Creating anomalous Floquet Chern insulators with magnetic quantum walks

We propose a realistic scheme to construct anomalous Floquet Chern topological insulators using spin-$\frac{1}{2}$ 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 paper, 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.

Figure 1

Magnetic quantum walks on a 2D lattice. (a) Left: block representation of the unitary operators constituting a single time step of the evolution $\widehat{W}$, as defined in Eq. (1). (b) A single time step of a walker prepared initially in a spin-up state, represented on a square lattice, with the arrows denoting the two spin states. To simulate a homogeneous magnetic field $B$, with artificial vector potential $\mathit{A}=(0,Bx,0)$, a spin-dependent linear potential gradient is stroboscopically applied along the $x$ axis, imprinting onto the walker's wave function a spin-dependent linear phase gradient (block 5).

Figure 2

Time evolution of a walker in a weak magnetic field $\varphi =\frac{1}{1200}$. The walker is initially prepared in a Gaussian wave packet with a root-mean-square width of 15 sites and a well-defined momentum state chosen at a distance $\delta k=\pi /4$ from one of the four inequivalent Dirac points [see Fig. 3]. Its initial spin state is oriented such that only energy states in the upper Dirac cone are occupied. The color scale represents the spatial probability distribution after $n$ steps of the walk, with dark red indicating a high, white an intermediate, and blue a vanishing probability. According to semiclassical equations (see main text), we expect a cyclotron circular orbit of 150 sites radius (white circle) and a revolution period of 942 steps. An animation of the time evolution is provided in the Supplemental Material [50].

Figure 3

Quasienergy spectrum of magnetic quantum walks as a function of quasimomentum $(k_x,k_y)$. (a) In a zero magnetic field, the spectrum presents two bands, touching each other at four inequivalent Dirac points. The number of bands reflects the spin multiplicity. (b) For a high magnetic field, $\varphi =\frac{1}{3}$, the original gapless spectrum splits into nearly flat quasienergy bands, separated in some cases by a large gap. The Chern numbers of isolated bands are denoted by $C_i$ with $i$ denoting the respective band index, or set of indices in the case of two bands touching. The Floquet invariants $R_i$ associated with the $i\mathrm{th}$ quasienergy gap are also shown.

Figure 4

Floquet Hofstadter butterfly showing the quasienergy spectrum of magnetic quantum walks as a function of the magnetic flux $\varphi$. For each rational value of the flux $\varphi =p/q$, there are $2q$ bands reflecting the spin multiplicity and the size of the magnetic unit cell. The six colored vertical lines at $\varphi =\frac{1}{3}$ correspond to the quasienergy bands shown in Fig. 3. On the left-hand side, the overlaid parabolic red lines, $E=\sqrt{4\pi n\varphi }$, indicate the Landau levels for a massless Dirac particle in the weak-field limit, with $n\ge 0$ denoting the index of the Landau level [62]. Inset: a detail of the butterfly, illustrating its self-similarity.

Figure 5

Quasienergy spectrum as a function of quasimomentum $k_y$, computed for a quantum walk with the magnetic field inverted in a central stripe. Inset: schematic representation (not to scale) of the magnetic-field landscape, with $\varphi =\frac{1}{3}$ outside the stripe and $\varphi =-\frac{1}{3}$ inside. Main figure: the filled bands correspond to bulk states, whereas the mid-gap energy branches to topologically protected edge modes, with the solid (dotted) lines denoting the left (right) edge of the stripe. The net number of topologically protected edge modes (per edge) matches the difference between the gap topological invariants [cf. Fig. 3] of the two different topological domains. The spectrum is computed under realistic conditions, based on the Floquet phase-imprinting scheme detailed in Sec. 3c, assuming periodic boundary conditions along the $x$ direction, and considering 60 sites in total, with the left edge at $x=15$ and the right one at $x=45$.

Figure 6

Time evolution of a walker that is initially prepared in a single site close to the boundary separating two distinct, topologically nontrivial regions. The boundary (overlaid white curve) has the shape of a quarter of a circle with a radius of 40 sites. The magnetic flux $\varphi$ is $-\frac{1}{3}$ in the inner region, and $\frac{1}{3}$ in the outer one. The color scale, similarly as in Fig. 2, represents the spatial probability distribution after $n$ steps of the walk. The simulations are carried out numerically assuming realistic conditions, based on the Floquet phase-imprinting scheme detailed in Sec. 3c, using a simple lattice with a relative shift of 0.5 (cf. Fig. 10) with respect to the sawtooth intensity profile employed to imprint the Peierls phases. An animation showing the time evolution of the walker is provided in the Supplemental Material [50].

Figure 7

Schematic representation of Cs energy levels corresponding to the D1 transition, including the related hyperfine structure. A laser beam is detuned by $\pm {\mathrm{\Delta }}_{\text{HF}}/2$ from resonance with the $|\uparrow \rangle$ and $|\downarrow \rangle$ states, respectively, so that the induced optical potential shifts the two states by an equal amount $\mathrm{\Delta }U$, but in opposite directions. Short light pulses, thus, imprint a purely differential phase shift onto the two states. A similar scheme exists for the D2 transition.

Figure 8

Probability to create a motional excitation to a higher-energy band of the optical lattice, after stroboscopically applying for a time $\tau$ the spin-dependent linear potential gradient that realizes the magnetic-field operator $\widehat{F}$. We assume here a lattice depth $V_0\sim 850E_R$, a field flux $\varphi =\frac{1}{3}$, and that the atom is initially prepared in the motional ground state. The two curves show the probability obtained through numerical integration of the time-dependent Schrödinger equation (thick red curve), and the probability to excite an atom to the first motional state obtained using Eq. (9) (thin black curve). The small discrepancies between the two curves are ascribed to the anharmonicity of the optical lattice potential.

Figure 9

Floquet phase imprint realizing the magnetic-field operator $\widehat{F}$. Only the phase for one spin component is shown in the figure, the other one being equal in magnitude, but opposite in sign. The graph shows the representative situation of a magnetic flux $\varphi =\frac{1}{3}$, which is realized in the Landau gauge $\mathit{A}=(0,Bx,0)$. Instead of imprinting a linear phase gradient (dashed orange line), a sawtooth-shaped phase pattern (dashed green line) lying within the first Floquet zone, $[0,2\pi ]$, is imprinted onto the atoms (note the different definition here of the Floquet zone with respect to Sec. 2a). The modulated phase pattern is produced by a sawtooth intensity profile, which is imaged onto the atoms through a high-numerical-aperture objective lens $\text{NA}=0.92$ [81]. Due to the $2\pi$ periodicity of the imprinted phases, the two phase patterns are effectively identical in the proximity of the atoms (blue dots). The folded phase pattern is also shown in the idealized case of no optical diffraction (solid orange line).

Figure 10

Quasienergy gap width of the bulk spectrum as a function of the relative shift between the sawtooth intensity profile and the optical lattice. All quasienergy gaps turn out to be identical. The graph is shown for the representative case of a magnetic flux $\varphi =\frac{1}{3}$. The three curves refer to the case of a simple lattice ($•$), a superlattice with a twofold longer lattice constant ($\blacksquare$), and a superlattice with a fourfold longer lattice constant ($▴$). The relative shift is expressed in units of the rescaled lattice constant. The quasienergy gap width is maximum at a relative shift of 0.5, when the sawtooth intensity profile is aligned with respect to the lattice sites as shown in the illustration in Fig. 9.

Figure 11

Spectral flow $R$ calculated [65] as a function of quasienergy $E$ for a magnetic quantum walk with magnetic flux $\varphi =\frac{1}{3}$. Using Eq. (G3), we obtain the spectral flow induced by a small change of an additional, fictitious magnetic field $\beta :0\to 2\pi /15$, assuming a fixed quasimomentum $k_x=k_y=\frac{1}{10}$, since the result does not depend on the chosen quasimomenutm. The density of states, normalized to the total number of states, is overlaid (colored areas) to indicate the regions of bulk gaps and quasienergy bands. The spectral flow is only defined in the band gaps (thick black lines), where it yields the (integer) RLBL topological invariant $R$.