Holographic Complexity of Einstein-Maxwell-Dilaton Gravity

TL;DR

This work investigates holographic complexity in Einstein-Maxwell-Dilaton gravity with hyperscaling violation, applying both the complexity=action (CA) and complexity=volume (CV) dualities. It derives a late-time CA growth rate $\frac{\delta I}{\delta t}=2E\left(1+\frac{z-1}{d-\theta}\right)$, showing an enhancement for $z>1$ and a violation of the standard bound, while CV growth exhibits a different temperature scaling. The authors construct simple tensor-network models, including $(d-\theta)$-dimensional networks and branching MERA, that reproduce the hyperscaling-violating growth behavior and connect RG flow to complexity. They also demonstrate the switchback effect in shockwave geometries and discuss subtleties arising from null-surface discontinuities in the action computation. Overall, CA and CV respond differently to hyperscaling violation, and the work links holographic complexity to concrete tensor-network constructions, suggesting avenues for UV-complete realizations and refined bounds.

Abstract

We study the holographic complexity of Einstein-Maxwell-Dilaton gravity using the recently proposed "complexity = volume" and "complexity = action" dualities. The model we consider has a ground state that is represented in the bulk via a so-called hyperscaling violating geometry. We calculate the action growth of the Wheeler-DeWitt patch of the corresponding black hole solution at non-zero temperature and find that, in the presence of violations of hyperscaling, there is a parametric enhancement of the action growth rate. We partially match this behavior to simple tensor network models which can capture aspects of hyperscaling violation. We also exhibit the switchback effect in complexity growth using shockwave geometries and comment on a subtlety of our action calculations when the metric is discontinuous at a null surface.

Figures (6)

• Figure 1: The shaded part of the Penrose diagram shows a WDW patch of a Cauchy surface that intersects $r=\infty$ at A and B. The past and future horizon are represented by dashed lines, while the past and future singularity are represented by wave lines.
• Figure 2: Illustration of the change of the WDW patch after evolving $t_L$ by $\delta t$. The Penrose diagram for an EMD black hole shares a similar structure to that of the AdS-Schwartzschild black hole, and the deviation does not affect the analysis hereafter.
• Figure 3: An illustration of a system described in Section \ref{['stack']} with $d=3$, $\theta=1$. The two dimensional networks are aligned along the one dimensional vertical line. The dots are the sites of the lattice on which the Hamiltonian acts, and the ellipses denote the intermediate layers that are not drawn in the figure.
• Figure 4: An illustration of a branching tensor network with $\theta=2, s=4$. The circles at each layer represent copies of the thermal density matrix of the corresponding Hamiltonian. After $n=\log_2 \left(\frac{\xi}{a}\right)$ RG steps, we get $2^{n\theta}$ copies of the density matrices $\rho(H^{(\xi)})$, corresponding to the Hamiltonian on a lattice length $\xi$, where $\xi$ is the correlation length.
• Figure 5: Illustration of the Penrose diagram for the shockwave geometry. Left (\ref{['nonintersect']}): the patch only intersects with future singularity; Right (\ref{['intersect']}): the patch intersects with both singularities. The bulk regions whose actions are time-independent are colored in green while those whose actions are time-dependent are colored in blue.