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
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.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving
N Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The
N=1 model with a single supercharge has unbroken supersymmetry at large
N, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large
N equation, and also by performing numerical computations for small values of
N. We also compute the large
N spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an
N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large
N computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanical many body system with random all-to-all interactions on fermionic N sites (N>>1). This model is shown to saturate the known maximal chaos bound of many body system and then based on this observation it is conjectured to be dual to a quantum black hole in the sense of the AdS/CFT correspondence. In this dissertation, we show that the large N physics of the SYK model is systematically described by a single bi-local field. In particular, we emphasize the appearance of the emergent conformal reparametrization symmetry at the critical IR fixed point and the corresponding divergent contribution of the symmetry modes in the propagator of the bi-local field. We discuss non-linear-level derivation of the zero modes effective action, which is given by the Schwarzian derivative for finite reparametrization symmetry. Besides the symmetry modes, which correspond to the dilaton-gravity sector in the dual AdS theory, the SYK model also predicts an infinite tower of matter fields in AdS_2. We demonstrate that this infinite spectrum can be nicely packaged into a single field in 3-dimensional space-time. Finally, we consider the question of identifying the dual space-time of the SYK model. Focusing on the signature of emergent space-time of the (Euclidean) model, we explain the need for non-local (Radon-type) transformations on external legs of n-point Green's functions. This results in a dual theory with Euclidean AdS signature with additional leg-factors. We speculate that these factors incorporate the coupling of additional bulk states similar to the discrete states of 2D string theory.