Exact holographic mapping and emergent space-time geometry

TL;DR

This paper introduces an exact holographic mapping from boundary lattice systems to bulk geometries, revealing how different boundary states produce various emergent space-time structures, including wormholes, with potential applications to interacting systems.

Contribution

It presents a novel exact unitary holographic mapping that constructs emergent bulk geometries from boundary lattice states, including entangled and thermal states.

Findings

Bulk geometry derived from correlation functions varies with boundary state.

Entangled chains produce worm-hole geometries in the bulk.

Quantum quenches map to worm-hole dynamics.

Abstract

In this paper, we propose an {\it exact holographic mapping} which is a unitary mapping from the Hilbert space of a lattice system in flat space (boundary) to that of another lattice system in one higher dimension (bulk). By defining the distance in the bulk system from two-point correlation functions, we obtain an emergent bulk space-time geometry that is determined by the boundary state and the mapping. As a specific example, we study the exact holographic mapping for $(1+1)$-dimensional lattice Dirac fermions and explore the emergent bulk geometry corresponding to different boundary states including massless and massive states at zero temperature, and the massless system at finite temperature. We also study two entangled one-dimensional chains and show that the corresponding bulk geometry consists of two asymptotic regions connected by a worm-hole. The quantum quench of the coupled chains is mapped to dynamics of the worm-hole. In the end we discuss the general procedure of applying this approach to interacting systems, and other open questions.