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Diffstat (limited to 'topology.tex')
| -rw-r--r-- | topology.tex | 26 |
1 files changed, 13 insertions, 13 deletions
diff --git a/topology.tex b/topology.tex index f56a2c7..f353b74 100644 --- a/topology.tex +++ b/topology.tex @@ -85,7 +85,7 @@ out-of-equilibrium dynamics. This work is a submission to SciPost Physics. \newline License information to appear upon publication. \newline Publication information to appear upon publication.} - \end{minipage} & \begin{minipage}{0.4\textwidth} + \end{minipage} & \begin{minipage}{0.4\textwidth} {\small Received Date \newline Accepted Date \newline Published Date}% \end{minipage} \end{tabular}} @@ -286,19 +286,19 @@ effective action defined by \mathcal S_\chi(m) =-\frac\alpha2\bigg[ \log\left( - 1-\frac{f(1)}{f'(1)}\frac{1+\frac m{R_*}}{1-m^2} + 1-\frac{f(1)}{f'(1)}\frac{1+\frac m{R_m}}{1-m^2} \right) +\frac{V_0^2}{f(1)}\left( - 1-\frac{f'(1)}{f(1)}\frac{1-m^2}{1+\frac m{R_*}} + 1-\frac{f'(1)}{f(1)}\frac{1-m^2}{1+\frac m{R_m}} \right)^{-1} \bigg] - +\frac12\log\left(-\frac m{R_*}\right) + +\frac12\log\left(-\frac m{R_m}\right) \end{equation} Here we have introduced the ratio $\alpha=M/N$ between the number of equations -and the number of variables, and $R_*$ is a function of $m$ given by +and the number of variables, and $R_m$ is a function of $m$ given by \begin{equation} \label{eq:rs} \begin{aligned} - R_* + R_m \equiv\frac{-m(1-m^2)}{2[f(1)-(1-m^2)f'(1)]^2} \Bigg[ \alpha V_0^2f'(1) @@ -996,9 +996,9 @@ order in $N$ by a saddle point approximation. To get the formula \eqref{eq:S.m} in the main text, we first extremize this expression with respect to $R$, $D$, and $\hat m$, which take the saddle-point values \begin{align} - R=R^* + R=R_m && - D=-\frac{m+R_*}{1-m^2}R_* + D=-\frac{m+R_m}{1-m^2}R_m && \hat m=0 \end{align} @@ -1306,9 +1306,9 @@ $C_{12}=1$. We restrict ourselves to cases with $f(0)=0$, which correspond to co \hspace{2em} R_{22}=R_{11}^\dagger \hspace{2em} - R_{11}=R_* + R_{11}=R_m \end{align} -where $\dagger$ denotes the complex conjugate and $R_*$ is the saddle point +where $\dagger$ denotes the complex conjugate and $R_m$ is the saddle point solution of \eqref{eq:rs}. Upon substituting these solutions into the expressions above, we find in both cases that \begin{equation} @@ -1333,7 +1333,7 @@ employed in our derivation ensure that the resulting saddle point is never at a true maximum with respect to some combinations of variables. We rather look for places where the stability of this matrix changes, indicating another solution branching from the existing one. However, we must neglect the -branching of trivial solutions, which occur when $R_*$ goes from real- to +branching of trivial solutions, which occur when $R_m$ goes from real- to complex-valued. By examination of the results, it appears that nontrivial \textsc{rsb} @@ -1347,14 +1347,14 @@ correlations between the two copies of the system. We can therefore find the \te evaluated at the $m=0$ solution described above. The resulting expression is usually quite heinous and we will not reproduce it in its general form in the text, but there is a regime where a dramatic simplification is possible. The instability always occurs along the direction -$R_{21}=R_{12}^\dagger$, but when $R_*$ is real, $R_{11}=R_{22}$ and the instability occurs along +$R_{21}=R_{12}^\dagger$, but when $R_m$ is real, $R_{11}=R_{22}$ and the instability occurs along the direction $R_{21}=R_{12}$. This allows us to examine a simpler action, and we find the determinant is proportional to two nontrivial factors, with \begin{equation} \label{eq:stab.det} \det\partial\partial\mathcal S_{\chi^2}=-\frac{2B_1B_2}{[r_*f'(1)]^3[(1+r_*)f(1)-r_*f'(1)]^7} \end{equation} If we define -$r_*\equiv\lim_{m\to0}R_*/m$, then the factors $B_1$ and $B_2$ are +$r_*\equiv\lim_{m\to0}R_m/m$, then the factors $B_1$ and $B_2$ are \begin{align} B_1=[(1+r_*)f(1)]^3-3r_*[(1+r_*)f(1)]^2f'(1) +\alpha V_0^2\big[2(1+r_*)f'(0)^2+r_*f'(1)f''(0)\big] |