From ff835f43bdb2986bdd0a8f7dc16431b1afff7584 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 7 Jun 2019 14:43:03 -0400
Subject: added description of Ginzburg critereon and the values of the
q-dependent elastic constants
---
hidden-order.tex | 45 +++++++++++++++++++++++++++++++++++----------
1 file changed, 35 insertions(+), 10 deletions(-)
diff --git a/hidden-order.tex b/hidden-order.tex
index ca61105..52918e8 100644
--- a/hidden-order.tex
+++ b/hidden-order.tex
@@ -62,19 +62,19 @@ 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
\[
\tilde a_{\text{X, disordered}}=\frac{\partial^2\bar f}{\partial\epsilon_{\mathrm X}^2}\bigg|_{\epsilon=0}=a_{\mathrm X}+\frac{\partial^2\bar f_\eta}{\partial\epsilon_{\mathrm X}^2}\bigg|_{\epsilon=0}
- =a_{\mathrm X}-b^2/2r
+ =a_{\mathrm X}-b^2/4r
\]
for the unordered phase, and
\[
- \tilde a_{\text{X, modulated}}=a_{\mathrm X}-b^2d/(c_\perp^2-2dr)
+ \tilde a_{\text{X, modulated}}=a_{\mathrm X}-b^2d/2(c_\perp^2-2dr)
\]
for the modulated phase. As a function of $t=r-r_c=r-c_\perp^2/4d$, this is
\[
\tilde a_{\text{X, disordered--modulated}}=a_{\mathrm X}-\begin{cases}
- 2b^2d/(c_\perp^2-4dt)&t<0\\
- 2b^2d/(c_\perp^2+4dt)&t>0
+ b^2d/(c_\perp^2-4dt)&t<0\\
+ b^2d/(c_\perp^2+4dt)&t>0
\end{cases}
- =a_{\mathrm X}-2b^2d/(c_\perp^2+4d|t|)
+ =a_{\mathrm X}-b^2d/(c_\perp^2+4d|t|)
\]
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.
@@ -86,20 +86,20 @@ The effective elastic constant for the component of strain coupled to the order
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
\[
- \tilde a_{\text{X, ordered}}=a_{\mathrm X}+b^2/4r
+ \tilde a_{\text{X, ordered}}=a_{\mathrm X}+b^2/8r
\]
As a function of $t=r-r_c=r$, this is
\[
\tilde a_{\text{X, disordered--ordered}}=a_{\mathrm X}-\begin{cases}
- b^2/4|t|&t<0\\
- b^2/2|t|&t>0
+ b^2/8|t|&t<0\\
+ b^2/4|t|&t>0
\end{cases}
\]
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
\[
- \tilde a_{\mathrm{X,\ ordered}}-\tilde a_{\mathrm{X,\ modulated}}\Big|_{r=-(2+\sqrt6)c_\perp^2/4d}=\frac{8+3\sqrt6}{14+6\sqrt6}\frac{b^2d}{c_\perp^2}
+ \tilde a_{\mathrm{X,\ ordered}}-\tilde a_{\mathrm{X,\ modulated}}\Big|_{r=-(2+\sqrt6)c_\perp^2/4d}=\frac{8+3\sqrt6}{28+12\sqrt6}\frac{b^2d}{c_\perp^2}
\]
A phase diagram is shown below. The ordered phase is metastable for all $r<0$, while the modulated phase is metastable for $c_\perp<0$.
@@ -118,11 +118,36 @@ Adding this interaction is akin to simply shifting $r\to r+e\epsilon_{\mathrm A_
\tilde a_{\text{X, modulated}}=a_{\mathrm X}-\frac{b^2d}{c_\perp^2-2dr}
&&
\tilde a_{\text{$\mathrm A_1$, modulated}}=a_{\mathrm A_1}-\frac{e^2}{3u} \\
- \tilde a_{\text{X, ordered}}=a_{\mathrm X}+\frac{b^2}{4r}+\frac{be}{2^{3/2}(-ru)^{1/2}}
+ \tilde a_{\text{X, ordered}}=a_{\mathrm X}+\frac{b^2}{4r}
&&
\tilde a_{\text{$\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.
+The Ginzburg criterion for the validity of mean field theory gives the crossover temperature difference as
+\[
+ t_G=\frac{k_B^2}{32\pi^2(\Delta c_V)^2\xi_0^6}
+\]
+where $\Delta c_V$ is the jump in the specific heat at the transition and $\xi_0$ is the bare coherence length defined by $\xi\sim\xi_0|(T-T_c)/T_c|^{-\nu}$, e.g., the critical amplitude of the correlation length.
+
+The $q$-dependent elastic response can be calculated the same way as above, but with a $q$-dependent strain and order response. The results are
+\[
+ \tilde a_{\text{X, disordered--modulated}}
+ =a_{\mathrm X}-\frac{2b^2d}{(c_\perp+2dq^2)^2+4d|t|}
+\]
+\[
+ \tilde a_{\text{X, disordered--ordered}}=a_{\mathrm X}-\begin{cases}
+ \frac{b^2}{4(c_\perp q^2+dq^4+|t|)}&t<0\\
+ \frac{b^2}{4(c_\perp q^2+dq^4+2|t|)}&t>0
+ \end{cases}
+\]
+
+
+The conditions for $\eta$ being a stationary function of $F$ is
+\[
+ 0=\frac{\delta F}{\delta\eta}=\frac{\partial f}{\partial\eta}-\partial_i\frac{\partial f}{\partial(\partial_i\eta)}+\partial_i^2\frac{\partial f}{\partial(\partial_i^2\eta)}
+ =r\eta-c_\parallel\nabla_\parallel^2\eta-c_\perp\partial_3^2\eta+D\nabla^4\eta+2u\eta^3
+\]
+
\end{document}
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