Chiral Floquet phases support quantized chiral flow along their boundaries even when the bulk is fully localized. In this Letter, we establish a complementary phenomenon, which we call Hilbert-space pumping: under a boundary condition twisted by time translation, a part of the boundary Hilbert space, of quantized dimension, is transferred between the opposite boundaries. The twist is the temporal analogue of Laughlin's flux insertion in the quantum Hall effect. For noninteracting fermions, the number of transferred boundary modes is determined by the winding number, as we show analytically for the boundary dynamics and numerically for the anomalous Floquet Anderson insulator. For interacting systems whose boundary dynamics is a quantum cellular automaton, the transferred dimension is set by the Gross-Nesme-Vogts-Werner (GNVW) index: the twisted boundary Floquet operator is the Floquet operator on a tilted time slice, whose Hilbert-space dimension is multiplied by the index for each period of twist. We derive this relation for matrix product unitaries and test it numerically in random unitary circuits. Applied to the matrix product operators of non-invertible symmetries, the same twisted rank returns the quantum dimension of the symmetry operator in addition to its flow index.
We construct and classify chiral topological phases in driven (Floquet) systems of strongly interacting bosons, with finite-dimensional site Hilbert spaces, in two spatial dimensions.The construction proceeds by introducing exactly soluble models with chiral edges, which in the presence of many-body localization (MBL) in the bulk are argued to lead to stable chiral phases.These chiral phases do not require any symmetry and in fact owe their existence to the absence of energy conservation in driven systems.Surprisingly, we show that they are classified by a quantized many-body index, which is well defined for any MBL Floquet system.The value of this index, which is always the logarithm of a positive rational number, can be interpreted as the entropy per Floquet cycle pumped along the edge, formalizing the notion of quantum-information flow.We explicitly compute this index for specific models and show that the nontrivial topology leads to edge thermalization, which provides an interesting link between bulk topology and chaos at the edge.We also discuss chiral Floquet phases in interacting fermionic systems and their relation to chiral bosonic phases.
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 topological order. We evaluate our invariant for the edge of such a system with eigenstate
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.
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.
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.
Chiral topological phases, like quantum Hall states, are characterized by anomalous unidirectional edge transport arising from a nontrivial insulating bulk. Recent developments suggest that they can be extended to periodically driven (Floquet) systems, where quantum information is pumped in lieu of charge or energy. The authors uncover a new chiral Floquet phase of quantum dynamics, whose edge pumps quantized units of fractionalized excitations. These phases are ``radical,'' in that the quantum dimension pumped is the square root of an integer. Unexpectedly, such edge borders a bulk with dynamical anyon transmutation, demonstrating a novel form of bulk-boundary correspondence.
In two dimensions, interacting Floquet topological phases may arise even in the absence of any protecting symmetry, exhibiting chiral edge transport that is robust to local perturbations. We explore a similar class of Floquet topological phases in three dimensions, with translational invariance but no other symmetry, which also exhibit anomalous transport at a boundary surface. By studying the space of local 2D unitary operators, we show that the boundary behavior of such phases falls into equivalence classes, each characterized by an infinite set of reciprocal lattice vectors. In turn, this provides a classification of the 3D bulk, which we conjecture is complete. We demonstrate that such phases may be generated by exactly solvable ``exchange drives'' in the bulk. In the process, we show that the edge behavior of a general exchange drive in two or three dimensions can be deduced from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.