In this thesis, we study generalizations of well-known Majorana fermion models, including the SYK model and the Klebanov-Tarnopolsky tensor model. The models are compared at finite and large N, where we find that the models simplify considerably and can even become solvable.
In chapter 2, we study quantum mechanical models in which the dynamical degrees of freedom are real fermionic tensors of rank five and higher. For the tensors of rank five, there is a unique O(N)5 symmetric sixth-order Hamiltonian leading to a solvable large N limit dominated by the melonic diagrams. We solve the large N Schwinger-Dyson (SD) equations for higher rank Majorana tensor models and show that they match those of the corresponding SYK models exactly.
In chapter 3, we study a family of tensor models of complex fermions, with a six fermion interaction whose index structure resembles the topology of a prism. The model is dominated by melonic diagrams in the large N limit after introducing an auxiliary field. We consider interactions that preserve the U(1) global symmetry, solving the SD equations at large N and examining the bilinear spectrum. We find a complex scaling dimension in the U(1) charged sector in addition to an O(1) gap between the ground and first excited states. This model has a negative charge compressibility.
In chapter 4, we present a class of Hamiltonians H for which a sector of the Hilbert space invariant under a Lie group G, which is not a symmetry of H, possesses the essential properties of many-body scar states. These include the absence of thermalization and the ‘revivals’ of special initial states. Our study of an extended 2D tJU model illustrates the properties of the invariant scars and supports our findings.