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Diffstat (limited to 'stokes.tex')
| -rw-r--r-- | stokes.tex | 93 |
1 files changed, 84 insertions, 9 deletions
@@ -771,9 +771,9 @@ stationary point of index $k$ has $k$ real eigenvectors and $D-k$ purely imaginary eigenvectors that contribute to its thimble. The matrix of eigenvectors can therefore be written $U=i^kO$ for an orthogonal matrix $O$, and with all eigenvectors canonically oriented $\det O=1$. We therefore have -$\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvectors change: by a factor of $e^{-i\phi/2}$. Therefore, the contribution more generally is $(e^{-i\phi/2})^Di^k$. We therefore have, for real stationary points of a real action, +$\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvectors change: by a factor of $e^{-i\phi/2}$. Therefore, the contribution more generally is $(e^{-i\phi/2})^Di^k$. We therefore have, for real stationary points of a real action before any Stokes points, \begin{equation} - Z_\sigma(\beta)\simeq\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}\prod_{\lambda_0>0}\lambda_0^{-\frac12}e^{-\beta\mathcal S(s_\sigma)} + Z_\sigma(\beta)\simeq\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma}|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta\mathcal S(s_\sigma)} \end{equation} \begin{eqnarray} @@ -1071,17 +1071,17 @@ imaginary energy join. \begin{eqnarray} Z(\beta) - &=\int_{S^{N-1}}ds\,e^{-\beta H(s)} + &=\int_{S^{N-1}}ds\,e^{-\beta H_2(s)} =\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta H_2(s)} \\ &\simeq\sum_{k=0}^D2i^k\left(\frac{2\pi}\beta\right)^{D/2}e^{-\beta H_2(s_\sigma)}\prod_{\lambda_0>0}\lambda_0^{-1/2} \\ &=2\sum_{k=0}^D\exp\left\{ - i\frac\pi2k+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon_k-\frac12\sum_{\ell\neq k}\log\frac12|\epsilon_k-\epsilon_\ell| + i\frac\pi2k+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon_k-\frac12\sum_{\ell\neq k}\log2|\epsilon_k-\epsilon_\ell| \right\} \end{eqnarray} \begin{eqnarray} Z(\beta) &=2\int d\epsilon\,\rho(\epsilon)\exp\left\{ - i\frac\pi2k_\epsilon+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon-\frac D2\int d\epsilon'\,\rho(\epsilon')\log\frac12|\epsilon-\epsilon'| + i\frac\pi2k_\epsilon+\frac D2\log\frac{2\pi}\beta-N\beta\epsilon-\frac D2\int d\epsilon'\,\rho(\epsilon')\log2|\epsilon-\epsilon'| \right\} \end{eqnarray} @@ -1100,12 +1100,12 @@ The index as a function of energy level is given by the cumulative density funct \end{equation} Finally, the product over the singular values corresponding to descending directions gives \begin{equation} - \frac12\int d\epsilon'\,\rho(\epsilon')\log\frac12|\epsilon-\epsilon'| - =\log\frac{\epsilon_{\mathrm{th}}}2-\frac14+\frac12\left(\frac{\epsilon}{\epsilon_{\mathrm{th}}}\right)^2 + \frac12\int d\epsilon'\,\rho(\epsilon')\log2|\epsilon-\epsilon'| + =-\frac14+\frac12\left(\frac{\epsilon}{\epsilon_{\mathrm{th}}}\right)^2 \end{equation} for $\epsilon<\epsilon_{\mathrm{th}}$. Then \begin{equation} - \operatorname{Re}f=-\epsilon\operatorname{Re}\beta-\log\frac12+\frac14-\frac12\epsilon^2 + \operatorname{Re}f=-\epsilon\operatorname{Re}\beta+\frac14-\frac12\epsilon^2 \end{equation} \begin{equation} \operatorname{Im}f=-\epsilon\operatorname{Im}\beta+\frac12\left( @@ -1442,11 +1442,86 @@ function of $\Delta$ and $\arg\delta$. } \end{figure} +\subsection{Pure $p$-spin: is analytic continuation possible?} + +\begin{eqnarray} + Z(\beta) + &=\sum_{\sigma\in\Sigma_0}n_\sigma Z_\sigma(\beta) \\ + &=\sum_{\sigma\in\Sigma_0}\left(\frac{2\pi}\beta\right)^{D/2}i^{k_\sigma} + |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12} + e^{-\beta\mathcal S(s_\sigma)} \\ + &\simeq\sum_{k=0}^D\int d\epsilon\,\mathcal N_\mathrm{typ}(\epsilon,k) + \left(\frac{2\pi}\beta\right)^{D/2}i^k + \left(|\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}\bigm|\mathcal S(s_\sigma)=N\epsilon,k_\sigma=k\right) e^{-\beta N\epsilon} +\end{eqnarray} +Following Derrida \cite{Derrida}, +\begin{equation} + \mathcal N_\mathrm{typ}(\epsilon,k)=\overline\mathcal N(\epsilon,k)+\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2} +\end{equation} +where $\eta$ is a random, sample-dependant number of order one. This gives two +contributions to the partition function, but as we shall see it is the +fluctuations in the number of stationary points at a given energy level that +governs things at large $|\beta|$, not its total. This gives two terms to the typical partition function +\begin{equation} + Z_\mathrm{typ}=Z_A+Z_B +\end{equation} +\begin{eqnarray} + Z_A + &\simeq\sum_{k=0}^D\int d\epsilon\,\overline\mathcal N(\epsilon,k) + \left(\frac{2\pi}\beta\right)^{D/2}i^k + |\det\operatorname{Hess}\mathcal S|^{-\frac12} e^{-\beta N\epsilon} + =\int d\epsilon\,e^{Nf_A(\epsilon)} + \\ + Z_B + &\simeq\sum_{k=0}^D\int d\epsilon\,\eta(\epsilon,k)\overline\mathcal N(\epsilon,k)^{1/2} + \left(\frac{2\pi}\beta\right)^{D/2}i^k + |\det\operatorname{Hess}\mathcal S(s_\sigma)|^{-\frac12}e^{-\beta N\epsilon} + =\int d\epsilon\,\tilde\eta(\epsilon)e^{Nf_B(\epsilon)} +\end{eqnarray} +for +\begin{eqnarray} + f_A + &=-\beta\epsilon+\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta + +i\frac\pi2P(\lambda<0\mid\epsilon) \\ + f_B + &=-\beta\epsilon+\frac12\Sigma(\epsilon)-\frac12\int d\lambda\,\rho(\lambda\mid\epsilon)|\lambda|+\frac12\log\frac{2\pi}\beta + +i\frac\pi2P(\lambda<0\mid\epsilon) +\end{eqnarray} +Each integral will be dominated by its value near the maximum of the real part of the exponential argument. Assuming that $\epsilon<\epsilon_\mathrm{th}$, this maximum occurs at +\begin{equation} + 0=-\operatorname{Re}\beta-\frac12\frac{3p-4}{p-1}\epsilon+\frac12\frac p{p-1}\sqrt{\epsilon^2-\epsilon_\mathrm{th}^2} +\end{equation} +\begin{equation} + 0=-\operatorname{Re}\beta-\epsilon^2 +\end{equation} +As with the 2-spin model, the integral over $\epsilon$ is oscillatory and can only be reliably evaluated with a saddle point when either the period of oscillation diverges \emph{or} when the maximum lies at a cusp. We therefore expect changes in behavior when $\epsilon=\epsilon_0$, the ground state energy. The temperature at which this happens is +\begin{eqnarray} + \operatorname{Re}\beta_A&=-\frac12\frac{3p-4}{p-1}\epsilon_0+\frac12\frac p{p-1}\sqrt{\epsilon_0^2-\epsilon_\mathrm{th}^2}\\ + \operatorname{Re}\beta_B&=-\epsilon_0 +\end{eqnarray} +which for all $p\geq2$ has $\operatorname{Re}\beta_A>\operatorname{Re}\beta_B$. +Therefore, the emergence of zeros in $Z_A$ does not lead to the emergence of +zeros in the partition function as a whole, because $Z_B$ still produces a +coherent result (despite the unknown constant factor $\eta(\epsilon_0)$). It is +only at $\operatorname{Re}\beta_B=-\epsilon_0$ where both terms contributing to +the partition function at large $N$ involve incoherent integrals near the +maximum, and only here where the density of zeros is expected to become +nonzero. + +In fact, in the limit of $|\beta|\to\infty$, $\operatorname{Re}\beta_B$ is +precisely the transition found in \cite{Obuchi_2012_Partition-function} between +phases with and without a density of zeros. This value is an underestimate for +the transition for finite $|\beta|$, which likely results from the invalidity +of our large-$\beta$ approximation. More of the phase diagram might be +constructed by continuing the series for individual thimbles to higher powers +in $\beta$, which would be equivalent to allowing non-constant terms in the +Jacobian over the thimble. + + \section{The $p$-spin spherical models: numerics} To confirm the presence of Stokes lines under certain processes in the $p$-spin, we studied the problem numerically. - \bibliographystyle{unsrt} \bibliography{stokes} |