From 740356e2c23caa67f70e60ca4e86bc85a4e9ee3a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Tue, 17 Dec 2019 10:53:17 -0500
Subject: small edits from jim's suggestions

---
 main.tex | 43 +++++++++++++++++++++----------------------
 1 file changed, 21 insertions(+), 22 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index c02215a..8af828b 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 phenomenological model of order parameters of
+general symmetry, linearly coupled to strain. We identify yet another mechanism
+that is best 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.,
-- 
cgit v1.2.3-70-g09d2


From 7b444f163a155d85eaf9e6db8e278493c11d056e Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 18 Dec 2019 17:07:54 -0500
Subject: wording change for brad

---
 main.tex | 15 +++++++--------
 1 file changed, 7 insertions(+), 8 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 8af828b..bb3dad6 100644
--- a/main.tex
+++ b/main.tex
@@ -118,14 +118,13 @@ theories~\cite{kambe_odd-parity_2018, haule_arrested_2009,
 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 analyzes a phenomenological model of order parameters of
-general symmetry, linearly coupled to strain. We identify yet another mechanism
-that is best 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}
+with \ho. This work analyzes a phenomenological model with order parameters of
+general symmetry, linearly coupled 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}
 
 Recent resonant ultrasound spectroscopy (\rus) measurements were used to
 examine the thermodynamic discontinuities in the elastic moduli at
-- 
cgit v1.2.3-70-g09d2


From 2781b05409a1360b062e613ae9daa608b43adc4a Mon Sep 17 00:00:00 2001
From: bradramshaw undefined <bradramshaw@cornell.edu>
Date: Wed, 18 Dec 2019 14:29:29 +0000
Subject: Update on Overleaf.

---
 main.tex | 8 +++-----
 1 file changed, 3 insertions(+), 5 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 8af828b..2c6682b 100644
--- a/main.tex
+++ b/main.tex
@@ -119,7 +119,7 @@ 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 analyzes a phenomenological model of order parameters of
-general symmetry, linearly coupled to strain. We identify yet another mechanism
+general symmetry that couple linearly to strain. We identify yet another mechanism
 that is best compatible with two experimental observations: first, the $\Bog$
 ``nematic" elastic susceptibility $(C_{11}-C_{12})/2$ softens anomalously from
 room temperature down to
@@ -547,10 +547,8 @@ pressure, where the depth of the cusp in the $\Bog$ modulus should deepen
 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
+\urusi\ with Rhodium,\cite{yanagisawa_ultrasonic_2014} the magnetic rhodium dopants likely promote magnetic phases. A non-magnetic dopant such as phosphorous may more faithfully explore
+the transition out of the HO phase. 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
-- 
cgit v1.2.3-70-g09d2


From f4360074c4cef0dd2bcbf1a044467caf6af6db6a Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 19 Dec 2019 15:08:07 -0500
Subject: tiny text change from mike

---
 main.tex | 15 ++++++++-------
 1 file changed, 8 insertions(+), 7 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 78ae8a5..fbb5e48 100644
--- a/main.tex
+++ b/main.tex
@@ -118,13 +118,14 @@ theories~\cite{kambe_odd-parity_2018, haule_arrested_2009,
 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 analyzes a phenomenological model 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}
+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}
 
 Recent resonant ultrasound spectroscopy (\rus) measurements were used to
 examine the thermodynamic discontinuities in the elastic moduli at
-- 
cgit v1.2.3-70-g09d2


From 0ca5ed156c788dfdd2e2738ec19de4e43439bd73 Mon Sep 17 00:00:00 2001
From: bradramshaw undefined <bradramshaw@cornell.edu>
Date: Tue, 7 Jan 2020 19:28:57 +0000
Subject: Update on Overleaf.

---
 hidden_order.bib | 11 +++++++++++
 library.bib      | 11 +++++++++++
 main.tex         | 38 +++++++++++++++-----------------------
 3 files changed, 37 insertions(+), 23 deletions(-)

(limited to 'main.tex')

diff --git a/hidden_order.bib b/hidden_order.bib
index cc404a0..45bb709 100644
--- a/hidden_order.bib
+++ b/hidden_order.bib
@@ -770,4 +770,15 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature
   file = {/home/pants/.zotero/data/storage/UJTH89KV/Yanagisawa - 2014 - Ultrasonic study of the hidden order and heavy-fer.pdf}
 }
 
+@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 fbb5e48..9d086c4 100644
--- a/main.tex
+++ b/main.tex
@@ -496,7 +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
@@ -546,44 +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 rhodium dopants likely promote magnetic phases. A non-magnetic dopant such as phosphorous may more faithfully explore
-the transition out of the HO phase. 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
@@ -613,10 +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}
-- 
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