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@@ -364,6 +364,98 @@ occur order-one times, one could reasonably hope to perform such a procedure. If they occur exponentially often in the system size, there is little hope of keeping track of the resulting weights, and analytic continuation is intractable. +\subsection{Gradient flow} + +The `dynamics' describing thimbles is defined by gradient descent on the real +part of the action, with a given thimble incorporating all trajectories which +asymptotically flow to its associated stationary point. Since our phase space +is not necessary flat (as for the \emph{spherical} $p$-spin models), we will +have to do a bit of differential geometry to work out their form. Gradient +descent on a complex (Kähler) manifold is given by +\begin{equation} \label{eq:flow.coordinate.free} + \dot s + =-\operatorname{grad}\operatorname{Re}\beta\mathcal S + =-\left(\frac\partial{\partial s^*}\operatorname{Re}\beta\mathcal S\right)^\sharp + =-\frac{\beta^*}2\frac{\partial\mathcal S^*}{\partial s^*}g^{-1}\frac\partial{\partial s} +\end{equation} +where $g$ is the metric and the holomorphicity of the action was used to set +$\partial\mathcal S/\partial s^*=0$. If the complex phase space is $\mathbb C^N$ and the +metric is diagonal, this means that the flow is proportional to the conjugate +of the gradient, or $\dot s\propto-\beta^*(\partial S/\partial s)^*$. + +In the cases we will consider here (namely, that of the spherical models), it +will be more convenient to work in terms of coordinates in a flat embedding +space than in terms of local coordinates in the curved space, e.g., in terms of +$z\in\mathbb C^N$ instead of $s\in S^{N-1}$. Let $z:\tilde\Omega\to\mathbb C^N$ +be an embedding of complex phase space into complex euclidean space. The +dynamics in the embedding space is given by +\begin{equation}\label{eq:flow.raw} + \dot z + =-\frac{\beta^*}2\frac{\partial\mathcal S^*}{\partial z^*}(Dz)^* g^{-1}(Dz)^T\frac\partial{\partial z} +\end{equation} +where $Dz=\partial z/\partial s$ is the Jacobian of the embedding. +The embedding induces a metric on $\tilde\Omega$ by $g=(Dz)^\dagger Dz$. +Writing $\partial=\partial/\partial z$, this gives +\begin{equation} \label{eq:flow} + \dot z=-\frac{\beta^*}2(\partial\mathcal S)^\dagger(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T + =-\frac12(\partial \mathcal S)^\dagger P +\end{equation} +which is nothing but the projection of $(\partial\mathcal S)^*$ into the +tangent space of the manifold, with the projection operator +$P=(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T$. Note that $P$ is hermitian. For the spherical models, where $\tilde\Omega$ is the complex phase spaced defined by all points $z\in\mathbb C^N$ such that $z^Tz=1$, the projection operator is given by +\begin{equation} + P=I-\frac{zz^\dagger}{|z|^2} +\end{equation} +something that we be worked out in detail in a following section. One can +quickly verify that this operator indeed projects the dynamics onto the +manifold: its tangent at any point $z$ is given by $\partial(z^Tz)=z$, and +$Pz=z-z|z|^2/|z|^2=0$. For any vector $u$ perpendicular to $z$, i.e., +$z^\dagger u=0$, $Pu=u$. + +\begin{figure} + \includegraphics{figs/thimble_flow.pdf} + + \caption{Example of gradient descent flow on the action $\mathcal S$ featured + in Fig.~\ref{fig:example.action} in the complex-$\theta$ plane, with + $\arg\beta=0.4$. Symbols denote the stationary points, while thick blue and + red lines depict the thimbles and antithimbles, respectively. Streamlines + of the flow equations are plotted in a color set by their value of + $\operatorname{Im}\beta\mathcal S$; notice that the color is constant along + each streamline. + } \label{fig:flow.example} +\end{figure} + +Gradient descent on $\operatorname{Re}\beta\mathcal S$ is equivalent to +Hamiltonian dynamics with the Hamiltonian $\operatorname{Im}\beta\mathcal S$ +and conjugate coordinates given by the real and imaginary parts of each complex +coordinate. This is because $(\tilde\Omega, g)$ is Kähler and therefore admits +a symplectic structure, but that the flow conserves +$\operatorname{Im}\beta\mathcal S$ can be shown using \eref{eq:flow} and the +holomorphic property of $\mathcal S$: +\begin{eqnarray} + \frac d{dt}\operatorname{Im}\beta\mathcal S + &=\dot z\partial\operatorname{Im}\beta\mathcal S+\dot z^*\partial^*\operatorname{Im}\beta\mathcal S \\ + &=\frac i4\left( + (\beta\partial \mathcal S)^\dagger P\beta\partial\mathcal S-(\beta\partial\mathcal S)^TP^*(\beta\partial\mathcal S)^* + \right) \\ + &=\frac{i|\beta|^2}4\left( + (\partial\mathcal S)^\dagger P\partial\mathcal S-[(\partial\mathcal S)^\dagger P\partial\mathcal S]^* + \right) \\ + &=\frac{i|\beta|^2}4\left( + \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2 + \right)=0. +\end{eqnarray} +A consequence of this conservation is that the flow in the action takes a +simple form: +\begin{equation} + \dot{\mathcal S} + =\dot z\partial\mathcal S + =-\frac{\beta^*}2(\partial\mathcal S)^\dagger P\partial\mathcal S + =-\frac{\beta^*}2\|\partial\mathcal S\|^2. +\end{equation} +In the complex-$\mathcal S$ plane, dynamics is occurs along straight lines in +a direction set by the argument of $\beta$. + \subsection{The structure of stationary points} The shape of each thimble in the vicinity of a stationary point can be @@ -371,9 +463,16 @@ described using an analysis of the hessian of the real part of the action at the stationary point. Here we'll review some general properties of this hessian, which because the action is holomorphic has rich structure. -First, consider the problem as one of $2N$ real variables $x,y\in\mathbb R^N$ -with $z=x+iy$. The hessian of the real part of the action with respect to these -real variables is +For the moment we will work in flat space, where the notation is nicer. +Everything follows for a curved manifold embedded in a higher-dimensional flat +space by projecting out the directions normal to the manifold, e.g., +$P\partial\partial\mathcal S P^T$ for the projection operator $P$ from the +previous section. + +First, consider +the problem as one of $2N$ real variables $x,y\in\mathbb R^N$ with $z=x+iy$. +The hessian of the real part of the action with respect to these real variables +is \begin{equation} \label{eq:real.hessian} \operatorname{Hess}_{x,y}\operatorname{Re}\beta\mathcal S =\left[\matrix{ @@ -501,97 +600,9 @@ of the real hessian are the singular values of the complex matrix $\beta\partial\partial\mathcal S$, and their eigenvectors coincide up to a constant complex factor. -\subsection{Gradient flow and the structure of thimbles} +\subsection{The conditions for Stokes points} -The `dynamics' describing thimbles is defined by gradient descent on the real -part of the action, with a given thimble incorporating all trajectories which -asymptotically flow to its associated stationary point. Since our phase space -is not necessary flat (as for the \emph{spherical} $p$-spin models), we will -have to do a bit of differential geometry to work out their form. Gradient -descent on a complex (Kähler) manifold is given by -\begin{equation} \label{eq:flow.coordinate.free} - \dot s - =-\operatorname{grad}\operatorname{Re}\beta\mathcal S - =-\left(\frac\partial{\partial s^*}\operatorname{Re}\beta\mathcal S\right)^\sharp - =-\frac{\beta^*}2\frac{\partial\mathcal S^*}{\partial s^*}g^{-1}\frac\partial{\partial s} -\end{equation} -where $g$ is the metric and the holomorphicity of the action was used to set -$\partial\mathcal S/\partial s^*=0$. If the complex phase space is $\mathbb C^N$ and the -metric is diagonal, this means that the flow is proportional to the conjugate -of the gradient, or $\dot s\propto-\beta^*(\partial S/\partial s)^*$. - -In the cases we will consider here (namely, that of the spherical models), it -will be more convenient to work in terms of coordinates in a flat embedding -space than in terms of local coordinates in the curved space, e.g., in terms of -$z\in\mathbb C^N$ instead of $s\in S^{N-1}$. Let $z:\tilde\Omega\to\mathbb C^N$ -be an embedding of complex phase space into complex euclidean space. The -dynamics in the embedding space is given by -\begin{equation}\label{eq:flow.raw} - \dot z - =-\frac{\beta^*}2\frac{\partial\mathcal S^*}{\partial z^*}(Dz)^* g^{-1}(Dz)^T\frac\partial{\partial z} -\end{equation} -where $Dz=\partial z/\partial s$ is the Jacobian of the embedding. -The embedding induces a metric on $\tilde\Omega$ by $g=(Dz)^\dagger Dz$. -Writing $\partial=\partial/\partial z$, this gives -\begin{equation} \label{eq:flow} - \dot z=-\frac{\beta^*}2(\partial\mathcal S)^\dagger(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T - =-\frac12(\partial \mathcal S)^\dagger P -\end{equation} -which is nothing but the projection of $(\partial\mathcal S)^*$ into the -tangent space of the manifold, with the projection operator -$P=(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T$. Note that $P$ is hermitian. For the spherical models, where $\tilde\Omega$ is the complex phase spaced defined by all points $z\in\mathbb C^N$ such that $z^Tz=1$, the projection operator is given by -\begin{equation} - P=I-\frac{zz^\dagger}{|z|^2} -\end{equation} -something that we be worked out in detail in a following section. One can -quickly verify that this operator indeed projects the dynamics onto the -manifold: its tangent at any point $z$ is given by $\partial(z^Tz)=z$, and -$Pz=z-z|z|^2/|z|^2=0$. For any vector $u$ perpendicular to $z$, i.e., -$z^\dagger u=0$, $Pu=u$. - -\begin{figure} - \includegraphics{figs/thimble_flow.pdf} - - \caption{Example of gradient descent flow on the action $\mathcal S$ featured - in Fig.~\ref{fig:example.action} in the complex-$\theta$ plane, with - $\arg\beta=0.4$. Symbols denote the stationary points, while thick blue and - red lines depict the thimbles and antithimbles, respectively. Streamlines - of the flow equations are plotted in a color set by their value of - $\operatorname{Im}\beta\mathcal S$; notice that the color is constant along - each streamline. - } \label{fig:flow.example} -\end{figure} - -Gradient descent on $\operatorname{Re}\beta\mathcal S$ is equivalent to -Hamiltonian dynamics with the Hamiltonian $\operatorname{Im}\beta\mathcal S$ -and conjugate coordinates given by the real and imaginary parts of each complex -coordinate. This is because $(\tilde\Omega, g)$ is Kähler and therefore admits -a symplectic structure, but that the flow conserves -$\operatorname{Im}\beta\mathcal S$ can be shown using \eref{eq:flow} and the -holomorphic property of $\mathcal S$: -\begin{eqnarray} - \frac d{dt}\operatorname{Im}\beta\mathcal S - &=\dot z\partial\operatorname{Im}\beta\mathcal S+\dot z^*\partial^*\operatorname{Im}\beta\mathcal S \\ - &=\frac i4\left( - (\beta\partial \mathcal S)^\dagger P\beta\partial\mathcal S-(\beta\partial\mathcal S)^TP^*(\beta\partial\mathcal S)^* - \right) \\ - &=\frac{i|\beta|^2}4\left( - (\partial\mathcal S)^\dagger P\partial\mathcal S-[(\partial\mathcal S)^\dagger P\partial\mathcal S]^* - \right) \\ - &=\frac{i|\beta|^2}4\left( - \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2 - \right)=0. -\end{eqnarray} -A consequence of this conservation is that the flow in the action takes a -simple form: -\begin{equation} - \dot{\mathcal S} - =\dot z\partial\mathcal S - =-\frac{\beta^*}2(\partial\mathcal S)^\dagger P\partial\mathcal S - =-\frac{\beta^*}2\|\partial\mathcal S\|^2. -\end{equation} -In the complex-$\mathcal S$ plane, dynamics is occurs along straight lines in -a direction set by the argument of $\beta$. +As we have seen in the previous sections, gradient descent dynamics results in flow that enters and leaves each thimble What does the topology of the space of thimbles look like? Let us consider the generic case, where the critical points of $\beta\mathcal S$ have distinct @@ -619,15 +630,7 @@ Breaking into real and imaginary parts gives \right) \end{eqnarray} Therefore, in the complex plane defined by each eigenvector of -$(\partial\partial\mathcal S)^\dagger P$ there is a separatrix flow of the form in -Figure \ref{fig:local_flow}. The effect of these separatrices in each complex -direction of the tangent space $T_{z_0}M$ is to separate that space into four -quadrants: two disconnected pieces with greater imaginary part than the -critical point, and two with lesser imaginary part. This partitioning implies -that the level set of $\operatorname{Im}\mathcal S=C$ for -$C\neq\operatorname{Im}\mathcal S(z_0)$ is split into two disconnected pieces, one -lying in each of two quadrants corresponding with its value relative to that at -the critical point. +$(\partial\partial\mathcal S)^\dagger P$ there is a separatrix flow. Continuing to `insert' critical points whose imaginary energy differs from $C$, one repeatedly partitions the space this way with each insertion. Therefore, @@ -666,15 +669,80 @@ energies tend to be nearby, Stokes lines will proliferate, while in regions where critical points with the same energies tend to be distant compared to those with different energies, Stokes lines will be rare. -\section{Analytic continuation} +\subsection{Evaluating thimble integrals} + +After all the work of decomposing an integral into a sub over thimbles, one +eventually wants to actually evaluate it. For large $|\beta|$ and in the +absence of any Stokes points, one can come to a nice asymptotic expression. +Suppose that $\sigma\in\Sigma$ is a critical point at $s_\sigma\in\tilde\Omega$ +with a thimble $\mathcal J_\sigma$ that is not involved in any upstream Stokes points. +Define its contribution to the partition function (neglecting the integer +weight) as +\begin{equation} + Z_\sigma(\beta)=\oint_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)} +\end{equation} +To evaluate this contour integral in the limit of large $|\beta|$, we will make +use of the saddle point method, since the integral will be dominated by its +value at and around the critical point, where the real part of the action is by +construction at its minimum on the thimble and the integrand is therefore +largest. + +We will make a change of coordinates $u(s)\in\mathbb R^N$ such that $\beta\mathcal +S(s)=\beta\mathcal S(s_\sigma)+\frac{|\beta|}2 u(s)^Tu(s)$ \emph{and} the direction of each $\partial u/\partial +s$ is along the direction of the contour. This is possible because the thimble... + + +Then +\begin{equation} +Z_\sigma(\beta)=e^{-\beta\mathcal S(s_\sigma)}\int du\,\det\frac{ds}{du}e^{-\frac{|\beta|}2u^Tu} +\end{equation} +\begin{equation} + Z_\sigma(\beta)\simeq e^{-\beta\mathcal S(s_\sigma)}\left.\det\frac{ds}{du}\right|_{s=s_\sigma}\int du\,e^{-\frac{|\beta|}2u^Tu} + =e^{-\beta\mathcal S(s_\sigma)}\left.\det\frac{ds}{du}\right|_{s=s_\sigma}\left(\frac{2\pi}{|\beta|}\right)^{D/2} +\end{equation} +We are left with evaluating the determinant of the coordinate transformation. The eigenvectors of the hessian corresponding to positive eigenvalues provide a basis for the thimble globally, and locally they provide the coordinates $u$ by +\begin{equation} + s(u)=s_\sigma+\sum_{i=1}^{D/2}\sqrt{\frac{|\beta|}{\lambda^{(i)}}}v^{(i)}u_i+O(u^2) +\end{equation} \begin{eqnarray} - Z(\beta) - &=\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)} \\ - &=\sum_{\sigma\in\Sigma_0}(-1)^{k_\sigma}\int_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)} \\ - &=\sum_{\sigma\in\Sigma_0}(-1)^{k_\sigma}\left(\frac{2\pi}{\beta}\right)^{N/2}(\det\partial\partial\mathcal S(s_\sigma))^{-N/2}e^{-\beta\mathcal S_(s_\sigma)} \\ - &=\sum_k(-1)^k\int d\epsilon\,\mathcal N_k(\epsilon)\left(\frac{2\pi}{\beta}\right)^{N/2}\exp\left\{-\beta N\epsilon-\frac N2\int_0^\infty d\lambda\,\rho(\lambda\mid\epsilon)\log\lambda\right\} + \beta\mathcal S(s(u)) + &=\beta\mathcal S(s_\sigma) + +\frac12(s(u)-s_\sigma)^T(\beta\partial\partial\mathcal S)(s(u)-s_\sigma)+O((s(u)-s_\sigma)^3) \\ + &=\beta\mathcal S(s_\sigma) + +\frac{|\beta|}2\sum_i\sum_j\frac{v^{(i)}_k}{\sqrt{\lambda^{(i)}}}(\beta\partial_k\partial_\ell\mathcal S)\frac{v^{(j)}_\ell}{\sqrt{\lambda^{(j)}}}u_iu_j+\cdots \\ + &=\beta\mathcal S(s_\sigma) + +\frac{|\beta|}2\sum_i\sum_j\frac{v^{(i)}_k}{\sqrt{\lambda^{(i)}}}\frac{\lambda^{(j)}(v^{(j)}_k)^*}{\sqrt{\lambda^{(j)}}}u_iu_j+\cdots \\ + &=\beta\mathcal S(s_\sigma) + +\frac{|\beta|}2\sum_i\sum_j\frac{\sqrt{\lambda^{(j)}}}{\sqrt{\lambda^{(i)}}}\delta_{ij}u_iu_j+\cdots \\ + &=\beta\mathcal S(s_\sigma) + +\frac{|\beta|}2\sum_iu_i^2+\cdots \\ \end{eqnarray} +\begin{equation} + \frac{\partial s_i}{\partial u_j}=\lambda_0^{(j)}v^{(j)}_i +\end{equation} +This is the product of a diagonal matrix of positive eigenvalues with the unitary matrix of their associated eigenvectors. Therefore, its determinant is +\begin{equation} + \det\frac{\partial s}{\partial u}=\det_{ij}v_i^{(j)}\prod_{\lambda>0}\lambda_0 +\end{equation} +In circumstances you may be used to, only the absolute value of the determinant +from the coordinate transformation is relevant, and since the determinant of a +unitary matrix is always magnitude one, it doesn't enter the computation. +However, because we are dealing with a path integral, the directions matter, +and there is not an absolute value around the determinant. Therefore, we must +determine the phase that it contributes. + +This is difficult in general, but for real critical points it can be reasoned +out easily. Take the same convention we used earlier, that the direction of +contours along the real line is in the conventional directions. Then, a +critical 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 critical points of a real action, +\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)} +\end{equation} \section{The \textit{p}-spin spherical models} |