diff options
| author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2024-09-04 01:23:01 +0200 |
|---|---|---|
| committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2024-09-04 01:23:01 +0200 |
| commit | 032584fe152ca1e91d0d8152e063b447211ff636 (patch) | |
| tree | ed5344ceab81d4e36bf5b991ed12c93ec3855d80 | |
| parent | 8cff4a74e2792b6037364aa63bd7d78085724ca5 (diff) | |
| download | SciPostPhys_18_158-032584fe152ca1e91d0d8152e063b447211ff636.tar.gz SciPostPhys_18_158-032584fe152ca1e91d0d8152e063b447211ff636.tar.bz2 SciPostPhys_18_158-032584fe152ca1e91d0d8152e063b447211ff636.zip | |
Lots of writing and work.
| -rw-r--r-- | figs/regime_1.pdf | bin | 13821 -> 13370 bytes | |||
| -rw-r--r-- | figs/regime_2.pdf | bin | 10467 -> 10011 bytes | |||
| -rw-r--r-- | figs/regime_3.pdf | bin | 11196 -> 10867 bytes | |||
| -rw-r--r-- | figs/regime_4.pdf | bin | 0 -> 10035 bytes | |||
| -rw-r--r-- | figures.nb | 1106 | ||||
| -rw-r--r-- | topology.tex | 357 |
6 files changed, 1174 insertions, 289 deletions
diff --git a/figs/regime_1.pdf b/figs/regime_1.pdf Binary files differindex 04a8da5..a7e6830 100644 --- a/figs/regime_1.pdf +++ b/figs/regime_1.pdf diff --git a/figs/regime_2.pdf b/figs/regime_2.pdf Binary files differindex f3473cc..ce14910 100644 --- a/figs/regime_2.pdf +++ b/figs/regime_2.pdf diff --git a/figs/regime_3.pdf b/figs/regime_3.pdf Binary files differindex 255b960..6d312a3 100644 --- a/figs/regime_3.pdf +++ b/figs/regime_3.pdf diff --git a/figs/regime_4.pdf b/figs/regime_4.pdf Binary files differnew file mode 100644 index 0000000..3f18260 --- /dev/null +++ b/figs/regime_4.pdf @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 696527, 13283] -NotebookOptionsPosition[ 692566, 13211] -NotebookOutlinePosition[ 692965, 13227] -CellTagsIndexPosition[ 692922, 13224] +NotebookDataLength[ 729520, 13943] +NotebookOptionsPosition[ 725323, 13867] +NotebookOutlinePosition[ 725718, 13883] +CellTagsIndexPosition[ 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This is -equivalent to asking that the constraints $V_k$ all have a stationary point at -the same place. When the $V_k$ are independent and random, this is vanishingly -unlikely, requiring $NM+1$ independent equations to be simultaneously satisfied. -This means that different connected components of the set of solutions do not + $0=\frac\partial{\partial\mathbf x}V_k(\mathbf x)$ for all $k$. This is + equivalent to asking that the constraints $V_k$ all have a stationary point at + the same place. When the $V_k$ are independent and random, this is vanishingly + unlikely, requiring $NM+1$ independent equations to be simultaneously satisfied. + This means that different connected components of the set of solutions do not intersect, nor are there self-intersections, without extraordinary fine-tuning.} We study the topology of the manifold $\Omega$ by two related means: its average Euler characteristic, and the average number of stationary points of a @@ -190,9 +190,7 @@ direct sum of the Betti numbers of $\Omega$. We find that for the varied cases we study, these two always coincide at the largest exponential order in $N$, putting strong constraints on the resulting topology and geometry. -\section{Methods} - -\subsection{The average Euler characteristic} +\section{The average Euler characteristic} The Euler characteristic $\chi$ of a manifold is a topological invariant \cite{Hatcher_2002_Algebraic}. It is perhaps most familiar in the context of connected compact orientable surfaces, where it @@ -204,14 +202,14 @@ by an alternating sum over the number of cells of increasing dimension, which for 2-manifolds corresponds to the number of vertices, minus the number of edges, plus the number of faces. -Morse theory offers another way to compute the Euler characteristic using the -statistics of stationary points of a function $H:\Omega\to\mathbb R$ \cite{Audin_2014_Morse}. For +Morse theory offers another way to compute the Euler characteristic of a manifold $\Omega$ using the +statistics of stationary points in a function $H:\Omega\to\mathbb R$ \cite{Audin_2014_Morse}. For functions $H$ without any symmetries with respect to the manifold, the surfaces of gradient flow between adjacent stationary points form a complex. The alternating sum over cells to compute $\chi$ becomes an alternating sum over the count of stationary points of $H$ with increasing index, or \begin{equation} - \chi=\sum_{i=0}^N(-1)^i\mathcal N_H(\text{index}=i) + \chi(\Omega)=\sum_{i=0}^N(-1)^i\mathcal N_H(\text{index}=i) \end{equation} Conveniently, we can express this abstract sum as an integral over the manifold using a small variation on the Kac--Rice formula for counting stationary @@ -227,7 +225,7 @@ of the determinant is a nuisance that one must take pains to preserve We need to choose a function $H$ for our calculation. Because $\chi$ is a topological invariant, any choice will work so long as it does not share some -symmetry with the underlying manifold, i.e., that it $H$ satisfies the Smale condition. Because our manifold of random +symmetry with the underlying manifold, i.e., that $H$ satisfies the Smale condition. Because our manifold of random constraints has no symmetries, we can take a simple height function $H(\mathbf x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with $\|\mathbf x_0\|^2=N$. $H$ is a height function because when $\mathbf x_0$ is @@ -255,10 +253,12 @@ can be applied to calculate it. To evaluate the average of $\chi$ over the constraints, we first translate the $\delta$ functions and determinant to integral form, with \begin{align} + \label{eq:delta.exp} \delta\big(\partial L(\mathbf x,\pmb\omega)\big) &=\int\frac{d\hat{\mathbf x}}{(2\pi)^N}\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}} e^{i[\hat{\mathbf x},\hat{\pmb\omega}]\cdot\partial L(\mathbf x,\pmb\omega)} \\ + \label{eq:det.exp} \det\partial\partial L(\mathbf x,\pmb\omega) &=\int d\bar{\pmb\eta}\,d\pmb\eta\,d\bar{\pmb\gamma}\,d\pmb\gamma\, e^{-[\bar{\pmb\eta},\bar{\pmb\gamma}]^T\partial\partial L(\mathbf x,\pmb\omega)[\pmb\eta,\pmb\gamma]} @@ -272,11 +272,11 @@ therefore be averaged over, and the resulting expression treated with standard methods. Details of this calculation can be found in Appendix~\ref{sec:euler}. The result is the reduction of the average Euler characteristic to an expression of the form -\begin{equation} +\begin{equation} \label{eq:pre-saddle.characteristic} \overline{\chi(\Omega)} - =\int dR\,dD\,dm\,d\hat m\,g(R,D,m,\hat m)\,e^{N\mathcal S_\Omega(R,D,m,\hat m)} + =\left(\frac N{2\pi}\right)^2\int dR\,dD\,dm\,d\hat m\,g(R,D,m,\hat m)\,e^{N\mathcal S_\Omega(R,D,m,\hat m)} \end{equation} -where $g$ is a prefactor subexponential in $N$, and $\mathcal S_\Omega$ is an effective action defined by +where $g$ is a prefactor of $o(N^0)$, and $\mathcal S_\Omega$ is an effective action defined by \begin{equation} \begin{aligned} \mathcal S_\Omega(R,D,m,\hat m\mid\alpha,V_0) @@ -300,7 +300,7 @@ The remaining order parameters defined by the scalar products \hat m=-i\frac1N\hat {\mathbf x}\cdot\mathbf x_0 \end{align} -This integral can be evaluated by a saddle point method. For reasons we will +This integral can be evaluated to leading order by a saddle point method. For reasons we will see, it is best to extremize with respect to $R$, $D$, and $\hat m$, leaving a new effective action of $m$ alone. This can be solved to give \begin{equation} @@ -313,11 +313,11 @@ new effective action of $m$ alone. This can be solved to give \Bigg[ \alpha V_0^2f'(1) +(2-\alpha)f(1)\left(\frac{f(1)}{1-m^2}-f'(1)\right) \\ - &\quad+\operatorname{sgn}(m)\alpha\sqrt{ + &\quad+\alpha\sqrt{ \tfrac{4V_0^2}\alpha f(1)f'(1)\left[\tfrac{f(1)}{1-m^2}-f'(1)\right] +\left[\tfrac{f(1)^2}{1-m^2}-\big(V_0^2+f(1)\big)f'(1)\right]^2 } - \Bigg] + \Bigg] \end{aligned} \end{equation} \begin{equation} @@ -339,10 +339,10 @@ $\mathbf x_0$. Therefore, the value $m^*$ that maximizes this action can be understood as the latitude on the sphere where most of the contribution to the Euler characteristic is made. -The action $\mathcal S_\Omega$ is extremized with respect to $m$ at $m^*=0$ or $m^*=-R^*$. -In the latter case, $m*$ takes the value +The action $\mathcal S_\Omega$ is extremized with respect to $m$ at $m=0$ or $m=\pm m^*=\mp R^*(m^*)$. +In the latter case, $m^*$ takes the value \begin{equation} - m^*=\pm\sqrt{1-\frac{\alpha}{f'(1)}\big(V_0^2+f(1)\big)} + m^*=\sqrt{1-\frac{\alpha}{f'(1)}\big(V_0^2+f(1)\big)} \end{equation} and $\mathcal S_\Omega(m^*)=0$. If this solution was always well-defined, it would vanish when the argument of the square root vanishes for \begin{equation} @@ -403,7 +403,7 @@ relying on the saddle point of an exponential integral would be invalid. In the regime of $\mathcal S_\Omega(m)$ where the action is complex-valued, is this breakdown the result of a negative average characteristic, or is it the result of another reason? It is difficult to answer this with the previous calculation -alone. However, in the next subsection we will make a complementary calculation +alone. However, in the next section we will make a complementary calculation that rules out the negative Euler characteristic picture. Instead, we will see that in the complex region, the large-deviation function in $m$ that $\mathcal S_\Omega(m)$ represents breaks down due to the vanishingly small probability of @@ -411,7 +411,7 @@ finding any stationary points. -\subsection{Complexity of a linear test function} +\section{Complexity of a linear test function} One way to definitely narrow possible interpretations of the average Euler characteristic is to compute a complementary average. The Euler characteristic @@ -445,7 +445,7 @@ function around the determinant. Following \cite{Fyodorov_2004_Complexity}, we make use of the identity \begin{equation} \begin{aligned} - |\det A| + |\det A| &=\lim_{\epsilon\to0}\frac{(\det A)^2}{\sqrt{\det(A+i\epsilon I)}\sqrt{\det(A-i\epsilon I)}} \\ &=\frac1{(2\pi)^N}\int d\bar{\pmb\eta}_1\,d\pmb\eta_1\,d\bar{\pmb\eta}_2\,d\pmb\eta_2\,d\mathbf a\,d\mathbf b\, @@ -453,7 +453,7 @@ make use of the identity -\bar{\pmb\eta}_1^TA\pmb\eta_1-\bar{\pmb\eta}_2^TA\pmb\eta_2 -\frac12\mathbf a^T(A+i\epsilon I)\mathbf a-\frac12\mathbf b^T(A-i\epsilon I)\mathbf b } - \end{aligned} +\end{aligned} \end{equation} for an $N\times N$ matrix $A$. Here $\bar{\pmb\eta}_1$, $\pmb\eta_1$, $\bar{\pmb\eta}_2$, and $\pmb\eta_2$ are Grassmann vectors and $\mathbf a$ and @@ -465,7 +465,7 @@ Appendix~\ref{sec:complexity.details}. The result is that, to largest order in $N$, the logarithm of the average number of stationary points is the same as the logarithm of the average Euler characteristic. -\subsection{How to interpret these calculations} +\section{Implications for the topology of solutions} It is not straightforward to directly use the average Euler characteristic to infer something about the number of connected components in the set of @@ -514,13 +514,15 @@ have characteristics of either 0 or 4. \begin{figure} \includegraphics{figs/regime_1.pdf} - \hspace{-3em} + \hspace{-3.5em} \includegraphics{figs/regime_2.pdf} - \hspace{-3em} + \hspace{-3.5em} \includegraphics{figs/regime_3.pdf} + \hspace{-3.5em} + \includegraphics{figs/regime_4.pdf} \caption{ - \textbf{Behavior of the action in our three nontrivial regimes.} The effective action $\mathcal S_\Omega$ as a function of overlap $m$ with + \textbf{Behavior of the action in four regimes.} The effective action $\mathcal S_\Omega$ as a function of overlap $m$ with the height axis for our model with $f(q)=\frac12q^3$ and $\alpha=\frac12$ at three different target values $V_0$. \textbf{Left: the connected regime.} The action is maximized with $\mathcal S_\Omega(m^*)=0$, and no @@ -533,7 +535,7 @@ have characteristics of either 0 or 4. } \end{figure} -\paragraph{The connected regime: \boldmath{$V_0^2<V_\text{on}$}.} +\paragraph{The connected regime: \boldmath{$V_0^2<V_\text{on}^2$}.} In our calculation above, $\overline{\chi(\Omega)}=2$ could mean a fine-tuned average like this, or it could indicate the presence of manifold homeomorphic @@ -543,7 +545,7 @@ latitude $m^*$. Randomly chosen points on the sphere have a typical nearest overlap $m^*$ with the solution manifold, but can never have a smaller overlap than $m_\text{min}$, indicating that the manifold is extensive. -\paragraph{The onset regime: \boldmath{$V_\text{on}^2<V_0^2<V_\text{sh}$}.} +\paragraph{The onset regime: \boldmath{$V_\text{on}^2<V_0^2<V_\text{sh}^2$}.} In this regime $\log\overline{\chi(\Omega)}=O(N)$ but a minimum overlap $m_\text{min}>0$ still exists. The minimum overlap indicates that the solution @@ -568,12 +570,10 @@ components to the solution manifold. While most minima and maxima of the height are located at the equator $m=0$, they are found in exponential number up to the latitude $m_\text{max}$. -\section{Results} - -\subsection{Topology of solutions to many equations and the satisfiability transition} +\paragraph{The \textsc{unsat} regime: \boldmath{$V_\text{\textsc{sat}}^2<V_0^2$}.} -The results of the previous sections indicate the following picture for the -topology of the solution manifold. +In this regime $\log\overline{\chi(\Omega)}<0$, indicating that in most +realizations of the functions $V_k$ the set $\Omega$ is empty. \begin{figure} \includegraphics[width=0.245\textwidth]{figs/connected.pdf} @@ -618,7 +618,7 @@ topology of the solution manifold. } \end{figure} -\subsection{Topology of level sets of the spherical spin glasses and the dynamic threshold} +\section{Implications for dynamic thresholds in the spherical spin glasses} \cite{Folena_2020_Rethinking, Folena_2021_Gradient} @@ -681,12 +681,22 @@ JK-D is supported by a \textsc{DynSysMath} Specific Initiative of the INFN. \section{Details of the calculation of the average Euler characteristic} \label{sec:euler} -To make the calculation compact, we introduce superspace coordinates. Define the supervectors -\begin{equation} +Our starting point is the expression \eqref{eq:kac-rice.lagrange} with the +substitutions of the $\delta$-function and determinant \eqref{eq:delta.exp} and +\eqref{eq:det.exp} made. To make the calculation compact, we introduce +superspace coordinates. Introducing the Grassmann indices $\bar\theta_1$ +and $\theta_1$, we define the supervectors +\begin{align} \pmb\phi(1)=\mathbf x+\bar\theta_1\pmb\eta+\bar{\pmb\eta}\theta_1+\bar\theta_1\theta_1i\hat{\mathbf x} - \qquad + && \sigma_k(1)=\omega_k+\bar\theta_1\gamma_k+\bar\gamma_k\theta_1+\bar\theta_1\theta_1i\hat\omega_k -\end{equation} +\end{align} +with associated measures +\begin{align} + d\pmb\phi=d\mathbf x\,\frac{d\hat{\mathbf x}}{(2\pi)^N}\,d\bar{\pmb\eta}\,d\pmb\eta + && + d\pmb\sigma=d\pmb\omega\,\frac{d\hat{\pmb\omega}}{(2\pi)^{M+1}}\,d\bar{\pmb\gamma}\,d\pmb\gamma +\end{align} The Euler characteristic can be expressed using these supervectors as \begin{align} &\chi(\Omega) @@ -699,7 +709,9 @@ The Euler characteristic can be expressed using these supervectors as \right] \right\} \notag \end{align} -Since this is an exponential integrand linear in the functions $V_k$, we can average over the functions to find +where $d1=d\bar\theta_1\,d\theta_1$ is the integral over the Grassmann +indices. Since this is an exponential integrand linear in the Gaussian +functions $V_k$, we can take their average to find \begin{equation} \begin{aligned} \overline{\chi(\Omega)} @@ -709,11 +721,11 @@ Since this is an exponential integrand linear in the functions $V_k$, we can ave +\frac12\sigma_0(1)\big(\|\pmb\phi(1)\|^2-N\big) -V_0\sum_{k=1}^M\sigma_k(1) \right] \\ - -\frac12\int d1\,d2\,\sum_{k=1}^M\sigma_k(1)\sigma_k(2)f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right) + +\frac12\int d1\,d2\,\sum_{k=1}^M\sigma_k(1)\sigma_k(2)f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right) \Bigg\} \end{aligned} \end{equation} -This is a super-Gaussian integral in the Lagrange multipliers with $1\leq k\leq M$. +This is a super-Gaussian integral in the super-Lagrange multipliers $\sigma_k$ with $1\leq k\leq M$. Performing that integral yields \begin{equation} \begin{aligned} @@ -724,13 +736,15 @@ Performing that integral yields +\frac12\sigma_0(1)\big(\|\pmb\phi(1)\|^2-N\big) \right] \\ &\hspace{5em}-\frac M2V_0^2\int d1\,d2\,f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right)^{-1} - -\frac M2\log\operatorname{sdet}f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right) - \Bigg\} + -\frac M2\log\operatorname{sdet}f\left(\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N\right) + \Bigg\} \end{aligned} \end{equation} The supervector $\pmb\phi$ enters this expression as a function only of the scalar product with itself and with the vector $\mathbf x_0$ inside the -function $H$. We therefore change variables to the superoperator $\mathbb Q$ and the supervector $\mathbb M$ defined by +function $H(\mathbf x)=\mathbf x_0\cdot\mathbf x$. We therefore make a change +of variables to the superoperator $\mathbb Q$ and the supervector $\mathbb M$ +defined by \begin{equation} \mathbb Q(1,2)=\frac{\pmb\phi(1)\cdot\pmb\phi(2)}N \qquad @@ -738,21 +752,22 @@ function $H$. We therefore change variables to the superoperator $\mathbb Q$ and \end{equation} These new variables can replace $\pmb\phi$ in the integral using a generalized Hubbard--Stratonovich transformation, which yields \begin{align} - \overline{\chi(\Omega)} + \overline{\chi(\Omega)} &=\int d\mathbb Q\,d\mathbb M\,d\sigma_0\, - \left[g(\mathbb Q,\mathbb M)+O(N^{-1})\right] + \left[(\operatorname{sdet}\mathbb Q)^\frac12+O(N^{-1})\right] \,\exp\Bigg\{ N\int d1\left[ \mathbb M(1) +\frac12\sigma_0(1)\big(\mathbb Q(1,1)-1\big) \right] \notag \\ &\hspace{3em}-\frac M2V_0^2\int d1\,d2\,f(\mathbb Q)^{-1}(1,2) - -\frac M2\log\operatorname{sdet}f(\mathbb Q) - +\frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T) - \Bigg\} + -\frac M2\log\operatorname{sdet}f(\mathbb Q) + +\frac N2\log\operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T) + \Bigg\} + \label{eq:post.hubbard-strat} \end{align} -where $g$ is a function of $\mathbb Q$ and $\mathbb M$ independent of $N$ and -$M$, detailed in Appendix~\ref{sec:prefactor}. To move on from this expression, +where we show the asymptotic value of the prefactor in Appendix~\ref{sec:prefactor}. +To move on from this expression, we need to expand the superspace notation. We can write \begin{equation} \begin{aligned} @@ -763,20 +778,29 @@ we need to expand the superspace notation. We can write &\qquad +(\bar\theta_1+\bar\theta_2)H +\bar H(\theta_1+\theta_2) - -(\bar\theta_1\theta_1\bar\theta_2+\bar\theta_2\theta_2\bar\theta_1)\hat H + -(\bar\theta_1\theta_1\bar\theta_2+\bar\theta_2\theta_2\bar\theta_1)i\hat H -\bar{\hat H}(\theta_1\bar\theta_2\theta_2+\theta_1\bar\theta_1\theta_1) \end{aligned} \end{equation} and \begin{equation} \mathbb M(1) - =m+\bar\theta_1H_0+\bar H_0\theta_1-\hat m\bar\theta_1\theta_1 + =m+\bar\theta_1H_0+\bar H_0\theta_1+i\hat m\bar\theta_1\theta_1 + \label{eq:ops} \end{equation} +with associated measures +\begin{align} + d\mathbb Q + =dC\,dR\,dG\,\frac{dD}{(2\pi)^2}\,d\bar H\,dH\,d\bar{\hat H}\,d\hat H + && + d\mathbb M=dm\,\frac{d\hat m}{2\pi}\,d\bar H_0\,dH_0 + \label{eq:op.measures} +\end{align} The order parameters $C$, $R$, $G$, $D$, $m$, and $\hat m$ are ordinary numbers defined by \begin{align} C=\frac{\mathbf x\cdot\mathbf x}N && - R=-i\frac{\mathbf x\cdot\hat{\mathbf x}}N + R=i\frac{\mathbf x\cdot\hat{\mathbf x}}N && G=\frac{\bar{\pmb\eta}\cdot\pmb\eta}N && @@ -784,7 +808,7 @@ The order parameters $C$, $R$, $G$, $D$, $m$, and $\hat m$ are ordinary numbers && m=\frac{\mathbf x_0\cdot\mathbf x}N && - \hat m=-i\frac{\mathbf x_0\cdot\hat{\mathbf x}}N + \hat m=i\frac{\mathbf x_0\cdot\hat{\mathbf x}}N \end{align} while $\bar H$, $H$, $\bar{\hat H}$, $\hat H$, $\bar H_0$ and $H_0$ are Grassmann numbers defined by \begin{align} @@ -792,22 +816,22 @@ while $\bar H$, $H$, $\bar{\hat H}$, $\hat H$, $\bar H_0$ and $H_0$ are Grassman && H=\frac{\pmb\eta\cdot\mathbf x}N && - \bar{\hat H}=\frac{\bar{\pmb\eta}\cdot\hat{\mathbf x}}N + \bar{\hat H}=i\frac{\bar{\pmb\eta}\cdot\hat{\mathbf x}}N && - \hat H=\frac{\pmb\eta\cdot\hat{\mathbf x}}N + \hat H=i\frac{\pmb\eta\cdot\hat{\mathbf x}}N && \bar H_0=\frac{\bar{\pmb\eta}\cdot\mathbf x_0}N && H_0=\frac{\pmb\eta\cdot\mathbf x_0}N \end{align} We can treat the integral over $\sigma_0$ immediately. It gives -\begin{equation} +\begin{equation} \label{eq:sigma0.integral} \int d\sigma_0\,e^{N\int d1\,\frac12\sigma_0(1)(\mathbb Q(1,1)-1)} - =2\pi i\,\delta(C-1)\,\delta(G+R)\,\bar HH + =2\times2\pi\delta(C-1)\,\delta(G+R)\,\bar HH \end{equation} This therefore sets $C=1$ and $G=-R$ in the remainder of the integrand, as well -as setting everything depending on $\bar H$ and $H$ to zero. With these solutions inserted, the remaining remaining terms in the exponential give -\begin{equation} +as removing all dependence on $\bar H$ and $H$. With these solutions inserted, the remaining terms in the exponential give +\begin{equation} \label{eq:sdet.q} \begin{aligned} \operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T) &=1+\frac{(1-m^2)D+\hat m^2-2Rm\hat m}{R^2} @@ -820,12 +844,12 @@ as setting everything depending on $\bar H$ and $H$ to zero. With these solution \right] \end{aligned} \end{equation} -\begin{equation} +\begin{equation} \label{eq:sdet.fq} \operatorname{sdet}f(\mathbb Q) =1+\frac{Df(1)}{R^2f'(1)} +\frac{2f(1)}{R^3f'(1)}\bar{\hat H}\hat H \end{equation} -\begin{equation} +\begin{equation} \label{eq:inv.q} \int d1\,d2\,f(\mathbb Q)^{-1}(1,2) =\left(f(1)+\frac{R^2f'(1)}{D}\right)^{-1} +2\frac{Rf'(1)}{(Df(1)+R^2f'(1))^2}\bar{\hat H}\hat H @@ -834,6 +858,207 @@ as setting everything depending on $\bar H$ and $H$ to zero. With these solution \section{Calculation of the prefactor of the average Euler characteristic} \label{sec:prefactor} + +Because of our convention of including the appropriate factors of $2\pi$ in the +superspace measure, super-Gaussian integrals do not produce such factors in our +derivation. Prefactors to our calculation come from three sources: the +introduction of $\delta$-functions to define the order parameters, integrals +over Grassmann order parameters, and from the saddle point approximation to the +large-$N$ integral. + +\subsection{Contribution from the Hubbard--Stratonovich transformations} +\label{sec:prefactor.hs} + +First, we examine the factors arising from the definition of order parameters. This begins by introducing to the integral the factor of one +\begin{equation} + 1=(2\pi)^3\int d\mathbb Q\,d\mathbb M\, + \delta\big(N\mathbb Q(1,2)-\pmb\phi(1)\cdot\pmb\phi(2)\big)\, + \delta\big(N\mathbb M(1)-\mathbf x_0\cdot\pmb\phi(1)\big) +\end{equation} +where three factors of $2\pi$ come from the measures as defined in +\eqref{eq:op.measures}. Converting the $\delta$-function into an exponential +integral yields +\begin{equation} + \begin{aligned} + 1=\int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\,d\tilde{\mathbb M}\, + \exp\Bigg\{ + \frac12\int d1\,d2\,\tilde{\mathbb Q}(1,2)\big(N\mathbb Q(1,2)-\pmb\phi(1)\cdot\pmb\phi(2)\big) \qquad \\ + +\int d1\,i\tilde{\mathbb M}(1)\big(N\mathbb M(1)-\mathbf x_0\cdot\pmb\phi(1)\big) + \Bigg\} + \end{aligned} +\end{equation} +where the supervectors and measures for $\tilde{\mathbb Q}$ and $\tilde{\mathbb M}$ are defined +analogously to those of $\mathbb Q$ and $\mathbb M$. This is now a super-Gaussian integral in $\pmb\phi$, which can be performed to yield +\begin{equation} + \begin{aligned} + \int d\pmb\phi\,1=\int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\,d\tilde{\mathbb M}\, + \exp\Bigg\{ + \frac N2\int d1\,d2\,\tilde{\mathbb Q}(1,2)\mathbb Q(1,2) + +N\int d1\,i\tilde{\mathbb M}(1)\mathbb M(1) \qquad \\ + -\frac N2\log\operatorname{sdet}\tilde{\mathbb Q} + -\frac N2\int d1\,d2\,\tilde{\mathbb M}(1)\tilde{\mathbb Q}^{-1}(1,2)\tilde{\mathbb M}(2) + \Bigg\} + \end{aligned} +\end{equation} +We can perform the remaining Gaussian integral in $\tilde{\mathbb M}$ to find +\begin{equation} + \begin{aligned} + \int d\pmb\phi\,1= + \int d\mathbb Q\,d\mathbb M\,d\tilde{\mathbb Q}\, + (\operatorname{sdet}\tilde{\mathbb Q}^{-1})^{-\frac12} + \exp\Bigg\{ + -\frac N2\log\operatorname{sdet}\tilde{\mathbb Q} + \hspace{5em} \\ + \frac N2\int d1\,d2\,\tilde{\mathbb Q}(1,2)\big[ + \mathbb Q(1,2)-\mathbb M(1)\mathbb M(2) + \big] + \Bigg\} + \end{aligned} +\end{equation} +The integral over $\tilde{\mathbb Q}$ can be evaluated to leading order using the saddle point +method. The integrand is stationary at $\tilde{\mathbb Q}=(\mathbb Q-\mathbb +M\mathbb M^T)^{-1}$, whose substitution results in the term +$\frac12\log\det(\mathbb Q-\mathbb M\mathbb M^T)$ in the effective action from +\eqref{eq:post.hubbard-strat}. The saddle point also yields a prefactor of the form +\begin{equation} + \left(\operatorname{sdet}_{\{1,2\},\{3,4\}}\frac{\partial^2\frac12\log\operatorname{sdet}\tilde{\mathbb Q}}{\partial\tilde{\mathbb Q}(1,2)\partial\tilde{\mathbb Q}(3,4)}\right)^{-\frac12} + =\left(\operatorname{sdet}_{\{1,2\},\{3,4\}}\tilde{\mathbb Q}^{-1}(3,1)\tilde{\mathbb Q}^{-1}(2,4)\right)^{-\frac12} + =1 +\end{equation} +The Hubbard--Stratonovich transformation therefore contributes a factor of +\begin{equation} + \operatorname{sdet}(\mathbb Q-\mathbb M\mathbb M^T)^{\frac12} + =[(C-m^2)(D+\hat m^2)+(R-m\hat m)^2]^\frac12G^{-1} +\end{equation} +to the prefactor. + +\subsection{Sign of the prefactor} + +The superspace notation papers over some analytic differences between branches +of the logarithm that are not important for determining the saddle point but +are important to getting correctly the sign of the prefactor. For instance, consider the superdeterminant of $\mathbb Q$ from \eqref{eq:ops} (dropping the fermionic order parameters for a moment for brevity), +\begin{equation} + \operatorname{sdet}\mathbb Q=\frac{CD+R^2}{G^2} +\end{equation} +The numerator and denominator arise from the determinant in the sector of number and Grassmann number basis elements for the superoperator, respectively. In our calculation, such superdeterminants appear after Gaussian integrals, which produce +\begin{equation} + \int d\pmb\phi\,\exp\left\{-\frac12\int d1\,d2\,\pmb\phi(1)\mathbb Q(1,2)\pmb\phi(2)\right\} + =(\operatorname{sdet}\mathbb Q)^{-\frac12} + =(CD+R^2)^{-\frac12}G +\end{equation} +Here we emphasize that in the expanded result of the integral, the term from +the denominator of the square root enters not as $(G^2)^{\frac12}=|G|$ but as +$G$, including its sign. Therefore, when we write in the effective action +$\frac12\log\operatorname{sdet}\mathbb Q$, we should really be writing +\begin{equation} + \int d\pmb\phi\,\exp\left\{-\frac12\int d1\,d2\,\pmb\phi(1)\mathbb Q(1,2)\pmb\phi(2)\right\} + =\operatorname{sign}(G)e^{-\frac12\log\operatorname{sdet}\mathbb Q} +\end{equation} +In our calculation in Appendix \ref{sec:euler} we elide this several times, +and accumulate $M$ factors of $\operatorname{sign}(-Gf'(C))=\operatorname{sign}(-G)$ from the Gaussian +integral over Lagrange multipliers and $N$ factors of $\operatorname{sign}(-G)$ +from the Hubbard--Stratonovich transformation. Since at all saddle points $G=-R$, we have +\begin{equation} + \operatorname{sign}(R)^{N+M}e^{N\mathcal S_\Omega(\tilde{\mathbb Q},\mathbb Q,\mathbb M)} +\end{equation} + +\subsection{Contribution from integrating the Grassmann order parameters} + +After integrating out the Lagrange multiplier enforcing the spherical +constraint in \eqref{eq:sigma0.integral}, the Grassmann variables $\bar H$ and +$H$ are eliminated from the integrand. This leaves dependence on $\bar{\hat +H}$, $\hat H$, $\bar H_0$, and $H_0$. Expanding the contributions from +\eqref{eq:sdet.q}, \eqref{eq:sdet.fq}, and \eqref{eq:inv.q}, the contribution to the action is given by +\begin{equation} + \int d\bar{\hat H}\,d\hat H\,d\bar H_0\,dH_0\,\exp\left\{ + N\begin{bmatrix} + \bar{\hat H} & \bar H_0 + \end{bmatrix} + \begin{bmatrix} + h_1 & h_2 \\ + h_2 & h_3 + \end{bmatrix} + \begin{bmatrix} + \hat H \\ + H_0 + \end{bmatrix} + +Nh_4\bar{\hat H}\hat H\bar H_0H_0 + \right\} + =N^2(h_1h_3-h_2^2)+Nh_4 +\end{equation} +where +\begin{align} + &h_1=\frac1R\left( + \frac{1-m^2}{D(1-m^2)+R^2-2Rm\hat m+\hat m^2} + -\alpha\frac{Df(1)^2+R^2f'(1)[V_0^2+f(1)]}{[Df(1)+R^2f'(1)]^2} + \right) + \\ + &h_2=\frac1R\frac{Rm-\hat m}{D(1-m^2)+R^2-2Rm\hat m+\hat m^2} + \qquad + h_3=-\frac1R\frac{D+R^2}{D(1-m^2)+R^2-2Rm\hat m+\hat m^2} + \\ + &h_4=-\frac1{R^2}\frac1{D(1-m^2)+R^2-2Rm\hat m+\hat m^2} +\end{align} +The contribution at leading order in $N$ is therefore +\begin{equation} + \frac{N^2}{R^2[D(1-m^2)+R^2-2Rm\hat m+\hat m^2]}\left( + \alpha\frac{(D+R^2)[Df(1)^2+R^2f'(1)[V_0^2+f(1)]]}{ + [Df(1)+R^2f'(1)]^2 + } + -1 + \right) +\end{equation} + +\subsection{Contribution from the saddle point approximation} + +We now want to evaluate the prefactor for the asymptotic value of +$\overline{\chi(\Omega)}$. From the previous sections, the definition of the +measures $d\mathbb Q$ and $d\mathbb M$ in \eqref{eq:op.measures}, and the +integral over $\sigma_0$ of \eqref{eq:sigma0.integral}, we can now see that the +function $g(R,D,m,\hat m)$ of \eqref{eq:pre-saddle.characteristic} is given by +\begin{equation} + \begin{aligned} + g(R,D,m,\hat m) + =- + \frac{\operatorname{sign}(R)^{N+M}}{R^3[D(1-m^2)+R^2-2Rm\hat m+\hat m^2]^{\frac12}} + \hspace{3pc} \\ + \times\left( + \alpha\frac{(D+R^2)[Df(1)^2+R^2f'(1)[V_0^2+f(1)]]}{ + [Df(1)+R^2f'(1)]^2 + } + -1 + \right) + \end{aligned} +\end{equation} +In the connected regime, there are two saddle points of the integrand that +contribute to the asymptotic value of the integral, at $m=\pm m^*$ with $R=-m^*$, $D=0$, and $\hat m=0$. At this saddle point $\mathcal S_\Omega=0$. We can therefore write +\begin{equation} + \overline{\chi(\Omega)} + =\sum_{m=\pm m^*} + g(-m,0,m,0)\big[\det\partial\partial\mathcal S_\Omega(-m,0,m,0)\big]^{-\frac12} +\end{equation} +where here $\partial=[\frac{\partial}{\partial R},\frac{\partial}{\partial D},\frac\partial{\partial m},\frac\partial{\partial \hat m}]$ is the +vector of derivatives with respect to the remaining order parameters. For both of the two saddle points, the determinant of the Hessian of the effective action evaluates to +\begin{equation} + \det\partial\partial\mathcal S_\Omega + =\left[\frac1{(m^*)^4}\left(1-\frac{\alpha[V_0^2+f(1)]}{f'(1)}\right)\right]^2 +\end{equation} +whereas +\begin{equation} + g(\mp m^*,0,\pm m^*,0) + =\frac{(\mp1)^{N+M+1}}{|m^*|^4}\left(1-\frac{\alpha[V_0^2+f(1)]}{f'(1)}\right) +\end{equation} +The saddle point at $m=-m^*$, characterized by minima of the height function, always contributes with a positive term. On the other hand, the saddle point with $m=+m^*$, characterized by maxima of the height function, contributes with a sign depending on if $N+M+1$ is even or odd. This follows from the fact that minima, with an index of 0, have a positive contribution to the sum over stationary points, while maxima, with an index of $N-M-1$, have a contribution that depends on the dimension of the manifold. + +We have finally that, in the connected regime, +\begin{equation} + \overline{\chi(\Omega)}=1+(-1)^{N+M+1}+O(N^{-1}) +\end{equation} +When $N+M+1$ is odd, this evaluates to zero. In fact it must be zero to all +orders in $N$, since for odd-dimensional manifolds the Euler characteristic is +always zero. When $N+M+1$ is even, we have $\overline{\chi(\Omega)}=2$ to +leading order in $N$, as specified in the main text. + \section{Details of the calculation of the average number of stationary points} \label{sec:complexity.details} |