The Banks-Fischler-Shenker-Susskind (BFSS) model is one of the popular microscopic models of black hole dynamics, capable of explaining how black holes scramble information. Studies of the BFSS model suggest that a microscopic mechanism of information scrambling by black holes is related to their intrinsically chaotic dynamics. In particular, BFSS model at high temperatures becomes very similar to dimensionally reduced Yang-Mills theory, a strongly nonlinear theory of strong interactions between nuclei. Strong nonlinearity of Yang-Mills theory gives rise to chaotic classical dynamics, with the distance between any two trajectories diverging exponentially with time – a generic phenomenon known as Lyapunov instability.

In quantum mechanics, a counterpart of classical-mechanical Lyapunov instability is the exponential growth of the so-called out-of-time-order correlators (OTOCs) of quantum-mechanical operators.

In this project, we studied OTOCs in a supersymmetric quantum-mechanical system very similar to the BFSS model. Our particular focus was the low-temperature regime, where classical-mechanical description becomes inaccurate, and the non-supersymmetric Yang-Mills theory is known to be in a trivial gapped phase.

Using exact diagonalization techniques, we were able to demonstrate the chaotic behaviour OTOCs of our BFSS-like model down to lowest temperatures, in agreement with the expected duality with black holes. Furthermore, we demonstrated an expected transition between the graviton gas, the Schwarzschild black hole, and the D0-brane regimes in the model. To this end, we looked at fluctuations in the spacing ΔEi between energy level, considered as a statistical system for a large number of levels. Characterizing these fluctuations in terms of r-ratios ri = min(ΔEi-1, ΔEi)/max(ΔEi-1, ΔEi), with 0 < r < 1, we identified 3 different regimes as a function of energy E: a) the graviton gas regime at low energies, with no signatures of chaos in level spacing (E < 1 on the plot above),b) the intermediate Schwarzschild black hole regime with 1 < E < 100, where statistical fluctuations of r set in, and c) the high-energy black-brane regime with a fully developed classical chaos.

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