Hierarchical generalization of dual unitarity

Xie-Hang Yu, Zhiyuan Wang, and Pavel Kos

Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany

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Quantum dynamics with local interactions in lattice models display rich physics, but is notoriously hard to study. Dual-unitary circuits allow for exact answers to interesting physical questions in clean or disordered one- and higher-dimensional quantum systems. However, this family of models shows some non-universal features, like vanishing correlations inside the light-cone and instantaneous thermalization of local observables. In this work we propose a generalization of dual-unitary circuits where the exactly calculable spatial-temporal correlation functions display richer behavior, and have non-trivial thermalization of local observables. This is achieved by generalizing the single-gate condition to a hierarchy of multi-gate conditions, where the first level recovers dual-unitary models, and the second level exhibits these new interesting features. We also extend the discussion and provide exact solutions to correlators with few-site observables and discuss higher-orders, including the ones after a quantum quench. In addition, we provide exhaustive parametrizations for qubit cases, and propose a new family of models for local dimensions larger than two, which also provides a new family of dual-unitary models.

The dynamics of extended systems with local interactions is the focal topic of research of different communities, such as statistical physics, condensed matter physics, quantum chaos, and high energy physics. The computational complexity of these dynamics necessitates the development of new solvable models to unravel many-body behaviors. Some of the most important models employed to this end are so-called dual-unitary circuits, which remain physical upon changing the roles of space and time. However, they still manifest certain non-universal features, including vanishing correlation functions inside the light cone and instantaneous thermalization of local observables.

To address these limitations, our work relaxes the dual-unitary condition into a hierarchy of conditions containing more and more gates where the dual-unitary circuit is the first level. Higher levels maintain a level of solvability and show more generic physical behavior. Thus our work paves the way for a deeper understanding of quantum chaotic dynamics and inspires the development of more intricate solvable models.

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Cited by

[1] Chuan Liu and Wen Wei Ho, "Solvable entanglement dynamics in quantum circuits with generalized dual unitarity", arXiv:2312.12239, (2023).

[2] Grace M. Sommers, Sarang Gopalakrishnan, Michael J. Gullans, and David A. Huse, "Zero-temperature entanglement membranes in quantum circuits", arXiv:2404.02975, (2024).

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[4] Alessandro Foligno, Pavel Kos, and Bruno Bertini, "Quantum information spreading in generalised dual-unitary circuits", arXiv:2312.02940, (2023).

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