Holographic entanglement renormalisation for fermionic quantum matter

TL;DR

This paper explores how a holographic dimension emerges in 2D Dirac fermions through entanglement scaling under RG transformations, revealing topological transitions, quantum wormholes, and connections to physical properties like the Luttinger volume.

Contribution

It analytically demonstrates the emergence of a holographic space, topological transitions, and geometric structures in fermionic quantum matter using entanglement and RG analysis.

Findings

Holographic dimension emerges in 2D Dirac fermions via entanglement scaling.

Topological transition involves a critical Fermi surface and quantum wormhole formation.

Entanglement measures relate to physical properties like Luttinger volume and Chern number.

Abstract

We demonstrate the emergence of a holographic dimension in a system of 2D non-interacting Dirac fermions placed on a torus, by studying the scaling of multipartite entanglement measures under a sequence of renormalisation group (RG) transformations applied in momentum space. Geometric measures defined in this emergent space can be related to the RG beta function of the spectral gap, hence establishing a holographic connection between the spatial geometry of the emergent spatial dimension and the entanglement properties of the boundary quantum theory. We prove, analytically, that changing the boundedness of the holographic space involves a topological transition accompanied by a critical Fermi surface in the boundary theory. We go on to show that this results in the formation of a quantum wormhole geometry that connects the UV and the IR of the emergent dimension. The additional conformal symmetry at the transition also supports a relation between the emergent metric and the stress-energy tensor. In the presence of an Aharonov-Bohm flux, the entanglement gains a geometry-independent piece which is shown to be topological, sensitive to changes in boundary conditions, and related to the Luttinger volume of the system. Upon the insertion of a strong transverse magnetic field, we show that the Luttinger volume is linked to the Chern number of the occupied single-particle Landau levels.