Abstract
We study the complexity of holographic superconductors (Einstein-Maxwell-complex scalar actions in d + 1 dimension) by the “complexity = volume” (CV) conjecture. First, it seems that there is a universal property: the superconducting phase always has a smaller complexity than the unstable normal phase below the critical temperature, which is similar to a free energy. We investigate the temperature dependence of the complexity. In the low temperature limit, the complexity (of formation) scales as Tα, where α is a function of the complex scalar mass m2, the U(1) charge q, and dimension d. In particular, for m2 = 0, we find α = d−1, independent of q, which can be explained by the near horizon geometry of the low temperature holographic superconductor. Next, we develop a general numerical method to compute the time-dependent complexity by the CV conjecture. By this method, we compute the time-dependent complexity of holographic superconductors. In both normal and superconducting phase, the complexity increases as time goes on and the growth rate saturates to a temperature dependent constant. The higher the temperature is, the bigger the growth rate is. However, the growth rates do not violate the Lloyd’s bound in all cases and saturate the Lloyd’s bound in the high temperature limit at a late time.
Highlights
In the CV conjecture, the complexity of a state |ψ(tL, tR) corresponds to the maximal volume of the codimension-one surface connecting the codimension-two time slices at two AdS boundaries:
We study the complexity of holographic superconductors (Einstein-Maxwellcomplex scalar actions in d + 1 dimension) by the “complexity = volume” (CV) conjecture
In the low temperature limit, the complexity of formation scales as T α, where α is a function of the complex scalar mass m2, the U(1) charge q, and dimension d
Summary
In the CV conjecture, the complexity of a state |ψ(tL, tR) corresponds to the maximal volume of the codimension-one surface connecting the codimension-two time slices (denoted by tL and tR) at two AdS boundaries:. The holographic time-dependent complexity for the AdSd+1-Schwarzschild or AdSd+1-RN blackhole geometry has been studied in [64] These works focus on the late time behavior and rely on the analytic solutions. The growth rates do not violate the Lloyd’s bound (intuitively meaning the fastest computing time [70,71,72]) in all cases and saturates the Lloyd’s bound in the high temperature limit at late time. This result is a nontrivial support for the conjecture that the Schwarzshield black is the fastest quantum computer of the same energy. Let us recall that the absolute value of the complexity growth rate in the CA conjecture can be arbitrarily large and does not satisfy the Lloyd’s bound [38, 64].3 our results indicate that, from the perspective of the Lloyd’s bound, the CV conjecture is more suitable than the CA conjecture as a definition of the holographic complexity
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