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@@ -360,6 +360,96 @@ 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{The structure of stationary points} + +\begin{eqnarray} + \operatorname{Hess}\operatorname{Re}\beta\mathcal S + &=\left[\matrix{ + \partial_x\partial_x\operatorname{Re}\beta\mathcal S & + \partial_y\partial_x\operatorname{Re}\beta\mathcal S \cr + \partial_x\partial_y\operatorname{Re}\beta\mathcal S & + \partial_y\partial_y\operatorname{Re}\beta\mathcal S + }\right] \\ + &=\left[\matrix{ + \hphantom{-}\operatorname{Re}\beta\partial\partial\mathcal S & + -\operatorname{Im}\beta\partial\partial\mathcal S \cr + -\operatorname{Im}\beta\partial\partial\mathcal S & + -\operatorname{Re}\beta\partial\partial\mathcal S + }\right] +\end{eqnarray} + +The eigenvalues and eigenvectors of the Hessian are important for evaluating +thimble integrals, because those associated with upward directions provide a +local basis for the surface of the thimble. Suppose that $v_x,v_y\in\mathbb +R^N$ are such that +\begin{equation} + (\operatorname{Hess}\operatorname{Re}\beta\mathcal S)\left[\matrix{v_x \cr v_y}\right] + =\lambda\left[\matrix{v_x \cr v_y}\right] +\end{equation} +where the eigenvalue $\lambda$ must be real because the hessian is real symmetric. The problem can be put into a more obviously complex form by a change of basis. Writing $v=v_x+iv_y$, we find +\begin{eqnarray} + &\left[\matrix{0&-i(\beta\partial\partial\mathcal S)^*\cr i\beta\partial\partial\mathcal S&0}\right] + \left[\matrix{v \cr iv^*}\right]\\ + &\qquad=\left[\matrix{1&i\cr i&1}\right] + (\operatorname{Hess}\operatorname{Re}\beta\mathcal S) + \left[\matrix{1&i\cr i&1}\right]^{-1} + \left[\matrix{1&i\cr i&1}\right] + \left[\matrix{v_x \cr v_y}\right] \\ + &\qquad=\lambda\left[\matrix{1&i\cr i&1}\right]\left[\matrix{v_x \cr v_y}\right] + =\lambda\left[\matrix{v \cr iv^*}\right] +\end{eqnarray} +It therefore follows that the eigenvalues and vectors of the real hessian satisfy the equation +\begin{equation} \label{eq:generalized.eigenproblem} + \beta\partial\partial\mathcal S v=\lambda v^* +\end{equation} +a sort of generalized +eigenvalue problem. If we did not know the eigenvalues were real, we could +still see it from the second implied equation, $(\beta\partial\partial\mathcal +S)^*v^*=\lambda v$, which is the conjugate of the first if $\lambda^*=\lambda$. + +Something somewhat hidden in the structure of the real hessian but more clear +in its complex form is that each eigenvalue comes in a pair, since +\begin{equation} + \beta\partial\partial\mathcal S(iv)=i\lambda v^*=-\lambda(iv) +\end{equation} +Therefore, if $\lambda$ is an eigenvalue of the hessian with eigenvector $v$, +than so is $-\lambda$, with associated eigenvector $iv$, rotated in the complex +plane. It follows that each stationary point has an equal number of descending +and ascending directions, e.g. the index of each stationary point is $N$. For a +stationary point in a real problem this might seem strange, because there are +clear differences between minima, maxima, and saddles of different index. +However, we can quickly see here that for a such a stationary point, its $N$ +real eigenvectors which determine its index in the real problem are accompanied +by $N$ purely imaginary eigenvectors, pointing into the complex plane and each +with the negative eigenvalue of its partner. + +These eigenvalues and vectors can be further related to properties of the +complex symmetric matrix $\beta\partial\partial\mathcal S$. Suppose that +$u\in\mathbb R^N$ satisfies the eigenvalue equation +\begin{equation} + (\beta\partial\partial S)^\dagger(\beta\partial\partial S)u + =\sigma u +\end{equation} +for some positive real $\sigma$ (real because $(\beta\partial\partial +S)^\dagger(\beta\partial\partial S)$ is self-adjoint). The square root of these +numbers, $\sqrt{\sigma}$, are the definition of the \emph{singular values} of +$\beta\partial\partial\mathcal S$. A direct relationship between these singular +values and the eigenvalues of the hessian immediately follows by taking an +eigenvector $v\in\mathbb C$ that satisfies \eref{eq:generalized.eigenproblem}, +and writing +\begin{eqnarray} + \sigma v^\dagger u + &=v^\dagger(\beta\partial\partial S)^\dagger(\beta\partial\partial S)u + =(\beta\partial\partial Sv)^\dagger(\beta\partial\partial S)u\\ + &=(\lambda v^*)^\dagger(\beta\partial\partial S)u + =\lambda v^T(\beta\partial\partial S)u + =\lambda^2 v^\dagger u +\end{eqnarray} +Thus if $v^\dagger u\neq0$, $\lambda^2=\sigma$. It follows that the eigenvalues +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} The `dynamics' describing thimbles is defined by gradient descent on the real |