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arXiv:1706.05362 · Journal of High Energy Physics

More on supersymmetric and 2d analogs of the SYK model

Jeff Murugan, Douglas Stanford, Edward Witten

Published Aug 1, 2017• 220 citations
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Abstract

In this paper, we explore supersymmetric and 2d analogs of the SYK model. We begin by working out a basis of (super)conformal eigenfunctions appropriate for expanding a four-point function. We use this to clarify some details of the 1d supersymmetric SYK model. We then introduce new bosonic and supersymmetric analogs of SYK in two dimensions. These theories consist of N fields interacting with random q-field interactions. Although models built entirely from bosons appear to be problematic, we find a supersymmetric model that flows to a large N CFT with interaction strength of order one. We derive an integral formula for the four-point function at order 1/N , and use it to compute the central charge, chaos exponent and some anomalous dimensions. We describe a problem that arises if one tries to find a 2d SYK-like CFT with a continuous global symmetry.

DOI
10.1007/jhep08(2017)146

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In this paper, we explore supersymmetric and 2d analogs of the SYK model. We begin by working out a basis of (super)conformal eigenfunctions appropriate for expanding a four-point function. We use this to clarify some details of the 1d supersymmetric SYK model. We then introduce new bosonic and supersymmetric analogs of SYK in two dimensions. These theories consist of N fields interacting with random q-field interactions. Although models built entirely from bosons appear to be problematic, we find a supersymmetric model that flows to a large N CFT with interaction strength of order one. We derive an integral formula for the four-point function at order 1/N , and use it to compute the central charge, chaos exponent and some anomalous dimensions. We describe a problem that arises if one tries to find a 2d SYK-like CFT with a continuous global symmetry.
N=2 \mathcal{N}=2 N=2 SYK model in the superspace formalism
Abstract We use superspace methods to study an SYK-like model with N=2 \mathcal{N}=2 N=2 N = 2 supersymmetry in one dimension, and an analog of this model in two dimensions. We find the four-point function as an expansion in the basis of eigenfunctions of the Casimir of su(1, 1|1). We also find retarded kernels and Lyapunov exponents for both cases.
Correlators in the N = 2 N=2 \mathcal{N}=2 N=2 supersymmetric SYK model
We study correlation functions in the one-dimensional N=2 \mathcal{N}=2 N=2 supersymmetric SYK model. The leading order 4-point correlation functions are computed by summing over ladder diagrams expanded in a suitable basis of conformal eigenfunctions. A novelty of the N=2 \mathcal{N}=2 N=2 model is that both symmetric and antisymmetric eigenfunctions are required. Although we use a component formalism, we verify that the operator spectrum and 4-point functions are consistent with N=2 \mathcal{N}=2 N=2 supersymmetry. We also confirm the maximally chaotic behavior of this model and comment briefly on its 6-point functions.
Abstract We investigate a model closely related to both the original Sachdev-Ye-Kitaev (SYK) model and the N \mathcal{N} N N = 1 supersymmetric SYK model. It consists of N real Majorana fermions and M auxiliary bosons with Yukawa interactions. We consider the large N and M limit and keep the ratio M/N fixed. The model has two branches characterized by the conformal dimensions of fields, which we compute as a function of the ratio M/N. One of the branches contains the supersymmetric saddle for M = N. As we take the limit M/N → ∞ both branches coincide and we obtain the same conformal dimensions as SYK. Furthermore, we determine the Lyapunov exponent of the model and find maximal chaos independent of M/N.
Abstract In this paper, we introduce an N=1 \mathcal{N}=1 N=1 N = 1 supersymmetric SYK model with SO(q) global symmetry. We study the large N expansion of the bi-local collective action of our model. At strong coupling limit, this model exhibits a super-reparametrization symmetry, and the SO(q) global symmetry is enhanced to a SO^(q) \widehat{\mathrm{SO}}(q) SO(q) SO ^ q local symmetry. The corresponding symmetry algebra is the semi-direct product of the super-Virasoro and the super-Kac-Moody algebras. These emergent symmetries are spontaneously and explicitly broken, which leads to a low energy effective action: super-Schwarzian action plus an action of a super-particle on the SO(q) group manifold. We analyze the zero mode contributions to the chaotic behavior of four point functions in various SO(q) channels. In singlet channel, we show that the out-of-time-ordered correlators related to bosonic bi-locals exhibit the saturation of the chaos bound as in the non-SUSY SYK model. On the other hand, we find that the ones with fermionic bi-locals in the singlet channel have πβ \frac{\pi }{\beta } βπ​ π β Lyapunov exponent. In the anti-symmetric channel, we demonstrate that the out-of-time-ordered correlator related to a SO(q) generator grows linearly in time. We also compute the non-zero mode contributions which give consistent corrections to the leading Lyapunov exponents from the zero modes.
In this thesis, we discuss the Sachdev-Ye-Kitaev (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 two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold 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.
We investigate a model closely related to both the original Sachdev-Ye-Kitaev (SYK) model and the mathcalN=1\\mathcal{N}=1mathcalN=1 supersymmetric SYK model. It consists of NNN real Majorana fermions and MMM auxiliary bosons with Yukawa interactions. We consider the large NNN and MMM limit and keep the ratio M/NM/NM/N fixed. The model has two branches characterized by the conformal dimensions of fields, which we compute as a function of the ratio M/NM/NM/N. One of the branches contains the supersymmetric saddle for M=NM=NM=N. Furthermore, we determine the Lyapunov exponent of the model and find maximal chaos independent of M/NM/NM/N.