Creating anomalous Floquet Chern insulators with magnetic quantum walks

Abstract

We propose a realistic scheme to construct anomalous Floquet Chern topological insulators using spin-1/2 particles carrying out a discrete-time quantum walk in a two-dimensional lattice. By Floquet engineering the quantum-walk protocol, an Aharonov-Bohm geometric phase is imprinted onto closed-loop paths in the lattice, thus realizing an abelian gauge field the analog of a magnetic flux threading a two-dimensional electron gas. We show that in the strong field regime, when the flux per plaquette is a sizable fraction of the flux quantum, magnetic quantum walks give rise to nearly flat energy bands featuring nonvanishing Chern numbers. Furthermore, we find that because of the nonperturbative nature of the periodic driving, a second topological number the so-called RLBL invariant is necessary to fully characterize the anomalous Floquet topological phases of magnetic quantum walks and to compute the number of topologically protected edge modes expected at the boundaries between different phases. In the second part of this article, we discuss an implementation of this scheme using neutral atoms in two-dimensional spin-dependent optical lattices, which enables the generation of arbitrary magnetic-field landscapes, including those with sharp boundaries. The robust atom transport, which is observed along boundaries separating regions of different field strength, reveals the topological character of the Floquet Chern bands. arXiv:1808.08923v2 [quant-ph]

Quantum walks in external gauge fields

“…To give a concrete example of the purely magnetic case, let us consider the simplest setting in which magnetic fields can occur, i.e. a two-dimensional lattice [31,35]. On 2 (Z 2 ) ⊗ C 2 we consider the decomposed walk…”

“…General results on propagation, or the analogs of Landau orbits do not seem to exist yet. Fascinating trapping behaviour of the boundary between two regions with different magnetic field, characterized by distinct Chern numbers, is predicted in [35].…”

“…Of course, our scheme is gauge invariant in the sense that, up to a gauge transformation, the result depends only on the field, and not on the vector potentials. While there are several examples in the literature [13,16,20,35,37] which describe an external electromagnetic field in accordance with our general scheme, the general case has not been formulated and studied at its natural level of generality. Hence we give a unifying background to the works mentioned, and also help to correct some faulty approaches [no cite].…”

“…For example, electric fields have been realized by accelerating the lattice [21]. A discussion of the options for magnetic fields in 2D is found in [35]. In any case the justification of calling such a system magnetic is in the realization of the structure we describe, or some version thereof.…”

Topological quantum walks: Theory and experiments

“…The Rudner winding number is able to identify phenomena that cannot be characterised by the Chern number, such as the appearance of robust chiral edge states in two-dimensional driven systems even though the Chern numbers of all the bulk Floquet bands are zero [50]. By constructing an Abelian gauge field-the analogue of a magnetic flux threading a two-dimensional electron gas, the two-dimensional discrete-time magnetic quantum walk is proposed to be able to simulate anomalous Floquet Chern topological insulators, where Rudner winding number is also necessary to fully characterise the corresponding anomalous Floquet topological phases and to compute the number of topologically protected edge modes [14].…”