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| author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2024-09-13 17:08:53 +0200 |
|---|---|---|
| committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2024-09-13 17:08:53 +0200 |
| commit | b953eb189459b1756f630cf8c528c09227df9106 (patch) | |
| tree | fa36dc1c92769efcc6c6a65603584067e264ad23 | |
| parent | 28cfd58481308e5d670be280e91ab5d458b86c9b (diff) | |
| download | SciPostPhys_18_158-b953eb189459b1756f630cf8c528c09227df9106.tar.gz SciPostPhys_18_158-b953eb189459b1756f630cf8c528c09227df9106.tar.bz2 SciPostPhys_18_158-b953eb189459b1756f630cf8c528c09227df9106.zip | |
Some new writing.
| -rw-r--r-- | figs/alt_sols_1.pdf | bin | 16778 -> 16778 bytes | |||
| -rw-r--r-- | figs/alt_sols_2.pdf | bin | 8690 -> 8693 bytes | |||
| -rw-r--r-- | figures.nb | 9483 | ||||
| -rw-r--r-- | topology.tex | 43 |
4 files changed, 5352 insertions, 4174 deletions
diff --git a/figs/alt_sols_1.pdf b/figs/alt_sols_1.pdf Binary files differindex 2207fcd..dcec747 100644 --- a/figs/alt_sols_1.pdf +++ b/figs/alt_sols_1.pdf diff --git a/figs/alt_sols_2.pdf b/figs/alt_sols_2.pdf Binary files differindex 076f0e1..386de3b 100644 --- a/figs/alt_sols_2.pdf +++ b/figs/alt_sols_2.pdf @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 10555001, 193025] -NotebookOptionsPosition[ 10539747, 192780] -NotebookOutlinePosition[ 10540142, 192796] -CellTagsIndexPosition[ 10540099, 192793] +NotebookDataLength[ 10596956, 194164] +NotebookOptionsPosition[ 10579546, 193890] +NotebookOutlinePosition[ 10579944, 193906] +CellTagsIndexPosition[ 10579901, 193903] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -25,15 +25,25 @@ Cell["Definitions", "Section", 3.9337634781959476`*^9}},ExpressionUUID->"bbc7c15f-5b7d-4690-8a22-\ e1e38c3edf32"], +Cell[CellGroupData[{ + +Cell[TextData[{ + "Extensive 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\label{eq:euler.action} \begin{aligned} \mathcal S_\chi(R,D,m,\hat m\mid\alpha,V_0) - &=\hat m-\frac\alpha2\left[ + &=-\hat m-\frac\alpha2\left[ \log\left(1+\frac{f(1)D}{f'(1)R^2}\right) +\frac{V_0^2}{f(1)}\left(1+\frac{f'(1)R^2}{f(1)D}\right)^{-1} \right] \\ @@ -389,7 +389,46 @@ However, the action becomes complex in the region $m^2<m_\text{min}^2$ for 2V_0^2f(1)(2-\alpha)-\alpha(V_0^4+f(1)^2) } \end{equation} -When $m_*^2<m_\text{min}^2$, this solution is no longer valid. Likewise, when $m_\text{min}^2>0$, the solution at $m=0$ is also not valid. In fact, it is not clear what the average value of the Euler characteristic should be at all +When $m_*^2<m_\text{min}^2$, this solution is no longer valid. Likewise, when +$m_\text{min}^2>0$, the solution at $m=0$ is also not valid. In fact, it is not +clear what the average value of the Euler characteristic should be at all when +there is some range $-m_\text{min}<m<m_\text{min}$ where the effective action +is complex. Such a result could arise from the breakdown of the large-deviation +principle behind the calculation of the effective action, or it could be the +result of a negative Euler characteristic. + +To address this problem, we compute also the average of the square of the Euler +characteristic, $\overline{\chi(\Omega)^2}$. This has the benefit of always +being positive, so that the saddle-point approach to the calculation at large +$N$ does not produce complex values even when $\overline{\chi(\Omega)}$ is +negative. We find three saddle points that could contribute to the value of +$\overline{\chi(\Omega)^2}$: two at $\pm m^*$ where +$\frac1N\log\overline{\chi(\Omega)^2}=\frac1N\log\overline{\chi(\Omega)}\simeq0$, and one at $m=0$ where +\begin{equation} + \frac1N\log\overline{\chi(\Omega)^2}=2\operatorname{Re}\mathcal S_\chi(0) +\end{equation} + +We therefore have four possible topological regimes: +\begin{itemize} + \item \textbf{Simple connected: + \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}=0$, + $\frac1N\log\overline{\chi(\Omega)}=0$}.} This regime occurs when $m_*^2>0$ + and $\operatorname{Re}\mathcal S_\chi(0)<0$. Here, + $\overline{\chi(\Omega)}=2+o(1)$ for even $N-M-1$, strongly indicating a + topology homeomorphic to the $S^{N-M-1}$ sphere. + \item \textbf{Complex connected: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}>0$, $\overline{\chi(\Omega)}<0$}.} + This regime occurs when $m_\text{min}^2>0$ and $\operatorname{Re}\mathcal + S_\chi(0)>0$. Here the average Euler characteristic is large and negative. + While the topology of the manifold is not necessarily connected in this + regime, holes are more numerous than components. + \item \textbf{Disconnected: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}>0$, $\overline{\chi(\Omega)}>0$}.} + This regime occurs when $m_\text{min}^2<0$ and $\mathcal S(0)>0$. Here the + average Euler characteristic is large and positive. Large connected + components of the manifold may or may not exist, but disconnected + components outnumber holes. + \item \textbf{\textsc{Unsat}: \boldmath{$\frac1N\log\overline{\chi(\Omega)^2}<0$}.} + There is typically not a manifold at all, indicating that the equations cannot be satisfied. +\end{itemize} However, when the magnitude of $V_0$ is sufficiently large, with \begin{equation} |