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arXiv:2408.08548 · Physical review. B./Physical review. B

Local and energy-resolved topological invariants for Floquet systems

Arnob Kumar Ghosh, R. Arouca, Annica M. Black‐Schaffer

Published Dec 17, 2024• 7 citations
arXivHTMLPDF

Abstract

Periodically driven systems offer a perfect breeding ground for out-of-equilibrium engineering of topological boundary states at zero energy (0 mode), as well as finite energy ( π mode), with the latter having no static analog. The Floquet operator and the effective Floquet Hamiltonian, which encapsulate the stroboscopic features of the driven system, capture both spectral and localization properties of the 0 and π modes but sometimes fail to provide complete topological characterization, especially when 0 and π modes coexist. In this work, we utilize the spectral localizer, a powerful local probe that can provide numerically efficient, spatially local, and energy-resolved topological characterization. In particular, we apply the spectral localizer to the effective Floquet Hamiltonian for driven one- and two-dimensional topological systems with no or limited symmetries and are able to assign topological invariants, or local markers, that characterize the 0- and the π -boundary modes individually and unambiguously. Due to the spatial resolution, we also demonstrate that the extracted topological invariants are suitable for studying driven disordered systems and can even capture disorder-induced phase transitions. Published by the American Physical Society 2024

DOI
10.1103/physrevb.110.245306

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Periodically driven systems offer a perfect breeding ground for out-of-equilibrium engineering of topological boundary states at zero energy (0 mode), as well as finite energy ( π mode), with the latter having no static analog. The Floquet operator and the effective Floquet Hamiltonian, which encapsulate the stroboscopic features of the driven system, capture both spectral and localization properties of the 0 and π modes but sometimes fail to provide complete topological characterization, especially when 0 and π modes coexist. In this work, we utilize the spectral localizer, a powerful local probe that can provide numerically efficient, spatially local, and energy-resolved topological characterization. In particular, we apply the spectral localizer to the effective Floquet Hamiltonian for driven one- and two-dimensional topological systems with no or limited symmetries and are able to assign topological invariants, or local markers, that characterize the 0- and the π -boundary modes individually and unambiguously. Due to the spatial resolution, we also demonstrate that the extracted topological invariants are suitable for studying driven disordered systems and can even capture disorder-induced phase transitions. Published by the American Physical Society 2024
We show how second-order Floquet engineering can be employed to realize systems in which many-body localization coexists with topological properties in a driven system. This allows one to implement and dynamically control a symmetry protected topologically ordered qubit even at high energies, overcoming the roadblock that the respective states cannot be prepared as ground states of nearest-neighbor Hamiltonians. Floquet engineering-the idea that a periodically driven nonequilibrium system can effectively emulate the physics of a different Hamiltonian-is exploited to approximate an effective three-body interaction among spins in one dimension, using time-dependent two-body interactions only. In the effective system, emulated topology and disorder coexist, which provides an intriguing insight into the interplay of many-body localization that defies our standard understanding of thermodynamics and into the topological phases of matter, which are of fundamental and technological importance. We demonstrate explicitly how combining Floquet engineering, topology, and many-body localization allows one to harvest the advantages (time-dependent control, topological protection, and reduction of heating, respectively) of each of these subfields while protecting them from their disadvantages (heating, static control parameters, and strong disorder).
Coherent control via periodic modulation, also known as Floquet engineering, has emerged as a powerful experimental method for the realization of novel quantum systems with exotic properties. In particular, it has been employed to study topological phenomena in a variety of different platforms. In driven systems, the topological properties of the quasienergy bands can often be determined by standard topological invariants, such as Chern numbers, which are commonly used in static systems. However, due to the periodic nature of the quasienergy spectrum, this topological description is incomplete and new invariants are required to fully capture the topological properties. Most prominently, there exist two-dimensional anomalous Floquet systems that exhibit robust chiral edge modes, despite all Chern numbers are equal to zero. Here, we realize such a system with bosonic atoms in a periodically-driven hexagonal lattice and infer the complete set of topological invariants from energy gap measurements and local Hall deflections.
The classification of topological Floquet systems with time-periodic Hamiltonians transcends that of static systems. For example, spinless fermions in periodically driven two-dimensional lattices are not completely characterized by the Chern numbers of the quasienergy bands, but rather by a set of winding numbers associated with the gaps. We propose a feasible scheme for measuring these winding numbers in a periodically driven optical lattice efficiently and directly. It is based on the construction of a one-parameter family of drives, continuously connecting the Floquet system of interest to a trivial reference system. The winding numbers are then determined by the identification and the tomography of the band-touching singularities occurring on the way. As a by-product, we also propose a method for probing spectral properties of time evolution operators via a time analog of crystallography.
This dissertation focus on the classification of topological matters in static and periodically driven systems.First, we build a complete topological classification of local unitary operators in free fermionic systems. This result can be encoded in a periodic table with a period of eight, similar to the periodic table of topological insulators. In our classification, we define two local unitaries in a certain symmetry class to be topologically equivalent if they can be connected via finite time evolutions (locally generated unitaries) without breaking the symmetries of the symmetry class while maintaining locality. In this way, the classification we derive will allow us to distinguish locally generated unitaries from those that cannot be locally generated.Besides, we also how to find possible generating Hamiltonians for locally generated unitaries. These results can be used to study the edge behaviors of Floquet systems.Second, we propose bulk invariants and edge invariants that are locally computable and improve existing topological invariants by being applicable to systems with the disorder. Afterward, we set up a rigorous connection between bulk and edge invariants for Floquet systems belonging to Class AIII and class AII of the Altland-Zirnbauer symmetry classification.Last, after treating a static system as a Floquet system generated by a constant Hamiltonian, we employ the tools we developed in Floquet systems to classify the edge behaviors in static systems.
Periodically driven quantum systems host exotic phenomena that often do not have any analog in undriven systems. Floquet prethermalization and dynamical freezing of certain observables, via the emergence of conservation laws, are realized by controlling the drive frequency. Recent experimental developments in synthetic quantum matter, such as superconducting qubits and cold atoms, have opened avenues for implementing local Floquet engineering which can achieve spatially modulated quantum control of states. Here, we uncover the novel memory effects of local periodic driving in a nonintegrable spin-half staggered Heisenberg chain. For a boundary-driven protocol at the dynamical freezing frequency, we show the formation of an approximate strong zero mode, a prethermal quasilocal operator, due to the emergence of a discrete global Z2 symmetry. This is captured by constructing an accurate effective Floquet Hamiltonian using a higher-order partially resummed Floquet-Magnus expansion. The lifetime of the boundary spin can be exponentially enhanced by enlarging the set of suitably chosen driven sites. We demonstrate that in the asymptotic limit, achieved by increasing the number of driven sites, a strong zero mode emerges, where the lifetime of the boundary spin grows exponentially with system size. The nonlocal processes in the Floquet Hamiltonian play a pivotal role in the total freezing of the boundary spin in the thermodynamic limit. The novel dynamics of the boundary spin is accompanied by a rich structure of entanglement in the Floquet eigenstates where specific bipartitions yield an area-law scaling while the entanglement for random bipartitions scales as a volume-law. Our work addresses the long-standing question of the existence of a strong zero mode in a nonintegrable model and elucidates the complex nature of thermalization in locally driven systems. Published by the American Physical Society 2024
A d-dimensional, nth-order topological insulator or superconductor has localized eigenmodes at its (d−n)-dimensional boundaries (n≤d). In this work, we apply periodic driving fields to two-dimensional superconductors, and obtain a wide variety of Floquet second-order topological superconducting (SOTSC) phases with many Majorana corner modes at both zero and π quasienergies. Two distinct Floquet SOTSC phases are found to be separated by three possible kinds of transformations, i.e., a topological phase transition due to the closing/reopening of a bulk spectral gap, a topological phase transition due to the closing/reopening of an edge spectral gap, or an entirely different phase in which the bulk spectrum is gapless. Thanks to the strong interplay between driving and intrinsic energy scales of the system, all the found phases and transitions are highly controllable via tuning a single hopping parameter of the system. Our discovery not only enriches the possible forms of Floquet SOTSC phases, but also offers an efficient scheme to generate many coexisting Majorana zero and π corner modes, which may find applications in Floquet quantum computation.