On the time dependence of holographic complexity

Abstract

Abstract:We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd's bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. For either conjecture, we find that the late time limit for the rate of change of complexity is saturated at times of the order of the inverse temperature. Adding a charge to the eternal black holes washes out the early time behaviour, i.e. complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.

On complexity of holographic flavors

“…where M is the mass of the system (check references [2]- [5] for some violations of this bound). Recently a holographic recipe has been proposed [5,6] to compute complexity for thermofield double states in strongly coupled quantum field theories.…”

Complexity growth in flat spacetimes

“…The null joint term has an ambiguity due to the normalization constant of the null vectors a. This ambiguity is the same as the ambiguity in the AdS case [33] and reveals the existence of the new length in BMSFT.…”

“…The corresponding computation of the rate of complexity growth in the context of AdS=CFT has been done in [33] (see also [34][35][36][37][38]). Therein, the gravitational action is evaluated in the background of eternal two-sided black holes.…”

“…The final answer of [33] for the rate of complexity growth is given in terms of bulk parameters. It is not difficult to take the flat-space limit from these results.…”

“…It is not difficult to take the flat-space limit from these results. One can check that taking the flat-space limit from the results of [33] yields the following expressions for the rate of complexity growth 2 :…”