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In quantum many-body systems, complex dynamics delocalize the physical degrees of freedom. This spreading of information throughout the system has been extensively studied in relation to quantum thermalization, scrambling, and chaos. Locality is typically defined with respect to a tensor product structure (TPS) which identifies the local subsystems of the quantum system. In this paper, we investigate a simple geometric measure of operator spreading by quantifying the distance of the space of local operators from itself evolved under a unitary channel. We show that this TPS distance is related to the scrambling properties of the dynamics between the local subsystems and coincides with the entangling power of the dynamics in the case of a symmetric bipartition. Additionally, we provide sufficient conditions for the maximization of the TPS distance and show that the class of 2-unitaries provides examples of dynamics that achieve this maximal value. For Hamiltonian evolutions at short times, the characteristic timescale of the TPS distance depends on scrambling rates determined by the strength of interactions between the local subsystems. Beyond this short-time regime, the behavior of the TPS distance is explored through numerical simulations of prototypical models exhibiting distinct ergodic properties, ranging from quantum chaos and integrability to Hilbert space fragmentation and localization.
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From zero to positive entropy
In the sciences in general, the phrase “route to chaos” has come to refer to a metaphor when some physical, biological, economic, or social system transitions from one exhibiting order to one displaying randomness (or chaos). Sometimes the goal is to understand which universal mechanisms explain that transition, and how one can describe systems that operate in a region between order and complete chaos. In other words, the goal is to understand the mathematical processes by which a system evolves from one whose recurrent set is finite towards another one exhibiting chaotic behavior as parameters governing the behavior of the system are varied. This has only been understood for one-dimensional dynamics. The present note exposes new approaches that allow one to move away from those limitations. A tentative global framework toward describing a large class of two-dimensional dynamics, inspired partially by the developments in the one-dimensional theory of interval maps is discussed. More precisely, we present a class of intermediate smooth dynamics between one and higher dimensions. In this setting, it could be possible to develop a similar one-dimensional type approach and in particular to understand the transition from zero entropy to positive entropy.
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- Award ID(s):
- 1956022
- PAR ID:
- 10349972
- Editor(s):
- Daniela De Silva
- Date Published:
- Journal Name:
- Notices of the American Mathematical Society
- Volume:
- 69
- Issue:
- 5
- ISSN:
- 0002-9920
- Page Range / eLocation ID:
- 748-761
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/https://doi.org/10.1090/noti2478
