Floquet engineering offers a unique approach to generate nonequilibrium topological phases in which the unbounded nature of a quasienergy band allows two kinds of topological edge modes, one of them traversing the 0 gap and another one traversing the $\ensuremath{\pi}$ gap. Characterizing of these two modes is the main topic of Floquet topological insulators, where they are usually characterized by different topological invariants. However, in this paper, for a specific protocol of Floquet engineering where the Dirac mass term of the Chern insulator is periodically kicked, its topological phases are characterized with the Floquet Chern number ${C}_{F}$ . Specifically, in our illustrative example, the periodically kicked Qi-Wu-Zhang model, there are six different topological phases in total, denoted as ${C}_{F}={\ensuremath{-}{1}_{0},\ensuremath{-}2,\ensuremath{-}{1}_{\ensuremath{\pi}},{1}_{\ensuremath{\pi}},2,{1}_{0}}$. Topological phases with larger topological number are observed, i.e., ${C}_{F}=\ifmmode\pm\else\textpm\fi{}2$, where the chiral edge modes traversing the 0 gap and those traversing $\ensuremath{\pi}$ gap have the opposite chirality. The mechanism of its topology is revealed by studying the corresponding low-energy effective Dirac Hamiltonian, and the phase boundaries between different topological phases are explicitly found. Additionally, we investigate the orders of phase transitions between different topological phases by studying the von Neumann entropy of the Floquet steady state (FSS), where the FSS corresponds to a stationary state of the Floquet system that is coupled to a Markovian environment. The hallmark of a periodically kicked Dirac mass term is uncovered in this paper, which may inspire further explorations of the physical effects of periodically kicked Dirac mass terms in other systems.Floquet engineering offers a unique approach to generate nonequilibrium topological phases in which the unbounded nature of a quasienergy band allows two kinds of topological edge modes, one of them traversing the 0 gap and another one traversing the $\ensuremath{\pi}$ gap. Characterizing of these two modes is the main topic of Floquet topological insulators, where they are usually characterized by different topological invariants. However, in this paper, for a specific protocol of Floquet engineering where the Dirac mass term of the Chern insulator is periodically kicked, its topological phases are characterized with the Floquet Chern number ${C}_{F}$. Specifically, in our illustrative example, the periodically kicked Qi-Wu-Zhang model, there are six different topological phases in total, denoted as ${C}_{F}={\ensuremath{-}{1}_{0},\ensuremath{-}2,\ensuremath{-}{1}_{\ensuremath{\pi}},{1}_{\ensuremath{\pi}},2,{1}_{0}}$. Topological phases with larger topological number are observed, i.e., ${C}_{F}=\ifmmode\pm\else\textpm\fi{}2$, where the chiral edge modes traversing the 0 gap and those traversing $\ensuremath{\pi}$ gap have the opposite chirality. The mechanism of its topology is revealed by studying the corresponding low-energy effective Dirac Hamiltonian, and the phase boundaries between different topological phases are explicitly found. Additionally, we investigate the orders of phase transitions between different topological phases by studying the von Neumann entropy of the Floquet steady state (FSS), where the FSS corresponds to a stationary state of the Floquet system that is coupled to a Markovian environment. The hallmark of a periodically kicked Dirac mass term is uncovered in this paper, which may inspire further explorations of the physical effects of periodically kicked Dirac mass terms in other systems.

Physical review. B./Physical review. BMay 7, 20242 citations

Realization of unconventional topological quantum phases by Floquet engineering

Long Zhang, Xiong-Jun Liu

Chinese Science Bulletin (Chinese Version)May 1, 20230 citations

Dissipative preparation of a Floquet topological insulator in an optical lattice via bath engineering

Alexander Schnell, Christof Weitenberg, André Eckardt

Floquet engineering is an important tool for realizing topologically nontrivial band structures for charge-neutral atoms in optical lattices. However, the preparation of a topological-band-insulator-type state of fermions, with one nontrivial quasi-energy band filled completely and the others empty, is challenging as a result of both driving induced heating as well as imperfect adiabatic state preparation (with the latter induced by the unavoidable gap closing when passing the topological transition). An alternative procedure that has been proposed is to prepare such states dissipatively, i.e. as a steady state that emerges when coupling the system to reservoirs. Here we discuss a concrete scheme that couples the system to a weakly interacting Bose condensate given by second atomic species acting as a heat bath. Our strategy relies on the engineering of the potential for the bath particles, so that they occupy weakly coupled tubes perpendicular to the two-dimensional system. Using Floquet-Born-Markov theory, we show that the resulting nonequilibrium steady state of the driven-dissipative system approximates a topological insulator. We even find indications for the approximate stabilization of an anomalous Floquet topological insulator, a state that is impossible to realize in equilibrium.Floquet engineering is an important tool for realizing topologically nontrivial band structures for charge-neutral atoms in optical lattices. However, the preparation of a topological-band-insulator-type state of fermions, with one nontrivial quasi-energy band filled completely and the others empty, is challenging as a result of both driving induced heating as well as imperfect adiabatic state preparation (with the latter induced by the unavoidable gap closing when passing the topological transition). An alternative procedure that has been proposed is to prepare such states dissipatively, i.e. as a steady state that emerges when coupling the system to reservoirs. Here we discuss a concrete scheme that couples the system to a weakly interacting Bose condensate given by second atomic species acting as a heat bath. Our strategy relies on the engineering of the potential for the bath particles, so that they occupy weakly coupled tubes perpendicular to the two-dimensional system. Using Floquet-Born-Markov theory, we show that the resulting nonequilibrium steady state of the driven-dissipative system approximates a topological insulator. We even find indications for the approximate stabilization of an anomalous Floquet topological insulator, a state that is impossible to realize in equilibrium.

arXiv:2307.03739SciPost PhysicsAug 14, 20244 citations

Floquet topological phases coupled to environments

Szabolcs Vajna, B. Horovitz

Bulletin of the American Physical SocietyMar 17, 20160 citations

Adiabatic realization of anomalous Floquet topological systems

Luca Asteria, Marcel N. Kosch, Henrik P. Zahn +4 more

Topology has emerged as a central concept for classifying phases of matter. The situation is especially rich in periodically driven systems, where anomalous Floquet topological phases break the usual bulk-boundary correspondence between Chern number and edges modes of two-dimensional systems. These phases were so far realized by periodic modulation of the tunneling elements at frequencies near-resonant with respect to the system's bandwidth, a regime where Floquet heating plays a significant role in interacting systems. Here we show that such anomalous Floquet topological phases can also be realized by means of an adiabatic protocol, where the system is always in the instantaneous ground state of the cyclic path in parameter space, like in a Thouless charge pump. We experimentally realize such a state using ultracold atoms in a hexagonal lattice where we adiabatically modulate the lattice geometry, including the sublattice offset. To infer the topology, we use the micromotion area in real-space, which was recently identified as a proxy for the winding number. This way of realizing anomalous phases avoids resonant Floquet heating and imperfect loading into the target state. We demonstrate the robustness of the adiabatic construction by observing the anomalous phase even in the presence of mean-field interactions of magnitude comparable to all other energy scales. These findings are promising for engineering novel topological states in a more robust way.Topology has emerged as a central concept for classifying phases of matter. The situation is especially rich in periodically driven systems, where anomalous Floquet topological phases break the usual bulk-boundary correspondence between Chern number and edges modes of two-dimensional systems. These phases were so far realized by periodic modulation of the tunneling elements at frequencies near-resonant with respect to the system's bandwidth, a regime where Floquet heating plays a significant role in interacting systems. Here we show that such anomalous Floquet topological phases can also be realized by means of an adiabatic protocol, where the system is always in the instantaneous ground state of the cyclic path in parameter space, like in a Thouless charge pump. We experimentally realize such a state using ultracold atoms in a hexagonal lattice where we adiabatically modulate the lattice geometry, including the sublattice offset. To infer the topology, we use the micromotion area in real-space, which was recently identified as a proxy for the winding number. This way of realizing anomalous phases avoids resonant Floquet heating and imperfect loading into the target state. We demonstrate the robustness of the adiabatic construction by observing the anomalous phase even in the presence of mean-field interactions of magnitude comparable to all other energy scales. These findings are promising for engineering novel topological states in a more robust way.

arXiv:2606.16942Jun 15, 20260 citations

{C}_{F}

. Specifically, in our illustrative example, the periodically kicked Qi-Wu-Zhang model, there are six different topological phases in total, denoted as ${C}_{F}={\ensuremath{-}{1}_{0},\ensuremath{-}2,\ensuremath{-}{1}_{\ensuremath{\pi}},{1}_{\ensuremath{\pi}},2,{1}_{0}}$. Topological phases with larger topological number are observed, i.e., ${C}_{F}=\ifmmode\pm\else\textpm\fi{}2$, where the chiral edge modes traversing the 0 gap and those traversing $\ensuremath{\pi}$ gap have the opposite chirality. The mechanism of its topology is revealed by studying the corresponding low-energy effective Dirac Hamiltonian, and the phase boundaries between different topological phases are explicitly found. Additionally, we investigate the orders of phase transitions between different topological phases by studying the von Neumann entropy of the Floquet steady state (FSS), where the FSS corresponds to a stationary state of the Floquet system that is coupled to a Markovian environment. The hallmark of a periodically kicked Dirac mass term is uncovered in this paper, which may inspire further explorations of the physical effects of periodically kicked Dirac mass terms in other systems.