Irreversibility, the butterfly effect, and scrambling

Thermalization of a closed system means, in particular, that any non-equilibrium state eventually evolves towards a state that is by no practical means distinguishable from a thermal state. Besides the fact that physical observables attain thermal expectation values, genuine thermalization requires irreversibility, because a thermal state contains no information about its past apart from a few conserved quantities. It is, however, an open question in the context of quantum many-body systems (especially those without a classical limit), what is a useful definition of irreversibility and which systems exhibit irreversible dynamics. As an outcome of my PhD project supervised by Stefan Kehrein we proposed to define irreversibility based on the decay of observable echoes under imperfect effective time reversal, which is realized, e.g., in nuclear magnetic resonance experiments or quantum simulators [1,2]. In these detailed numerical and analytical investigations, we revealed that in genuinely interacting quantum magnets practically inevitable perturbations in the time reversal protocol restrict the recovery of the initial condition to short times, whereas the dynamics of non-interacting systems is well reversible. In a follow-up collaboration with Dries Sels and Anatoli Polkovnikov we extended our approach to study a paradigmatic model for quantum chaos, the Sachdev-Ye-Kitaev model [3]. Using a novel semi-classical approach for fermionic systems, we established that our echo protocol probes the high temperature limit of the chaotic behavior, which had previously been identified through so-called out-of-time-order correlators.

In a second line of thought we investigated the scrambling of information in quantum lattice models [4]. The concept of scrambling originates in the discussion of the black hole information paradox and it captures how initially local information becomes inaccessible, because it is delocalizated under unitary dynamics. By tailoring a suited numerical scheme we were able systematically study with unprecedented system sizes the dynamics of tripartite information, a measure that indicates how much information is truly non- locally distributed. On this basis, we identified criteria for genuine scrambling and we found that the presence of interactions is sufficient to fully scramble the information, which is compatible with our findings from the studies of echo dynamics.

[1] M. Schmitt and S. Kehrein, Effective time reversal and echo dynamics in the transverse field Ising model, EPL 115 50001 (2016)

[2] M. Schmitt and S. Kehrein, Irreversible dynamics in quantum many-body systems, Phys. Rev. B 98, 180301(R) (2018)

[3] M. Schmitt, D. Sels, S. Kehrein, and A. Polkovnikov, Semiclassical echo dynamics in the Sachdev-Ye-Kitaev model, Phys. Rev. B 99, 134301 (2019)

[4] O. Schnaack, N. Bölter, S. Paeckel, S. R. Manmana, S. Kehrein, M. Schmitt, Tripartite information, scrambling, and the role of Hilbert space partitioning in quantum lattice models, Phys. Rev. B 100, 224302 (2019)