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diff --git a/marginal.tex b/marginal.tex index f46ed38..038f214 100644 --- a/marginal.tex +++ b/marginal.tex @@ -1820,42 +1820,45 @@ full-RSB setting. Using the $\mathbb R^{N|2}$ superfields \begin{equation} - \pmb\phi_a(1)=\mathbf x+\bar\theta_1\pmb\eta+\bar{\pmb\eta}\theta_1+\bar\theta_1\theta_1\hat{\mathbf x}, + \pmb\phi_a(1)=\mathbf x_a+\bar\theta_1\pmb\eta_a+\bar{\pmb\eta}_a\theta_1+\bar\theta_1\theta_1\hat{\mathbf x}_a, \end{equation} the replicated count of stationary points can be written \begin{equation} \begin{aligned} &\mathcal N(E,\mu)^n - =\int\prod_{a=1}^nd\hat\beta_a\,d\pmb\phi_a\, - \\ - &\qquad\times\exp\left[ - \hat\beta_a E-\frac12\int d1\,B_a(1)\sum_{k=1}^MV^k(\pmb\phi_a(1))^2 - \right] + =\int d\hat\beta\prod_{a=1}^n\,d\pmb\phi_a\, + \exp\bigg[ + N\hat\beta E \\ + &\qquad-\frac12\int d1\,\left( + B(1)\sum_{k=1}^MV_k(\pmb\phi_a(1))^2 + -\mu\big(\|\pmb\phi_a(1)\|^2-N\big) + \right) + \bigg] \end{aligned} \end{equation} -for $B_a(1)=1-\hat\beta_a\bar\theta_1\theta_1$. +for $B(1)=1-\hat\beta\bar\theta_1\theta_1$. The derivation of the complexity follows from here nearly identically to that in Appendix A.2 of \citeauthor{Fyodorov_2022_Optimization} with superoperations replacing standard ones \cite{Fyodorov_2022_Optimization}. First we insert -Dirac $\delta$ functions to fix each of the $M$ energies $V^k(\pmb\phi_a(1))$ as +Dirac $\delta$ functions to fix each of the $M$ energies $V_k(\pmb\phi_a(1))$ as \begin{equation} \label{eq:Vv.delta} \begin{aligned} - &\int dv^k_a\,\delta\big(V^k(\pmb\phi_a(1))-v^k_a(1)\big) + &\delta\big(V_k(\pmb\phi_a(1))-v_{ka}(1)\big) \\ - &\quad=\int dv^k_a\,d\hat v^k_a\,\exp\left[i\int d1\,\hat v^k_a(1)\big(V^k(\pmb\phi_a(1))-v^k_a(1)\big)\right] + &=\int d\hat v_{ka}\,\exp\left[i\int d1\,\hat v_{ka}(1)\big(V_k(\pmb\phi_a(1))-v_{ka}(1)\big)\right] \end{aligned} \end{equation} -The squared $V^k$ appearing in the energy can now be replaced by the variables -$v^k$, leaving the only remaining dependence on the disordered $V$ in the +The squared $V_k$ appearing in the energy can now be replaced by the variables +$v_k$, leaving the only remaining dependence on the disordered $V$ in the contribution of \eqref{eq:Vv.delta}, which is linear. The average over the disorder can then be computed, which yields \begin{equation} \begin{aligned} - &\overline{\sum_{k=1}^M\sum_{a=1}^n\exp\left[i\int d1\,\hat v^k_a(1)V^k(\pmb\phi_a(1))\right]} + &\overline{\sum_{k=1}^M\sum_{a=1}^n\exp\left[i\int d1\,\hat v_{ka}(1)V_k(\pmb\phi_a(1))\right]} \\ & =\exp\left[ - -\frac12\sum_{k=1}^M\sum_{ab=1}^n\int d1\,d2\,\hat v_a^k(1)f\left(\frac{\pmb\phi_a(1)^T\pmb\phi_b(2)}N\right)\hat v_b^k(2) + -\frac12\sum_{k=1}^M\sum_{ab}^n\int d1\,d2\,\hat v_{ka}(1)f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right)\hat v_{kb}(2) \right] \end{aligned} \end{equation} @@ -1863,24 +1866,26 @@ The result is factorized in the indices $k$ and Gaussian in the superfields $v$ and $\hat v$ with kernel \begin{equation} \begin{bmatrix} - B_a(1)\delta_{ab}\delta(1,2) & i\delta_{ab}\delta(1,2) \\ - i\delta_{ab}\delta(1,2) & f\left(\frac{\pmb\phi_a(1)^T\pmb\phi_b(2)}N\right) + B(1)\delta_{ab}\delta(1,2) & i\delta_{ab}\delta(1,2) \\ + i\delta_{ab}\delta(1,2) & f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right) \end{bmatrix} \end{equation} -Making the $M$ independent Gaussian integrals, we therefore have +Making the $M$ independent Gaussian integrals, we find \begin{equation} \begin{aligned} &\mathcal N(E,\mu)^n - =\int\left(\prod_{a=1}^nd\hat\beta_a\,d\pmb\phi_a\right) + =\int d\hat\beta\left(\prod_{a=1}^nd\pmb\phi_a\right) \exp\bigg[ - \sum_a^n\hat\beta_aE \\ - &\qquad-\frac M2\log\operatorname{sdet}\left( - \delta_{ab}\delta(1,2)+B_a(1)f\left(\frac{\pmb\phi_a(1)^T\pmb\phi_b(2)}N\right) + nN\hat\beta E+\frac\mu2\sum_a^n\int d1\,\|\pmb\phi_a\|^2 \\ + &\quad-\frac M2\log\operatorname{sdet}\left( + \delta_{ab}\delta(1,2)+B(1)f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right) \right) \bigg] \end{aligned} \end{equation} -We make a change of variables from the fields $\pmb\phi$ to matrices $\mathbb Q_{ab}(1,2)=\frac1N\pmb\phi_a(1)^T\pmb\phi_b(2)$. This transformation results in a change of measure of the form +We make a change of variables from the fields $\pmb\phi$ to matrices $\mathbb +Q_{ab}(1,2)=\frac1N\pmb\phi_a(1)\cdot\pmb\phi_b(2)$. This transformation results +in a change of measure of the form \begin{equation} \prod_{a=1}^n d\pmb\phi_a=d\mathbb Q\,(\operatorname{sdet}\mathbb Q)^\frac N2 =d\mathbb Q\,\exp\left[\frac N2\log\operatorname{sdet}\mathbb Q\right] @@ -1889,18 +1894,20 @@ We therefore have \begin{equation} \begin{aligned} &\mathcal N(E,\mu)^n - =\int\left(\prod_{a=1}^nd\hat\beta_a\right)\,d\mathbb Q\, - \exp\bigg[ - \sum_a^n\hat\beta_aE - +\frac N2\log\operatorname{sdet}\mathbb Q + =\int d\hat\beta\,d\mathbb Q\, + \exp\bigg\{ + nN\hat\beta E+N\frac\mu2\operatorname{sTr}\mathbb Q + +\frac N2\log\operatorname{sdet}\mathbb Q \\ - &\qquad-\frac M2\log\operatorname{sdet}\left( - \delta_{ab}\delta(1,2)+B_a(1)f(\mathbb Q_{ab}(1,2)) - \right) - \bigg] + &\qquad + -\frac M2\log\operatorname{sdet}\left[ + \delta_{ab}\delta(1,2)+B(1)f(\mathbb Q_{ab}(1,2)) + \right] + \bigg\} \end{aligned} \end{equation} -We now need to blow up our supermatrices into our physical order parameters. We have that +We now need to blow up our supermatrices into our physical order parameters. We +have from the definition of $\pmb\phi$ and $\mathbb Q$ that \begin{equation} \begin{aligned} &\mathbb Q_{ab}(1,2) @@ -1913,7 +1920,7 @@ where $C$, $R$, $D$, and $G$ are the matrices defined in \eqref{eq:order.parameters}. Other possible combinations involving scalar products between fermionic and bosonic variables do not contribute at physical saddle points \cite{Kurchan_1992_Supersymmetry}. Inserting this expansion into -the expression above and evaluating the superdeterminants, we find +the expression above and evaluating the superdeterminants and supertrace, we find \begin{equation} \mathcal N(E,\mu)^n=\int d\hat\beta\,dC\,dR\,dD\,dG\,e^{nN\mathcal S_\mathrm{KR}(\hat\beta,C,R,D,G)} \end{equation} @@ -1921,63 +1928,76 @@ where the effective action is given by \begin{widetext} \begin{equation} \begin{aligned} - &\mathcal S_\mathrm{KR}(\hat\beta,C,R,D,G) - =\hat\beta E-\frac1n\operatorname{Tr}(G+R)\mu - +\frac1n\frac12\Big(\log\det(CD+R^2)-\log\det G^2\Big) + \mathcal S_\mathrm{KR}(\hat\beta,C,R,D,G) + &=\hat\beta E+\lim_{n\to0}\frac1n\Bigg(-\mu\operatorname{Tr}(G+R) + +\frac12\log\det\big[G^{-2}(CD+R^2)\big] + +\alpha\log\det\big[I+G\odot f'(C)\big] \\ - &-\frac1n\frac\alpha2\left\{\log\det\left[ + &\qquad-\frac\alpha2\log\det\left[ \Big( - f'(C)\odot D-\hat\beta I+(G\odot G-R\odot R)\odot f''(C) + f'(C)\odot D-\hat\beta I+(G^{\circ2}-R^{\circ2})\odot f''(C) \Big)f(C) +(I-R\odot f'(C))^2 - \right]-\log\det(I+G\odot f'(C))^2\right\} + \right]\Bigg) \end{aligned} \end{equation} -where $\odot$ gives the Hadamard or componentwise product between the matrices, while other products and powers are matrix products and powers. +where $\odot$ gives the Hadamard or componentwise product between the matrices +and $A^{\circ n}$ gives the Hadamard power of $A$, while other products and +powers are matrix products and powers. -In the case where $\mu$ is not specified, the model has a BRST symmetry whose -Ward identities give $D=\hat\beta R$ and $G=-R$ -\cite{Annibale_2004_Coexistence, Kent-Dobias_2023_How}. Using these relations, -the effective action becomes particularly simple: +In the case where $\mu$ is not specified, we can make use of the BRST symmetry +of Appendix~\ref{sec:brst} whose Ward identities give $D=\hat\beta R$ and +$G=-R$. Using these relations, the effective action becomes particularly +simple: \begin{equation} - \mathcal S(\hat\beta, C, R) + \mathcal S_\mathrm{KR}(\hat\beta, C, R) = \hat\beta E - +\lim_{n\to0}\frac1n\left[ - -\frac\alpha2\log\det\left[ - I-\hat\beta f(C)(I-R\odot f'(C))^{-1} + +\frac12\lim_{n\to0}\frac1n\Big( + \log\det(I+\hat\beta CR^{-1}) + -\alpha\log\det\left[ + I-\hat\beta f(C)\big(I-R\odot f'(C)\big)^{-1} \right] - +\frac12\log\det(I+\hat\beta CR^{-1}) - \right] + \Big) \end{equation} -This effective action is general for arbitrary matrices $C$ and $R$. When using -a replica symmetric ansatz of $C_{ab}=\delta_{ab}+c_0(1-\delta_{ab})$ and +This effective action is general for arbitrary matrices $C$ and $R$, and +therefore arbitrary \textsc{rsb} order. When using a replica symmetric ansatz +of $C_{ab}=\delta_{ab}+c_0(1-\delta_{ab})$ and $R_{ab}=r\delta_{ab}+r_0(1-\delta_{ab})$, the resulting function of $\hat\beta$, $c_0$, $r$, and $r_0$ is \begin{equation} \begin{aligned} - \mathcal S= + &\mathcal S_\mathrm{KR}(\hat\beta,c_0,r,r_0)= \hat\beta E - -\frac\alpha 2\left[ - \log\left(1-\frac{\hat\beta\big(f(1)-f(c_0)\big)}{1-rf'(1)+r_0f'(c_0)}\right) - -\frac{\hat\beta f(c_0)+r_0f'(c_0)}{ - 1-\hat\beta\big(f(1)-f(c_0)\big)-rf'(1)+rf'(c_0) - }+\frac{r_0f'(c_0)}{1-rf'(1)+r_0f'(c_0)} - \right] \\ +\frac12\left[ \log\left(1+\frac{\hat\beta(1-c_0)}{r-r_0}\right) +\frac{\hat\beta c_0+r_0}{\hat\beta(1-c_0)+r-r_0} -\frac{r_0}{r-r_0} \right] + \\ + &\qquad-\frac\alpha 2\left[ + \log\left(1-\frac{\hat\beta\big(f(1)-f(c_0)\big)}{1-rf'(1)+r_0f'(c_0)}\right) + -\frac{\hat\beta f(c_0)+r_0f'(c_0)}{ + 1-\hat\beta\big(f(1)-f(c_0)\big)-rf'(1)+rf'(c_0) + }+\frac{r_0f'(c_0)}{1-rf'(1)+r_0f'(c_0)} + \right] \end{aligned} \end{equation} +\end{widetext} When $f(0)=0$ as in the cases directly studied in this work, this further -simplifies as $c_0=r_0=0$. Extremizing this expression with respect to the +simplifies as $c_0=r_0=0$. The effective action is then +\begin{equation} + \mathcal S_\mathrm{KR}(\hat\beta,r)= + \hat\beta E + +\frac12 + \log\left(1+\frac{\hat\beta}{r}\right) + -\frac\alpha 2 + \log\left(1-\frac{\hat\beta f(1)}{1-rf'(1)}\right) +\end{equation} +Extremizing this expression with respect to the order parameters $\hat\beta$ and $r$ produces the red line of dominant minima shown in Fig.~\ref{fig:ls.complexity}. -\end{widetext} - \bibliography{marginal} \end{document} |