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

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.

• The paper introduces a flux-controlled Floquet quantum walk that creates a logical 0/π boundary subspace using coin-dependent drift and momentum-dependent mixing.
• It demonstrates dual-gap topology by quantifying independent winding numbers and achieving robust 2T period doubling in edge responses.
• The work offers an experimentally accessible platform for realizing topological quantum information processing through precise bulk dynamical signatures.

Flux-Controlled Anomalous Floquet Quantum Walk: Boundary Logical Subspace and Bulk Dynamical Probes

Floquet Walk Construction and Microscopic Realization

The paper "Boundary $0/\pi$ logical subspace and bulk dynamical probes in flux-controlled anomalous Floquet quantum walks" (2605.25792) introduces a one-dimensional quantum walk protocol, meticulously defined via a coin-dependent drift and momentum-dependent mixing. The Floquet unitary is $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$, with $\theta_1(k)$ encapsulating the flux $\phi$ as a spatial drift and $\theta_2(k)$ encoding coin mixing. A bipartite lattice, driven with periodic modulation and Peierls phase control, directly realizes this protocol at the microscopic level; the mapping between lattice Hamiltonians and quantum walk operations is exact.

Importantly, this setup admits two chiral symmetric time frames, enabling separate quantization of quasienergy gaps at $0$ and $\pi/T$, characterized by winding numbers $W_1$ and $W_2$. The dual-gap structure organizes four topological phases within the $(M, \phi)$ parameter space: trivial, $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$0-only, $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$1-only, and coexistence ($$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$2 and $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$3 edge modes).

Bulk Topological Structure

In the bulk, the Floquet operator is always represented as a $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$4 unitary matrix, which allows explicit computation of the quasienergy bands:

$$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$5

with gap closings determined by integer values of $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$6 and $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$7 at critical $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$8. The chiral symmetry, $$U(k) = e^{-i\theta_2(k)\sigma_x}e^{-i\theta_1(k)\sigma_z}$$9, requires reading the topology in two symmetric time frames ($\theta_1(k)$0, $\theta_1(k)$1). The winding numbers correspond to independent indices for the $\theta_1(k)$2 and $\theta_1(k)$3 gaps, yielding:

$\theta_1(k)$4

(Figure 1)

Figure 1: Bulk topological phase diagram reveals domains differentiated by winding numbers $\theta_1(k)$5 and $\theta_1(k)$6, marking the presence of edge modes at $\theta_1(k)$7 and $\theta_1(k)$8.

Boundary $\theta_1(k)$9 Logical Subspace and Doubled-Period Response

Under open boundary conditions, the coexistence region features both $\phi$0 and $\phi$1 edge modes localized at the same boundary. These modes, denoted $\phi$2 and $\phi$3, define a two-dimensional logical subspace $\phi$4, protected by chiral symmetry and dual gap topology. Floquet evolution acts as a relative phase operation within this subspace: a single period alternates the sign of the $\phi$5 mode, resulting in robust $\phi$6 oscillations in local boundary observables when both modes are coherently occupied.

Figure 2: Open-boundary edge spectrum and time-domain site-one probability illustrate the $\phi$7 doubled-period response for the coexistence sector, with modes at $\phi$8 and $\phi$9 on the same edge.

Numerical evidence is provided for four sectors; only the coexistence case manifests pronounced period doubling. Optimized superpositions of the edge modes explicitly maximize this effect, showing alternating local expectation values at the boundary.

Bulk Dynamical Probes: Mean Chiral Displacement and Critical Dynamics

The bulk dynamical signatures of dual-gap topology are accessed through two complementary probes:

• Frame-resolved mean chiral displacement (MCD): Running time averages of $\theta_2(k)$0 in each symmetric time frame approach the winding number targets, $\theta_2(k)$1, $\theta_2(k)$2, over pre-reflection time windows. This quantitatively resolves the dual-gap structure dynamically in the bulk.

  Figure 3: Frame-resolved mean chiral displacement traces for bulk representative points converge toward their respective winding numbers, validating the bulk topology from real-space dynamics.

• Benchmark critical dynamics: Local stroboscopic return probabilities at analytically tractable $\theta_2(k)$3 and $\theta_2(k)$4 gap-closing points expose qualitative differences. For $\theta_2(k)$5 gap benchmark, the Floquet evolution produces $\theta_2(k)$6 period alternation, yielding sharp odd-even separation in local observables. The $\theta_2(k)$7 gap case lacks such staggering, as justified by leading-order expansions of the Floquet operators.

  Figure 4: Short-time return probabilities show strong odd-even alternation for $\theta_2(k)$8 gap critical points, but weak staggering for $\theta_2(k)$9 gap critical points, illustrating the dynamical distinction between gap types.

Distinct bulk and boundary responses are tied to the same microscopic flux-controlled quantum walk protocol. The appearance of robust $0$0 signals is not generic, but strictly contingent on simultaneous edge mode occupancy.

Implications and Outlook

The coupling of boundary logical structure and bulk dynamical diagnostics within a flux-tunable Floquet protocol provides a symmetry-protected, minimal platform for quantum-walk information primitives. The logical ($0$1) boundary sector, supporting coherent superpositions and doubled-period responses, is of particular relevance for stroboscopic edge qubit manipulations—notably without invoking non-Abelian anyonic statistics.

Bulk probes such as MCD offer practical, real-space dynamic readout of topological invariants, directly accessible in driven lattice experiments (optical lattices, photonic quantum walks, coupled resonator arrays). The distinction between $0$2 and $0$3 gap criticality in dynamical diagnostics should motivate further exploration of Floquet-engineered critical phases and their applications in topological quantum information platforms.

The theoretical framework articulates a unified, experimentally tractable approach: driven bipartite lattices with position-resolved detection yield both logical edge subspaces and bulk topology via the same Floquet protocol, suggesting future developments in programmable topological Floquet matter, including hybrid quantum information and robust probe schemes.

Conclusion

This study establishes a flux-controlled anomalous Floquet quantum walk with explicit microscopic realization in driven bipartite lattices, organizing both boundary logical subspaces and bulk dynamical probes under a unified protocol. The coexistence region delivers symmetry-protected $0$4 boundary modes with stroboscopic period doubling, while bulk chiral displacement and critical dynamics enable direct dynamical readout of dual-gap topology. Experimental platforms already possess the necessary ingredients for implementation and measurement, so the presented framework is poised for near-term realization and further extension into programmable Floquet architectures for information processing and topological control.