A theory of minimal updates in holography

TL;DR

The paper develops a theory of minimal updates for holographic MERA representations when local changes are made to the Hamiltonian, showing that holographic descriptions can be adjusted only within the causal cone of the modified region. By introducing directed influence in RG flow and connecting it to Wilson's impurity RG, it provides a principled, scale-aware mechanism for updating only the relevant part of the holographic network, yielding modular MERA descriptions for defects and interfaces. The work integrates the Wilson chain formalism with MERA to justify localized updates and demonstrates their accuracy and efficiency, with broader implications for AdS/CFT and practical simulations of critical systems with impurities. Overall, it offers a concrete framework for translating local Hamiltonian perturbations into localized holographic changes, enabling scalable and modular descriptions of inhomogeneous quantum many-body systems.

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 $\left.| ψ\right\rangle$ and $\left.| \tildeψ \right\rangle$. We observe that, even though $\left.| ψ\right\rangle$ and $\left.| \tildeψ \right\rangle$ 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.

Figures (11)

• Figure 1: (a) MERA tensor network for the ground state $\hbox{$| \psi \rangle$}$ of a lattice Hamiltonian $H$ in $d=1$ space dimensions (modified binary scheme of Ref. 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 $\hbox{$| \tilde{\psi} \rangle$}$ of the Hamiltonian $\tilde{H} = H + H_{\mathcal{R}}^{\hbox{\tiny imp}}$ 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.
• Figure 2: 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. BoundaryMERA. More complex systems, such as (b) an interface, and (c) a $Y$-junction, can be similarly described by a 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 LongCFT2.
• Figure 3: (a) Tensors inside the causal cone $\mathcal{C}(\mathcal{R})$ in $d=1$ dimensions. Site $s$ of the Wilson chain $\mathcal{L}^{\hbox{\tiny W}}_{\mathcal{R}}$ 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 $H^{\hbox{\tiny W}}_{\mathcal{R}}$. (b) Equivalent construction in $d=2$ dimensions. In this case, 12 effective sites at scale $s$ become a single site of $\mathcal{L}^{\hbox{\tiny W}}_{\mathcal{R}}$, whereas each MPS tensor $v$ corresponds to five tensors $(u,w,w,w,w)$. (c)-(d) Directed influence: changing the Wilson chain Hamiltonian $H^{\hbox{\tiny W}}_{\mathcal{R}}$ on site $s^{*}$ results in a new ground state MPS where only tensors $v_s$ for scales $s\geq s^*$ are updated.
• Figure 4: (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 $\mathcal{L}^{\hbox{\tiny W}}_{\mathcal{S}}$. (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 $H^{\hbox{\tiny W}}_{\mathcal{S}}$ and $\tilde{H}^{\hbox{\tiny W}}_{\mathcal{S}}$ 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 $s\geq s^{*}$.
• Figure 5: (a) A region of $l_s \gg 1$ sites is coarse-grained under a layer of MERA to a smaller region of $l_{s+1} \approx l_s /2$ sites. (b) A region of $l_s =2$ sites is coarse-grained under a layer of MERA to a region of equivalent width, i.e. $l_{s+1} = l_s = 2$. (c) The width of the causal cone $\mathcal{C}(\mathcal{R})$ of a region $\mathcal{R}$ comprised of $l_0 \gg 1$ sites shrinks with increasing scale $s$ until the crossover scale $s^{\textrm{c}}\approx \log_2(l_0)$ is reached, after which it remains stationary.