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Diffstat (limited to 'hidden-order.tex')
| -rw-r--r-- | hidden-order.tex | 7 |
1 files changed, 4 insertions, 3 deletions
diff --git a/hidden-order.tex b/hidden-order.tex index 364602e..b326d0d 100644 --- a/hidden-order.tex +++ b/hidden-order.tex @@ -69,7 +69,8 @@ Consider a generic order parameter $\eta$. To write down its free energy, we in \[ f_{\X}=\tfrac12b^{(i)}\epsilon_{\X}^{(i)}\cdot\eta +\tfrac12e^{(i)}\epsilon_{\Aog}^{(i)}\eta^2 -V\] + +\tfrac12h^{(ij)}\epsilon_{\Aog}^{(i)}\epsilon_{\X}^{(j)}\eta +\] The total free energy is \[ F=\int d^3x\,(f_{\mathrm e}+f_{\mathrm o}+f_{\X}) @@ -90,12 +91,12 @@ The most general quartic free energy density (discounting total derivatives) is independent of the symmetry of $\eta$. In principle we could have $D_\parallel\neq D_\perp$, but this does not affect the physics at hand. This is the free energy for a Lifshitz point, and so we expect to see that phenomenology in $\eta$. Before doing anything, we can minimize the free energy with respect to strain alone to find the strain in terms of $\eta$ exactly. We have \[ - 0=\frac{\delta F}{\delta\epsilon_{\mathrm X}^{(1)}(x)}=\lambda_{\mathrm X}^{(11)}\epsilon_{\mathrm X}^{(1)}(x)+\frac12b^{(1)}\eta(x) + 0=\frac{\delta F}{\delta\epsilon_{\mathrm X}^{(1)}(x)}=\lambda_{\mathrm X}^{(11)}\epsilon_{\mathrm X}^{(1)}(x)+\frac12b^{(1)}\eta(x)+\frac12h^{(i)}\epsilon_{\Aog}^{(i)}\eta \] whence we immediately have $\epsilon_{\mathrm X}^{(1)}=-\frac{b^{(1)}}{2\lambda_{\mathrm X}^{(11)}}\eta(x)$. We also have \[ 0=\frac{\delta F}{\delta\epsilon_{\Aog}^{(i)}(x)} - =\lambda_{\Aog}^{(ij)}\epsilon_{\Aog}^{(j)}(x)+\frac12 e^{(i)}\eta^2(x) + =\lambda_{\Aog}^{(ij)}\epsilon_{\Aog}^{(j)}(x)+\frac12 e^{(i)}\eta^2(x)+\frac12h^{(i)}\epsilon_\X\eta \] which is a linear system whose solutions are \begin{align*} |