From 480b88160062d0669dbf7d7ec2a65851b2efff59 Mon Sep 17 00:00:00 2001
From: jaron-overleaf <jaron-overleaf@kent-dobias.com>
Date: Mon, 3 Jun 2019 21:01:28 +0000
Subject: Update on Overleaf.

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+
+\documentclass[fleqn]{article}
+
+\usepackage{amsmath,amssymb}
+\usepackage{fullpage,graphicx}
+
+\title{Elastic Lifshitz point}
+\author{Jaron Kent-Dobias}
+
+\begin{document}
+
+\maketitle
+
+The elastic free energy density for a tetragonal system is given by
+\[
+  f_{\mathrm e}=\frac12\Big[a^{(1)}_{\mathrm A_1}(\epsilon_{11}+\epsilon_{22})^2+
+      a_{\mathrm A_1}^{(2)}\epsilon_{33}^2+
+      a_{\mathrm A_1}^{(3)}(\epsilon_{11}+\epsilon_{22})\epsilon_{33}+
+      a_{\mathrm B_1}(\epsilon_{11}-\epsilon_{22})^2+
+      a_{\mathrm B_2}\epsilon_{12}^2+
+      a_{\mathrm E}(\epsilon_{13}^2+\epsilon_{23}^2)\Big]
+\]
+Consider a generic one-component order parameter $\eta$. The most general quartic free energy density (discounting total derivatives) is
+\[
+  f_{\mathrm o}=\frac12\bigg(r\eta^2+
+                c_\parallel(\nabla_\parallel\eta)^2+
+                c_\perp(\partial_3\eta)^2+
+                D\big[(\nabla_\parallel^2+\partial_3^2)\eta\big]^2+
+              u\eta^4\bigg)
+\]
+The irreducible representation that $\eta$ transforms with determines the allowed coupling to elasticity. Every possible option for a single-component order parameter is to lowest order is
+\begin{align*}
+  f_{\mathrm A_1}=\frac12\eta\big[b_1(\epsilon_{11}+\epsilon_{22})+b_2\epsilon_{33}\big] &&
+  f_{\mathrm B_1}=\frac12b\eta(\epsilon_{11}-\epsilon_{22}) &&
+  f_{\mathrm B_2}=\frac12b\eta\epsilon_{12}
+\end{align*}
+which can generically be written $f_{\mathrm i}=\frac12B\eta$. The total $\eta$-dependent free energy is
+\[
+  F_\eta=\int d^3x\,(f_{\mathrm o}+f_{\mathrm i})
+\]
+We expect to see regimes where the order parameter is zero, constant, and modulated (perhaps about a constant for nonzero $B$), so we make the ansatz that $\eta=\eta_0+\eta_q\cos(qx)$. The resulting free energy per unit volume is
+\[
+  \bar f_\eta=\lim_{L\to\infty}\frac1{L^3}\int_\Box d^3x\,(f_{\mathrm o}+f_{\mathrm i})
+  =\frac12\bigg(r\eta_0^2+u\eta_0^4+\frac12c_\perp q^2\eta_q^2+\frac12dq^4\eta_q^4+\frac12r\eta_q^2+3u\eta_0^2\eta_q^2+\frac38u\eta_q^4+B\eta_0\bigg)
+\]
+where the integral is over a cube of side-length $L$.
+We will now minimize this trial free energy with respect to $\eta_0$, $\eta_q$, and $q$. For $c_\perp>0$ and $r<0$ or $c_\perp<0$ and $r<-(2+\sqrt6)c_\perp^2/4d$, and small $B$, the global minimizer is
+\begin{align*}
+  \eta_0=-\sqrt{-r/2u}+B/4r+O(B^2)&&\eta_q=0&& q=0
+\end{align*}
+which corresponds to an ordered phase.
+For $c_\perp>0$ and $r>0$, or $c_\perp<0$ and $r>c_\perp^2/4d$, the global minimizer is
+\begin{align*}
+  \eta_0=-B/2r+O(B^2)&&\eta_q=0&&q=0
+\end{align*}
+which corresponds to a disordered phase.
+For $c_\perp<0$ and $-(2+\sqrt6)c_\perp^2/4d<r<c_\perp^2/4d$, the global minimizer is
+\begin{align*}
+  \eta_0=-dB/(c_\perp^2-2dr)+O(B^2)\\\eta_q=\frac1{\sqrt{6du/(c_\perp^2-4dr)}}-\frac{2\sqrt6d^5uB^2}{(c_\perp^2-2dr)^2\sqrt{d^5(c_\perp^2-4dr)u}}+O(B^4)&&q=\sqrt{-c_\perp/2d}
+\end{align*}
+which corresponds to a modulated phase.
+We are interested in the behavior of the effective elastic constants as the second order transition between the disordered and modulated phases is crossed. We have
+\[
+  \tilde a_{\text{X, disordered}}=\frac{\partial^2\bar f}{\partial\epsilon_{\mathrm X}^2}\bigg|_{\epsilon=0}=a_{\mathrm X}+\frac{\partial^2\bar f_\eta}{\partial\epsilon_{\mathrm X}^2}\bigg|_{\epsilon=0}
+  =a_{\mathrm X}-b^2/2r
+\]
+for the unordered phase, and
+\[
+  \tilde a_{\text{X, modulated}}=a_{\mathrm X}-b^2d/(c_\perp^2-2dr)
+\]
+for the modulated phase. As a function of $t=r-r_c=r-c_\perp^2/4d$, this is
+\[
+  \tilde a_{\text{X, disordered--modulated}}=a_{\mathrm X}-\begin{cases}
+    2b^2d/(c_\perp^2-4dt)&t<0\\
+    2b^2d/(c_\perp^2+4dt)&t>0
+  \end{cases}
+  =a_{\mathrm X}-2b^2d/(c_\perp^2+4d|t|)
+\]
+The effective elastic constant for the component of strain coupled to the order parameter thus has a cusp at the disordered--modulated transition which is a local minimum. All other components are unaffected by the transition.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.6\textwidth]{cusp}
+  \caption{The effective elastic constant near the disordered--modulated transition.}
+\end{figure}
+
+What happens at the second order disordered--ordered transition? We already have the effective elastic constant for the disordered phase---for the ordered phase, we have
+\[
+  \tilde a_{\text{X, ordered}}=a_{\mathrm X}+b^2/4r
+\]
+As a function of $t=r-r_c=r$, this is
+\[
+  \tilde a_{\text{X, disordered--ordered}}=a_{\mathrm X}-\begin{cases}
+    b^2/4|t|&t<0\\
+    b^2/2|t|&t>0
+  \end{cases}
+\]
+Thus the elastic constant vanishes at this critical point, with an amplitude ratio of 2.
+
+Finally, between the ordered and modulated phases there is a first order transition. Here, we expect a jump in the effective elastic constant of
+\[
+  \tilde a_{\mathrm{X,\ ordered}}-\tilde a_{\mathrm{X,\ modulated}}\Big|_{r=-(2+\sqrt6)c_\perp^2/4d}=\frac{8+3\sqrt6}{14+6\sqrt6}\frac{b^2d}{c_\perp^2}
+\]
+A phase diagram is shown below. The ordered phase is metastable for all $r<0$, while the modulated phase is metastable for $c_\perp<0$.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.5\textwidth]{phases}
+  \caption{The phase diagram for this model. The white region is the disordered phase, the blue is ordered, and the orange is modulated.}
+\end{figure}
+
+We now extend this analysis to the next order in the interaction. At third order, since $\eta^2$ transforms like $\mathrm A_1$ for any order parameter symmetry, the only possible interaction is
+\[
+  f_{\mathrm X}^{(3)}=\frac12\eta^2\big[e_1(\epsilon_{11}+\epsilon_{22})+e_2\epsilon_{33}\big]
+\]
+Adding this interaction is akin to simply shifting $r\to r+e\epsilon_{\mathrm A_1}$, and all the solutions for the mean fields follow with this substitution made. This changes the strain dependence of the solutions, however. The disordered effective elastic constants are unchanged, but the modulated and ordered constants become
+\begin{align*}
+  \tilde a_{\text{X, modulated}}=a_{\mathrm X}-\frac{b^2d}{c_\perp^2-2dr}
+  &&
+  \tilde a_{\text{$\mathrm A_1$, modulated}}=a_{\mathrm A_1}-\frac{e^2}{3u} \\ 
+  \tilde a_{\text{X, ordered}}=a_{\mathrm X}+\frac{b^2}{4r}+\frac{be}{2^{3/2}(-ru)^{1/2}}
+  &&
+  \tilde a_{\text{$\mathrm A_1$, ordered}}=a_{\mathrm A_1}-\frac{e^2}{2u}+\frac{be}{2^{3/2}(-ru)^{1/2}}
+\end{align*}
+The form of the cusp at the disordered--modulated transition in the elastic component with the symmetry of the order parameter is unchanged, but there is now a discontinuity of magnitude $-e^2/3u$ in the $\mathrm A_1$ elastic constant. At the disordered--ordered transition there now appears a subleading divergence in the elastic constant with the symmetry of the order parameter, and both a discontinuity and a $|r|^{-1/2}$ divergence in the $\mathrm A_1$ elastic constant. The magnitude of the jump at the abrupt ordered--modulated transition is changed.
+
+\end{document}
+
-- 
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