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| -rw-r--r-- | marginal.tex | 87 |
1 files changed, 78 insertions, 9 deletions
diff --git a/marginal.tex b/marginal.tex index e9dc5a8..f06bc04 100644 --- a/marginal.tex +++ b/marginal.tex @@ -73,7 +73,7 @@ Taking the average over $B$, we have e^{NG_\sigma(\omega)} =\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\prod_{a=1}^n\left[d\mathbf x_a\,\delta(N-\mathbf x_a^T\mathbf x_a)\right] \exp\left\{-Nn\beta\omega+N\lambda\omega+\frac{\sigma^2}{N}\left[\beta^2\sum_{ab}^n(\mathbf x_a^T\mathbf x_b)^2 - +2\beta\lambda\sum_a^n(\mathbf x_a^T\mathbf x_1)^2 + -2\beta\lambda\sum_a^n(\mathbf x_a^T\mathbf x_1)^2 +\lambda^2N^2 \right]\right\} \end{equation} @@ -83,7 +83,7 @@ We make the Hubbard--Stratonovich transformation to the matrix field $Q_{ab}=\fr =\lim_{\beta\to\infty}\lim_{n\to0}\int d\lambda\,dQ\, \exp N\left\{ -n\beta\omega+\lambda\omega+\sigma^2\left[\beta^2\sum_{ab}^nQ_{ab}^2 - +2\beta\lambda\sum_a^nQ_{1a}^2 + +-\beta\lambda\sum_a^nQ_{1a}^2 +\lambda^2 \right]+\frac12\log\det Q\right\} \end{equation} @@ -109,17 +109,58 @@ Inserting these expressions and taking the limit of $n$ to zero, we find \begin{equation} \mathcal S(q_0,\tilde q_0,\lambda)=\lambda\omega+\sigma^2\left[ 2\beta^2(q_0^2-\tilde q_0^2)-2\beta\lambda(1-\tilde q_0^2)+\lambda^2 - \right]-\log(1-q_0)+\frac12\log(1-2q_0+\tilde q_1^2) + \right]-\log(1-q_0)+\frac12\log(1-2q_0+\tilde q_0^2) \end{equation} The integral is then given by its value at the stationary point of this -expression with respect to its three arguments. Making the extremization and taking the limit of $\beta$ to infinity, we find +expression with respect to its three arguments. +The extremal conditions are +\begin{align} + 0&=\frac{\partial\mathcal S}{\partial q_0} + =\frac1{1-q_0}-\frac1{1-2q_0+\tilde q_0^2}+4\beta^2\sigma^2q_0 \\ + 0&=\frac{\partial\mathcal S}{\partial \tilde q_0} + =\frac{\tilde q_0}{1-2q_0+\tilde q_0^2}-4\sigma^2(\beta^2-\beta\lambda)\tilde q_0 \\ + 0&=\frac{\partial\mathcal S}{\partial\lambda} + =\omega+2\sigma^2\big(\lambda-\beta(1-\tilde q_0^2)\big) +\end{align} +We expect the overlaps to concentrate on one as $\beta$ goes to infinity. We therefore take +\begin{align} + q_0=1-y\beta^{-1}-z\beta^{-2}+O(\beta^{-3}) + && + \tilde q_0=1-\tilde y\beta^{-1}-\tilde z\beta^{-2}+O(\beta^{-3}) +\end{align} +The first equations expanded in $\beta$ give +\begin{align} + &0=4\sigma^2\beta^2+\bigg(\frac1{y}-\frac12\frac1{y-\tilde y}-4y\sigma^2\bigg)\beta+O(\beta^0) \\ + &0=-4\sigma^2\beta^2+\bigg(\frac12\frac1{y-\tilde y}+4\sigma^2(\lambda+\tilde y)\bigg)\beta+O(\beta^0) +\end{align} +One cannot satisfy this equation order-by-order in $\beta$. However, a solution +suggests itself: the expansion is singular when $\tilde y=y$. Making this +identification, we find instead +\begin{align} + &0=\left(4\sigma^2-\frac1{y^2+2(z-\tilde z)}\right)\beta^2+\left(\frac1y+\frac{2y\tilde z}{(y^2+2(z-\tilde z))^2}-4\sigma^2y\right)\beta+O(\beta^0) \\ + &0=\left(-4\sigma^2+\frac1{y^2+2(z-\tilde z)}\right)\beta^2+\left(-\frac y{y^2+2(z-\tilde z)}-\frac{2y\tilde z}{(y^2+2(z-\tilde z))^2}+4\sigma^2(y+\lambda)\right)\beta+O(\beta^0) +\end{align} +Finally, expanding the equation for $\lambda$ to lowest order, we have +\begin{equation} + 0=\omega+2\sigma^2(\lambda-2y)+O(\beta^{-1}) +\end{equation} +Simultaneously solving these five equations stemming from the coefficients of $\beta$ for $y$, $z$, $\tilde z$, and $\lambda$, we have +\begin{align} + \lambda=-\frac1\sigma\sqrt{\frac{\omega^2}{(2\sigma)^2}-1} + && + y=\frac1{2\sigma}\left(\frac{\omega}{2\sigma}-\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}\right) + \\ + z=\frac1{2\sigma^2}\left(1-\frac{\omega^2}{(2\sigma)^2}\right) + && + \tilde z=\frac1{4\sigma^2}\left(1-\frac{\omega}{2\sigma}\left(\frac\omega{2\sigma}+\sqrt{\frac{\omega^2}{(2\sigma)^2}-1}\right)\right) +\end{align} +Inserting this solution into $\mathcal S$ and taking the limit of $\beta$ to zero, we find \begin{equation} G_\sigma(\omega)=-\frac{\omega}{2\sigma}\sqrt{\frac{\omega^2}{(2\sigma)^2}-1} - -\log\left[ - \frac{\omega}{2\sigma}-\sqrt{\frac{\omega^2}{(2\sigma)^2}-1} + +\log\left[ + \frac{\omega}{2\sigma}+\sqrt{\frac{\omega^2}{(2\sigma)^2}-1} \right] \end{equation} -where the branch of the square roots is the same as the sign of $\omega-2\sigma$. This function is plotted in Fig. For $\omega<2\sigma$ $G_\sigma(\omega)$ has an imaginary part, which makes any additional integral over $\omega$ highly oscillatory. This indicates that the existence of a marginal minimum for this @@ -260,12 +301,40 @@ $\Omega=S^{N-1}\times S^{N-1}$ \end{equation} \begin{equation} - \mathcal S_1(C^{11},R^{11},D^{11},\hat\beta,\omega_1)+\mathcal S_2(C^{22},R^{22},D^{22},\hat\beta,\omega_2) - -\epsilon r_{12}-\epsilon r_{21}-\omega_1r^{11}_d-\omega_2r^{22}_d+\hat\beta E + \mathcal S(C,R,D,W,\hat\beta,\omega) + =\frac12\frac1n + \sum_{ab}\left( + \hat\beta^2f(C_{ab})+(2\hat\beta R_{ab}-D_{ab})f'(C_{ab})+(R_{ab}^2-W_{ab}^2)f''(C_{ab}) + \right) +\end{equation} + +\begin{equation} + \mathcal S(C^{11},R^{11},D^{11},W^{11},\hat\beta)+\mathcal S(C^{22},R^{22},D^{22},W^{22},\hat\beta) + -\epsilon(r_{12}+r_{21})-\omega_1(r^{11}_d-w^{11}_d)-\omega_2(r^{22}_d-w^{22}_d)+\hat\beta E +\frac12\log\det\begin{bmatrix}C^{11}&iR^{11}\\iR^{11}&D^{11}\end{bmatrix} +\frac12\log\det\left( \begin{bmatrix}C^{22}-q_{12}^2C^{11}&iR^{22}\\iR^{22}&D^{22}\end{bmatrix} \right) + -\log\det(W^{11}W^{22}+W^{12}W^{21}) +\end{equation} + +\begin{equation} + \begin{aligned} + &\sum_a^n\left[\hat q_a(Q^{11}_{aa}+Q^{22}_{aa}-1)-\beta(\omega_1Q^{11}_{aa}+\omega_2Q^{22}_{aa}+2\epsilon Q^{12}_{aa})\right] + +\lambda(\omega_1Q^{11}_{11}+\omega_2Q^{22}_{11}+2\epsilon Q^{12}_{11}) \\ + &+\sum_{i=1,2}f_i''(1)\left[\beta^2\sum_{ab}^n(Q^{ii}_{ab})^2-2\beta\lambda\sum_a^n(Q^{ii}_{1a})^2+\lambda^2(Q^{ii}_{11})^2\right] + +\frac12\log\det\begin{bmatrix} + Q^{11}&Q^{12}\\ + Q^{12}&Q^{22} + \end{bmatrix} + \end{aligned} +\end{equation} +\begin{equation} + \log\det\begin{bmatrix} + Q^{11}&Q^{12}\\ + Q^{12}&Q^{22} + \end{bmatrix} + +\log\det(Q^{11}Q^{22}-Q^{12}Q^{12}) \end{equation} \section{Multi-species spherical model} |