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Register a new userAlmost strong 0, {pi} edge modes in clean, interacting 1D Floquet systems
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Certain periodically driven quantum many-particle systems in one dimension are known to exhibit edge modes that are related to topological properties and lead to approximate degeneracies of the Floquet spectrum. A similar situation occurs in spin chains, where stable edge modes were shown to exist at all energies in certain integrable spin chains. Moreover, these edge modes were found to be remarkably stable to perturbations. Here we investigate the stability of edge modes in interacting, periodically driven, clean systems. We introduce a model that features edge modes that persist over times scales well in excess of the time needed for the bulk of the system to heat to infinite temperatures.
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Floquet spin chains have been a venue for understanding topological states of matter that are qualitatively different from their static counterparts by, for example, hosting $pi$ edge modes that show stable period-doubled dynamics. However the stability of these edge modes to interactions has traditionally required the system to be many-body localized in order to suppress heating. In contrast, here we show that even in the absence of disorder, and in the presence of bulk heating, $pi$ edge modes are long lived. Their lifetime is extracted from exact diagonalization and is found to be non-perturbative in the interaction strength. A tunneling estimate for the lifetime is obtained by mapping the stroboscopic time-evolution to dynamics of a single particle in Krylov subspace. In this subspace, the $pi$ edge mode manifests as the quasi-stable edge mode of an inhomogeneous Su-Schrieffer-Heeger model whose dimerization vanishes in the bulk of the Krylov chain.
Integrable Floquet spin chains are known to host strong zero and $pi$ modes which are boundary operators that respectively commute and anticommute with the Floquet unitary generating stroboscopic time-evolution, in addition to anticommuting with a discrete symmetry of the Floquet unitary. Thus the existence of strong modes imply a characteristic pairing structure of the full spectrum. Weak interactions modify the strong modes to almost strong modes that almost commute or anticommute with the Floquet unitary. Manifestations of strong and almost strong modes are presented in two different Krylov subspaces. One is a Krylov subspace obtained from a Lanczos iteration that maps the Heisenberg time-evolution generated by the Floquet Hamiltonian onto dynamics of a single particle on a fictitious chain with nearest neighbor hopping. The second is a Krylov subspace obtained from the Arnoldi iteration that maps the Heisenberg time-evolution generated directly by the Floquet unitary onto dynamics of a single particle on a fictitious chain with longer range hopping. While the former Krylov subspace is sensitive to the branch of the logarithm of the Floquet unitary, the latter obtained from the Arnoldi scheme is not. The effective single particle models obtained in the two Krylov subspaces are discussed, and the topological properties of the Krylov chain that ensure stable $0$ and $pi$ modes at the boundaries are highlighted. The role of interactions is discussed. Expressions for the lifetime of the almost strong modes are derived in terms of the parameters of the Krylov subspace, and are compared with exact diagonalization.
Certain disorder-free Hamiltonians can be non-ergodic due to a emph{strong fragmentation} of the Hilbert space into disconnected sectors. Here, we characterize such systems by introducing the notion of `statistically localized integrals of motion (SLIOM), whose eigenvalues label the connected components of the Hilbert space. SLIOMs are not spatially localized in the operator sense, but appear localized to sub-extensive regions when their expectation value is taken in typical states with a finite density of particles. We illustrate this general concept on several Hamiltonians, both with and without dipole conservation. Furthermore, we demonstrate that there exist perturbations which destroy these integrals of motion in the bulk of the system, while keeping them on the boundary. This results in statistically localized emph{strong zero modes}, leading to infinitely long-lived edge magnetizations along with a thermalizing bulk, constituting the first example of such strong edge modes in a non-integrable model. We also show that in a particular example, these edge modes lead to the appearance of topological string order in a certain subset of highly excited eigenstates. Some of our suggested models can be realized in Rydberg quantum simulators.
Detection and manipulation of excitations with non-Abelian statistics, such as Majorana fermions, are essential for creating topological quantum computers. To this end, we show the connection between the existence of such localized particles and the phenomenon of unitary subharmonic response (SR) in periodically driven systems. In particular, starting from highly nonequilibrium initial states, the unpaired Majorana modes exhibit spin oscillations with twice the driving period, are localized, and can have exponentially long lifetimes in clean systems. While the lifetime of SR is limited in translationally invariant systems, we show that disorder can be engineered to stabilize the subharmonic response of Majorana modes. A viable observation of this phenomenon can be achieved using modern multiqubit hardware, such as superconducting circuits and cold atomic systems.
Almost strong edge-mode operators arising at the boundaries of certain interacting 1D symmetry protected topological phases with (Z_2) symmetry have infinite temperature lifetimes that are non-perturbatively long in the integrability breaking terms, making them promising as bits for quantum information processing. We extract the lifetime of these edge-mode operators for small system sizes as well as in the thermodynamic limit. For the latter, a Lanczos scheme is employed to map the operator dynamics to a one dimensional tight-binding model of a single particle in Krylov space. We find this model to be that of a spatially inhomogeneous Su-Schrieffer-Heeger model with a hopping amplitude that increases away from the boundary, and a dimerization that decreases away from the boundary. We associate this dimerized or staggered structure with the existence of the almost strong mode. Thus the short time dynamics of the almost strong mode is that of the edge-mode of the Su-Schrieffer-Heeger model, while the long time dynamics involves decay due to tunneling out of that mode, followed by chaotic operator spreading. We also show that competing scattering processes can lead to interference effects that can significantly enhance the lifetime.
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