0\bigg[1+\frac{b^2}{C_\X^0}\big(D_\perp q_*^4+|\Delta\tilde r|\big)^{-1}\bigg]^{-1}.
@@ -508,7 +497,7 @@ high temperature phase, suggesting that \ho\ can be described as a
 $\Bog$-nematic phase that is modulated at finite $q$ along the $c-$axis. The
 predicted softening appears over hundreds of Kelvin; Figures
 \ref{fig:data}(c--d) show the background modulus $C_\Bog^0$ and the
-\op--induced response isolated from each other.
+\op--induced response isolated from each other. 
 
 We have seen that the mean-field theory of a $\Bog$ \op\ recreates the topology
 of the \ho\ phase diagram and the temperature dependence of the $\Bog$ elastic
@@ -533,10 +522,10 @@ these two regimes, the ordering wavevector should shrink by jumping between
 ever-closer commensurate values in the style of the devil's staircase
 \cite{bak_commensurate_1982}. 
 
-This motivates future \rus\ experiments done at
+This motivates future ultrasound experiments done under
 pressure, where the depth of the cusp in the $\Bog$ modulus 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. 
+as $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. 
 {\color{blue}
     Moreover, 
 }
@@ -578,7 +567,7 @@ what one would expect for a generic magnetic transition.  The agreement of this
 data in the $t\sim0.1$--10 range with the mean field exponent suggests that
 this region is outside the Ginzburg region, but an experiment may begin to see
 deviations from mean field behavior within around several degrees Kelvin of the
-critical point. A \rus\ experiment with more precise temperature resolution
+critical point. An ultrasound 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 one-component \op\ is $\mathrm O(2)$
-- 
cgit v1.3

