[Paper Review] Local Hilbert Space Fragmentation and Out-of-Time-Ordered Crystals

Berislav Buča|arXiv (Cornell University)|Aug 30, 2021

Quantum many-body systems2 references4 citations

TL;DR

This paper introduces 'OTOC crystals'—quantum many-body systems exhibiting genuine many-body continuous time translation symmetry breaking via persistent oscillations in generalized out-of-time-ordered correlation functions (OTOCs). Using strictly local dynamical symmetries, it establishes a lower bound on these oscillations and demonstrates that such time-translation symmetry breaking is stable under local unitary and dissipative perturbations, with an XYZ Creutz ladder as a concrete example.

ABSTRACT

Quantum many-body models with both Hilbert space fragmentation and non-stationarity have recently been identified. Hilbert space fragmentation does not immediately imply non-stationarity. However, strictly local dynamical symmetries directly imply non-stationarity. It is demonstrated here that these symmetries are equivalent to local fragmentation into spatially localized blocks. Using strictly local dynamical symmetries, a lower bound is given here for persistent oscillations of generalised out-of-time-ordered correlation functions (OTOCs). A novel notion of genuinely many-body continuous time translation symmetry breaking is provided by demanding non-trivial spatial modulation of the Fourier transform of the OTOC. Such non-trivial spatial-temporal dynamics stems from a perpetual backflow of quantum scrambling. Here we call systems with time-translation symmetry breaking in the OTOC, OTO crystals. This breaking cannot be realised by systems with a single effective degree of freedom (e.g. spin precession). Furthermore, the breaking is stable to all local unitary and dissipative perturbations. An XYZ Creutz ladder is presented as an example.

Motivation & Objective

• To identify conditions under which Hilbert space fragmentation leads to non-stationary dynamics in quantum many-body systems.
• To establish a connection between strictly local dynamical symmetries and local Hilbert space fragmentation into spatially localized blocks.
• To define and characterize a novel form of continuous time translation symmetry breaking in many-body systems through non-trivial spatial modulation of the OTOC's Fourier transform.
• To prove the stability of this time-translation symmetry breaking under all local unitary and dissipative perturbations.
• To provide a concrete realization of such a system using an XYZ Creutz ladder model.

Proposed method

• Identifies strictly local dynamical symmetries as the origin of non-stationarity in fragmented Hilbert spaces.
• Demonstrates equivalence between local dynamical symmetries and local Hilbert space fragmentation into spatially localized blocks.
• Derives a lower bound on persistent oscillations of generalized OTOCs using the structure of local dynamical symmetries.
• Introduces a criterion for genuine many-body continuous time translation symmetry breaking based on non-trivial spatial modulation in the Fourier transform of the OTOC.
• Analyzes the stability of the time-translation symmetry breaking under arbitrary local unitary and dissipative perturbations.
• Constructs and analyzes an XYZ Creutz ladder as a concrete example of a system exhibiting OTOC crystals.

Experimental results

Research questions

• RQ1 Under what conditions does Hilbert space fragmentation lead to non-stationary dynamics in quantum many-body systems?
• RQ2 How are strictly local dynamical symmetries related to local Hilbert space fragmentation and persistent OTOC oscillations?
• RQ3 What defines a genuine many-body continuous time translation symmetry breaking in quantum systems beyond single-degree-of-freedom models?
• RQ4 Why is the proposed time-translation symmetry breaking stable under all local unitary and dissipative perturbations?
• RQ5 Can the OTOC crystal phase be realized in a concrete lattice model with tunable interactions?

Key findings

• Strictly local dynamical symmetries are shown to be equivalent to local Hilbert space fragmentation into spatially localized blocks.
• A lower bound is established for persistent oscillations in generalized out-of-time-ordered correlation functions (OTOCs) due to local dynamical symmetries.
• A novel phase—'OTOC crystals'—is defined by non-trivial spatial modulation in the Fourier transform of the OTOC, indicating genuine many-body continuous time translation symmetry breaking.
• This time-translation symmetry breaking cannot be realized in systems with a single effective degree of freedom, such as spin precession.
• The OTOC crystal phase is stable to all local unitary and dissipative perturbations, indicating robustness against local decoherence and control.
• An XYZ Creutz ladder is presented as a concrete model realizing the OTOC crystal phase with the predicted persistent OTOC oscillations and non-trivial spatial-temporal dynamics.