  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.

Quantum Physics;Strongly Correlated Electrons