From 672a84bf8e24408060509b24a5f53a41c597e90f Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 28 Jun 2019 14:18:01 -0400
Subject: added ginzburg criterion stuff

---
 main.tex | 54 ++++++++++++++++++++++++++++++++++++++++++++++--------
 1 file changed, 46 insertions(+), 8 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index 794f30d..c92400b 100644
--- a/main.tex
+++ b/main.tex
@@ -1,6 +1,6 @@
-\documentclass[aps,prl,reprint]{revtex4-2}
+\documentclass[aps,prl,reprint]{revtex4-1}
 \usepackage[utf8]{inputenc}
-\usepackage{amsmath,graphicx}
+\usepackage{amsmath,graphicx,upgreek,amssymb}
 
 % Our mysterious boy
 \def\urusi{URu$_2$Si$_2\ $}
@@ -27,6 +27,16 @@
 \def\X{\mathrm X}
 \def\Y{\mathrm Y}
 
+% Units
+\def\J{\mathrm J}
+\def\m{\mathrm m}
+\def\K{\mathrm K}
+\def\GPa{\mathrm{GPa}}
+\def\A{\mathrm{\c A}}
+
+% Other
+\def\G{\mathrm G} % Ginzburg
+
 \begin{document}
 
 \title{\urusi mft}
@@ -112,7 +122,7 @@ where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and $\n
 \end{equation}
 gives $\epsilon_\X(x)=-(b/2\lambda_\X)\eta(x)$. Upon substitution into the free energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$, with $r\to\tilde r=r-b^2/4\lambda_\X$.
 
-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, hornreich_lifshitz_1980}. For a scalar order parameter ($\Bog$ or $\Btg$) it is traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. 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. 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
+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 scalar order parameter ($\Bog$ or $\Btg$) it is traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. 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. 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}
 \end{equation}
@@ -122,7 +132,7 @@ with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The tra
 \end{equation}
 The uniform--modulated transition is now continuous. The schematic phase diagrams for this model are shown in Figure \ref{fig:phases}.
 
-\begin{figure}
+\begin{figure}[htpb]
   \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em}
   \includegraphics[width=0.51\columnwidth]{phases_vector}
   \caption{Schematic phase diagrams for this model. Solid lines denote
@@ -156,6 +166,7 @@ Near the unordered--modulated transition this yields
   &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2+|\tilde r-\tilde r_c|} \\
   &=\frac1{D_\perp}\frac{\xi_\perp^4}{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^4(q_*^2-q_\perp^2)^2},
   \end{aligned}
+  \label{eq:susceptibility}
 \end{equation}
 with $\xi_\perp=(|\tilde r-\tilde r_c|/D_\perp)^{-1/4}$ and $\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$.
 
@@ -203,7 +214,7 @@ whose Fourier transform follows from \eqref{eq:inv.func} as
 \end{equation}
 At $q=0$, which is where the stiffness measurements used here were taken, this predicts a cusp in the elastic susceptibility of the form $|\tilde r-\tilde r_c|^\gamma$ for $\gamma=1$.
 
-\begin{figure}
+\begin{figure}[htpb]
   \centering
   \includegraphics[width=0.49\columnwidth]{stiff_a11.pdf}
   \includegraphics[width=0.49\columnwidth]{stiff_a22.pdf}
@@ -216,17 +227,44 @@ At $q=0$, which is where the stiffness measurements used here were taken, this p
     for the six independent components of strain from ultrasound. The vertical
     dashed lines show the location of the hidden order transition.
   }
+  \label{fig:data}
 \end{figure}
 
-\begin{figure}
+We have seen that mean field theory predicts that whatever component of strain transforms like the order parameter will see a $t^{-1}$ softening in the stiffness that ends in a cusp. Ultrasound experiments \textbf{[Elaborate???]} yield the strain stiffness for various components of the strain; this data is shown in Figure \ref{fig:data}.  The $\Btg$ and $\Eg$ stiffnesses don't appear to have any response to the presence of the transition, exhibiting the expected linear stiffening with a low-temperature cutoff \textbf{[What's this called? Citation?]}. The $\Bog$ stiffness has a dramatic response, softening over the course of roughly $100\,\K$. There is a kink in the curve right at the transition. While the low-temperature response is not as dramatic as the theory predicts, mean field theory---which is based on a small-$\eta$ expansion---will not work quantitatively far below the transition where $\eta$ has a large nonzero value and higher powers in the free energy become important. The data in the high-temperature phase can be fit to the theory \eqref{eq:elastic.susceptibility}, with a linear background stiffness $\lambda_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure \ref{fig:fit}. The data and theory appear consistent.
+
+\begin{figure}[htpb]
   \includegraphics[width=\columnwidth]{cusp}
   \caption{
     Strain stiffness data for the $\Bog$ component of strain (solid) along with
-    a fit of \eqref{eq:elastic.susceptibility} (dashed).
-  }
+    a fit of \eqref{eq:elastic.susceptibility} to the data above $T_c$ (dashed). The fit gives $\lambda_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp q_*^4\simeq0.0038\,\K^{-1}$.
   }
+  \label{fig:fit}
 \end{figure}
 
+Mean field theory neglects the effect of fluctuations on critical behavior, yet also predicts the magnitude of those fluctuations. This allows a mean field theory to undergo an internal consistency check to ensure the predicted 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
+\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 $\mathcal I$ is defined by
+\begin{equation}
+  \mathcal I(x)=\frac1\pi\int_{-\infty}^\infty dy\,\frac{\sin\tfrac y2}y\bigg(\frac1{1+(y^2-x^2)^2}-\frac{K_1(\sqrt{1+(y^2-x^2)^2})}{\sqrt{1+(y^2-x^2)^2}}\bigg)
+\end{equation}
+For large argument, $\mathcal I(x)\sim x^{-4}$, yielding
+\begin{equation}
+  t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4}
+\end{equation}
+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 \textbf{[Why???]}. 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???]}.
+
 \begin{acknowledgements}
 
 \end{acknowledgements}
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