Generalized 1 + 0-dimensional Liouvillean dynamics describing deformations of the Sachdev–Ye–Kitaev (SYK) model, as well as the various 1 + 1-dimensional dilaton and Horava–Lifshitz gravity theories, can all be mapped onto the single-particle quantum mechanics of a non-relativistic charge propagating in a (generally, curved) 2d space and subject to a (generally, non-uniform) magnetic field. The latter description sets a stage for the phenomenon of quantum Hall effect (QHE), thereby elucidating the intrinsically topological nature of the pertinent gravity theories and demystifying their (pseudo)holographic connection to a broad class of the SYK-like models.
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 out of equilibrium dynamics of the Sachdev-Ye-Kitaev model (SYK), comprising
N Majoranas with random all-to-all four-body interactions, minimally coupled to a Markovian bath modeled by the Lindblad formalism, displays intriguing nontrivial features. In particular, the decay rate towards the steady state is a non-monotonic function of the bath coupling
μ, and an analogue of the Loschmidt echo for dissipative quantum systems undergoes a first order dynamical phase transitions that eventually becomes a crossover for sufficiently large
μ. We provide evidence that these features have their origin in the presence of exceptional points in the purely real eigenvalues of the SYK Liouvillian closest to the zero eigenvalue associated with the steady state. An analytic calculation at small
N, supported by numerical results for larger
N, reveals that the value of
μ∼0.1 at which the exceptional point corresponding to the longest living modes occurs is close to a local maximum of the decay rate. This value marks the start of a region of anomalous equilibration where the relaxation rate diminishes as the coupling to the bath becomes stronger. Moreover, the mentioned change from transition to crossover in the Loschmidt echo occurs at a larger
μ∼0.3 corresponding with a proliferation of exceptional points in the low energy limit of the Liouvillian spectrum. We expect these features to be generic in the approach to equilibrium in quantum strongly interacting many-body Liouvillians.
Abstract We study the chaos exponent of some variants of the Sachdev-Ye-Kitaev (SYK) model, namely, the
N N = 1 supersymmetry (SUSY)-SYK model and its sibling, the (N|M)-SYK model which is not supersymmetric, for arbitrary interaction strength. We find that for large q the chaos exponent of these variants, as well as the SYK and the
N N = 2 SUSY-SYK model, all follow a single-parameter scaling law. By quantitative arguments we further make a conjecture, i.e. that the found scaling law might hold for general one-dimensional (1D) SYK-like models with large q. This points out a universal route from maximal chaos towards completely regular or integrable motion in the SYK model and its 1D variants.
The Sachdev-Ye-Kitaev (SYK) model is a quantum-mechanical model of
N Majorana fermions which displays a number of appealing features---solvability in the strong coupling regime, near-conformal invariance and maximal chaos---which make it a suitable model for black holes in the context of the
AdS/CFT holography. In this paper, we show for the colored SYK model and several of its tensor model cousins that the next-to-leading order in the large-
N expansion preserves the conformal invariance of the two-point function in the strong-coupling regime, up to the contribution of the pseudo-Goldstone bosons due to the explicit breaking of the symmetry which are already seen in the leading-order four-point function. We also comment on the composite field approach for computing correlation functions in colored tensor models.