Complexity growth with Lifshitz scaling and hyperscaling violation

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.

Holographic complexity in Vaidya spacetimes. Part I

“…However, as noted above, transient violations of the proposed bound were already identified in studying the time evolution of complexity in an eternal black hole background [48]. Further, even stronger violations were found in the dual of a noncommutative gauge theory [47] and in hyperscaling violating geometries [51,52]. 31 Therefore, while the proposed bound can not be universal, it remains an interesting question to understand the situations when it does apply and when not, and the underlying reasons for this.…”

“…We might observe that a similar statement holds for the calculations with the CV proposal (1.1), where the extremal volume was found by evaluating separately the corresponding equations of motion inside and outside of the shell. Further in passing, we note that the vanishing of the gravitational action for the spacetime region containing the null fluid shell, was an implicit assumption in various previous studies of holographic complexity, e.g., [50][51][52].…”

“…In all of these cases, it was assumed that the contribution of the (infinitely thin) null shell was zero, as we explicitly demonstrated in section 2.2, and so the WDW action was determined by adding together the action evaluated separately on the regions above and below the shell, as in our calculation. It is particularly interesting to compare [51] and [52], which both studied holographic complexity in hyperscaling violating geometries, but the first did not include the counterterm while the second did. The same simple but ad hoc prescription for the normalization of the null boundary normals was chosen in [51] as in [50], i.e.,α = α.…”

“…In some of the other recent studies of the CA proposal with null shells, the counterterm (2.30) was included [52,93] but in other, it was not [51,55]. In all of these cases, it was assumed that the contribution of the (infinitely thin) null shell was zero, as we explicitly demonstrated in section 2.2, and so the WDW action was determined by adding together the action evaluated separately on the regions above and below the shell, as in our calculation.…”

Complexity growth in flat spacetimes

“…In higher dimensions, this rate is not a constant and approaches the Lloyd's bound from above. This result is similar to the higher dimensional asymptotically AdS black holes and different from the Lifshitz and the hyperscaling violating geometries that this bound has violated [53]. This shows that although BMSFTs are ultrarelativistic theories, they are more similar to the relativistic theories than to the nonrelativistic ones.…”