\section{Introduction}
The renormalization group (RG) \cite{RGA, RGB, RGC}, fundamental to our conceptual understanding of quantum field theory and critical phenomena, is also the basis of important approaches to many-body problems. In RG methods, the microscopic Hamiltonian of an extended system is simplified through a sequence of coarse-graining transformations until a fixed point of the RG flow is reached. From this scale invariant fixed point, the universal, low energy properties of the phase can then be extracted. The RG is also at the core of certain holographic constructions, where the many-body system is regarded as the boundary of another system in one additional dimension corresponding to scale. A prominent example is the AdS/CFT duality \cite{AdSCFTA, AdSCFTB, AdSCFTC} of string theory, where a conformal field theory (CFT) in $d+1$ space-time dimensions is dual to a gravity theory in anti-de Sitter (AdS) space-time in $d+2$ dimensions.

Entanglement renormalization \cite{ERA, ERB} is a modern formulation of real-space RG for quantum systems on a lattice, based on the removal of short-range entanglement at each coarse-graining step. By concatenating coarse-graining transformations, one obtains the multi-scale entanglement renormalization ansatz (MERA) \cite{MERA}, an efficient tensor network representation of the many-body ground state, see Fig. \ref{fig:DirectedMERA}(a). The MERA spans an additional dimension corresponding to RG scale and is thus regarded as a lattice realization of holography \cite{WhatIsHolography, Swingle}. Importantly, this tensor network (and generalizations thereof \cite{BranchingA, BranchingB}) is expected to produce a holographic description of \textit{any} many-body system and, in particular, it is not restricted to operate in the so-called \textit{strong coupling, large-$N$} regime that produces a \textit{weakly coupled, semi-classical} gravity dual --- as required in many practical applications of the AdS/CFT correspondence \cite{AdSCFTA, AdSCFTB, AdSCFTC}. As a result, the MERA is a promising tool to gain insights into the structure of holography for a generic many-body system \cite{Swingle, Swingle2, BranchingA, BranchingB, Metric, AllA, AllB, AllC, AllD, AllE, Singh, LongCFT2}, regardless of whether it has e.g. a weakly coupled, semi-classical gravity dual. For instance, in Ref. \onlinecite{LongCFT2} the authors already used the MERA to explore the \textit{modular} character of holography ---namely the possibility of building a holographic description of a complex system by stitching together pieces (or modules) corresponding to the holographic description of simpler systems--- and apply it to the study of critical systems with impurities, boundaries, interfaces, and $Y$-junctions.

In this paper we propose a theory of \textit{minimal updates} in holography. Specifically, we address the following question: Given the ground states $\ket{\psi}$ and $\ket{\tilde{\psi}}$ of two Hamiltonians $H$ and $\tilde{H}$ that only differ in a region $\mathcal{R}$ of a $d$-dimensional lattice $\mathcal{L}$ \cite{dVersusdPlus1}, how much do we have to modify the holographic description of $\ket{\psi}$ in order to produce a holographic description of $\ket{\tilde{\psi}}$? We claim that the answer to this question can be formulated in simple geometric terms: \textit{ A holographic description for $\ket{\tilde{\psi}}$ can be obtained by modifying that of $\ket{\psi}$ only in the causal cone $\mathcal{C}(\mathcal{R})$ of region $\mathcal{R}$}, where $\mathcal{C}(\mathcal{R})$ is the part of the holographic description that traces the evolution of the region $\mathcal{R}$ under coarse-graining. This claim, supported by abundant numerical evidence \cite{LongCFT2}, will be justified here theoretically in terms of a notion of \textit{directed influence} in the RG flow, which we argue to also underpin the success of Wilson's numerical renormalization group (NRG) \cite{RGA, RGB, RGC, NRGreview} for impurity problems. Directed influence leads to an extremely compact, accurate holographic representation of a critical system with an impurity by minimally updating the MERA of a homogeneous system. More generally, as argued in Ref. \onlinecite{LongCFT2}, directed influence implies the modular character of holography.

\begin{figure}[!tbph]
\begin{center}
\includegraphics[width=8.5cm]{F1DirectedMERA.eps}
\caption{ (a) MERA tensor network for the ground state $\ket{\psi}$ of a lattice Hamiltonian $H$ in $d=1$ space dimensions (modified binary scheme of Ref. \onlinecite{LongCFT1}). Scale and translation invariance result in a compact description: two tensors ($u,w$) are repeated throughout the infinite tensor network.
(b) The ground state $\ket{\tilde{\psi}}$ of the Hamiltonian $\tilde{H} = H + H_{\mathcal{R}}^{\imp}$ is represented by a MERA with the same tensors ($u,w$) outside the causal cone $\mathcal C(\mathcal R)$ (shaded). Inside, scale invariance implies again a very compact description: two new tensors ($\tilde{u},\tilde{w}$) repeated throughout the semi-infinite causal cone. (c-d) The same illustrations, without drawing the tensors of the network.}
\label{fig:DirectedMERA}
\end{center}
\end{figure}

For concreteness, let us consider a hypercubic lattice $\mathcal{L}$ in $d$ space dimensions, and a particular MERA for the ground states $\ket{\psi}$ and $\ket{\tilde{\psi}}$ of $H$ and $\tilde{H}$ based on a coarse-graining transformation that maps a hyper-cubic block of $2^d$ sites into one effective site, as illustrated for $d=1$ in Fig. \ref{fig:DirectedMERA}(a) \cite{ManyMERASchemes}. We emphasize that the MERA describes both the ground state of the system and a sequence of coarse-graining transformations, where the latter are labeled with a scale parameter $s$, with $s\in \{0,1,2,\cdots\}$. To simplify the notation, we will assume that $H$ is a translation invariant, quantum critical Hamiltonian corresponding to a fixed-point of the RG flow, so that $\ket{\psi}$ is invariant both under translations and changes of scale. Accordingly the MERA for $\ket{\psi}$ can be completely specified by a single pair ($u,w$) of tensors that are repeated throughout the entire tensor network \cite{Giovannetti, Pfeifer, LongCFT1}. [However, the proposed minimal updates do not require translation or scale invariance.]

\section{Causal cones}
The causal cone $\mathcal{C}(\mathcal{R})$ of a region $\mathcal{R}$ of the lattice $\mathcal{L}$, see Fig. \ref{fig:DirectedMERA}(b), was originally defined as the part of the holographic tensor network that can affect the properties of the state $\ket{\psi}$ in region $\mathcal{R}$ \cite{MERA}. The peculiar structure of causal cones in the MERA is the key reason why one can \textit{efficiently} compute expectation values of local observables from this tensor network \cite{Algorithms}. Here we argue that the causal cone $\mathcal{C}(\mathcal{R})$ also defines the region of the MERA that needs to be updated in order to account for a change of the Hamiltonian in region $\mathcal{R}$, Fig. \ref{fig:DirectedMERA}(c)-(d). We emphasize that this new role of the causal cones, of clear physical significance and (as we will argue) ultimately connected to the existence of different energy scales in the Hamiltonian $H$, is unrelated to the computational considerations that guided the design of the MERA \cite{MERA,Algorithms}.
Geometrically, the causal cone $\mathcal{C}(\mathcal{R})$ is the part of the tensor network that contains the evolution of region $\mathcal{R}$ under successive coarse-graining transformations. To further simplify the analysis, we will assume that $\mathcal{R}$ is a hyper-cubic region $\mathcal{R}$ made of $2^d$ sites \cite{ModularityAlsoForLargeR}. This region can be see to be mapped into an identical hyper-cubic region with $2^d$ sites under coarse-graining transformations (see Appendix \ref{ApdxA}).

\section{Minimal update}
Let us now consider the ground state $\ket{\tilde{\psi}}$ of Hamiltonian $\tilde{H} = H + H_{\mathcal{R}}^{\imp}$, where $H_{\mathcal{R}}^{\imp}$ accounts for an impurity on the hyper-cubic region $\mathcal{R}$ made of $2^d$ sites. Our claim is that a MERA for $\ket{\tilde{\psi}}$ can be obtained from the MERA for the ground state $\ket{\psi}$ in the absence of the impurity by simply replacing, inside the causal cone $\mathcal{C}(\mathcal{R})$, the tensors ($u$, $w$) with new tensors. Specifically, if the impurity is itself already a new RG fixed-point (e.g. a conformal defect in a CFT \cite{ConformalDefect}), which implies that $\ket{\tilde{\psi}}$ is still scale invariant, then the entire causal cone $\mathcal{C}(\mathcal{R})$ can be completely specified by a single new pair ($\tilde{u}$, $\tilde{w}$) of tensors, see Fig. \ref{fig:DirectedMERA}(b).

In Ref. \onlinecite{LongCFT2,BoundaryMERA} we have presented abundant numerical evidence supporting the validity of the proposed minimal update, and have argued that this construction naturally reproduces: ($i$) the power-law scaling of expectation values of local observables (e.g. of the local magnetization in the case of a magnetic impurity) with the distance to the impurity; and ($ii$) the set of new scaling operators and scaling dimensions attached to the impurity \cite{ConformalDefect}. The above compact description in terms of just two pairs of tensors $\{(u, w), (\tilde{u}, \tilde{w})\}$, valid even in the thermodynamic limit, is somewhat surprising. After all, one would expect that coarse-graining the impurity system, which is not translation invariant, would require the use of different coarse-graining tensors $(u(x), w(x))$ at different locations $x$ of lattice $\mathcal{L}$. Accordingly, the number of variational parameters, proportional to the number of different tensors in the MERA, would grow linearly in the size of the system. Instead, by only updating the causal cone $\mathcal{C}(\mathcal{R})$ [which amounts to exploiting the translation invariance of Hamiltonian $H$ to describe the ground state of $\tilde{H}$] we can address an impurity system directly in the thermodynamic limit, and thus avoid finite size effects when extracting the universal properties of the critical impurity \cite{BoundaryMERA,LongCFT2}. 

Similar constructions are also possible for more complex systems, including systems with a boundary, an interface or a $Y$-junction, see Fig. \ref{fig:Defects}, for which recursive application of minimal updates leads to the modular MERA, as discussed in \cite{LongCFT2}. Below we shall argue that the validity of the proposed minimal update follows a more fundamental property of RG flows, that we call directed influence. In order to discuss the latter, we must first introduce an effective lattice model that describes the causal cone $\mathcal{C}(\mathcal{R})$.

\begin{figure}[!tbph]
\begin{center}
\includegraphics[width=8.5cm]{F2Defects.eps}
\caption{
Updating only the causal cone of $\mathcal{R}$ also produces a simple holographic description of a scale-invariant boundary, in terms of tensors $(u,w)$ in the bulk and a boundary tensor $\tilde{w}$, as described in Ref. \onlinecite{BoundaryMERA}. More complex systems, such as (b) an interface, and (c) a $Y$-junction, can be similarly described by a \textit{modular} MERA, consisting of a bulk tensors $(u_{\alpha}, w_{\alpha})$ for each type of material $\alpha$ ($\alpha = A,B,\cdots$) and defect tensors $(\tilde{u},\tilde{w})$ that glue the different modules together \cite{LongCFT2}.
}
\label{fig:Defects}
\end{center}
\end{figure}

\section{Wilson chain}
We call the \textit{Wilson chain} of region $\mathcal{R}$, denoted $\mathcal{L}^{\Wilson}_{\mathcal{R}}$, the semi-infinite, one-dimensional lattice built by coarse-graining the $d$-dimensional lattice $\mathcal{L}$ by all the tensors in the MERA that lay outside the causal cone $\mathcal{C}(\mathcal{R})$. More precisely, each site of the Wilson chain $\mathcal{L}^{\Wilson}_{\mathcal{R}}$ is uniquely labeled by a value of the scale parameter $s$ and it collects together all the effective sites at scale $s$ obtained through the above coarse-graining, see Fig. \ref{fig:DirectedMPS}(a)-(b). By construction, site $s$ of $\mathcal{L}^{\Wilson}_{\mathcal{R}}$ effectively represents the $O(2^{ds})$ sites of $\mathcal{L}$ located roughly at a distance [measured in lattice spacing] $2^s$ away from region $\mathcal{R}$. Thus, progressing from site $s$ to site $s+1$ of the Wilson chain $\mathcal{L}^{\Wilson}_{\mathcal{R}}$ corresponds to simultaneously increasing length scale and moving away from region $\mathcal{R}$.

\begin{figure}[!tbph]
\begin{center}
\includegraphics[width=8.5cm]{F3DirectedMPS.eps}
\caption{
(a) Tensors inside the causal cone $\mathcal{C}(\mathcal{R})$ in $d=1$ dimensions. Site $s$ of the Wilson chain $\mathcal{L}^{\Wilson}_{\mathcal{R}}$ corresponds to the two effective sites at scale $s$. By replacing three tensors $(u,w,w)$ with a single tensor $v$, we obtain an MPS representation of the ground state of $H^{\Wilson}_{\mathcal{R}}$.
(b) Equivalent construction in $d=2$ dimensions. In this case, 12 effective sites at scale $s$ become a single site of $\mathcal{L}^{\Wilson}_{\mathcal{R}}$, whereas each MPS tensor $v$ corresponds to five tensors $(u,w,w,w,w)$.
(c)-(d) Directed influence: changing the Wilson chain Hamiltonian $H^{\Wilson}_{\mathcal{R}}$ on site $s^{*}$ results in a new ground state MPS where only tensors $v_s$ for scales $s\geq s^*$ are updated.
}
\label{fig:DirectedMPS}
\end{center}
\end{figure}

The Wilson chain is equipped with an effective Hamiltonian $H^{\Wilson}_{\mathcal{R}}$, obtained by coarse-graining $H$, of the form
\begin{equation}\label{eq:hWilson}
    H^{\Wilson}_{\mathcal{R}} = h_{[0]} + \sum_{s=0}^{\infty} \Lambda^{-s} h^{\Wilson}_{[s,s+1]}.
\end{equation}
The nearest neighbor term $\Lambda^{-s} h^{\Wilson}_{[s,s+1]}$ consists of a two-site hermitian operator $h^{\Wilson}_{[s,s+1]}$ that is \textit{independent} of $s$, multiplied by a negative power of an amplitude $\Lambda > 1$, which takes the value $\Lambda=2^{z}$, where $z$ is the dynamic critical exponent of $H$ (e.g. $z=1$ for Lorentz invariant quantum critical points), see Appendix \ref{ApdxD} for details and also Refs. \onlinecite{LongCFT2,BoundaryMERA} for complimentary derivations of the effective Hamiltonian for the Wilson chain.

The structure of the one-dimensional Hamiltonian $H^{\Wilson}_{\mathcal{R}}$, with exponentially decaying nearest-neighbor terms, is similar to that obtained by Wilson as part of his resolution of the Kondo impurity problem -- a single impurity in a three dimensional bath of three fermions \cite{RGA, RGB, RGC}. However, we note that while having a free fermion bath was key in Wilson's derivation of an effective one-dimensional lattice model, here we use the MERA to (at least in principle) address non-perturbatively any type of $d$-dimensional bath.

\section{Directed influence} Following Wilson's NRG method \cite{RGA, RGB, RGC, NRGreview} (see Appendix \ref{ApdxB}), the ground state of $H^{\Wilson}_{\mathcal{R}}$ can be obtained by identifying, progressing iteratively over $s$, the low energy subspace $\mathbb{H}_s$ of the first $s+1$ sites of $\mathcal{L}^{\Wilson}_{\mathcal{R}}$,
\begin{equation}\label{eq:Hsub}
    \mathbb{H}_s \subseteq \mathbb{V}_0 \otimes \mathbb{V}_1 \otimes \cdots \otimes \mathbb{V}_s,
\end{equation}
where $\mathbb{V}_s$ is the vector space of site $s$ in $\mathcal{L}^{\Wilson}_{\mathcal{R}}$. More specifically, $\mathbb{H}_s$ is chosen (by means of a suitable energy minimization) to be the low energy subspace of $\mathbb{H}_{s-1} \otimes \mathbb{V}_{s}$, and is characterized by a linear map $v_s$,
\begin{equation}\label{eq:v}
    v_s:\mathbb{H}_s \rightarrow \mathbb{H}_{s-1} \otimes \mathbb{V}_s.
\end{equation}
Then the tensors $\{v_1,v_2, \cdots\}$ form a \textit{matrix product state} (MPS) \cite{MPSA, MPSB} representation of the ground state of $H^{\Wilson}_{\mathcal{R}}$.

For the present purposes, the most important feature of the NRG method is that the low energy subspace $\mathbb{H}_s$ (equivalently, tensor $v_s$) only depends on the restriction of the Hamiltonian to sites $\{0,1, \cdots, s\}$,
\begin{equation}\label{eq:terms}
    h_{[0]} + \Lambda^0 h^{\Wilson}_{[0,1]} + \Lambda^{-1} h^{\Wilson}_{[1,2]} + \cdots + \Lambda^{-s+1} h^{\Wilson}_{[s-1,s]},
\end{equation}
and not on the Hamiltonian terms related to larger length scales. In other words, if we modify the Hamiltonian at some site $s^*$, then NRG produces an MPS representation of the new ground state where only the tensors $v_s$ for $s\geq s^*$ are modified, see Fig. \ref{fig:DirectedMPS}(c)-(d). That is, assuming the validity of the NRG approach, changes in the Hamiltonian at length scale $s^{*}$ only affect the ground state representation at larger length scales, a property that we refer to as \textit{directed influence} in the RG flow. We emphasize that the validity of Wilson's NRG, and thus also directed influence, relies heavily on the factor $\Lambda^{-s}$ to induce a separation of energy scales in the problem. When such a separation of energy scales is present, then the treatment of one energy scale at a time as prescribed by the NRG approach can be justified from perturbation theory (see Appendix \ref{ApdxB}). In the absence of such a factor the NRG approach would typically fail \cite{DMRG}, such that directed influence would also fail, and a change in the Hamiltonian at length scale $s^{*}$ could affect the ground state representation at all length scales $s$.

We are finally ready to show that the validity of the proposed minimal update in the MERA follows from assuming the validity of directed influence in Wilson chains. Let us modify the Hamiltonian from $H$ to $\tilde{H} = H + H^{\imp}_{\mathcal{R}}$ to account for an impurity, and study how the ground state MPS for different Wilson chains (corresponding to different regions of lattice $\mathcal{L}$) must react to this change according to directed influence.

\begin{figure}[!tbph]
\begin{center}
\includegraphics[width=8.5cm]{F4MERAproof.eps}
\caption{
(a) The causal cones $\mathcal{C}(\mathcal{R})$ and $\mathcal{C}(\mathcal{S})$ for two regions $\mathcal{R}$ and $\mathcal{S}$ separated by $r$ sites become coincident at scale $s^* \approx \log_2(r)$.
(b) MPS representation of the ground state in the Wilson chain $\mathcal{L}^{\Wilson}_{\mathcal{S}}$.
(c) In the presence of an impurity in region $\mathcal{R}$, a minimal update change only the tensors inside $\mathcal{C}(\mathcal{R})$. This amounts to changing the tensors inside $\mathcal{C}(\mathcal{S})$ only at scales $s\geq s^{*}$.
(d) Directed influence justifies this minimal update of tensors in the MERA: Hamiltonians $H^{\Wilson}_{\mathcal{S}}$ and $\tilde{H}^{\Wilson}_{\mathcal{S}}$ only differ at scale $s^{*}$, and therefore the tensors in the causal cone $\mathcal{C}(\mathcal{S})$ indeed only need to be updated at length scales $s\geq s^{*}$.
}
\label{fig:MERAproof}
\end{center}
\end{figure}

First, we notice that the effective Hamiltonian of the Wilson chain $\mathcal{L}^{\Wilson}_{\mathcal{R}}$ for the causal cone $\mathcal{C}(\mathcal{R})$ is modified from $H^{\Wilson}_{\mathcal{R}}$ in Eq. \ref{eq:hWilson} to
\begin{equation}\label{eq:htildeWilson}
    \tilde{H}^{\Wilson}_{\mathcal{R}} = \tilde{h}_{[0]} + \sum_{s=0}^{\infty} \Lambda^{-s} h^{\Wilson}_{[s,s+1]}.
\end{equation}
where $\tilde{h}_{[0]} = h_{[0]} + H^{\imp}_{\mathcal{R}}$ includes the impurity Hamiltonian. In this case, directed influence tells us to change all the tensors in the MPS representation of the ground state of $\tilde{H}^{\Wilson}_{\mathcal{R}}$. This is equivalent to the announced modification of all the tensors in the causal cone $\mathcal{C}(\mathcal{R})$ of the MERA representation of the ground state $\ket{\tilde{\psi}}$ of $\tilde{H}$.

Second, let us consider the causal cone $\mathcal{C}(\mathcal{S})$ of another small region $\mathcal{S}$ of the original lattice $\mathcal{L}$ that is diplaced $\vec r$ from $\mathcal R$, and the corresponding Wilson chain $\mathcal{L}^{\Wilson}_{\mathcal{S}}$, see Fig. \ref{fig:MERAproof}. Here we see that the effective Hamiltonians $H^{\Wilson}_{\mathcal{S}}$ and $\tilde{H}^{\Wilson}_{\mathcal{S}}$, corresponding to the homogeneous and impurity systems respectively, only differ at the length scale $s^* \sim \log_2 |\vec r|$, where the causal cones $\mathcal{C}(\mathcal{R})$ and $\mathcal{C}(\mathcal{S})$ become coincident. Directed influence implies that the MPS representations of the ground states of $H^{\Wilson}_{\mathcal{S}}$ and $\tilde{H}^{\Wilson}_{\mathcal{S}}$ only need to differ at scales $s\geq s^*$, i.e. that the tensors in $\mathcal C(\mathcal S)$ at scales $s<s^*$, which lie outside of the causal cone $\mathcal C(\mathcal R)$, may be left unchanged. This argument is general for any local region $\mathcal S$, thus justifying the proposal of minimal updates in MERA.

\section{Discussion} The holographic description of a many-body system based on real-space RG is not unique. Since the MERA is built by concatenating several coarse-graining transformations, there is indeed some freedom as to how we choose to coarse-grain the system at a given length scale, provided that we compensate for our choice when coarse-graining the system at larger length scales. The minimal update discussed in this paper corresponds to a particular choice of this freedom in coarse-graining. By restricting the update to the causal cone $\mathcal{C}(\mathcal{R})$ of region $\mathcal{R}$, an impurity that is initially localized in space remains localized in space under coarse-graining, and this leads to a very efficient holographic description of $\ket{\tilde{\psi}}$ \cite{NoDMRG}. We conclude with remarks on how the structure of minimal updates in the MERA may translate into a property of the AdS/CFT correspondence \cite{AdSCFTA, AdSCFTB, AdSCFTC}. It is natural to speculate that the non-uniqueness of MERA descriptions, originating in the freedom existing in real-space coarse-graining, is closely related to diffeomorphism invariance in the bulk of the gravity dual. Accordingly, the minimal updates discussed in this paper would be possible also in the gravity dual of the AdS/CFT correspondence after a proper choice of gauge. However, making these ideas more concrete may first require a better understanding of the bulk metric in the MERA \cite{Metric,Swingle2}.

The authors thank Davide Gaiotto, Rob Myers, and Brian Swingle for insightful comments. G.E. is supported by the Sherman Fairchild Foundation.

