A theory of minimal updates in holography

TL;DR

This paper explores how minimal changes in quantum Hamiltonians affect their holographic ground state representations, revealing that differences are confined to a specific causal region, which has implications for understanding influence in renormalization group flows.

Contribution

It introduces the concept of directed influence in holographic renormalization, showing that minimal Hamiltonian updates only alter the holographic description within a specific causal cone.

Findings

Holographic descriptions differ only inside the past causal cone of the region.

Directed influence relates to Wilson's numerical renormalization group for impurity problems.

Translation invariance can be exploited despite local Hamiltonian differences.

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 $\left.| \psi \right\rangle$ and $\left.| \tilde{\psi} \right\rangle$. We observe that, even though $\left.| \psi \right\rangle$ and $\left.| \tilde{\psi} \right\rangle$ 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.