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diff --git a/topology.tex b/topology.tex index 212a20d..ee658a0 100644 --- a/topology.tex +++ b/topology.tex @@ -1391,78 +1391,87 @@ The effective action then has the form \end{equation} It is not helpful to write out this entire expression, which is quite large. However, we only find saddle points of this action with $\bar -J=J=G_{12}=G_{21}=G_-=\hat X_1=\hat X_2=m_1=m_2=A_{12}=0$. Setting these variables to zero, we find +J=J=G_-=\hat X_1=\hat X_2=m_1=m_2=A_{12}=0$ and $G_{21}=G_{12}$. Setting these variables to zero, we find \begin{align} \notag \mathcal S =\hat m - +\frac12i\epsilon A_- -\alpha\frac12\log\frac{ - f'(1)(Df(1)+R^2f'(1))-\frac12((R-\frac12A_+)^2+\frac12 A_-^2)f(1)f''(1) + f'(1)(Df(1)+R^2f'(1))-(\frac12(R-A_+)^2+A_-^2-2G_{12}^2)f(1)f''(1) }{ - \frac14(R+\frac12A_+)^2f'(1)^2 + [\frac12(R+A_+)^2-2G_{12}^2]f'(1)^2 } \\ \notag - -\alpha\frac12\log\frac{ - \epsilon^2+i\epsilon f'(1)A_-+\frac14(A_+^2-A_-^2)f'(1)^2 + +\frac12(1-\alpha)\log\frac{ + 4(A_+^2-A_-^2) }{ - \frac14(R+\frac12A_+)^2f'(1)^2 + \frac12(R+A_+)^2-2G_{12}^2 + } + +\frac12\log\frac{D(1-m^2)+R^2-2Rm\hat m+\hat m^2}{ + \frac12(R+A_+)^2-2G_{12}^2 } - +\frac12\log\frac{D(1-m^2)+R^2-2Rm\hat m+\hat m^2}{\frac14(R+\frac12A_+)^2} \\ - +\frac12\log\frac{A_+^2-A_-^2}{(R+\frac12A_+)^2} -\frac12\alpha V_0^2\left( - f(1)+\frac{R^2f'(1)^2}{Df'(1)-\frac12((R-\frac12A_+)^2+\frac12A_-^2)f''(1)} + f(1)+\frac{R^2f'(1)^2}{Df'(1)-(\frac12(R-A_+)^2+A_-^2-2G_{12}^2)f''(1)} \right)^{-1} \end{align} -One solution to these equations at $\epsilon=0$ is $A_-=0$ and $A_+=2R^*$ with -$D=\hat m=0$ and $R=R^*$, exactly as for the Euler characteristic. The resulting effective -action as a function of $m$ is also exactly the same. We find two other -solutions that differ from this one, but with formulae too complex to share in -the text. One alternate solution has $A_-=0$ and one has $A_-\neq0$. - -\begin{figure} - \includegraphics{figs/alt_sols_1.pdf} - \hfill - \includegraphics{figs/alt_sols_2.pdf} +One solution to these equations is $A_-=G_{12}=0$ and $A_+=R^*$ with +$D$, $\hat m$, and $R$ exactly as for the Euler characteristic. The resulting effective +action as a function of $m$ is also exactly the same. There is another solution, this time with $A_+=R$, $A_-=G_{12}$, and +\begin{align} + &R=\frac{\alpha mf'(1)}{b^2}\big[ + V_0^2f'(1)^2(1-m^2)-f(1)b + \big] + \\ + &G_{12}^2 + =\frac{f'(1)mR}{f''(1)(1-m^2)b^2} + \big[\alpha V_0^2f'(1)^2f''(1)(1-m^2)^2-b^2-\alpha bf'(1)(f(1)-(1-m^2)f'(1)) + \big] +\end{align} +where we have defined the constant +\begin{equation} + b=(1-m^2)[f''(1)f(1)+f'(1)^2]-f(1)f'(1) +\end{equation} +The resulting form for the action is +\begin{align} + \mathcal S_\mathcal N(m) + =\frac12(1-\alpha)\log\left( + \frac{f''(1)(1-m^2)}{f'(1)(1-\alpha)} + \right) + -\frac12\alpha\log\left( + \frac{\alpha f'(1)V_0^2}b + \right) + -\frac12\alpha\frac{V_0^2\big[f''(1)(1-m^2)-f'(1)\big]+b}b +\end{align} +This solution is plotted alongside the solution that coincides with that of the +Euler characteristic in Fig. It is clear from this plot that the new solution +cannot be valid in the entire range of $m$, since it diverges as $m$ goes to 1 +where we know there are vanishingly few stationary points. However, there is a +single point $m_c$ where the two solutions coincide, and they have the +possibility of trading stability. This is given by +\begin{equation} + m_c^2 + =1-\frac{2(1-\alpha)f(1)f'(1)}{ + (2-\alpha)f(1)f''(1)+2(1-\alpha)f'(1)^2 + -\sqrt{\alpha f''(1)}\sqrt{4V_0^2(1-\alpha)f'(1)^2+\alpha f(1)^2f''(1)} + } +\end{equation} - \caption{ - \textbf{Choosing the correct saddle-point for the complexity.} In the - calculation of the complexity of stationary points in the height function, - there are three plausible solutions in some regimes. \textbf{Left:} Plot of - the three solutions for $M=1$, $f(q)=\frac12q^3$, and - $E=E_\text{on}=3^{-1/2}$. The inset shows detail of the region where these - solutions are nonnegative. The solid and dashed lines show the values of - $m$ where the $A_-=0$ and $A_-\neq0$ alternative solutions are maximized, - respectively. The value $m^*$ maximizing the action corresponding also to - $\mathcal S_\chi$ is also marked. \textbf{Right:} Numeric measurement of the - average value of $m$ for stationary points found using Newton's method as a - function of model size $N$. The thick solid line is a power-law fit to - $m-m^*$, while the thin solid and dashed lines show the same values of $m$ - as in the lefthand plot. - } \label{fig:alt.sols} -\end{figure} +\begin{equation} + \mathcal S_\mathcal N(m) + =\begin{cases} + \mathcal S_\chi(m) & m^2\geq m_c^2 \\ + \mathcal S_{\ast}(m) & m^2<m_c^2 + \end{cases} +\end{equation} +This formula remains valid also in the regime when $m_c^2<0$, when the +complexity is given by the function $\mathcal S_\chi$ in the entire range +$m\in(-1,1)$. -For both alternative solutions, there is a regime of small $V_0$ or $E$ for -which they are unphysical. In this case the action is everywhere negative -despite our knowledge from the calculation of the Euler characteristic that the -manifold exists. However, there is a second regime where they cross the action -$\mathcal S_\chi$ and become positive, both reaching values of $m$ that are -less than $m_\text{min}$. An example of this regime for the case with -$f(q)=\frac12q^3$ and $M=1$ (the spherical spin glass) is shown in the left -panel of Fig.~\ref{fig:alt.sols}. In order to investigate which solution is -valid in this regime, we turn to comparison with numeric experiments. We create -samples of this model at the specified parameters, choose random axis $\mathbf -x_0$, and use Newton's method with mild damping to find stationary points of -the Lagrangian starting from random initial conditions. When we find a -stationary point, we record the value of the overlap $m=\frac1N\mathbf -x\cdot\mathbf x_0$ between it and the height axis. The right panel of -Fig.~\ref{fig:alt.sols} shows that the average value of $m$ attained by this -method as a function of $N$ asymptotically approaches $m^*$, the value implied -by the Euler characteristic action $\mathcal S_\chi$, and not either of the -values implied by the alternative solutions. This indicates that $\mathcal -S_\mathcal N(m)=\mathcal S_\chi(m)$ in all regimes. +\begin{equation} + V_\text{on}^2=\frac{(1-\alpha)f'(1)^2-\alpha f(1)f''(1)}{\alpha f''(1)} +\end{equation} \section{The quenched shattering energy} \label{sec:1frsb} |