We compute the time-dependent complexity of the thermofield double states by four different proposals: two holographic proposals based on the “complexity-action” (CA) conjecture and “complexity-volume” (CV) conjecture, and two quantum field theoretic proposals based on the Fubini-Study metric (FS) and Finsler geometry (FG). We find that four different proposals yield both similarities and differences, which will be useful to deepen our understanding on the complexity and sharpen its definition. In particular, at early time the complexity linearly increase in the CV and FG proposals, linearly decreases in the FS proposal, and does not change in the CA proposal. In the late time limit, the CA, CV and FG proposals all show that the growth rate is 2E/(πℏ) saturating the Lloyd’s bound, while the FS proposal shows the growth rate is zero. It seems that the holographic CV conjecture and the field theoretic FG method are more correlated.

We provide a detailed examination of a thermal out-of-time-order correlator (OTOC) growing exponentially in time in systems without chaos. The system is a one-dimensional quantum mechanics with a potential whose part is an inverted harmonic oscillator. We numerically observe the exponential growth of the OTOC when the temperature is higher than a certain threshold. The Lyapunov exponent is found to be of the order of the classical Lyapunov exponent generated at the hilltop, and it remains non-vanishing even at high temperature. We adopt various shape of the potential and find these features universal. The study confirms that the exponential growth of the thermal OTOC does not necessarily mean chaos when the potential includes a local maximum. We also provide a bound for the Lyapunov exponent of the thermal OTOC in generic quantum mechanics in one dimension, which is of the same form as the chaos bound obtained by Maldacena, Shenker and Stanford.

We study a notion of operator growth known as Krylov complexity in free and interacting massive scalar quantum field theories in d-dimensions at finite temperature. We consider the effects of mass, one-loop self-energy due to perturbative interactions, and finite ultraviolet cutoffs in continuous momentum space. These deformations change the behavior of Lanczos coefficients and Krylov complexity and induce effects such as the “staggering” of the former into two families, a decrease in the exponential growth rate of the latter, and transitions in their asymptotic behavior. We also discuss the relation between the existence of a mass gap and the property of staggering, and the relation between our ultraviolet cutoffs in continuous theories and lattice theories.

We study out-of-time-order correlators (OTOCs) of rotating BTZ black holes using two different approaches: the elastic eikonal gravity approximation, and the Chern-Simons formulations of 3-dimensional gravity. Within both methods the OTOC is given as a sum of two contributions, corresponding to left and right moving modes. The contributions have different Lyapunov exponents, $$ {\lambda}_L^{\pm }=\frac{2\pi }{\beta}\frac{1}{1\mp \ell \Omega} $$ λ L ± = 2 π β 1 1 ∓ ℓ Ω , where Ω is the angular velocity and ℓ is the AdS radius. Since $$ {\lambda}_L^{-}\le \frac{2\pi }{\beta}\le {\lambda}_L^{+} $$ λ L − ≤ 2 π β ≤ λ L + , there is an apparent contradiction with the chaos bound. We discuss how the result can be made consistent with the chaos bound if one views the parameters β _± = β(1 ∓ ℓΩ) as the effective inverse temperatures of the left and right moving modes.

We investigate the Page curve for a nonstandard black hole which is asymptotically nonflat/AdS/dS. For this purpose, we apply the island prescription to the charged linear dilaton black holes and analyze, in detail, the entanglement entropy of Hawking radiation for both the nonextremal case and the extremal case. In the nonextremal case, we find the Page curve consistent with the unitarity principle: at early times the entanglement entropy grows linearly in time without the island and at late times it saturates to double of the Bekenstein-Hawking entropy in the presence of the island. We observe the Page time is universal for all different models studied by our method: $t_{\text{Page}}=\frac{3}{\pi c}\frac{S_{\mathrm{BH}}}{T_H}$. For the extremal case, the island prescription provides the well-defined entanglement entropy only with the island, which cannot be obtained from the continuous limit of the nonextremal case. This implies that the Page curve may not be reproduced for the extremal case and further investigation is needed.