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| -rw-r--r-- | main.tex | 20 |
1 files changed, 8 insertions, 12 deletions
@@ -267,22 +267,18 @@ diagrams for this model are shown in Figure \ref{fig:phases}. The susceptibility is given by \begin{equation} \begin{aligned} - &\chi_{ij}^{-1}(x,x') - =\frac{\delta^2F}{\delta\eta_i(x)\delta\eta_j(x')} \\ - &\quad=\Big[\big(\tilde r-c_\parallel\nabla_\parallel^2 - -c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+4u\eta^2(x)\big)\delta_{ij} \\ - &\qquad\qquad +8u\eta_i(x)\eta_j(x)\Big]\delta(x-x'), + &\chi^{-1}(x,x') + =\frac{\delta^2F}{\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} \end{equation} or in Fourier space, \begin{equation} - \begin{aligned} - \chi_{ij}^{-1}(q) - &=8u\sum_{q'}\tilde\eta_i(q')\eta_j(-q')+\bigg(\tilde r - +c_\parallel q_\parallel^2-c_\perp q_\perp^2 \\ - &\qquad+D_\perp q_\perp^4+4u\sum_{q'}\tilde\eta_k(q')\tilde\eta_k(-q')\bigg) - \delta_{ij}. - \end{aligned} + \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'}. \end{equation} Near the unordered--modulated transition this yields \begin{equation} |