From c8107eff3164f09c86167d1363ff6ad3a8704bde Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 22 Aug 2019 16:57:21 -0400
Subject: redid Ginzburg criterion section

---
 main.tex | 66 ++++++++++++++++++++++++++++++++--------------------------------
 1 file changed, 33 insertions(+), 33 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index a245429..3f9c01f 100644
--- a/main.tex
+++ b/main.tex
@@ -33,7 +33,7 @@
 \def\m{\text m}
 \def\K{\text K}
 \def\GPa{\text{GPa}}
-\def\A{\text{\c A}}
+\def\A{\text{\r A}}
 
 % Other
 \def\G{\text G} % Ginzburg
@@ -349,7 +349,7 @@ r_c|^\gamma$ for $\gamma=1$.
   \caption{
     Measurements of the effective strain stiffness as a function of temperature
     for the six independent components of strain from ultrasound. The vertical
-    dashed lines show the location of the hidden order transition.
+    lines show the location of the hidden order transition.
   }
   \label{fig:data}
 \end{figure}
@@ -391,43 +391,43 @@ fluctuations are indeed negligible. This is typically done by computing the
 Ginzburg criterion \cite{ginzburg_remarks_1961}, which gives the proximity to
 the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to
 break down by comparing the magnitude of fluctuations in a correlation-length
-sized box to the magnitude of the field, or since the correlation function is
-$k_BT\chi(x,x')$,
-\begin{equation}
-  V_\xi^{-1}k_BT\int_{V_\xi}d^3x\,\chi(x,0)
-  =\langle\delta\eta^2\rangle_{V_\xi}
-  \lesssim\frac12\eta_*^2=\frac{|\Delta\tilde r|}{6u}
-\end{equation}
-with $V_\xi$ the correlation volume, which we will take to be a cylinder of
-radius $\xi_\parallel/2$ and height $\xi_\perp$. Upon substitution of
-\eqref{eq:susceptibility} and using the jump in the specific heat at the
-transition from
+sized box to the magnitude of the field. In the modulated phase the spatially
+averaged magnitude is zero, and so we will instead compare fluctuations in the
+\emph{amplitude} at $q_*$ to the magnitude of that amplitude. Defining the
+field $\alpha$ by $\eta(x)=\alpha(x)e^{-iq_*x_3}$, it follows that in the
+modulated phase $\alpha(x)=\alpha_0$ for $\alpha_0^2=|\delta \tilde r|/4u$. In
+the modulated phase, the $q$-dependant fluctuations in $\alpha$ are given by
+\[
+  G_\alpha(q)=k_BT\chi_\alpha(q)=\frac1{c_\parallel q_\parallel^2+D_\perp(4q_*^2q_\perp^2+q_\perp^4)+2|\delta r|},
+\]
+An estimate of the Ginzburg criterion is then given by the temperature at which
+$V_\xi^{-1}\int_{V_\xi}G(0,x)\,dx=\langle\delta\alpha^2\rangle\simeq\langle\alpha\rangle^2=\alpha_0^2$,
+where $V_\xi=\xi_\perp\xi_\parallel^2$ is a correlation volume. The parameter $u$
+can be replaced in favor of the jump in the specific heat at the transition
+using
 \begin{equation}
   c_V=-T\frac{\partial^2f}{\partial T^2}
-    =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c,\end{cases}
-\end{equation}
-this expression can be brought to the form
-\begin{equation}
-  \frac{2k_B}{\pi\Delta c_V\xi_{\perp0}\xi_{\parallel0}^2}
-    \mathcal I(\xi_{\perp0} q_*|t|^{-1/4})
-  \lesssim |t|^{13/4},
-\end{equation}
-where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I(x)\sim x^{-4}$ for large $x$, yielding
-\begin{equation}
-  t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4}
+    =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c.\end{cases}
 \end{equation}
+The integral over the correlation function $G_\alpha$ can be preformed up to one integral analytically using a Gaussian-bounded correlation volume, yielding $\langle\delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}\delta r^{-1}\mathcal I(\xi_\perp q_*)$ for
+\[
+  \mathcal I(x)=-2^{3/2}\pi^{5/2}\int dy\,e^{[2+(4x^2-1)y^2+y^4]/2}
+\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big)
+\]
+This gives a transcendental equation
+\[
+  \mathcal I(\xi_{\perp0}q_*\delta t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}\delta t_\G^{3/4}.
+\]
 Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$
 \cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp
-q_*^4/aT_c)^{1/4}\sim2$. We have reason to believe that at zero pressure, very
-far from the Lifshitz point, $q_*$ is roughly the inverse lattice spacing
+q_*^4/aT_c)^{1/4}\simeq2$. We have reason to believe that at zero pressure, very
+far from the Lifshitz point, the half-wavelength of the modulation should be commensurate with the lattice, giving $q_*\simeq0.328\,\A^{-1}$ 
 \cite{meng_imaging_2013}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$,
-we find $t_\G\sim0.04$, so that an experiment would need to be within
-$\sim1\,\K$ to detect a deviation from mean field behavior. An ultrasound
-experiment able to capture data over several decades within this vicinity of
-$T_c$ may be able to measure a cusp with $|t|^\gamma$ for
-$\gamma=\text{\textbf{???}}$, the empirical exponent \textbf{[Citation???]}.
-Our analysis has looked at behavior for $T-T_c>1\,\K$, and so it remains
-self-consistent.
+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
+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.
 
 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
-- 
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