diff options
Diffstat (limited to 'stokes.tex')
| -rw-r--r-- | stokes.tex | 335 |
1 files changed, 332 insertions, 3 deletions
@@ -15,6 +15,9 @@ \begin{document} \newcommand\eqref[1]{\eref{#1}} +\makeatletter +\renewcommand\tableofcontents{\@starttoc{toc}} +\makeatother \title{Analytic continuation over complex landscapes} @@ -44,6 +47,8 @@ \maketitle +\tableofcontents + \section{Preamble} Complex landscapes are basically functions of many variables having many minima and, inevitably, @@ -806,6 +811,122 @@ $\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvector =Z(\beta^*) \end{eqnarray} +\section{The ensemble of symmetric complex-normal matrices} + +The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to +the unconstrained Hessian with a constant added to its diagonal. The eigenvalue +distribution $\rho$ is therefore related to the unconstrained distribution +$\rho_0$ by a similar shift: $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The +Hessian of the unconstrained Hamiltonian is +\begin{equation} \label{eq:bare.hessian} + \partial_i\partial_jH_0 + =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}}, +\end{equation} +which makes its ensemble that of Gaussian complex symmetric matrices, when the +anomalous direction normal to the constraint surface is neglected. Given its variances +$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)r^{p-2}/2N$ and +$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is +constant inside the ellipse +\begin{equation} \label{eq:ellipse} + \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{r^{p-2}+|\kappa|}\right)^2+ + \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{r^{p-2}-|\kappa|}\right)^2 + <\frac{p(p-1)}{2r^{p-2}} +\end{equation} +where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue +spectrum of $\partial\partial H$ is therefore constant inside the same ellipse +translated so that its center lies at $-p\epsilon$. Examples of these +distributions are shown in the insets of Fig.~\ref{fig:spectra}. + +The eigenvalue spectrum of the Hessian of the real part is not the +spectrum $\rho(\lambda)$ of $\partial\partial H$, but instead the +square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$; +in other words, the singular value spectrum $\rho(\sigma)$ of $\partial\partial +H$. When $\kappa=0$ and the elements of $J$ are standard complex normal, this +is a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and +to our knowledge a closed form is not in the literature. We have worked out an +implicit form for the singular value spectrum using the replica method. + +Introducing replicas to bring the partition function into the numerator of the +Green function \cite{Livan_2018_Introduction} gives +\begin{equation} \label{eq:green.replicas} + G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)} + \exp\left\{ + \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma + -\operatorname{Re}\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right) + \right] + \right\}, +\end{equation} +with sums taken over repeated Latin indices. The average is then made over +$J$ and Hubbard--Stratonovich is used to change variables to the replica matrices +$N\alpha_{\alpha\beta}=(\zeta^{(\alpha)})^\dagger\zeta^{(\beta)}$ and +$N\chi_{\alpha\beta}=(\zeta^{(\alpha)})^T\zeta^{(\beta)}$, and a series of +replica vectors. The replica-symmetric ansatz leaves all replica vectors +zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$, +$\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is +\begin{equation}\label{eq:green.saddle} + \overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0 + \exp\left\{nN\left[ + 1+\frac{p(p-1)}{16}r^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2) + +\frac p4\operatorname{Re}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right) + \right]\right\}. +\end{equation} + +\begin{figure} + \centering + + \includegraphics{figs/spectra_0.0.pdf} + \includegraphics{figs/spectra_0.5.pdf}\\ + \includegraphics{figs/spectra_1.0.pdf} + \includegraphics{figs/spectra_1.5.pdf} + + \caption{ + Eigenvalue and singular value spectra of the Hessian $\partial\partial H$ + of the $3$-spin model with $\kappa=\frac34e^{-i3\pi/4}$. Pictured + distributions are for critical points at `radius' $r=\sqrt{5/4}$ and with + energy per spin (a) $\epsilon=0$, (b) + $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c) + $\epsilon=-|\epsilon_{\mathrm{th}}|$, and (d) + $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each + inset shows the support of the eigenvalue distribution \eqref{eq:ellipse}. + The solid line on each plot shows the distribution of singular values + \eqref{eq:spectral.density}, while the overlaid histogram shows the + empirical distribution from $2^{10}\times2^{10}$ complex normal matrices + with the same covariance and diagonal shift as $\partial\partial H$. + } \label{fig:spectra} +\end{figure} + +The argument of the exponential has several saddles. The solutions $\alpha_0$ +are the roots of a sixth-order polynomial, and the root with the smallest value +of $\operatorname{Re}\alpha_0$ gives the correct solution in all the cases we +studied. A detailed analysis of the saddle point integration is needed to +understand why this is so. Evaluated at such a solution, the density of +singular values follows from the jump across the cut, or +\begin{equation} \label{eq:spectral.density} + \rho(\sigma)=\frac1{i\pi N}\left( + \lim_{\operatorname{Im}\sigma\to0^+}\overline G(\sigma) + -\lim_{\operatorname{Im}\sigma\to0^-}\overline G(\sigma) + \right) +\end{equation} +Examples can be seen in Fig.~\ref{fig:spectra} compared with numeric +experiments. + +The formation of a gap in the singular value spectrum naturally corresponds to +the origin leaving the support of the eigenvalue spectrum. Weyl's theorem +requires that the product over the norm of all eigenvalues must not be greater +than the product over all singular values \cite{Weyl_1912_Das}. Therefore, the +absence of zero eigenvalues implies the absence of zero singular values. The +determination of the threshold energy---the energy at which the distribution +of singular values becomes gapped---is reduced to the geometry problem of +determining when the boundary of the ellipse defined in \eqref{eq:ellipse} +intersects the origin, and yields +\begin{equation} \label{eq:threshold.energy} + |\epsilon_{\mathrm{th}}|^2 + =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2r^{p-2}} + {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)} +\end{equation} +for $\delta=\kappa r^{-(p-2)}$. Notice that the threshold depends on both the +energy per spin $\epsilon$ on the `radius' $r$ of the saddle. + \section{The \textit{p}-spin spherical models} The $p$-spin spherical models are statistical mechanics models defined by the @@ -830,8 +951,6 @@ this manifold is given by \end{equation} where indeed $Pz=z-z|z|^2/|z|^2=0$ and $Pz'=z'$ for any $z'$ orthogonal to $z$. -To find critical points, we use the method of Lagrange multipliers. Introducing $\mu\in\mathbb C$, - \subsection{2-spin} The Hamiltonian of the $2$-spin model is defined for $z\in\mathbb C^N$ by @@ -901,7 +1020,15 @@ imaginary energy join. \begin{eqnarray} Z(\beta) - &=\int_{S^{N-1}}dx\,e^{-\beta H(x)} + &=\int_{S^{N-1}}ds\,e^{-\beta H(s)} + =\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta H_2(s)} \\ + &\simeq\sum_{k=0}^D2n_ki^k\left(\frac{2\pi}\beta\right)^{D/2}e^{-\beta H_2(s_\sigma)}\prod_{\lambda_0>0}\lambda_0^{-1/2} \\ + &=\sum_{k=0}^D2n_ki^k\left(\frac{2\pi}\beta\right)^{D/2}e^{-N\beta\epsilon_k}\prod_{\ell\neq k}\frac12|\epsilon_k-\epsilon_\ell| +\end{eqnarray} + +\begin{eqnarray} + Z(\beta) + &=\int_{S^{N-1}}ds\,e^{-\beta H(s)} =\int_{\mathbb R^N}dx\,\delta(x^2-N)e^{-\beta H(x)} \\ &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12\beta x^TJx-\lambda(x^Tx-N)} \\ &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12x^T(\beta J+\lambda I)x+\lambda N} \\ @@ -913,6 +1040,208 @@ imaginary energy join. \subsection{Pure \textit{p}-spin} +Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also +one of $\operatorname{Im}H$, and therefore of $H$ itself. Writing $z=x+iy$ for +$x,y\in\mathbb R^N$, $\operatorname{Re}H$ can be considered a real-valued +function of $2N$ real variables. The number of critical points of $H$ is thus given by the +usual Kac--Rice formula applied to $\operatorname{Re}H$: +\begin{equation} \label{eq:real.kac-rice} + \mathcal N(\kappa,\epsilon,R) + = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H) + \left|\det\operatorname{Hess}_{x,y}\operatorname{Re}H\right|. +\end{equation} +This expression is to be averaged over $J$ to give the complexity $\Sigma$ as +$N \Sigma= \overline{\log\mathcal N}$, a calculation that involves the replica +trick. Based on the experience from similar problems \cite{Castellani_2005_Spin-glass}, the +\emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is +expected to be exact wherever the complexity is positive. + +The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in +a manifestly complex way. With the Wirtinger derivative +$\partial\equiv\frac12(\partial_x-i\partial_y)$, one can write +$\partial_x\operatorname{Re}H=\operatorname{Re}\partial H$ and +$\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these +transformations through, one finds +\begin{equation} \label{eq:complex.kac-rice} + \mathcal N(\kappa,\epsilon,r) + = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) + |\det\operatorname{Hess}H|^2. +\end{equation} +This gives three equivalent expressions for the determinant of the Hessian: as +that of a $2N\times 2N$ real symmetric matrix, that of the $N\times N$ Hermitian +matrix $(\partial\partial H)^\dagger\partial\partial H$, or the norm squared of +that of the $N\times N$ complex symmetric matrix $\partial\partial H$. + +These equivalences belie a deeper connection between the spectra of the +corresponding matrices. Each positive eigenvalue of the real matrix has a +negative partner. For each such pair $\pm\lambda$, $\lambda^2$ is an eigenvalue +of the Hermitian matrix and $|\lambda|$ is a \emph{singular value} of the +complex symmetric matrix. The distribution of positive eigenvalues of the +Hessian is therefore the same as the distribution of singular values of +$\partial\partial H$, or the distribution of square-rooted eigenvalues of +$(\partial\partial H)^\dagger\partial\partial H$. + +A useful property of the Gaussian $J$ is that gradient and Hessian at fixed +energy $\epsilon$ are statistically independent \cite{Bray_2007_Statistics, +Fyodorov_2004_Complexity}, so that the $\delta$-functions and the Hessian may +be averaged independently. First we shall compute the spectrum of the Hessian, +which can in turn be used to compute the determinant. Then we will treat the +$\delta$-functions and the resulting saddle point equations. The results of +these calculations begin around \eqref{eq:bezout}. + + +We will now address the $\delta$-functions of \eqref{eq:complex.kac-rice}. +These are converted to exponentials by the introduction of auxiliary fields +$\hat z=\hat x+i\hat y$. The average over $J$ can then be performed. A +generalized Hubbard--Stratonovich allows a change of variables from the $4N$ +original and auxiliary fields to eight bilinears defined by $Nr=z^\dagger z$, +$N\hat r=\hat z^\dagger\hat z$, $Na=\hat z^\dagger z$, $Nb=\hat z^Tz$, and +$N\hat c=\hat z^T\hat z$ (and their conjugates). The result, to leading order +in $N$, is +\begin{equation} \label{eq:saddle} + \overline{\mathcal N}(\kappa,\epsilon,R) + = \int dr\,d\hat r\,da\,da^*db\,db^*d\hat c\,d\hat c^*e^{Nf(r,\hat r,a,b,\hat c)}, +\end{equation} +where the argument of the exponential is +\begin{equation} + f=2+\frac12\log\det\frac12\left[\matrix{ + 1 & r & b & a \cr + r & 1 & a^* & b^* \cr + b & a^* & \hat c & \hat r \cr + a & b^* & \hat r & \hat c^* + }\right] + +\int d\lambda\,d\lambda^*\rho(\lambda)\log|\lambda|^2 + +p\operatorname{Re}\left\{ + \frac18\left[\hat rr^{p-1}+(p-1)|b|^2r^{p-2}+\kappa(\hat c^*+(p-1)a^2)\right]-\epsilon a + \right\}. +\end{equation} +The spectrum $\rho$ is given in \eqref{eq:ellipse} and is dependant on $r$ alone. This function has an +extremum in $\hat r$, $a$, $b$, and $\hat c$ at which its value is +\begin{equation} \label{eq:free.energy.a} + f=1+\frac12\log\left(\frac4{p^2}\frac{r^2-1}{r^{2(p-1)}-|\kappa|^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2 + -2C_+[\operatorname{Re}(\epsilon e^{-i\theta})]^2-2C_-[\operatorname{Im}(\epsilon e^{-i\theta})]^2, +\end{equation} +where $\theta=\frac12\arg\kappa$ and +\begin{equation} + C_{\pm}=\frac{r^p(1+p(r^2-1))\mp r^2|\kappa|}{r^{2p}\pm(p-1)r^p(r^2-1)|\kappa|-r^2|\kappa|^2}. +\end{equation} +Notice that level sets of $f$ in energy $\epsilon$ also give ellipses, but of +different form from the ellipse in \eqref{eq:ellipse}. + +This expression is maximized for $r=R$, its value at the boundary, for +all values of $\kappa$ and $\epsilon$. Evaluating the complexity at this +saddle, in the limit of unbounded spins, gives +\begin{equation} \label{eq:bezout} + \lim_{R\to\infty}\log\overline{\mathcal N}(\kappa,\epsilon,R) + =N\log(p-1). +\end{equation} +This is, to leading order, precisely the Bézout bound, the maximum number of +solutions to $N$ equations of degree $p-1$ \cite{Bezout_1779_Theorie}. That we +saturate this bound is perhaps not surprising, since the coefficients of our +polynomial equations \eqref{eq:polynomial} are complex and have no symmetries. +Reaching Bézout in \eqref{eq:bezout} is not our main result, but it provides a +good check. Analogous asymptotic scaling has been found for the number of pure +Higgs states in supersymmetric quiver theories \cite{Manschot_2012_From}. + +\begin{figure}[htpb] + \centering + \includegraphics{figs/complexity.pdf} + \caption{ + The complexity of the 3-spin model as a function of the maximum `radius' + $R$ at zero energy and several values of $\kappa$. The dashed line shows + $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$. + } \label{fig:complexity} +\end{figure} + +For finite $R$, everything is analytically tractable at $\epsilon=0$: +\begin{equation} \label{eq:complexity.zero.energy} + \Sigma(\kappa,0,R) + =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2R^{-4(p-1)}}{1-R^{-4}}\right). +\end{equation} +This is plotted as a function of $R$ for several values of $\kappa$ in +Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to +negative infinity as $R\to1$, i.e., as the spins are restricted to the reals. +This is natural, since volume of configuration space vanishes in this limit +like $(R^2-1)^N$. However, when the result is analytically continued to +$\kappa=1$ (which corresponds to real $J$) something novel occurs: the +complexity has a finite value at $R=1$. This implies a $\delta$-function +density of critical points on the $r=1$ (or $y=0$) boundary. The number of +critical points contained there is +\begin{equation} + \lim_{R\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,R) + = \frac12N\log(p-1), +\end{equation} +half of \eqref{eq:bezout} and corresponding precisely to the number of critical +points of the real $p$-spin model. (Note the role of conjugation symmetry, +already underlined in \cite{Bogomolny_1992_Distribution}.) The full +$\epsilon$-dependence of the real $p$-spin is recovered by this limit as +$\epsilon$ is varied. + +\begin{figure}[b] + \centering + \includegraphics{figs/desert.pdf} + \caption{ + The value of bounding `radius' $R$ for which $\Sigma(\kappa,\epsilon,R)=0$ as a + function of (real) energy per spin $\epsilon$ for the 3-spin model at + several values of $\kappa$. Above each line the complexity is positive and + critical points proliferate, while below it the complexity is negative and + critical points are exponentially suppressed. The dotted black lines show + the location of the ground and highest exited state energies for the real + 3-spin model. + } \label{fig:desert} +\end{figure} + +In the thermodynamic limit, \eqref{eq:complexity.zero.energy} implies that most +critical points are concentrated at infinite radius $r$. For finite $N$ the +average radius of critical points is likewise finite. By differentiating +$\overline{\mathcal N}$ with respect to $R$ and normalizing, one obtains the +distribution of critical points as a function of $r$. This yields an average +radius proportional to $N^{1/4}$. One therefore expects typical critical +points to have a norm that grows modestly with system size. + +These qualitative features carry over to nonzero $\epsilon$. In +Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap in $r$ +close to one in which solutions are exponentially suppressed. When +$\kappa=1$---the analytic continuation to the real computation---the situation +is more interesting. In the range of energies where there are real solutions +this gap closes, which is only possible if the density of solutions diverges at +$r=1$. Outside this range, around `deep' real energies where real solutions are +exponentially suppressed, the gap remains. A moment's thought tells us that +this is necessary: otherwise a small perturbation of the $J$s could produce +an unusually deep solution to the real problem, in a region where this should +not happen. + +\begin{figure}[t] + \centering + + \includegraphics{figs/threshold_2.000.pdf} + \includegraphics{figs/threshold_1.325.pdf} \\ + \includegraphics{figs/threshold_1.125.pdf} + \includegraphics{figs/threshold_1.000.pdf} + + \caption{ + Energies at which states exist (green shaded region) and threshold energies + (black solid line) for the 3-spin model with + $\kappa=\frac34e^{-i3\pi/4}$ and (a) $r=\sqrt2$, (b) $r=\sqrt{1.325}$, (c) $r=\sqrt{1.125}$, + and (d) $r=1$. No shaded region is shown in (d) because no states exist at + any energy. + } \label{fig:eggs} +\end{figure} + +The relationship between the threshold and ground, or extremal, state energies +is richer than in the real case. In Fig.~\ref{fig:eggs} these are shown in the +complex-$\epsilon$ plane for several examples. Depending on the parameters, the +threshold might have a smaller or larger magnitude than the extremal state, or +cross as a function of complex argument. For sufficiently large $r$ the +threshold is always at a larger magnitude. If this were to happen in the real +case, it would likely imply our replica symmetric computation were unstable, +since having a ground state above the threshold implies a ground state Hessian +with many negative eigenvalues, a contradiction. However, this is not an +contradiction in the complex case, where the energy is not bounded from below. +The relationship between the threshold, i.e., where the gap appears, and the +dynamics of, e.g., a minimization algorithm, deformed integration cycle, or +physical dynamics, are a problem we hope to address in future work. + \begin{equation} H_p=\frac1{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p} \end{equation} |