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We consider limits of $\mathcal{N} = 4$ super-Yang-Mills (SYM) theory thatapproach BPS bounds. These limits result in non-relativistic theories thatdescribe the effective dynamics near the BPS bounds and upon quantization areknown as Spin Matrix Theories. The near-BPS theories can be obtained byreducing $\mathcal{N}=4$ SYM on a three-sphere and integrating out the fieldsthat become non-dynamical in the limits. In previous works we have consideredvarious SU(1,1) and SU(1,2) types of subsectors in this limit. In the currentwork, we will construct the remaining Spin Matrix Theories defined near the$\frac{1}{8}$-BPS subsectors, which include the PSU(1,1|2) and SU(2|3) cases.We derive the Hamiltonians by applying the spherical reduction algorithm andshow that they match with the spin chain result, coming from the loopcorrections to the dilatation operator. In the PSU(1,1|2) case, we prove thepositivity of the spectrum by constructing cubic supercharges using theenhanced PSU$(1|1)^2$ symmetry and show that they close to the interactingHamiltonian. We finally analyse the symmetry structure of the sectors in viewof an interpretation of the interactions in terms of fundamental blocks.

Spin Matrix Theory in near $\frac{1}{8}$-BPS corners of $\mathcal{N} = 4$ super-Yang-Mills