The effect of Dilaton on the holographic complexity growth

TL;DR

The paper studies how a dilaton field coupled to a Maxwell sector alters holographic complexity growth under CA duality in two settings: an AdS-dilaton black hole and a Lifshitz-like black brane. In AdS, the dilaton slows the complexity rate and the full-time evolution approaches the late-time bound from above, consistent with prior work but with new full-time dynamics. In Lifshitz backgrounds, the Lloyd bound is violated at late times, and the complete time evolution exhibits noteworthy features for large dynamical exponent z, including approach from above. The results illuminate how dilaton dynamics and anisotropic scaling influence CA-based complexity, with implications for Lloyd bound revisions, CV proposals, and the role of thermodynamic volume in holography.

Abstract

In this paper, we investigate the action growth in various backgrounds in Einstein-Maxwell-Dilaton theory. We calculate the full time evolution of action growth in AdS dilaton black hole and find it approaches the late time bound from above. We investigate the black hole which is asymptotically Lifshitz and obtain its late-time and full time behavior. We find the violation of Lloyd bound in late time limit and show the full time behavior exhibiting some new features for z sufficiently large.

Figures (4)

• Figure 1: WdW patch of the time before(left side) and after (right side) the critical time, we assume boundary time satisfy the relation $t_{L}=t_{R}$, and at the right(left) boundary, bulk time flows in the same(opposite) direction as the boundary, in calculating the bulk contribution of the total action ,we partition the spacetime into three regions
• Figure 2: There are two parameters, $y=\frac{\mu Q}{2M}$, $z=r_{+}/L$. For this picture, we fix $z=1$, the green line correspond to $y=0.1$, blue line $y=0.2$, red line $y=0.3$. we find the complexity growth rate approaches the late time bound from above.
• Figure 3: WdW patch of Lifshitz black hole
• Figure 4: The relation between complexity growth rate and boundary time, we choose the normalization factor $\alpha=L r_{+}^{z}$, and green/ blue/red lines correspond respectively to $z=3$/$z=2$/$z=1$, we find it approaches the late time bound from above, and for z sufficiently large, the complexity growth rate experience a decreasing period during early time