Results are presented for the entanglement entropy and spectrum of half-filled graphene following the switch on of a circularly polarized laser. The laser parameters are chosen to correspond to several different Floquet-Chern insulator phases. The entanglement properties of the unitarily evolved wave functions are compared with the state where one of the Floquet bands is completely occupied. The true states show a volume law for the entanglement, whereas the Floquet states show an area law. Qualitative differences are found in the entanglement properties of the off-resonant and on-resonant laser. Edge states are found in the entanglement spectrum corresponding to certain physical edge states expected in a Chern insulator. However, some edge states that would be expected from the Floquet band structure are missing from the entanglement spectrum. An analytic theory is developed for the long-time structure of the entanglement spectrum. It is argued that only edge states corresponding to off-resonant processes appear in the entanglement spectrum.
Results are presented for the entanglement entropy and spectrum of half-filled graphene following the switch on of a circularly polarized laser. The laser parameters are chosen to correspond to several different Floquet-Chern insulator phases. The entanglement properties of the unitarily evolved wave functions are compared with the state where one of the Floquet bands is completely occupied. The true states show a volume law for the entanglement, whereas the Floquet states show an area law. Qualitative differences are found in the entanglement properties of the off-resonant and on-resonant laser. Edge states are found in the entanglement spectrum corresponding to certain physical edge states expected in a Chern insulator. However, some edge states that would be expected from the Floquet band structure are missing from the entanglement spectrum. An analytic theory is developed for the long-time structure of the entanglement spectrum. It is argued that only edge states corresponding to off-resonant processes appear in the entanglement spectrum.
We study the many-body physics in twisted bilayer graphene coupled to periodic driving of a circularly polarized light when electron-electron interactions are taken into account. In the limit of high driving frequency \Omega Ω , we use Floquet theory to formulate the system by an effective static Hamiltonian truncated to the order of \Omega^{-2} Ω − 2 , which consists of a single-electron part and the screened Coulomb interaction. We numerically simulate this effective Hamiltonian by extensive exact diagonalization in the parameter space spanned by the twist angle and the driving strength. Remarkably, in a wide region of the parameter space, we identify Floquet fractional Chern insulator states in the partially filled Floquet valence bands. We characterize these topologically ordered states by ground-state degeneracy, spectral flow, and entanglement spectrum. In regions of the parameter space where fractional Chern insulator states are absent, we find topologically trivial charge density waves and band-dispersion-induced Fermi liquids which strongly compete with fractional Chern insulator states.
Entanglement is one of the most fundamental features of quantum systems. In this work, we obtain the entanglement spectrum and entropy of Floquet noninteracting fermionic lattice models and build their connections with Floquet topological phases. Topological winding and Chern numbers are introduced to characterize the entanglement spectrum and eigenmodes. Correspondences between the spectrum and topology of entanglement Hamiltonians under periodic boundary conditions and topological edge states under open boundary conditions are further established. The theory is applied to Floquet topological insulators in different symmetry classes and spatial dimensions. Our work thus provides a useful framework for the study of rich entanglement patterns in Floquet topological matter.
We study the entanglement spectrum of a Chern insulator on a cylinder geometry, with the cut separating the two partitions taken parallel to the cylinder edge, at varying distances from the edge. In contrast to similar studies on a torus, there is only one cut, and hence only one virtual edge mode in the entanglement spectrum. The entanglement spectrum has a gap when the cut is close enough to the physical edge of the cylinder such that the edge mode spatially extends over the cut. This effect is suppressed for parameter choices where the edge mode is sharply localized at the edge. In the extreme case of a perfectly localized edge mode, the entanglement spectrum is gapless even if the smaller partition consists of a single edge row. For the single-row cut, we construct the corresponding entanglement Hamiltonian, which is a one-dimensional tight-binding Hamiltonian with complex long-range hopping and interesting properties. We also study and explain the effect of two different schemes of flux insertion through a ring described by such an entanglement Hamiltonian.
The geometry of quantum states could offer indispensable insights for characterizing the topological properties, phase transitions, and entanglement nature of many-body systems. In this work, we reveal the quantum geometry and the associated entanglement entropy (EE) of Floquet topological states in one-dimensional periodically driven systems. The quantum metric tensors of Floquet states are found to show nonanalytic signatures at topological phase transition points. Away from the transition points, the bipartite geometric EE of Floquet states exhibits an area-law scaling vs the system size, which holds for a Floquet band at any filling fractions. For a uniformly filled Floquet band, the EE further becomes purely quantum geometric. At phase transition points, the geometric EE scales logarithmically with the system size and displays cusps in the nearby parameter ranges. These discoveries are demonstrated by investigating typical Floquet models including periodically driven spin chains, Floquet topological insulators, and superconductors. Our findings uncover the rich quantum geometries of Floquet states, unveiling the geometric origin of EE for gapped Floquet topological phases, and introducing information-theoretic means of depicting topological transitions in Floquet systems.
The entanglement properties of the time-periodic Kitaev chain with nearest-neighbor and next-nearest-neighbor hopping is studied. The cases of the exact eigenstate of the time-periodic Hamiltonian, referred to as the Floquet ground state (FGS), as well as a physical state obtained from time evolving an initial state unitarily under the influence of the time-periodic drive are explored. Topological phases are characterized by different numbers of Majorana zero (Z0) and $\ensuremath{\pi}$ (${\mathbb{Z}}_{\ensuremath{\pi}}$) modes, where the zero modes are present even in the absence of the drive, while the $\ensuremath{\pi}$ modes arise due to resonant driving. The entanglement spectrum (ES) of the FGS as well as the physical state show topological Majorana modes whose number is different from that of the quasienergy spectrum. The number of Majorana edge modes in the ES of the FGS vary in time from $|{\mathbb{Z}}_{0}\ensuremath{-}{\mathbb{Z}}_{\ensuremath{\pi}}|$ to ${\mathbb{Z}}_{0}+{\mathbb{Z}}_{\ensuremath{\pi}}$ within one drive cycle, with the maximal ${\mathbb{Z}}_{0}+{\mathbb{Z}}_{\ensuremath{\pi}}$ modes appearing at a special time-reversal-symmetric point of the cycle. For the physical state, on the other hand, only the modes inherited from the initial wave function, namely the Z0 modes, appear in the ES. The ${\mathbb{Z}}_{\ensuremath{\pi}}$ modes are absent in the physical state as they merge with the bulk excitations that are simultaneously created due to resonant driving. The topological properties of the Majorana zero and $\ensuremath{\pi}$ modes in the ES are also explained by mapping the parent wave function to a Bloch sphere.
We investigate topological phase transitions in Chern insulators within three-band models, focusing on the empty band and lowest band populated by spinless fermions. We consider Lieb and kagome lattices and notice phase transitions driven by the hopping integral between nearest-neighbors, which leads to the change of the lowest band Chern number from C=1 to C=−1. In the single-particle picture, different phases are examined by investigating corresponding entanglement spectra and the evolution of the entanglement entropy. Entanglement spectra reveal the spectral flow characteristic of topologically nontrivial systems before and after phase transitions. For the lowest band with 1/3 filling, Fractional Chern insulator (FCI) phases are identified by examining the ground state momenta, the spectral flow, and counting of the entanglement energy levels below the gap in the entanglement spectrum. A quasilinear dependence of the entanglement entropy α(nA) term is observed for FCI phase, similarly to linear behavior expected for Laughlin phase in the fractional quantum Hall effect. At the topological phase transition, for both empty and partially filled lowest bands, the energy gap closure and a discontinuity in the entanglement entropy are observed. This coincides with the divergence of a standard deviation of the Berry curvature. We notice also a phase transition driven by the nearest-neighbor interaction in the Lieb lattice, where the many-body energy gap closes. The phase transitions are shown to be stable for an arbitrary system size, thus predicted to be present in the thermodynamic limit. While our calculations are performed for an interaction energy far exceeding the gap between the two lowest energy bands, we note that the higher band does not affect the phase transitions, however destabilizes FCI phases.