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Uncolored Random Tensors, Melon Diagrams, and the SYK Models (1611.08915v5)

Published 27 Nov 2016 in hep-th and cond-mat.str-el

Abstract: Certain models with rank-$3$ tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large $N$ limit, where $g2 N3$ is held fixed. In this limit the perturbative expansion in the quartic coupling constant, $g$, is dominated by a special class of "melon" diagrams. We study "uncolored" models of this type, which contain a single copy of real rank-$3$ tensor. Its three indexes are distinguishable; therefore, the models possess $O(N)3$ symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large $N$ limit dominated by the melon diagrams. The quantum mechanics of a real anti-commuting tensor therefore has a similar large $N$ limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the $O(N)3$ symmetry in our quantum mechanical model removes the non-singlet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anti-commuting tensor, which has $U(N)2\times O(N)$ symmetry and argue that it is equivalent in the large $N$ limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension $d$. While the quartic interaction is not positive definite, we construct the large $N$ Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4-\epsilon$ expansion.

Citations (347)

Summary

  • The paper demonstrates that uncolored rank-3 tensor models achieve a large-N limit dominated by melon diagrams, mirroring SYK dynamics.
  • It shows that an anti-commuting tensor model is mathematically equivalent to the SYK model without requiring disorder averaging.
  • The study confirms melonic dominance via Schwinger-Dyson equations, indicating potential insights into quantum chaos and holographic duality.

Uncolored Random Tensors, Melon Diagrams, and the SYK Models: A Summary

This paper explores a class of quantum mechanical models and quantum field theories based on rank-3 tensors with uncolored and colored symmetries, inspired by developments in the understanding of the Sachdev-Ye-Kitaev (SYK) models. The focus is on models possessing a novel large NN limit, where g2N3g^2 N^3 is held constant, resulting in a perturbative expansion dominated by "melon" diagrams. This work follows key insights by Gurau and others, where uncolored models containing a single copy of real rank-3 tensors exhibit similar large NN behavior to the SYK model without the need for disorder.

Key Contributions

  1. Uncolored Tensor Models: The authors paper uncolored versions of models with a single real rank-3 tensor field, establishing that such models also show a large NN limit dominated by melon diagrams. These models are constructed to have O(N)3O(N)^3 symmetry and transform the tensor field in the tri-fundamental representation.
  2. Quantum Mechanical Models: They demonstrate the equivalence of an uncolored quantum mechanical model described by an anti-commuting rank-3 tensor with the SYK model at large NN. The identification of operators similar to those in the SYK model is discussed, such as a "single Regge trajectory" of two-particle operators.
  3. Complex Tensor Models: The paper extends the discussion to include complex tensor models with U(N)2×O(N)U(N)^2 \times O(N) symmetry, drawing parallels with versions of the SYK model featuring complex fermions.
  4. Large NN and Schwinger-Dyson Equations: The derivation and analysis of Schwinger-Dyson equations for the two-point functions in these models underline their consistency with conformal invariance. The melonic dominance, characteristic for the large NN limit, is further supported by perturbative checks using expansions like the 4ϵ4-\epsilon expansion.
  5. Implications for Quantum Chaos: By aligning with SYK models, these tensor models potentially exhibit quantum chaos properties, a subject ripe for exploration through numerical investigations at finite NN.

Theoretical and Practical Implications

The presented models provide a solid foundation for constructing theories that could serve as dual descriptions in the context of holographic duality, akin to the AdS/CFT correspondence. Another key implication is the simpler computation framework in uncolored models compared to random ensemble models like SYK, foregoing the need for disorder averaging. Models in higher dimensions, although exhibiting unbounded potentials, open avenues for exploring novel CFTs with melonic dominance.

Speculative Future Directions

The paper speculates on enhancing these constructions to supersymmetric settings, exploring quantum field theories in various dimensional settings, and potentially coupling these models with superfields. The development of a gravitational dual theory remains a tantalizing prospect, suggested to be rich and complex due to the multitude of invariant operators. Furthermore, numerically studying energy levels and thermal partition functions at finite NN may yield insights into the chaotic dynamics predicted by the melonic structure.

In conclusion, the paper contributes substantial evidence and frameworks for understanding tensor models alongside SYK-like behavior, presenting potential platforms for translating intricate configurations of quantum chaos and holographic dualities into tangible results within the theoretical physics landscape.