A Generalization of Sachdev-Ye-Kitaev (1610.01569v2)

Abstract: The SYK model: fermions with a $q$-body, Gaussian-random, all-to-all interaction, is the first of a fascinating new class of solvable large $N$ models. We generalize SYK to include $f$ flavors of fermions, each occupying $N_a$ sites and appearing with a $q_a$ order in the interaction. Like SYK, this entire class of models generically has an infrared fixed point. We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global $O(N_1) \times O(N_2) \times \ldots\times O(N_f)$ symmetry: a subgroup of the $O(N)$ symmetry of SYK; thereby giving a richer spectrum. We also elucidate aspects of the large $q$ limit and the OPE, and solve $q=2$ SYK at finite $N$.

• The paper generalizes the SYK model by incorporating multiple fermion flavors, each with unique interaction orders and symmetry groups.
• It employs Schwinger-Dyson equations to compute infrared dimensions, identifying a dimension-two operator that breaks conformal symmetry and signals maximal chaos.
• The findings enhance our understanding of quantum chaos and open up new avenues in holography, quantum simulation, and the study of complex interacting systems.

A Generalization of the Sachdev-Ye-Kitaev Model

The paper by David J. Gross and Vladimir Rosenhaus proposes a notable generalization of the Sachdev-Ye-Kitaev (SYK) model by introducing multiple flavors of fermions, each associated with its interaction order and symmetries. The SYK model becomes a special case within a wider class, expanding its insights into large $N$ quantum models characterized by all-to-all interactions and maximal chaos in the infrared regime.

Overview of the SYK Generalization

The Sachdev-Ye-Kitaev model, recognized for its solvability and complex dynamics in the large $N$ limit, employs $q$-body interactions of fermions with Gaussian-distributed randomness. Gross and Rosenhaus extend this framework by introducing $f$ flavors of fermions, where each flavor exhibits unique interaction orders and occupies a distinct number of sites. Post disorder-averaging, this model exhibits an $O(N_1) \times O(N_2) \times \ldots \times O(N_f)$ symmetry, indicating a more complex symmetry structure than SYK's original $O(N)$ symmetry.

Strong Numerical Results and Claims

The authors succeed in computing the infrared dimensions of fermions and demonstrate that there exists a dimension-two operator in the spectrum, which leads to the breaking of conformal invariance and maximal chaos, akin to the original SYK model. This feature underscores important characteristics like conformal symmetry breaking and the emergence of universal chaotic dynamics in these novel quantum systems.

Moreover, the paper presents equations for the two-point function, utilizing Schwinger-Dyson equations to illustrate that this generalized framework retains its robustness across various parameter spaces, particularly highlighting the infrared fixed points and operator dimensions computationally.

Implications and Future Directions

The expansion of the SYK model into multi-flavored domains carries promising theoretical and practical implications. The paper's insights contribute to enriched models that are sufficiently complex to be chaotic yet analytically tractable, manifesting in varied physical scenarios.

Theoretically, this generalized model augments our understanding of conformal field theories, holography, and possibly quantum simulations that Nature might construct at lower dimensions. As such, exploring its duality with string theories or applications within black holes' quantum horizons may provide novel connections with high-energy physics.

Practically, these models, through their complex yet solvable nature, offer potential in quantum computational applications, modeling non-trivial quantum dynamics and phase transitions, and perhaps even in fabricating robust quantum error correction schemes reflective of thermalization dynamics.

Speculations on Further Developments

The research might extend into probing the function of large $q$ limits for each fermion flavor, examining cases that might harbor novel symmetries or lead to additional applicable domains. Additionally, the interactions of such generalized SYK-like models with higher-dimensional theories may unveil unique stringy or non-local features, enriching the quantum mechanical toolkit available for nuclear and condensed matter systems.

This paper highlights the versatility and depth of interactions possible within the SYK framework and sets a foundation for further fostering knowledge of quantum chaotic systems, their symmetries, and their potential dual gravitational counterparts. Understanding these intricate systems more broadly could inevitably provide deeper insights into the complex nature of quantum systems at large.