Entanglement properties of the time-periodic Kitaev chain

Abstract

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 π (Zπ) modes, where the zero modes are present even in the absence of the drive, while the π 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 quasi-energy spectrum. The number of Majorana edge modes in the ES of the FGS vary in time from |Z0 − Zπ| to Z0 + Zπ within one drive cycle, with the maximal Z0 + Zπ 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 wavefunction, namely the Z0 modes, appear in the ES. The Zπ 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 π modes in the ES are also explained by mapping the parent wavefunction to a Bloch sphere.

Almost strong ( 0,π ) edge modes in clean interacting one-dimensional Floquet systems

“…It also would be interesting to explore eigenspectrum phases with (almost) strong edge modes in spin-1 chains and higher dimensional equilibrium as well as periodically driven systems. Other questions are whether the strong edge modes in all free Floquet SPTs 19,[31][32][33][47][48][49][50] are equally robust to adding interactions. It is also interesting to explore the connection between almost strong mode operators in interacting Floquet Hamiltonians and edge modes of interacting topological phases 51,52 .…”

Central Charge of Periodically Driven Critical Kitaev Chains

“…Thus, while there is one kind of zero mode in a static Hamiltonian and in the corresponding ES, there are two kinds of such modes in a Floquet system: 0 and π modes. Since the ES is not periodic, there is no clear analog of the π mode in the ES [8,19].A further wrinkle is that the Z×Z topological invariant and the quasienergy spectrum are properties of the full drive cycle, while the ES and EE are constructed from the instantaneous quantum state. They are therefore sensitive to which point in the drive cycle they are calculated.…”

“…Concretely, restricting the quasienergy spectrum to lie between −Ω/2, Ω/2, and noting that the chiral symmetry of the Floquet Hamiltonian causes the quasienergy spectra to come in pairs of ±| k |, the FGS corresponds to occupying with probability 1 all Floquet modes with negative quasienergy. This should be contrasted with a half filled state obtained from unitary time evolution under H(t) from an arbitrary initial state, where such a state will show volume law scaling of the EE at a steady state [8,19].We briefly explain how the ES and EE are studied numerically and analytically. The underlying principle is that for a system of free fermions, the eigenvalues of the reduced density matrix can be extracted from the eigenvalues of only the two-point correlation function, a consequence of Wick's theorem [25,26].…”

“…Thus, while there is one kind of zero mode in a static Hamiltonian and in the corresponding ES, there are two kinds of such modes in a Floquet system: 0 and π modes. Since the ES is not periodic, there is no clear analog of the π mode in the ES [8,19].…”

“…Concretely, restricting the quasienergy spectrum to lie between −Ω/2, Ω/2, and noting that the chiral symmetry of the Floquet Hamiltonian causes the quasienergy spectra to come in pairs of ±| k |, the FGS corresponds to occupying with probability 1 all Floquet modes with negative quasienergy. This should be contrasted with a half filled state obtained from unitary time evolution under H(t) from an arbitrary initial state, where such a state will show volume law scaling of the EE at a steady state [8,19].…”