Merge branch 'master' of https://git.overleaf.com/5cf56f861d72e9071d1a343c

5 files changed, 60 insertions, 49 deletions

diff --git a/fig-stiffnesses.gplot b/fig-stiffnesses.gplot
index ae3a2f8..24d4e17 100644
--- a/fig-stiffnesses.gplot
+++ b/fig-stiffnesses.gplot
@@ -53,7 +53,7 @@ set x2tics 50, 50, 250 offset 0,-0.5 mirror
 set xrange [0:300]
 
 set title '(a)' offset 5,-2.7
-set ylabel '\scriptsize $C_{\mathrm{B_{2\mathrm g}}} / \mathrm{GPa}$' offset 3.5
+set ylabel '\scriptsize $C_{\mathrm{B_{2\mathrm g}}} / \mathrm{GPa}$' offset 4.3
 set yrange [140:145]
 set ytics 141,1,144 offset 0.5 mirror
 plot "data/c66.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
@@ -77,7 +77,7 @@ set xtics 50, 50, 250 offset 0,0.5 mirror
 unset y2tics
 set ytics 63,1,72
 unset y2label
-set ylabel '\scriptsize $\textcolor{mathc3}{C_{\mathrm{B_{1\mathrm g}}}},\textcolor{mathc4}{C_{\mathrm{B_{1\mathrm g}}}^0} / \mathrm{GPa}$' offset 3
+set ylabel '\scriptsize $\textcolor{mathc3}{C_{\mathrm{B_{1\mathrm g}}}},\textcolor{mathc4}{C_{\mathrm{B_{1\mathrm g}}}^0} / \mathrm{GPa}$' offset 3.7
 
 set yrange [64.5:71.5]
 plot "data/c11mc12.dat" using 1:(100 * $2) with lines lw 3 lc rgb cc3, \
diff --git a/fig-stiffnesses.pdf b/fig-stiffnesses.pdf
index 2edab9b..660033d 100644
--- a/fig-stiffnesses.pdf
+++ b/fig-stiffnesses.pdf
diff --git a/hidden_order.bib b/hidden_order.bib
index 0a92c54..2cb1621 100644
--- a/hidden_order.bib
+++ b/hidden_order.bib
@@ -837,4 +837,15 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature
   number = {32-33}
 }
 
+@article{ghiringhelli2012long,
+  title={Long-range incommensurate charge fluctuations in (Y, Nd) Ba2Cu3O6+ x},
+  author={Ghiringhelli, G and Le Tacon, M and Minola, Matteo and Blanco-Canosa, S and Mazzoli, Claudio and Brookes, NB and De Luca, GM and Frano, A and Hawthorn, DG and He, F and others},
+  journal={Science},
+  volume={337},
+  number={6096},
+  pages={821--825},
+  year={2012},
+  publisher={American Association for the Advancement of Science}
+}
+
 
diff --git a/library.bib b/library.bib
index c944258..6544ee6 100644
--- a/library.bib
+++ b/library.bib
@@ -3222,3 +3222,14 @@ I don't see how this is any better for classical systems and I don't see how it
 	Year = {1985},
 	Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.32.6920},
 	Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevB.32.6920}}
+	
+@article{ghiringhelli2012long,
+  title={Long-range incommensurate charge fluctuations in (Y, Nd) Ba2Cu3O6+ x},
+  author={Ghiringhelli, G and Le Tacon, M and Minola, Matteo and Blanco-Canosa, S and Mazzoli, Claudio and Brookes, NB and De Luca, GM and Frano, A and Hawthorn, DG and He, F and others},
+  journal={Science},
+  volume={337},
+  number={6096},
+  pages={821--825},
+  year={2012},
+  publisher={American Association for the Advancement of Science}
+}
diff --git a/main.tex b/main.tex
index c02215a..9d086c4 100644
--- a/main.tex
+++ b/main.tex
@@ -115,12 +115,14 @@ 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_nodate,
+rau_hidden_2012, riggs_evidence_2015, hoshino_resolution_2013,
+ikeda_theory_1998, chandra_hastatic_2013, harrison_hidden_nodate,
 ikeda_emergent_2012} propose associating any of a variety of broken symmetries
-with \ho.  This work proposes yet another, motivated by two experimental
-observations: first, the $\Bog$ ``nematic" elastic susceptibility
-$(C_{11}-C_{12})/2$ softens anomalously from room temperature down to
+with \ho. This work analyzes a family of phenomenological models with order
+parameters of general symmetry that couple linearly to strain. Of these, only
+one is compatible with two experimental observations: first, the $\Bog$
+``nematic" elastic susceptibility $(C_{11}-C_{12})/2$ softens anomalously from
+room temperature down to
 $T_{\text{\ho}}=17.5\,\K$;\cite{de_visser_thermal_1986} and second, a $\Bog$
 nematic distortion is observed by x-ray scattering under sufficient pressure to
 destroy the \ho\ state.\cite{choi_pressure-induced_2018}
@@ -137,15 +139,14 @@ model-independent way, but doesn't differentiate between those that remain.
 
 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 remains 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.\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 then determine the effect of its phase
-transitions on the elastic response in different symmetry channels. 
+(depending on temperature), \urusi\ undergoes a $\Bog$ nematic distortion,
+which might 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.\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 then determine the effect of its phase transitions on
+the elastic response in different symmetry channels. 
 
 We find that only one \op\ representation reproduces the anomalous $\Bog$
 elastic modulus, which softens in a Curie--Weiss-like manner from room
@@ -161,7 +162,7 @@ which should find that the $(C_{11}-C_{12})/2$ modulus diverges as the uniform
 $\Bog$ strain of the high pressure phase is approached.
 
 
-\section{Model}
+\section{Model \& Phase Diagram}
 The point group of \urusi\ is \Dfh, and any 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
@@ -292,7 +293,7 @@ to $f_\op$ with the identification $r\to\tilde r=r-b^2/2C^0_\X$.
 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} The properties discussed in the remainder of this
-section can all be found in a standard text, e.g., Chaikin \&
+section can all be found in a standard text, e.g., in chapter 4 \S6.5  of Chaikin \&
 Lubensky.\cite{chaikin_principles_2000} For a one-component \op\ ($\Bog$ or
 $\Btg$) and positive $c_\parallel$, it is traditional to make the field ansatz
 $\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$,
@@ -319,10 +320,10 @@ 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 \urusi, whose \ho\ phase is bounded by a line of first order transitions at high pressure,
 and so we will henceforth neglect the possibility of a multicomponent order
-parameter. The schematic phase diagrams for this model are shown in
+parameter. Schematic phase diagrams for both the one- and two-component models are shown in
 Figure~\ref{fig:phases}.
 
-\section{Results}
+\section{Susceptibility \& Elastic Moduli}
 We will now derive the effective elastic tensor $C$ that results from the
 coupling of strain to the \op. The ultimate result, found in
 \eqref{eq:elastic.susceptibility}, is that $C_\X$ differs from its bare value
@@ -388,7 +389,7 @@ on the configuration of the strain. Since $\eta_\star$ is a functional of $\epsi
 alone, only the modulus $C_\X$ will be modified from its bare value $C^0_\X$.
 
 Though the differential equation for $\eta_\star$ cannot be solved explicitly, we
-can use the inverse function theorem to make us \eqref{eq:implicit.eta} anyway.
+can use the inverse function theorem to make use of \eqref{eq:implicit.eta} anyway.
 First, denote by $\eta_\star^{-1}[\eta]$ the inverse functional of $\eta_\star$
 implied by \eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$
 corresponding to each solution of \eqref{eq:implicit.eta} it receives. This we
@@ -481,9 +482,7 @@ corresponding modulus.
    \eqref{eq:static_modulus}. The fit gives
    $C^0_\Bog\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $D_\perp
    q_*^4/b^2\simeq0.16\,\GPa^{-1}$, and
-   $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of an additional
-   parameter to fit the standard bare modulus\cite{varshni_temperature_1970}
-   led to poorly constrained fits. (c) $\Bog$ modulus data and the fit of the
+   $a/b^2\simeq6.1\times10^{-4}\,\GPa^{-1}\,\K^{-1}$. Addition of a quadratic term in $C^0_\Bog$ was here not needed for the fit.\cite{varshni_temperature_1970} (c) $\Bog$ modulus data and the fit of the
    \emph{bare} $\Bog$ modulus. (d) $\Bog$ modulus data and the fit transformed
    by $[C^0_\Bog(C^0_\Bog/C_\Bog-1)]]^{-1}$, which is predicted from
    \eqref{eq:static_modulus} to equal $D_\perp q_*^4/b^2+a/b^2|T-T_c|$, e.g.,
@@ -497,7 +496,7 @@ corresponding modulus.
 
 \section{Comparison to experiment}
 \Rus\ experiments~\cite{ghosh_single-component_nodate} yield the individual
-elastic moduli broken into irrep symmetries; data for the $\Bog$ and $\Btg$
+elastic moduli broken into irreps; data for the $\Bog$ and $\Btg$
 components defined in \eqref{eq:strain-components} are shown in Figures
 \ref{fig:data}(a--b).  The $\Btg$ in Fig.~\ref{fig:data}(a) modulus doesn't
 appear to have any response to the presence of the transition, exhibiting the
@@ -547,46 +546,36 @@ pressure, where the depth of the cusp in the $\Bog$ modulus should deepen
 (perhaps with these commensurability jumps) at low pressure and approach zero
 as $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. 
 Alternatively, \rus\ done at ambient pressure might examine the heavy Fermi
-liquid to \afm\ transition by doping. Though previous \rus\ studies have doped
-\urusi\ with Rhodium,\cite{yanagisawa_ultrasonic_2014} the magnetic nature of
-Rhodium ions likely artificially promotes magnetic phases. A dopant like
-phosphorous that only exerts chemical pressure might more faithfully explore
-the pressure axis of the phase diagram. Our work also motivates experiments
+liquid to \afm\ transition by doping. Our work also motivates experiments
 that can probe the entire correlation function---like x-ray and neutron
 scattering---and directly resolve its finite-$q$ divergence.  The presence of
 spatial commensurability is known to be irrelevant to critical behavior at a
 one-component disordered to modulated transition, and therefore is not
-expected to modify the thermodynamic behavior
-otherwise.\cite{garel_commensurability_1976} 
+expected to otherwise modify the thermodynamic behavior.\cite{garel_commensurability_1976} 
 
 There are two apparent discrepancies between the orthorhombic strain in the
-phase diagram presented by recent x-ray data\cite{choi_pressure-induced_2018}
-and that predicted by our mean field theory when its uniform ordered phase is
+phase diagram presented by recent x-ray data\cite{choi_pressure-induced_2018},
+and that predicted by our mean field theory if its uniform $\Bog$ phase is
 taken to be coincident with \urusi's \afm.  The first is the apparent onset of
-the orthorhombic phase in the \ho\ state prior to the onset of \afm.  As the
-recent x-ray research\cite{choi_pressure-induced_2018} notes, this could be due
+the orthorhombic phase in the \ho\ state at slightly lower pressures than the onset of \afm.  As the
+recent x-ray research\cite{choi_pressure-induced_2018} notes, this misalignment of the two transitions as function of doping 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} We suspect
-that the high-temperature orthorhombic signature is not the result of a bulk
-phase, and could be due to the high energy (small-wavelength) nature of x-rays
-as an experimental probe: \op\ fluctuations should lead to the formation of
-orthorhombic regions on the order of the correlation length that become larger
-and more persistent as the transition is approached. 
+the onset of \afm. We note that magnetic susceptibility data sees no trace of another phase
+transition at these higher temperatures. \cite{inoue_high-field_2001} It is therefore possible that the high-temperature orthorhombic signature in x-ray scattering is not the result of a bulk thermodynamic phase, but instead marks the onset of short-range correlations, as it does in the high-T$_{\mathrm{c}}$ cuprates \cite{ghiringhelli2012long} (where the onset of CDW correlations also lacks a thermodynamic phase transition). 
 
 Three dimensions is below the upper critical dimension $4\frac12$ of a
-one-component disordered to modulated transition, and so mean field theory
+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
+ginzburg_remarks_1961} Magnetic phase transitions tend to have a Ginzburg
 temperature of order one.  Our fit above gives $\xi_{\perp0}q_*=(D_\perp
 q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of
 $q_*\simeq\pi/a_3$ puts the bare correlation length $\xi_{\perp0}$ at about
 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
+deviations from mean field behavior within approximately several Kelvin of the
 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
@@ -616,10 +605,10 @@ uniform $\Bog$ electronic order.
 The corresponding prediction of uniform $\Bog$ symmetry breaking in the high
 pressure phase is consistent with recent diffraction experiments,
 \cite{choi_pressure-induced_2018} except for the apparent earlier onset in
-temperature of the $\Bog$ symmetry breaking, which we believe to be due to
-fluctuating order above the actual phase transition.  This work motivates both
+temperature of the $\Bog$ symmetry breaking, which we believe may be due to
+fluctuating order at temperatures above the actual transition temperature.  This work motivates both
 further theoretical work regarding a microscopic theory with modulated $\Bog$
-order, and preforming \rus\ experiments at pressure that could further support
+order, and preforming symmetry-sensitive thermodynamic experiments at pressure, such as ultrasound, that could further support
 or falsify this idea.
 
 \begin{acknowledgements}