From c94d723e095fc34137bc50efe41a082949f926fd Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 23 Aug 2019 16:59:24 -0400
Subject: more notation updates, finished purging indicies

---
 main.tex | 131 ++++++++++++++++++++++++++++++++++-----------------------------
 1 file changed, 70 insertions(+), 61 deletions(-)

(limited to 'main.tex')

diff --git a/main.tex b/main.tex
index a53e0ff..895c9a5 100644
--- a/main.tex
+++ b/main.tex
@@ -6,9 +6,9 @@
 % Our mysterious boy
 \def\urusi{URu$_{\text2}$Si$_{\text2}$}
 
-\def\e{{\text e}} % "elastic"
-\def\o{{\text o}} % "order parameter"
-\def\i{{\text i}} % "interaction"
+\def\e{{\text{\textsc{Elastic}}}} % "elastic"
+\def\o{{\text{\textsc{Op}}}} % "order parameter"
+\def\i{{\text{\textsc{Int}}}} % "interaction"
 
 \def\Dfh{D$_{\text{4h}}$}
 
@@ -41,6 +41,7 @@
 \def\ho{\textsc{ho}} % hidden order
 \def\rus{\textsc{rus}} % Resonant ultrasound spectroscopy 
 \def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy 
+\def\recip{{\{-1\}}} % functional reciprocal
 
 \begin{document}
 
@@ -205,15 +206,19 @@ free energy density is
 \end{equation}
 where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and
 $\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$
-since this does not affect the physics at hand. Neglecting interaction terms
+since this does not affect the physics at hand. The full free energy functional of $\eta$ and $\epsilon$ is then
+\begin{equation}
+  F[\eta,\epsilon]=\int dx\,(f_\o+f_\e+f_\i)
+\end{equation}
+Neglecting interaction terms
 higher than quadratic order, the only strain relevant to the problem is
 $\epsilon_\X$, and this can be traced out of the problem exactly, since
 \begin{equation}
-  0=\frac{\delta F}{\delta\epsilon_{\X i}(x)}=C_\X\epsilon_{\X i}(x)
+  0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_{\X i}(x)}=C_\X\epsilon_{\X i}(x)
     +\frac12b\eta_i(x)
 \end{equation}
-gives $\epsilon_\X(x)=-(b/2C_\X)\eta(x)$. Upon substitution into the free
-energy, tracing out $\epsilon_\X$ has the effect of shifting $r$ in $f_\o$,
+gives $\epsilon_\X[\eta]=-(b/2C_\X)\eta$. Upon substitution into the free
+energy, the resulting effective free energy $F_\o[\eta]=F[\eta,\epsilon[\eta]]$ has a density identical to $f_\o$
 with $r\to\tilde r=r-b^2/4C_\X$.
 
 With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz
@@ -229,6 +234,7 @@ $q_*^2=-c_\perp/2D_\perp$ and
 \begin{equation}
   \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u}
     =\frac{\tilde r_c-\tilde r}{3u}
+    =\frac{\Delta\tilde r}{3u}
 \end{equation}
 with $\tilde r_c=c_\perp^2/4D_\perp$ and the system has modulated order. The
 transition between the uniform and modulated orderings is abrupt for a one-component
@@ -236,12 +242,8 @@ field and occurs along the line $c_\perp=-2\sqrt{-D_\perp\tilde r/5}$. For a
 two-component \op\ ($\Eg$) we must also allow a relative phase between the
 two components of the field. In this case the uniform ordered phase is only
 stable for $c_\perp>0$, and the modulated phase is now characterized by helical
-order with $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$ and
-\begin{equation}
-  \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{16D_\perp u}
-    =\frac{\tilde r_c-\tilde r}{4u}
-\end{equation}
-The uniform--modulated transition is now continuous. The schematic phase
+order with $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$.
+The uniform--modulated transition is now continuous. This already does not reproduce the physics of \ho, and so we will henceforth neglect this possibility. The schematic phase
 diagrams for this model are shown in Figure \ref{fig:phases}.
 
 \begin{figure}[htpb]
@@ -258,94 +260,101 @@ diagrams for this model are shown in Figure \ref{fig:phases}.
     field theory of a two-component ($\Eg$) Lifshitz point. Solid lines denote
     continuous transitions, while dashed lines denote abrupt transitions.
     Later, when we fit the elastic stiffness predictions for a $\Bog$ \op\ to
-    data along the zero (atmospheric) pressure line, we will take $\tilde
+    data along the zero (atmospheric) pressure line, we will take $\Delta\tilde r=\tilde
     r-\tilde r_c=a(T-T_c)$.
   }
   \label{fig:phases}
 \end{figure}
 
-The susceptibility is given by
+The susceptibility of the order parameter to a field linearly coupled to it is given by
 \begin{equation}
   \begin{aligned}
-    &\chi^{-1}(x,x')
-      =\frac{\delta^2F}{\delta\eta(x)\delta\eta(x')}
+    &\chi^\recip(x,x')
+      =\frac{\delta^2F_\o[\eta]}{\delta\eta(x)\delta\eta(x')}
       =\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2 \\
     &\qquad\qquad+D_\perp\nabla_\perp^4+12u\eta^2(x)\big)
     \delta(x-x'),
   \end{aligned}
+  \label{eq:sus_def}
 \end{equation}
-or in Fourier space,
+where $\recip$ indicates a \emph{functional reciprocal} in the sense that
+\[
+  \int dx''\,\chi^\recip(x,x'')\chi(x'',x')=\delta(x-x').
+\]
+Taking the Fourier transform and integrating over $q'$ we have
 \begin{equation}
-    \chi^{-1}(q)
-    =\tilde r+c_\parallel q_\parallel^2-c_\perp q_\perp^2+D_\perp q_\perp^4
-      +12u\sum_{q'}\tilde\eta_{q'}\tilde\eta_{-q'}.
+    \chi(q)
+    =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4
+    +12u\sum_{q'}\tilde\eta_{q'}\tilde\eta_{-q'}\big)^{-1}.
 \end{equation}
 Near the unordered--modulated transition this yields
 \begin{equation}
   \begin{aligned}
-    \chi_{ij}(q)
-    &=\frac{\delta_{ij}}{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
-      +|\tilde r-\tilde r_c|} \\
-    &=\frac{\delta_{ij}}{D_\perp}\frac{\xi_\perp^4}
+    \chi(q)
+    &=\frac1{c_\parallel q_\parallel^2+D_\perp(q_*^2-q_\perp^2)^2
+      +|\Delta\tilde r|} \\
+    &=\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
-$\xi_\parallel=(|\tilde r-\tilde r_c|/c_\parallel)^{-1/2}$.
+with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}$ and
+$\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}$. We must emphasize that this is \emph{not} the magnetic susceptibility because a $\Bog$ or $\Btg$ \op\ cannot couple linearly to a uniform magnetic field. The object defined in \eqref{eq:sus_def} is most readily interpreted as proportional to the two-point connected correlation function $\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$.
 
-The elastic susceptibility (inverse stiffness) is given in the same way: we
+The strain stiffness is given in a similar way to the inverse susceptibility: we
 must trace over $\eta$ and take the second variation of the resulting free
-energy. Extremizing over $\eta$ yields
+energy functional of $\epsilon$. Extremizing over $\eta$ yields
 \begin{equation}
-  0=\frac{\delta F}{\delta\eta_i(x)}=\frac{\delta F_\o}{\delta\eta_i(x)}
-    +\frac12b\epsilon_{\X i}(x),
+  0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}=\frac{\delta F_\o[\eta]}{\delta\eta(x)}
+    +\frac12b\epsilon_\X(x),
   \label{eq:implicit.eta}
 \end{equation}
-which implicitly gives $\eta$ as a functional of $\epsilon_\X$. Though this
-cannot be solved explicitly, we can make use of the inverse function theorem to
-write
+which implicitly gives $\eta[\epsilon]$ and $F_\e[\epsilon]=F[\eta[\epsilon],\epsilon]$. Since $\eta$ is a functional of $\epsilon_\X$ alone, only the stiffness $\lambda_\X$ is modified from its bare value $C_\X$. Though this
+cannot be solved explicitly, we can make use of the inverse function theorem.
+First, denote by $\eta^{-1}[\eta]$ the inverse functional of $\eta$ implied by
+\eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$ corresponding
+to each solution of \eqref{eq:implicit.eta} it receives. Now, we use the inverse function
+theorem to relate the functional reciprocal of the functional derivative of $\eta[\epsilon]$ with respect to $\epsilon_\X$ to the functional derivative of $\eta^{-1}[\eta]$ with respect to $\eta$, yielding
 \begin{equation}
   \begin{aligned}
-    \bigg(\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}\bigg)^{-1}
-    &=\frac{\delta\eta_j^{-1}[\eta](x)}{\delta\eta_i(x')}
-    =-\frac2b\frac{\delta^2F_\o}{\delta\eta_i(x)\delta\eta_j(x')} \\
-    &=-\frac2b\chi_{ij}^{-1}(x,x')-\frac{b}{2C_\X}\delta_{ij}\delta(x-x')
+    \bigg(\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}\bigg)^\recip
+    &=\frac{\delta\eta^{-1}[\eta](x)}{\delta\eta(x')}
+    =-\frac2b\frac{\delta^2F_\o[\eta]}{\delta\eta(x)\delta\eta(x')} \\
+    &=-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x'),
   \end{aligned}
   \label{eq:inv.func}
 \end{equation}
-It follows from \eqref{eq:implicit.eta} and \eqref{eq:inv.func} that the
-susceptibility of the material to $\epsilon_\X$ strain is given by
+where we have used what we already know about the variation of $F_\o[\eta]$ with respect to $\eta$.
+Finally, \eqref{eq:implicit.eta} and \eqref{eq:inv.func} can be used in concert with the ordinary rules of functional calculus to yield the strain stiffness
 \begin{widetext}
 \begin{equation}
   \begin{aligned}
-    \lambda_{\X ij}(x,x')
-    &=\frac{\delta^2F}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\
-    &=C_\X\delta_{ij}\delta(x-x')+
-    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
-    +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_{\X i}(x')\delta\epsilon_{\X j}(x'')} \\
-    &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o}{\delta\eta_k(x'')\delta\eta_\ell(x''')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')}
-    +\int dx''\,\frac{\delta F_\o}{\delta\eta_k(x'')}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)\delta\epsilon_{\X j}(x')} \\ 
-    &=C_\X\delta_{ij}\delta(x-x')+
-    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
-    -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta_k(x'')}{\partial\epsilon_{\X\ell}(x''')}\bigg)^{-1}\frac{\delta\eta_k(x'')}{\delta\epsilon_{\X i}(x)}\frac{\delta\eta_\ell(x''')}{\delta\epsilon_{\X j}(x')} \\ 
-    &=C_\X\delta_{ij}\delta(x-x')+
-    b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')}
-    -\frac12b\int dx''\,\delta_{i\ell}\delta(x-x'')\frac{\delta\eta_\ell(x'')}{\delta\epsilon_{\X j}(x')} 
-    =C_\X\delta_{ij}\delta(x-x')+
-    \frac12b\frac{\delta\eta_i(x)}{\delta\epsilon_{\X j}(x')},
+    \lambda_\X(x,x')
+    &=\frac{\delta^2F_\e[\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\
+    &=C_\X\delta(x-x')+
+    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
+    +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')} \\
+    &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o[\eta]}{\delta\eta(x'')\delta\eta(x''')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')}
+    +\int dx''\,\frac{\delta F_\o[\eta]}{\delta\eta(x'')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\ 
+    &=C_\X\delta(x-x')+
+    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
+    -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta(x'')}{\partial\epsilon_\X(x''')}\bigg)^{-1}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')} \\ 
+    &=C_\X\delta(x-x')+
+    b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}
+    -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta(x'')}{\delta\epsilon_\X(x')} 
+    =C_\X\delta(x-x')+
+    \frac12b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')},
   \end{aligned}
 \end{equation}
 \end{widetext}
 whose Fourier transform follows from \eqref{eq:inv.func} as
 \begin{equation}
-  \lambda_{\X ij}^{-1}(q)=\frac{\delta_{ij}}{C_\X}+\frac{b^2}{4C_\X^2}\chi_{ij}(q).
+  \lambda_\X(q)=C_\X\bigg(1+\frac{b^2}{4C_\X}\chi(q)\bigg)^{-1}.
   \label{eq:elastic.susceptibility}
 \end{equation}
+Though not relevant here, this result generalizes to multicomponent order parameters.
 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 $\gamma=1$.
-
+predicts a cusp in the elastic susceptibility of the form $|\Delta\tilde r|^\gamma$ for $\gamma=1$. 
 \begin{figure}[htpb]
   \centering
   \includegraphics[width=\columnwidth]{fig-stiffnesses}
@@ -404,7 +413,7 @@ 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
-$V_\xi^{-1}\int_{V_\xi}G(0,x)\,dx=\langle\delta\alpha^2\rangle\simeq\langle\alpha\rangle^2=\alpha_0^2$,
+$V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle\simeq\langle\alpha\rangle^2=\alpha_0^2$,
 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
@@ -412,7 +421,7 @@ using
   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}
-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
+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)
-- 
cgit v1.2.3-70-g09d2

