Complexity Growth with Lifshitz Scaling and Hyperscaling Violation

TL;DR

Problem: understand holographic complexity growth in nontrivial scaling geometries with Lifshitz dynamical exponent $z$ and hyperscaling violation $\theta$. Approach: apply the complexity=action proposal to compute the full time dependence of the WDW patch action for both two-sided and one-sided black branes in Einstein-Maxwell-Dilaton theory, including bulk, joint, boundary, and counterterm contributions. Key findings: Lloyd's bound is violated for these exponents; the complexity growth saturates to a mass-like scale $2E$, with $E>M$ for $z\neq 1$; two-sided geometries saturate from above while one-sided saturate from below, with distinct early- and late-time behaviors. Significance: reveals universal features of complexity growth in anisotropic, hyperscaling-violating holographic theories and emphasizes the importance of counterterms and an effective dimension in the UV structure and thermalization.

Abstract

Using complexity=action proposal we study the growth rate of holographic complexity for Lifshitz and hyperscaling violating geometries. We will consider both one and two sided black branes in an Einstein-Maxwell-Dilaton gravitational theory. We find that in either case Lloyd's bound is violated and the rate of growth of complexity saturates to a value which is greater than twice the mass of the corresponding black brane. This value reduces to the mass of the black brane in the isotropic case. We show that in two sided black brane the saturation happens from above while for one sided black brane it happens from below.

Figures (4)

• Figure 1: WDW patch of a two sided black brane, moving forward in time assuming $t_R=t_L$ .
• Figure 2: Rate of the complexity growth in a WDW patch for two sided black brane. Left (right) panel shows different values of dynamical exponent for $d_e=1 (d_e=2)$. For each curve on both panels the Lloyd's bound is violated at some time before the curve reaches unity on the vertical axes.
• Figure 3: WDW patch of a one sided black brane, i.e., Vaidya geometry. The orange line is the in-falling collapsing null shell located at $v=0$.
• Figure 4: Rate of the complexity growth in a WDW patch for Vaidya geometry with $d_e=1$ (left) and $d_e=2$ (right).