[Paper Review] Entanglement Renormalization and Holography

Brian Swingle|DSpace@MIT (Massachusetts Institute of Technology)|May 8, 2009

Black Holes and Theoretical PhysicsPhysics and Astronomy221 citations

TL;DR

This paper proposes a holographic framework for quantum many-body states using entanglement renormalization, where spatial entanglement structure gives rise to an emergent higher-dimensional geometry. It shows that quantum critical points yield discrete anti-de Sitter space, finite-temperature states include black hole-like objects, and correlation functions are geometrically encoded, linking entanglement renormalization to holographic duality.

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.

Motivation & Objective

• To establish a connection between entanglement renormalization and holographic duality in quantum many-body systems.
• To demonstrate how the entanglement structure of quantum states can generate an emergent higher-dimensional geometry.
• To show that quantum critical points naturally give rise to discrete anti-de Sitter (AdS) space in the geometric construction.
• To incorporate finite-temperature states with black hole-like features into the geometric framework.
• To reproduce gross features of equal-time correlation functions using the emergent geometric description.

Proposed method

• Uses entanglement renormalization to coarse-grain quantum states by scale, organizing degrees of freedom in logarithmic scale intervals.
• Defines entanglement entropy contributions from each scale using a measure proportional to $ d ext{log}z = dz/z $, leading to a geometric scaling law.
• Constructs a discrete bulk geometry from the entanglement structure of a many-body state, with scale-dependent contributions to entanglement entropy.
• Applies the method to the 1D quantum Ising model and extends it to higher dimensions and other phases like topological order and superconductivity.
• Introduces a metric tensor in the emergent geometry derived from the renormalization group flow of the state.
• Models finite-temperature states by introducing a thermal scale, which leads to black hole-like horizons in the geometric picture.

Experimental results

Research questions

• RQ1 Can entanglement renormalization generate a higher-dimensional geometric structure from quantum many-body states?
• RQ2 Does the geometric construction reproduce key features of quantum criticality, such as logarithmic violation of the boundary law?
• RQ3 How do finite-temperature states appear in the emergent geometry, and can they exhibit black hole-like behavior?
• RQ4 Can equal-time correlation functions be geometrically encoded in this framework?
• RQ5 What is the relationship between this geometric picture and established holographic dualities in quantum gravity?

Key findings

• Quantum critical points in one-dimensional systems yield a discrete version of anti-de Sitter (AdS) space in the emergent geometry, consistent with holographic duality.
• Finite-temperature states in the construction include black hole-like objects, with a stretched horizon that corresponds to a hydrodynamic scale distinct from the true horizon.
• The entanglement entropy of a region scales as $ dS \sim \frac{L^{d_s-1}}{z^{d_s-1}} \frac{dz}{z} $, leading to a geometric description of entanglement across scales.
• The framework successfully reproduces the gross features of equal-time correlation functions through the emergent metric structure.
• The construction applies broadly to various quantum phases, including topological order, frustrated magnets, and superconductors, beyond just the Ising model.
• The geometric description is invariant under bulk diffeomorphisms, suggesting a deeper connection to gravitational degrees of freedom in the bulk.