big reformulation, added computation of the Ginsburg criterion

2 files changed, 161 insertions, 92 deletions

diff --git a/cusp.pdf b/cusp.pdf
index 4727390..e566319 100644
--- a/cusp.pdf
+++ b/cusp.pdf
diff --git a/hidden-order.tex b/hidden-order.tex
index 52918e8..5031d69 100644
--- a/hidden-order.tex
+++ b/hidden-order.tex
@@ -9,145 +9,214 @@
 
 \begin{document}
 
+\def\Aog{\mathrm A_{1\mathrm g}}
+\def\Bog{\mathrm B_{1\mathrm g}}
+\def\Btg{\mathrm B_{2\mathrm g}}
+\def\Eg{\mathrm E_{\mathrm g}}
+\def\X{\mathrm X}
+
 \maketitle
 
-The elastic free energy density for a tetragonal system is given by
+The free energy density of our model has three terms, respectively corresponding to the strain, the order parameter, and the strain--order parameter coupling. The free energy density for the strain, taken to quadratic order under the assumption of small strains, is generically of the form $f_{\mathrm e}=\frac12\lambda_{ijk\ell}\epsilon_{ij}\epsilon_{k\ell}$. Symmetries of the strain tensor, which is symmetric, and those of the lattice restrict the form of $\lambda$. The most general form for a system with a tetragonal point group is
 \[
-  f_{\mathrm e}=\frac12\Big[a^{(1)}_{\mathrm A_1}(\epsilon_{11}+\epsilon_{22})^2+
-      a_{\mathrm A_1}^{(2)}\epsilon_{33}^2+
-      a_{\mathrm A_1}^{(3)}(\epsilon_{11}+\epsilon_{22})\epsilon_{33}+
-      a_{\mathrm B_1}(\epsilon_{11}-\epsilon_{22})^2+
-      a_{\mathrm B_2}\epsilon_{12}^2+
-      a_{\mathrm E}(\epsilon_{13}^2+\epsilon_{23}^2)\Big]
+  f_{\mathrm e}=\frac12\Big[\lambda_{1111}(\epsilon_{11}^2+\epsilon_{22}^2)+
+    \lambda_{3333}\epsilon_{33}^2+
+      2\lambda_{1133}(\epsilon_{11}+\epsilon_{22})\epsilon_{33}+
+      2\lambda_{1122}\epsilon_{11}\epsilon_{22}+
+      4\lambda_{1212}\epsilon_{12}^2+
+      4\lambda_{1313}(\epsilon_{13}^2+\epsilon_{23}^2)\Big]
 \]
-Consider a generic one-component order parameter $\eta$. The most general quartic free energy density (discounting total derivatives) is
+A very convenient way to come by this form is to use knowledge of the point group of the lattice and the way that strain transforms under it. The following linear combinations of strains transform like a particular irreducible representation of the point group:
+\begin{align*}
+  \epsilon_{\Aog}^{(1)}=\epsilon_{11}+\epsilon_{22}
+  &&
+  \epsilon_{\Aog}^{(2)}=\epsilon_{33}
+  \\
+  \epsilon_{\Bog}^{(1)}=\epsilon_{11}-\epsilon_{22}
+  &&
+  \epsilon_{\Btg}^{(1)}=\epsilon_{12}
+  &&
+  \epsilon_{\mathrm E_{\mathrm G}}^{(1)}=\{\epsilon_{11},\epsilon_{22}\}
+\end{align*}
+The most general quadratic free energy density is built from combinations of strains that transform like scalars, or $\Aog$, yielding
+\[
+  \begin{aligned}
+    f_{\mathrm e}&=\frac12\sum_{\X}\lambda_{\X}^{(ij)}\epsilon_{\X}^{(i)}\epsilon_{\X}^{(j)}\\
+    &=\frac12\Big[\lambda^{(11)}_{\Aog}(\epsilon_{\Aog}^{(1)})^2+
+      \lambda_{\Aog}^{(22)}(\epsilon_{\Aog}^{(2)})^2+
+      \lambda_{\Aog}^{(12)}\epsilon_{\Aog}^{(1)}\epsilon_{\Aog}^{(2)}+
+      \lambda_{\Bog}^{(11)}(\epsilon_{\Bog}^{(1)})^2+
+      \lambda_{\Btg}^{(11)}(\epsilon_{\Btg}^{(1)})^2+
+    \lambda_{\Eg}^{(11)}\epsilon_{\Eg}^{(1)}\cdot\epsilon_{\Eg}^{(1)}\Big]
+  \end{aligned}
+\]
+where the sum is over the irreducible representations of the point group.
+We can now identify
+\begin{align*}
+  \lambda_{\Aog}^{(1)}=\tfrac12(\lambda_{1111}+\lambda_{1122})
+  &&
+  \lambda_{\Aog}^{(2)}=\lambda_{3333}
+  &&
+  \lambda_{\Aog}^{(3)}=2\lambda_{1133}
+  \\
+  \lambda_{\Bog}^{(1)}=\tfrac12(\lambda_{1111}-\lambda_{1122})
+  &&
+  \lambda_{\Btg}^{(1)}=4\lambda_{1212}
+  &&
+  \lambda_{\Eg}^{(1)}=4\lambda_{1313}
+\end{align*}
+Consider a generic order parameter $\eta$. To write down its free energy, we in principle must know both how it transforms under symmetry operations of the lattice and how derivative operators transform. For derivative operators, the two independent combinations are $\nabla_\parallel=\{\partial_x,\partial_y\}$ which transforms like $\Eg$, and $\nabla_\perp=\partial_3$ which transform like $\Aog$. The most general quartic free energy density (discounting total derivatives) is
 \[
-  f_{\mathrm o}=\frac12\bigg(r\eta^2+
+  f_{\mathrm o}=\frac12\Big[r\eta^2+
                 c_\parallel(\nabla_\parallel\eta)^2+
                 c_\perp(\partial_3\eta)^2+
-                D\big[(\nabla_\parallel^2+\partial_3^2)\eta\big]^2+
-              u\eta^4\bigg)
+                D\big[(\nabla_\parallel^2+\partial_3^2)\eta\big]^2\Big]
+                +u\eta^4
+\]
+independent of the symmetry of $\eta$. This is the free energy for a Lifshitz point, and so we expect to see that phenomenology in $\eta$. The irreducible symmetry that $\eta$ transforms like determines its coupling to strain. To quadratic order we have
+\[
+  f_{\X}=\tfrac12b^{(i)}\epsilon_{\X}^{(i)}\cdot\eta
+  +\tfrac12e^{(i)}\epsilon_{\Aog}^{(i)}\eta^2
+\]
+The total  free energy is 
+\[
+  F=\int d^3x\,(f_{\mathrm e}+f_{\mathrm o}+f_{\X})
 \]
-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
+Replacing $\eta$ with its inverse Fourier transform, we have
+\[
+  F_{\mathrm e}=\frac V2\sum_q\sum_X\lambda_X^{(ij)}\tilde\epsilon_X^{(i)}(q)\tilde\epsilon_X^{(j)}(-q)
+\]
+\[
+  F_{\mathrm o}=\frac V2\sum_q\Big[r|\tilde\eta_q|^2+c_\parallel q_{\parallel}^2|\tilde\eta_q|^2
+  +c_\perp q_{\perp}^2|\tilde\eta_q|^2+Dq^4|\tilde\eta_q|^2\Big]
+  +Vu\sum_q\sum_{q'}\sum_{q''}\tilde\eta_q\tilde\eta_{q'}\tilde\eta_{q''}\tilde\eta_{-(q+q'+q'')}
+\]
+\[
+  F_{\X}=\frac V2\sum_q\Big[b^{(i)}\tilde\epsilon^{(i)}_{\X}(-q)\tilde\eta_q+\sum_qe^{(i)}\tilde\epsilon_{\Aog}^{(i)}(-(q+q'))\tilde\eta_q\tilde\eta_{q'}\Big]
+\]
+There are three distinct possibilities here: $\eta$ transforms like a one-component irreducible representation of the point group that is not $\Aog$ ($\Bog$ or $\Btg$), $\eta$ transforms like $\Aog$, and $\eta$ is a two-component vector that transforms like $\Eg$.
+
+We will first tackle the case of a non-$\Aog$, one-component order parameter. We will simply write $\epsilon=\epsilon_{\X}^{(1)}$, $\lambda=\lambda_{\X}^{(11)}$, and $\alpha^{(i)}=\epsilon_{\Aog}^{(i)}$ for the duration of this section.
+We will assume that our system orders at some specific $q_\perp=q^*$. This ansatz is equivalent to
 \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}
+  \tilde\eta_q=\tfrac12\eta_*\big[\delta_{q_\perp,q_*}+\delta_{q_\perp,-q_*}\big]\delta_{q_\parallel,0}
+  &&
+  \tilde\epsilon_q=\tfrac12\epsilon_*\big[\delta_{q_\perp,q_*}+\delta_{q_\perp,-q_*}\big]\delta_{q_\parallel,0}
+  &&
+  \tilde\alpha_q^{(i)}=\delta_{q,0}
 \end{align*}
-which can generically be written $f_{\mathrm i}=\frac12B\eta$. The total $\eta$-dependent free energy is
+where because of the coupling of $\alpha$ to $\eta$ we don't expect it to produce modulated order (this can also be confirmed directly by minimizing over a modulated $\alpha$).
+For $q_*\neq0$, we have
 \[
-  F_\eta=\int d^3x\,(f_{\mathrm o}+f_{\mathrm i})
+  F=\frac V2\bigg[\frac12(r+c_\perp q_*^2+Dq_*^4)\eta_*^2+\frac34u\eta_*^4+\frac12\lambda\epsilon_*^2+\lambda^{(ij)}\alpha^{(i)}_*\alpha^{(j)}_*+\frac12b\epsilon_*\eta_*+\frac12e^{(i)}\alpha_*^{(i)}\eta_*^2\bigg]
 \]
-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
+while for $q^*=0$ we have
 \[
-  \bar f_\eta=\lim_{L\to\infty}\frac1{L^3}\int_\Box d^3x\,(f_{\mathrm o}+f_{\mathrm i})
-  =\frac12\bigg(r\eta_0^2+u\eta_0^4+\frac12c_\perp q^2\eta_q^2+\frac12dq^4\eta_q^4+\frac12r\eta_q^2+3u\eta_0^2\eta_q^2+\frac38u\eta_q^4+B\eta_0\bigg)
+  F=\frac V2\big(r\eta_*^2+2u\eta_*^4+\lambda\epsilon_*^2+\lambda^{(ij)}\alpha^{(i)}_*\alpha^{(j)}_*+b\epsilon_*\eta_*+e^{(i)}\alpha_*^{(i)}\eta_*^2\big)
 \]
-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
+For $r>b^2/4\lambda$ and $c_\perp>0$ or $r>b^2/4\lambda+c_\perp^2/4D$ and $c_\perp<0$, there is a free energy minimizer with all fields zero. For $r<b^2/4\lambda$ there is a local minimum with $q_*=0$ and 
 \begin{align*}
-  \eta_0=-\sqrt{-r/2u}+B/4r+O(B^2)&&\eta_q=0&& q=0
+  \eta_*^2=\frac12|r-r_c|\bigg(2u-\frac{(e^{(1)})^2\lambda_{\Aog}^{(22)}+(e^{(2)})^2\lambda_{\Aog}^{(11)}-e^{(1)}e^{(2)}\lambda_{\Aog}^{(12)}}{4\lambda_{\Aog}^{(11)}\lambda_{\Aog}^{(22)}-(\lambda_{\Aog}^{(12)})^2}\bigg)^{-1}
 \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
+  \epsilon_*=-\frac b{2\lambda}\eta_*
+  &&
+  \epsilon^{(1)}_*=\frac12\frac{e^{(2)}\lambda_{\Aog}^{(12)}-2e^{(1)}\lambda_{\Aog}^{(22)}}{4\lambda_{\Aog}^{(11)}\lambda_{\Aog}^{(22)}-(\lambda_{\Aog}^{(12)})^2}\eta_*^2
+  &&
+  \epsilon^{(2)}_*=\frac12\frac{e^{(1)}\lambda_{\Aog}^{(12)}-2e^{(2)}\lambda_{\Aog}^{(11)}}{4\lambda_{\Aog}^{(11)}\lambda_{\Aog}^{(22)}-(\lambda_{\Aog}^{(12)})^2}\eta_*^2
 \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
+with $r_c=b^2/4\lambda$. For $c_\perp<0$ and $r<b^2/4\lambda+c_\perp^2/4D$ there is a local minimum with $q_*=\sqrt{-c_\perp/2D}$ and
 \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}
+  \eta_*^2=|r-r_c|\bigg(3u-\frac{(e^{(1)})^2\lambda_{\Aog}^{(22)}+(e^{(2)})^2\lambda_{\Aog}^{(11)}-e^{(1)}e^{(2)}\lambda_{\Aog}^{(12)}}{4\lambda_{\Aog}^{(11)}\lambda_{\Aog}^{(22)}-(\lambda_{\Aog}^{(12)})^2}\bigg)^{-1}
 \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
-\[
-  \tilde a_{\text{X, disordered}}=\frac{\partial^2\bar f}{\partial\epsilon_{\mathrm X}^2}\bigg|_{\epsilon=0}=a_{\mathrm X}+\frac{\partial^2\bar f_\eta}{\partial\epsilon_{\mathrm X}^2}\bigg|_{\epsilon=0}
-  =a_{\mathrm X}-b^2/4r
-\]
-for the unordered phase, and
-\[
-  \tilde a_{\text{X, modulated}}=a_{\mathrm X}-b^2d/2(c_\perp^2-2dr)
-\]
-for the modulated phase. As a function of $t=r-r_c=r-c_\perp^2/4d$, this is
+now with $r_c=b^2/4\lambda+c_\perp^2/4D$. Between the disordered and either of these phases is a continuous phase transition, and between the nontrivial phases is an abrupt transition.
+
+We are interested in the response of the system near the critical lines. The susceptibility can be found by adding an additional modulated field $h$ to the free energy linearly coupled to $\eta$ and computing
 \[
-  \tilde a_{\text{X, disordered--modulated}}=a_{\mathrm X}-\begin{cases}
-    b^2d/(c_\perp^2-4dt)&t<0\\
-    b^2d/(c_\perp^2+4dt)&t>0
-  \end{cases}
-  =a_{\mathrm X}-b^2d/(c_\perp^2+4d|t|)
+  \chi(q)=\frac{\delta^2F}{\delta\tilde h(q)^2}\bigg|_{\tilde h(q)=0}
 \]
-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
+The susceptibility in the disordered (trivial) phase is
 \[
-  \tilde a_{\text{X, ordered}}=a_{\mathrm X}+b^2/8r
+  \chi(q)=\frac12\big(r-b^2/4\lambda+c_\parallel q_\parallel^2+c_\perp q_\perp^2+Dq^4\big)^{-1}
 \]
-As a function of $t=r-r_c=r$, this is
+The susceptibility in the in ordered (uniform) phase is
 \[
-  \tilde a_{\text{X, disordered--ordered}}=a_{\mathrm X}-\begin{cases}
-    b^2/8|t|&t<0\\
-    b^2/4|t|&t>0
-  \end{cases}
+  \chi(q)=\frac12\big(b^2/2\lambda-2r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+Dq^4\big)^{-1}
 \]
-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
+The susceptibility in the modulated phase is
 \[
-  \tilde a_{\mathrm{X,\ ordered}}-\tilde a_{\mathrm{X,\ modulated}}\Big|_{r=-(2+\sqrt6)c_\perp^2/4d}=\frac{8+3\sqrt6}{28+12\sqrt6}\frac{b^2d}{c_\perp^2}
+  \chi(q)=\frac12\big(b^2/4\lambda+c_\perp^2/2D-r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+Dq^4\big)^{-1}
 \]
-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
+This makes the susceptibility near the disordered--ordered transition
 \[
-  f_{\mathrm X}^{(3)}=\frac12\eta^2\big[e_1(\epsilon_{11}+\epsilon_{22})+e_2\epsilon_{33}\big]
+  \chi(q)=\frac12\begin{cases}\big[Dq^4+c_\parallel q_\parallel^2+c_\perp q_\perp^2+|\Delta r|\big]^{-1}&r>r_c\\\big[Dq^4+c_\parallel q_\parallel^2+c_\perp q_\perp^2+2|\Delta r|\big]^{-1}&r<r_c\end{cases}=\frac1{2c_\perp}\frac{\xi_\perp^2}{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^2q_\perp^2+\xi_\perp^2(D/c_\perp)q^4}
 \]
-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
+for
 \begin{align*}
-  \tilde a_{\text{X, modulated}}=a_{\mathrm X}-\frac{b^2d}{c_\perp^2-2dr}
+  \xi_\perp=\begin{cases}(|\Delta r|/c_\perp)^{-1/2}&r<r_c\\(2|\Delta r|/c_\perp)^{-1/2}&r>r_c\end{cases}
   &&
-  \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}
-  &&
-  \tilde a_{\text{$\mathrm A_1$, ordered}}=a_{\mathrm A_1}-\frac{e^2}{2u}+\frac{be}{2^{3/2}(-ru)^{1/2}}
+  \xi_\parallel=\begin{cases}(|\Delta r|/c_\parallel)^{-1/2}&r<r_c\\(2|\Delta r|/c_\parallel)^{-1/2}&r>r_c\end{cases}
 \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
+and the susceptibility near the disordered--modulated transition is
 \[
-  t_G=\frac{k_B^2}{32\pi^2(\Delta c_V)^2\xi_0^6}
+  \chi(q)=\frac12\big[c_\parallel q_\parallel^2+D(q_0^2-q_\perp^2)^2+|\Delta r|\big]^{-1}=\frac1{2D}\frac{\xi_\perp^4}{1+\xi_\parallel^2q_\parallel^2+\xi_\perp^{4}(q_0^2-q^2)^2}
 \]
-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
+where $\xi_\perp=(|\Delta r|/D)^{-1/4}$ and $\xi_\parallel=(|\Delta r|/c_\parallel)^{-1/2}$. We're also interested in the elastic response, defined by
 \[
-  \tilde a_{\text{X, disordered--modulated}}
-  =a_{\mathrm X}-\frac{2b^2d}{(c_\perp+2dq^2)^2+4d|t|}
+  \frac12\tilde\lambda_{\X}^{(ij)}(q)=\frac{\delta^2F}{\delta\epsilon_{\mathrm X}^{(i)}(q)\delta\epsilon_{\mathrm X}^{(j)}(q)}\bigg|_{\epsilon=\epsilon_*}
 \]
+The effective elastic susceptibility for the symmetry component of our order parameter at the disordered--ordered transition has a response of the form
 \[
-  \tilde a_{\text{X, disordered--ordered}}=a_{\mathrm X}-\begin{cases}
-    \frac{b^2}{4(c_\perp q^2+dq^4+|t|)}&t<0\\
-    \frac{b^2}{4(c_\perp q^2+dq^4+2|t|)}&t>0
+  \frac{\lambda}{\tilde\lambda(q)}=1+\frac{b^2}{4\lambda}\begin{cases}\big[Dq^4+c_\perp q^2+c_\parallel q_\parallel^2+|\Delta r|\big]^{-1}&r>r_c\\
+    \big[Dq^4+c_\perp q_\perp^2+c_\parallel q_\parallel^2+2|\Delta r|\big]^{-1}&r<r_c
   \end{cases}
+  =1+\frac{b^2}{2\lambda}\chi(q)
+\]
+and the effective susceptibility for $\Aog$ has a response like
+\[
+  \tilde\lambda_{\Aog}^{(ij)}(q)-\lambda_{\Aog}^{(ij)}=\delta_{q,0}\begin{cases}0&r>r_c\\-e^{(i)}e^{(j)}/4u&r<r_c\end{cases}
+\]
+At the disordered--modulated transition the responses are of the form
+\[
+  \frac{\lambda}{\tilde\lambda(q)}=1+\frac{b^2}{4\lambda}\big[c_\parallel q_\parallel^2+D(q_0^2-q_\perp^2)^2+|\Delta r|\big]^{-1}=1+\frac{b^2}{2\lambda}\chi(q)
+\]
+and
+\[
+  \tilde\lambda_{\Aog}^{(ij)}(q)-\lambda_{\Aog}^{(ij)}=\delta_{q,0}\begin{cases}0&r>r_c\\-e^{(i)}e^{(j)}/6u&r<r_c\end{cases}
 \]
 
+The Ginzburg criterion gives the proximity $t_G$ of the critical point at which mean field theory is no longer self consistent, with
+\begin{align*}
+  t_G=\frac{k_B^2}{32\pi^2(\Delta c_V)^2\xi_0^6}
+\end{align*}
+Experiments give $\Delta c_V\sim1\times10^5\,\mathrm J\,\mathrm m^{-3}\,\mathrm K^{-1}$ \cite{fisher_specific_1990} and $T_c\sim17.5\,\mathrm K$. A fit of $\tilde\lambda$ to experimental data (Fig.~\ref{fig:B1g.fit}) yields
+\begin{align*}
+  \lambda=71\,\mathrm{GPa}-(0.10\,\mathrm{GPa}\,\mathrm{K}^{-1})T &&
+  \frac{b^2}{4\lambda Dq_0^4}=0.084&&\frac a{Dq_0^4}=0.0038\,\mathrm K^{-1}
+\end{align*}
+with $|r-r_c|=a|T-T_c|$. We suspect that the modulation at the transition at very low pressure is on the order of the lattice spacing, which would give $q_0^{-1}\sim9.568\,\text{\r A}$. Combined with our fit, this gives
+\[
+  \xi_0=(aT_c/D)^{-1/4}=\big[T_c(a/Dq_0^4)q_0^4\big]^{-1/4}\sim19\,\text{\r A}
+\]
+and therefore $t_G\sim1.2\times10^{-6}$, which means one would need to get within about $\Delta T=T_ct_G\sim20\,\mu\mathrm K$ of the critical point to see mean field theory break down.
+
+\begin{figure}
+  \centering
+  \includegraphics[width=0.5\textwidth]{cusp}
+  \caption{Experimental data for the $\Bog$ effective elastic constant $\tilde\lambda_{\Bog}$ (blue dots), along with a fit of the above theory (orange line).}
+  \label{fig:B1g.fit}
+\end{figure}
+
 
 The conditions for $\eta$ being a stationary function of $F$ is
 \[
   0=\frac{\delta F}{\delta\eta}=\frac{\partial f}{\partial\eta}-\partial_i\frac{\partial f}{\partial(\partial_i\eta)}+\partial_i^2\frac{\partial f}{\partial(\partial_i^2\eta)}
   =r\eta-c_\parallel\nabla_\parallel^2\eta-c_\perp\partial_3^2\eta+D\nabla^4\eta+2u\eta^3
 \]
+We can estimate the error in our ansatz by how much the total variation in $F$ differs from zero.
+
+\bibliographystyle{plain}
+\bibliography{hidden_order.bib}
 
 \end{document}