Order of fluctuations of the free energy in the positive semi-definite MSK model at critical temperature

In this note, we consider the multi-species Sherrington-Kirkpatrick spin glass model at its conjectured critical temperature, and we show that, when the variance profile matrix ${\mathrm{\Delta }}^2$ is positive semi-definite, the variance of the free energy is $O({\log }^{2}N)$. Furthermore, when one approaches this temperature threshold from the low temperature side at a rate of $O(N^{-\alpha })$ with $\alpha >0$, the variance is $O({\log }^{2}N+N^{1-\alpha })$. This result is a direct extension of the work of Chen and Lam (2019) who proved an analogous result for the SK model, and our proof methods are adapted from theirs.