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arXiv:2605.25792

Boundary $0/π$ logical subspace and bulk dynamical probes in flux-controlled anomalous Floquet quantum walks

WeiCheng Ning, YanSheng Liu, XiaoXue Zhang, XiZheng Zhang

quant-phSubmitted May 25, 2026• Announced May 26, 2026• 0 citations
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Abstract

We formulate a one-dimensional flux-controlled anomalous Floquet quantum walk and show that it admits a direct microscopic realization in a driven bipartite lattice. The walk consists of a coin-dependent drift step and a momentum-dependent coin mixing step, so the same evolution operator governs quasienergy bands, boundary modes, and bulk dynamics in real space. Because the walk is chiral, the quasienergy gaps at $0$ and $π/T$ carry independent topological information, which organizes trivial, $0$-only, $π$-only, and coexistence sectors in the $(M,φ)$ plane. In the coexistence sector, a $0$ mode and a $π$ mode reside on the same edge and span a natural boundary logical subspace. One Floquet period acts there as a relative phase operation and produces a clear $2T$ response in local boundary observables. In the bulk, the same anomalous Floquet structure is probed dynamically in two complementary ways. Frame-resolved mean chiral displacements approach the two winding numbers in the clean pre-reflection window of the symmetric time frames, while selected benchmark cuts at a representative $0$ gap closing and a representative $π$ gap closing exhibit distinct local stroboscopic responses, with the $π$ gap benchmark showing a much stronger odd-even alternation. The boundary logical subspace and the bulk dynamical probes are therefore organized within one flux-controlled anomalous Floquet quantum walk, suggesting a symmetry-protected route to quantum-walk information primitives in driven microstructured lattices.

Comments
11 pages, 4 figures

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Boundary 0/π0/π0/π logical subspace and bulk dynamical probes in flux-controlled anomalous Floquet quantum walks
We formulate a one-dimensional flux-controlled anomalous Floquet quantum walk and show that it admits a direct microscopic realization in a driven bipartite lattice. The walk consists of a coin-dependent drift step and a momentum-dependent coin mixing step, so the same evolution operator governs quasienergy bands, boundary modes, and bulk dynamics in real space. Because the walk is chiral, the quasienergy gaps at 000 and π/Tπ/Tπ/T carry independent topological information, which organizes trivial, 000-only, πππ-only, and coexistence sectors in the (M,φ)(M,φ)(M,φ) plane. In the coexistence sector, a 000 mode and a πππ mode reside on the same edge and span a natural boundary logical subspace. One Floquet period acts there as a relative phase operation and produces a clear 2T2T2T response in local boundary observables. In the bulk, the same anomalous Floquet structure is probed dynamically in two complementary ways. Frame-resolved mean chiral displacements approach the two winding numbers in the clean pre-reflection window of the symmetric time frames, while selected benchmark cuts at a representative 000 gap closing and a representative πππ gap closing exhibit distinct local stroboscopic responses, with the πππ gap benchmark showing a much stronger odd-even alternation. The boundary logical subspace and the bulk dynamical probes are therefore organized within one flux-controlled anomalous Floquet quantum walk, suggesting a symmetry-protected route to quantum-walk information primitives in driven microstructured lattices.
Driven Floquet systems can realize topological phases with no static counterparts. This so-called anomalous Floquet topology breaks the bulk-boundary correspondence based on the Chern number. The number of edge modes in each band gap is instead determined by another integer index, a winding number, which is calculated from the time evolution operator of the bulk states within one driving period. While topological Chern numbers can be inferred from real-space measurements, it is not straightforward to obtain information about the anomalous Floquet topological phase from bulk measurements. Here we consider two-band models and show that the area enclosed during a Floquet period by an initially localized particle signals the presence of an anomalous phase when it approaches half the unit cell area. In general, we show that for dispersionless dynamics during the micromotion, the enclosed area is quantized and an exact proportionality relation exists between the area and the winding number. Direct detection of anomalous topology in real space could be realized in several quantum simulation platforms, and could be useful for systems with disorder or interactions. Building on the connection between area and winding number, we also show a way to realize arbitrarily high winding numbers.
Discrete-time quantum walks have been shown to simulate all known topological phases in one and two dimensions. Being periodically driven quantum systems, their topological description, however, is more complex than that of closed Hamiltonian systems. We map out the topological phases of the particle-hole symmetric one-dimensional discrete-time quantum walk. We find that there is no chiral symmetry in this system: its topology arises from the particle-hole symmetry alone. We calculate the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological invariant in a simple way that is consistent with a general definition for one-dimensional periodically driven quantum systems. These results allow for a transparent interpretation of the edge states on a finite lattice via the the bulk-boundary correspondence. We find that the bulk Floquet operator does not contain all the information needed for the topological invariant. As an illustration to this statement, we show that in the split-step quantum walk, the edges between two bulks with the same Floquet operator can host topologically protected edge states.
We study the topological properties of one-dimensional discrete-time quantum walks with Fibonacci quasiperiodic modulation. Spectral analysis under open boundary conditions reveals isolated edge modes that coexist at both zero and πππ energies within bulk gaps. Using the mean chiral displacement (MCD) as a dynamical bulk probe, we obtain a fractal, butterfly-like phase diagram, indicating a nontrivial bulk topology. However, since the MCD involves wave-packet averaging, its direct correspondence with boundary-localized states remains ambiguous. To resolve this issue, we determine the integer topological invariant by employing the Schur function formalism, a scattering-based approach defined on the unit disk. This analysis identifies a topological phase with winding number W=2W=2W=2 inside the ``wings'' of the butterfly diagram, demonstrating that the coexistence of zero- and πππ-energy edge modes is an intrinsic bulk property. Moreover, we find that the topological index crucially depends on the phason degree of freedom: the winding number can take values of W=0W=0W=0, 222, and even 444 depending on the local surface termination. This behavior originates from the surface impedance, which yields an effective winding number ranges from complete masking by reflective boundaries (W=0W=0W=0) to enhancement through surface resonances (W=4W=4W=4). Finally, we show that although individual surface terminations lead to anisotropic phase diagrams, their ensemble average restores the overall geometric structure observed in the MCD phase diagram. Our results establish a complete bulk-edge correspondence in quasiperiodic quantum walks and provide a guiding principle for surface topological engineering, where edge transport can be controlled purely through boundary manipulation.
We show that a three-state discrete-time quantum walk (DTQW) on a square lattice, with an SU(3) coin, realizes stroboscopic Floquet phases that host dispersive topological bound states in the continuum (TBICs). Time-independent step operations act as the periodic drive and map the one-step unitary to an effective Floquet Hamiltonian. Chern-number calculations resolve three quasienergy bands and their phase boundaries as the rotation angles are varied, and bulk-boundary correspondence in semi-infinite and fully finite geometries accounts for the edge spectra. At interfaces between media with the same bandwise Chern numbers but different coin parameters, nonchiral edge modes become embedded yet localized TBICs with tunable group velocity and selective excitation, while numerics indicate robustness to moderate coin disorder. Together with feasible routes in trapped-ion, cold-atom, and photonic platforms and an explicit decoherence model, these results identify DTQWs as a controllable setting for protected quantum-state transport and storage, relevant to quantum information and communication technologies.
We construct a many-body quantized invariant that sharply distinguishes among two-dimensional nonequilibrium driven phases of interacting fermions. This is an interacting generalization of a band-structure Floquet quasienergy winding number and describes chiral pumping of quantum information along the edge. In particular, our invariant sharply distinguishes between a trivial and anomalous Floquet Anderson insulator in the interacting, many-body localized setting. It also applies more generally to models where only fermion parity is conserved, where it differentiates between trivial models and ones that pump Kitaev Majorana chains to the boundary, such as ones recently introduced in the context of emergent fermions arising from eigenstate Z2{\mathbb{Z}}_{2}Z2​ topological order. We evaluate our invariant for the edge of such a system with eigenstate Z2{\mathbb{Z}}_{2}Z2​ topological order, and show that it is necessarily nonzero when the Floquet unitary exchanges electric and magnetic excitations, proving a connection between bulk anyonic symmetry and edge chirality that was recently conjectured.
Abstract We propose a simple approach to realize two-dimensional Floquet topological superfluid by periodically tuning the depth of square optical lattice potentials. We show that the periodic driving can induce topological phase transitions between trivial superfluid and Floquet topological superfluid. For this systems we verify the anomalous bulk-boundary correspondence, namely that the robust chiral Floquet edge states can appear even when the winding number of all the bulk Floquet bands is zero. We establish the existence of two Floquet Majorana zero modes separated in the quasienergy space, with ε0,π = 0,π/T at the topological defects.