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

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 $N=3$.

24 pages, 5 figures, v2: minor improvements and references added, v3: matches published version

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• Conformal Symmetry and Composite Operators in the $O(N{)}^3$ Tensor Field Theory
• Hints of unitarity at large $N$ in the $O(N{)}^3$ tensor field theory
• On the large $D$ expansion of Hermitian multi-matrix models
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• On the large N limit of the Schwinger-Dyson equation of tensor field theory
• Diagrammatics of the quartic $O(N{)}^3$-invariant Sachdev-Ye-Kitaev-like tensor model
• Double scaling limit of multi-matrix models at large $D$
• Sextic tensor model in rank $3$ at next-to-leading order
• Duality of Orthogonal and Symplectic Random Tensor Models: General Invariants
• Double scaling limit for the $O(N{)}^3$-invariant tensor model
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• The Tensor Track VII: From Quantum Gravity to Artificial Intelligence
• Duality of O(N) and Sp(N) random tensor models: tensors with symmetries
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• Graph calculus and the disconnected-boundary Schwinger-Dyson equations of quartic tensor field theories
• New Limits for Large $N$ Matrix and Tensor Models: Large $D$, Melons and Applications
• Another proof of the $1/N$ expansion of the rank three tensor model with tetrahedral interaction
• Multiple scaling limits of $\mathrm{U}(N{)}^2\times \mathrm{O}(D)$ multi-matrix models
• Melonic large $N$ limit of $5$-index irreducible random tensors