Entanglement Renormalization and Holography

TL;DR

The paper presents a framework in which the entanglement structure of quantum many-body states defines an emergent higher-dimensional geometry, inspired by holographic duality. By combining entanglement renormalization with a tensor-network description, it shows how scale decomposition yields a discrete AdS-like geometry at criticality and black hole–like horizons at finite temperature, connecting entropy bounds to bulk geodesics. It bridges concrete lattice constructions (Ising model with disentanglers/isometries) and the gauge/gravity duality, suggesting that bulk fields and dynamics may be read from renormalization-group flows of the boundary theory. The approach provides qualitative and semi-quantitative links between entanglement, geometry, and holography, while outlining future directions to make the dual gravity description more explicit. Overall, it proposes a versatile, information-theoretic route to holography applicable to a wide range of quantum states beyond traditional conformal field theories.

Abstract

I show how recent progress in real space renormalization group methods can be used to define a generalized notion of holography inspired by holographic dualities in quantum gravity. The generalization is based upon organizing information in a quantum state in terms of scale and defining a higher dimensional geometry from this structure. While states with a finite correlation length typically give simple geometries, the state at a quantum critical point gives a discrete version of anti de Sitter space. Some finite temperature quantum states include black hole-like objects. The gross features of equal time correlation functions are also reproduced in this geometric framework. The relationship between this framework and better understood versions of holography is discussed.

Figures (5)

• Figure 1: The tensor network structure of entanglement renormalization. Circles are lattice sites at various coarse grained scales. Squares with four lines are unitary disentaglers and triangles with three lines are isometric coarse graining transformations. The network shown here represents a $2\rightarrow 1$ coarse graining scheme and has a characteristic fractal structure. In principle, each tensor can be different, but translation and scale invariance can provide strong constraints. This network implements a renormalization group transformation that is local in space and scale. This transformation has the important property that it coarse grains local operators into local operators.
• Figure 2: Curved boxes represent primitive "cells" of the higher dimensional bulk geometry.
• Figure 3: Causal cones of different blocks after a single layer of disentanglers and isometries. The causal cone of a large block decreases by roughly a factor of two in width (30 sites to 16 sites) after one coarse graining. On the other hand, causal cones for small blocks may grow slightly. Once the causal cone of a large block has reached a width of order one, it stops shrinking.
• Figure 4: Upper left: a piece of the causal cone of a small block. Upper right: reversing the flow to proceed from three sites to six sites. Lower left: shaded sites are outside the causal cone of the two site block and can be traced out. Lower right: four sites remain and we can now apply the next layer of disentanglers to reach the two site block of interest.
• Figure 5: Sketch of minimal curves for the zero temperature gapped and finite temperature critical geometries. The curve, defined by the boundary of the causal cone of a large block, quickly falls to the factorization scale and then runs along it. In the gapped case, the length is zero at the factorization scale, and the entropy has only a boundary contribution which saturates for large blocks. However, in the finite temperature case, the horizon scale has finite size and gives an extensive contribution to the entropy of a large block in the ultraviolet.