Holographic dual effective field theory in the Luttinger-Ward functional approach: Application to an SYK model

Abstract

We construct an emergent holographic dual description in the Luttinger-Ward functional approach, where the renormalization group (RG) flows of collective bi-local fields appear manifestly in the bulk effective action with an emergent extra dimension. This holographic dual effective field theory reproduces $1/N$ quantum corrections in a self-consistent manner when we take the UV limit in the bulk effective action. Going into the IR regime in the extra dimension, we observe that a partial class of the field theoretic $1/N$, $1/N^{2}$, ... quantum corrections are resummed in the all-loop order and reorganized to form a holographic dual effective field theory in a large $N$ fashion living on the one-higher dimensional spacetime. In this study, we apply this theoretical framework into an Sachdev-Ye-Kitaev (SYK) model. Taking the large $N$ limit in the holographic dual effective field theory, we obtain nonlinearly coupled second-order bulk differential equations of motion for the three bi-local order-parameter fields of fermion self-energy, Green's function, and polarization function. Here, both UV and IR boundary conditions are derived self-consistently from the boundary effective action. We solve these highly intertwined nonlinear differential equations based on the so called matching method. Our ansatz for the bi-local order-parameter fields coincide with the conformally invariant solution of the field theoretic large $N$ limit in the UV limit, but their overall coefficients $RG-flow$ along the extra dimensional space, respectively, reflecting effects of higher-order quantum corrections. As a result, we find an insulating behavior, where the self-energy diverges at IR. ...