The article by Juan Maldacena and Douglas Stanford studies the Sachdev-Ye-Kitaev (SYK) model, a quantum mechanical system with $N$ Majorana fermions interacting randomly. The model is tractable in the large $N$ limit, where the classical variable is a bilocal fermion bilinear. At low energies, the model develops an emergent conformal symmetry, which is spontaneously broken to $SL(2,R)$, leading to zero modes. These zero modes are lifted by a small explicit breaking, producing an enhanced contribution to the four-point function, which displays a maximal Lyapunov exponent in the chaos region. This feature is expected to be universal for large $N$ quantum mechanics systems with emergent reparametrization symmetry.
The authors review the two-point functions in the large $N$ limit and discuss the simplified large $q$ limit, where the classical equations can be solved analytically. They derive an explicit integral expression for the four-point function in the infrared limit, showing that it is infinite in the strict conformal limit due to Nambu-Goldstone bosons. The explicit breaking of the conformal symmetry lifts these modes, leading to an enhanced contribution to the four-point function that saturates the chaos bound.
The paper also explores the effective theory of reparametrizations, the spectrum of the model, and the bulk interpretation of the fermion fields. The authors find that the enhanced non-conformal contribution agrees with the four-point function expected in a theory of dilaton gravity, while the $\mathcal{J}$-independent and finite in the conformal limit contributions provide information about the bulk states. They speculate that the bulk interpretation involves a string attached to the boundary of a single fermion, rather than multiple fermions in the bulk.
Finally, the authors discuss the free energy and the corrections to the conformal propagator, providing numerical results for the free energy at $q=32$ and the leading correction to the conformal two-point function.The article by Juan Maldacena and Douglas Stanford studies the Sachdev-Ye-Kitaev (SYK) model, a quantum mechanical system with $N$ Majorana fermions interacting randomly. The model is tractable in the large $N$ limit, where the classical variable is a bilocal fermion bilinear. At low energies, the model develops an emergent conformal symmetry, which is spontaneously broken to $SL(2,R)$, leading to zero modes. These zero modes are lifted by a small explicit breaking, producing an enhanced contribution to the four-point function, which displays a maximal Lyapunov exponent in the chaos region. This feature is expected to be universal for large $N$ quantum mechanics systems with emergent reparametrization symmetry.
The authors review the two-point functions in the large $N$ limit and discuss the simplified large $q$ limit, where the classical equations can be solved analytically. They derive an explicit integral expression for the four-point function in the infrared limit, showing that it is infinite in the strict conformal limit due to Nambu-Goldstone bosons. The explicit breaking of the conformal symmetry lifts these modes, leading to an enhanced contribution to the four-point function that saturates the chaos bound.
The paper also explores the effective theory of reparametrizations, the spectrum of the model, and the bulk interpretation of the fermion fields. The authors find that the enhanced non-conformal contribution agrees with the four-point function expected in a theory of dilaton gravity, while the $\mathcal{J}$-independent and finite in the conformal limit contributions provide information about the bulk states. They speculate that the bulk interpretation involves a string attached to the boundary of a single fermion, rather than multiple fermions in the bulk.
Finally, the authors discuss the free energy and the corrections to the conformal propagator, providing numerical results for the free energy at $q=32$ and the leading correction to the conformal two-point function.