SYK-like tensor quantum mechanics with $\mathrm{Sp}(N)$ symmetry

TL;DR

This paper introduces a new family of tensor quantum-mechanical models with $ ext{Sp}(N)$ symmetry, supporting a melonic large $N$ limit similar to SYK models, and analyzes their properties through character formulas and numerical diagonalization.

Contribution

It presents the first $ ext{Sp}(N)$-symmetric tensor models with non-vanishing fermionic interactions and explores their large $N$ behavior and spectral properties.

Findings

Supports melonic large $N$ limit with $ ext{Sp}(N)$ symmetry

Provides character formulas for singlet states

Numerical diagonalization of specific models

Abstract

We introduce a family of tensor quantum-mechanical models based on irreducible rank-$3$ representations of $\mathrm{Sp}(N)$. In contrast to irreducible tensor models with $\mathrm{O}(N)$ symmetry, the fermionic tetrahedral interaction does not vanish and can therefore support a melonic large $N$ limit. The strongly-coupled regime has a very analogous structure as in the complex SYK model or in $\mathrm{U}(N)\times\mathrm{O}(N)\times\mathrm{U}(N)$ tensor quantum mechanics, the main difference being that the states are now singlets under $\mathrm{Sp}(N)$. We introduce character formulas that enumerate such singlets as a function of $N$, and compute their first values. We conclude with an explicit numerical diagonalization of the Hamiltonian in two simple examples: the symmetric model at $N=1$, and the antisymmetric traceless model at $N=3$.