A theory of minimal updates in holography

Abstract

Consider two quantum critical Hamiltonians $H$ and $\tilde{H}$ on a $d$-dimensional lattice that only differ in some region $\mathcal{R}$. We study the relation between holographic representations, obtained through real-space renormalization, of their corresponding ground states $|\psi ⟩$ and $|\tilde{\psi }⟩$. We observe that, even though $|\psi ⟩$ and $|\tilde{\psi }⟩$ disagree significantly both inside and outside region $\mathcal{R}$, they still admit holographic descriptions that only differ inside the past causal cone $\mathcal{C}(\mathcal{R})$ of region $\mathcal{R}$, where $\mathcal{C}(\mathcal{R})$ is obtained by coarse-graining region $\mathcal{R}$. We argue that this result follows from a notion of directed influence in the renormalization group flow that is closely connected to the success of Wilson's numerical renormalization group for impurity problems. At a practical level, directed influence allows us to exploit translation invariance when describing a homogeneous system with e.g. an impurity, in spite of the fact that the Hamiltonian is no longer invariant under translations.

main text: 5 pages, 4 figures, appendices: 7 pages, 7 figures. Revised for greater clarity

References in corpus (6)

• The numerical renormalization group method for quantum impurity systems
• A class of quantum many-body states that can be efficiently simulated
• Time Evolution of Entanglement Entropy from Black Hole Interiors
• Entanglement renormalization, scale invariance, and quantum criticality
• Quantum MERA Channels
• Algorithms for entanglement renormalization: boundaries, impurities and interfaces

Cited by in corpus (13)

• Surface/State Correspondence as a Generalized Holography
• Tensor network renormalization yields the multi-scale entanglement renormalization ansatz
• Tensor Networks from Kinematic Space
• Bidirectional holographic codes and sub-AdS locality
• Algorithms for tensor network renormalization
• Hyper-invariant tensor networks and holography
• Algorithms for entanglement renormalization: boundaries, impurities and interfaces
• A defect in holographic interpretations of tensor networks
• Matrix product state renormalization
• Conformal Properties of Hyperinvariant Tensor Networks
• Disentangling Interacting Systems with Fermionic Gaussian Circuits: Application to Quantum Impurity Models
• Entanglement compression in scale space: from the multiscale entanglement renormalization ansatz to matrix product operators
• Entanglement renormalization for quantum fields with boundaries and defects