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| -rw-r--r-- | hidden-order.tex | 11 |
1 files changed, 9 insertions, 2 deletions
diff --git a/hidden-order.tex b/hidden-order.tex index 4bf9bde..57ec702 100644 --- a/hidden-order.tex +++ b/hidden-order.tex @@ -121,14 +121,21 @@ For $r>b^2/4\lambda$ and $c_\perp>0$ or $r>b^2/4\lambda+c_\perp^2/4D$ and $c_\pe \begin{align*} \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^{(1)}_*=\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 + \epsilon^{(2)}_*=\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*} 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_*^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*} +\begin{align*} + \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*} 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 |