Operator growth bounds in a cartoon matrix model

Abstract

We study operator growth in a model of $N(N-1)/2$ interacting Majorana fermions, which live on the edges of a complete graph of $N$ vertices. Terms in the Hamiltonian are proportional to the product of $q$ fermions which live on the edges of cycles of length $q$. This model is a cartoon "matrix model": the interaction graph mimics that of a single-trace matrix model, which can be holographically dual to quantum gravity. We prove (non-perturbatively in $1/N$, and without averaging over any ensemble) that the scrambling time of this model is at least of order $\log N$, consistent with the fast scrambling conjecture. We comment on apparent similarities and differences between operator growth in our "matrix model" and in the melonic models.

arXiv comment17 pages, 1 figure

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• Optimal Frobenius light cone in spin chains with power-law interactions
• Operator Size Distribution in Large $N$ Quantum Mechanics of Majorana Fermions
• Scheme to Detect the Strong-to-weak Symmetry Breaking via Randomized Measurements
• Scrambling Transition in Free Fermion Systems Induced by a Single Impurity