Browsing by Author "Kuusela, Pyry"
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Kuusela, Pyry (2017)In this thesis, we discuss the SachdevYeKitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the twopoint function are all socalled iterated melonic diagrams. This allows us to derive a SchwingerDyson equation for the twopoint function, which, in turn, can be solved exactly in the infrared limit. We also consider the fourpoint function. In the large N limit, the leadingorder correction to the fourpoint function is given by socalled ladder diagrams. This allows us to derive an explicit expression for the fourpoint function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through qfold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.
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