Floquet Many-Body Cages

Abstract: Many-body cages have very recently emerged as a general route for nonergodic behaviour in quantum matter. Here, we show that new types of many-body cages can be engineered in Floquet circuits with the potential to realize novel nonequilibrium quantum states. For that purpose, we first identify an explicit, general construction of Floquet circuits capable of hosting many-body cages. We then present a generic strategy to engineer and structure Floquet many-body cages. We demonstrate the developed scheme for the quantum hard disk model as a generic constrained model system, realizable for instance in Rydberg atom arrays. We construct Floquet circuits yielding Floquet many-body cages with topological properties and $π$-quasienergy modes, implying `time crystalline' spatiotemporal order. Our results can be directly extended to general quantum circuits, thus providing a new tool to engineer nonequilibrium behaviour in driven systems.

• The paper introduces a framework for engineering nonergodic many-body cages using palindromic chiral-symmetric Floquet circuits in periodically driven quantum systems.
• It demonstrates that tailored Floquet drives enable robust localization features such as persistent Loschmidt echo oscillations and zero-/π-quasienergy bands in both IBRG and QHD models.
• The method paves the way for experimental quantum simulation, offering potential novel platforms for disorder-free time crystals and Fock-space topological order.

Floquet Many-Body Cages: Engineering Nonergodic States in Periodically Driven Quantum Systems

Introduction and Motivation

This paper presents a comprehensive framework for the construction, detection, and engineering of many-body cages (MBCs) in the context of Floquet quantum circuits. MBCs are special classes of many-body eigenstates, first identified in static (undriven) systems, that are localized on subgraphs of the many-body state (Fock) graph due to a combination of local constraints and quantum interference. These nonergodic states defy conventional thermalization, maintaining long-lived memory of initial conditions and supporting flat bands in the many-body spectrum.

Previous studies on nonergodic quantum matter have explored disorder-induced many-body localization (MBL) and Hilbert-space fragmentation, but MBCs introduce additional mechanisms for localization stemming from the structure of constrained quantum models. The central aim of this work is to establish both the existence and engineerability of MBCs in nonequilibrium, periodically driven (Floquet) systems—a challenge with profound implications for quantum simulation and control of nonthermal states.

Floquet Circuits and Chiral-Symmetric Drives

The main technical advance of the paper is a systematic approach to constructing Floquet drives that preserve the chiral symmetry essential for MBC formation. By composing periodic unitaries from layers of Hamiltonians each possessing chiral symmetry, and arranging them in a palindromic (time-symmetric) order, the authors guarantee that the effective Floquet Hamiltonian retains the requisite structural features.

Explicitly, the palindromic Floquet circuit

$U = U_1 U_2 \dots U_M \dots U_2 U_1$

ensures that odd-order commutator terms in the Baker-Campbell-Hausdorff expansion vanish, so the emergent Floquet operator possesses the same bipartite block structure as the static case. This is critical for two mechanisms supporting MBCs: chiral (bipartite imbalance) induced zero-modes, and the tree-grafting mechanism associated with compact localized states on network motifs ("dangling trees").

Prototypical Models: Imbalanced Bipartite Random Graphs and Quantum Hard-Disks

The theoretical formalism is instantiated in two classes of models:

• Imbalanced Bipartite Random Graphs (IBRGs): These provide a tunable platform to explore the impact of bipartite structure, sublattice imbalance, and sparsity on MBCs in Fock space. Floquet circuits built atop IBRGs manifest persistent flat bands and long-time memory signatures, robust against increased drive depth and large graph size.
• Quantum Hard-Disk Model (QHD): Realizable in Rydberg atom arrays, the QHD model features hard-core bosons on a 2D lattice with a nearest-neighbor exclusion constraint. The constrained Hilbert space admits a Fock-space bipartite structure, enabling the direct application of the palindromic Floquet construction. Experimental relevance is underscored by the existence of corresponding local observables and accessible Floquet protocols.

A key observable is the Loschmidt echo $\mathcal{L}(t)$, which quantifies return probability and, by extension, memory of the initial state. The presence of MBCs in Floquet systems is reflected in persistent nonzero values and oscillatory behavior of this echo, indicative of degenerate quasienergy bands and many-body Rabi oscillations.

Engineering Topological and Spatiotemporal Order in Floquet Cages

The paper goes beyond the mere demonstration of Floquet-stabilized MBCs by introducing methodologies for their topological and dynamical engineering. By adjusting the durations of individual layers within the Floquet cycle, specifically for the QHD model, the effective Fock-space graph can be modulated to support a hierarchy of localized states analogous to edge states in the Su-Schrieffer-Heeger (SSH) model.

A particularly notable construction introduces a swap Hamiltonian that maps the two end sites of a three-node tree motif. This extension yields Floquet eigenstates with quasienergy exactly at $\epsilon = \pi/T$, the defining property of a discrete time crystal (DTC). Unlike MBL-based DTCs, these caged DTCs do not require disorder; spatiotemporal order arises from symmetry and local constraints within the engineered Fock-space, a result only achievable via the Floquet-MBC framework presented.

Numerical Results and Diagnostic Observables

Strong numerical evidence supports the realization and stability of Floquet MBCs:

• For IBRGs, the state-averaged Loschmidt echo remains finite and largely insensitive to system size, establishing that a nonvanishing fraction of Fock space hosts nonergodic MBCs in the thermodynamic limit.
• In the QHD model, persistent oscillations of the Loschmidt echo and autocorrelation function corroborate the existence of both zero- and $\pi$-quasienergy bands. The engineered drives yield clear transitions in the band-overlap order parameter, systematically controlling localization properties.

The toy model with two grafted three-site trees further validates the controllability and predictable evolution of MBCs under Floquet engineering, demonstrating that topological manipulations in Fock space are not only possible but transparent to analytical treatment.

Implications and Future Directions

The implications of this research are substantial for both fundamental theory and experimental realization:

• Floquet prethermalization and thermalization avoidance: By encoding nonergodic subspaces via local constraints and symmetry-preserving drives, this framework delineates a class of systems that avoid Floquet heating to infinite temperature, sidestepping the generic tendency of driven systems to thermalize.
• Fock-space topology and artificial gauge fields: The demonstration that single-particle topological structures (e.g., SSH chains, edge modes) can be transplanted into Fock space foreshadows the possibility of more intricate Fock-space engineering, including artificial gauge fields and higher-dimensional topological order.
• Quantum simulation and many-body control: The presented protocols are compatible with advanced quantum simulators, including Rydberg atom arrays and digital quantum computers, making experimental validation plausible.

Further exploration may address the inclusion of many-body interactions beyond the hard-constraint paradigm, extensions to higher (and non-Abelian) symmetry classes, and the systematic classification of possible eigenspectrum orders in driven many-body systems.

Conclusion

This work establishes a robust and versatile platform for stabilizing and engineering many-body cages under periodic driving, generalizing the phenomenon of MBCs to the Floquet paradigm. The identification of techniques—palindromic chiral-symmetric Floquet drives—capable of controlling the spectral and topological properties of nonergodic many-body states, enables the realization of new forms of quantum order, including disorder-free time crystals and Fock-space topological motifs. The results underscore the expanding toolbox for nonequilibrium quantum engineering, with profound ramifications for both theoretical physics and quantum technologies.

Reference: "Floquet Many-Body Cages" (2604.13027)