Replica symmetry breaking condition for indefinite multi-species Sherrington–Kirkpatrick models

Let the multi-species Sherrington–Kirkpatrick model have an indefinite interaction matrix $\boldsymbol{D}^2$, and let $\boldsymbol{C}$, $\boldsymbol{C}'$, $\boldsymbol{L}$ and $\boldsymbol{93}'$ denote the matrices defined by the replica-symmetric solution. Write $\rho$ for the spectral radius, $\operatorname{spec}$ for the spectrum, and $\Re$ for the real part of an eigenvalue. Replica symmetry breaking condition. For general indefinite $\boldsymbol{D}^2$, if

$$\beta^2\max\left\{\rho(\boldsymbol{C}\boldsymbol{D}^2\boldsymbol{L}),\max_{\lambda\in \operatorname{spec}(\boldsymbol{\Gamma}'\boldsymbol{D}^2\boldsymbol{L})} \Re(\lambda)\right\}>1,$$

the system is in the replica symmetry breaking phase. This proposes the boundary of the replica-symmetric regime for indefinite interaction matrices; the preceding bipartite example gives evidence for the condition, but the general assertion is not established in the supplied text.