We introduce two disorder-free variants of the Sachdev-Ye-Kitaev (SYK) model, demonstrate their integrability, and study their static and dynamical properties. Unlike diagrammatic techniques, the integrability of these models allows us to obtain dynamical correlation functions even when the number of Majorana fermions is finite. From the solutions, we find that out-of-time-order correlators (OTOCs) in these models exhibit exponential growth at early times, resembling that of many-body quantum chaotic systems, such as those with disorder or external kick terms, despite their large N behavior differing from that of typical chaotic systems. Conversely, our analysis shows no evidence of random-matrix behavior in level spacing statistics or the spectral form factor. Our findings illustrate that the clean versions of the SYK models represent simple but nontrivial examples of disorder-free quantum many-body systems displaying chaoslike behavior of OTOCs.
We point out that a classical analog of the Sachdev-Ye-Kitaev (SYK) model, a solvable model of quantum many-body chaos, was studied long ago in the turbulence literature. Motivated by the Navier-Stokes equation in the turbulent regime and the nonlinear Schrödinger equation describing plasma turbulence, in which there is mixing between many different modes, the random coupling model has a Gaussian-random coupling between any four of a large number N of modes. The model was solved in the 1960s, before the introduction of large-N path-integral techniques, using a method referred to as the direct interaction approximation. We use the path integral to derive the effective action for the model. The large-N saddle gives an integral equation for the two-point function, which is very similar to the corresponding equation in the SYK model. The connection between the SYK model and the random coupling model may, on the one hand, provide new physical contexts in which to realize the SYK model and, on the other hand, suggest new models of turbulence and techniques for studying them.
Abstract The Sachdev–Ye–Kitaev (SYK) model is a strongly coupled, quantum many-body system that is chaotic, nearly conformally invariant, and exactly solvable. This remarkable and, to date, unique combination of properties have driven the intense activity surrounding the SYK model and its applications within both high energy and condensed matter physics. In this review we give an introduction to the SYK model and recent developments connected to it. We discuss: SYK and tensor models as a new class of large N quantum field theories, the near-conformal invariance in the infrared, the computation of correlation functions, generalizations of the SYK model, and applications to AdS/CFT and strange metals.
The SYK model of Sachdev, Ye and Kitaev is a recent model of N Majorana fermions coupled via quartic random interaction. It has the extraordinary properties of being exactly solvable at large N , having emergent conformal symmetry in the IR, and exhibiting maximal quantum chaos. The confluence of these properties make SYK a highly attractive candidate for a solvable AdS/CFT correspondence. What’s more, it was found to capture elements of non-Fermi-liquid behavior in strange metals. This thesis introduces the SYK model and discusses a recent variant, the SYK∗ model of Marcus and Vandoren. In the latter, the effective action is derived and branches of conformal solutions are discussed. The parameter space of SYK∗ is investigated for large and small M . The former reveals a phase separation and a N critical point in the T−M plane is claimed to exist.
The Sachdev-Ye-Kitaev (SYK) model is a quantum-mechanical model of fermions interacting with
q-body random couplings. For
q=2, it describes free particles and is nonchaotic in the many-body sense, while for
q>2 it is strongly interacting and exhibits many-body chaos. In this work we study the entanglement entropy (EE) of the
SYKq models for a bipartition of
N real or complex fermions into subsystems containing
2m real/
m complex fermions and $N\ensuremath{-}2m$/$N\ensuremath{-}m$ fermions in the remainder. For the free model
SYK2, we obtain an analytic expression for the EE, derived from the $\ensuremath{\beta}$-Jacobi random matrix ensemble. Furthermore, we use the replica trick and path-integral formalism to show that the EE is maximal when one subsystem is small, i.e., $m\ensuremath{\ll}N$, for arbitrary
q. We also demonstrate that the EE for the SYK4 model is noticeably smaller than the Page value when the two subsystems are comparable in size, i.e.,
m/N is
O(1). Finally, we explore the EE for a model with both SYK2 and SYK4 interactions and find a crossover from SYK2 (low-temperature) to SYK4 (high-temperature) behavior as we vary energy.