# Non-local Geometry inside Lifshitz Horizon

## TL;DR

This paper derives a holographic dual geometry for a fermionic vector model at finite charge density, revealing a Lifshitz horizon and a non-local interior that encodes the Fermi sea shape.

## Contribution

It introduces a novel holographic dual geometry with a Lifshitz horizon and non-local interior, derived from quantum renormalization group analysis of the fermionic vector model.

## Key findings

- Lifshitz geometry outside the horizon with higher-spin hair
- Horizon at finite radial coordinate due to RG flow obstruction
- Non-local algebraic structure inside the horizon encodes Fermi sea shape

## Abstract

Based on the quantum renormalization group, we derive the bulk geometry that emerges in the holographic dual of the fermionic U(N) vector model at a nonzero charge density. The obstruction that prohibits the metallic state from being smoothly deformable to the direct product state under the renormalization group flow gives rise to a horizon at a finite radial coordinate in the bulk. The region outside the horizon is described by the Lifshitz geometry with a higher-spin hair determined by microscopic details of the boundary theory. On the other hand, the interior of the horizon is not described by any Riemannian manifold, as it exhibits an algebraic non-locality. The non-local structure inside the horizon carries the information on the shape of the filled Fermi sea.