Majorana fermion quantum mechanics for higher rank tensors

Abstract

We study quantum mechanical models in which the dynamical degrees of freedom are real fermionic tensors of rank five and higher. They are the non-random counterparts of the Sachdev-Ye-Kitaev (SYK) models where the Hamiltonian couples six or more fermions. 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 for the complete energy spectrum of this model when $N=2$ and deduce exact expressions for all the eigenvalues. The subset of states which are gauge invariant exhibit degeneracies related to the discrete symmetries of the gauged model. We also study quantum chaos properties of the tensor model and compare them with those of the $q=6$ SYK model. For $q>6$ there is a rapidly growing number of $O(N)^{q-1}$ invariant tensor interactions. We focus on those of them that are maximally single-trace - their stranded diagrams stay connected when any set of $q-3$ colors is erased. We present a general discussion of why the tensor models with maximally single-trace interactions have large $N$ limits dominated by the melonic diagrams. We solve the large $N$ Schwinger-Dyson equations for the higher rank Majorana tensor models and show that they match those of the corresponding SYK models exactly. We also study other gauge invariant operators present in the tensor models.