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arXiv:2609.00975

Topological charges and parity selection at Floquet quasienergy degeneracies

Sigmund Kohler, David Guéry-Odelin

cond-mat.mes-hallquant-phSubmitted Sep 1, 2026• Announced Sep 2, 2026
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Abstract

The quasienergy spectrum of a strongly driven two-level system as a function of the driving parameters exhibits conical intersections, which are enabled by hidden time-nonlocal symmetries. We show that each such crossing carries a quantized topological charge: the Floquet--Berry phase acquired along an adiabatic loop around a cone is equal to a $\mathbb{Z}_2$-valued charge. We further identify a second family of degeneracies that occurs at vanishing driving amplitude, when the level splitting matches $m$ energy quanta of the field. Along the Stark-shifted resonance line, the minimum quasienergy gap opens as $|A|^m$, and the charge is nontrivial only for odd $m$. We analytically derive both results from a perturbative reduction to a spin-$1/2$ in an effective two-dimensional magnetic field and confirm them numerically through the Bargmann invariant. Moreover, we propose a chirality-based protocol that cancels the dynamical phase to isolate the geometric one, and an ancilla-based Ramsey readout that renders the topological charge directly observable.

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10 pages

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The quasienergy spectrum of a strongly driven two-level system as a function of the driving parameters exhibits conical intersections, which are enabled by hidden time-nonlocal symmetries. We show that each such crossing carries a quantized topological charge: the Floquet--Berry phase acquired along an adiabatic loop around a cone is equal to a Z2\mathbb{Z}_2Z2​-valued charge. We further identify a second family of degeneracies that occurs at vanishing driving amplitude, when the level splitting matches mmm energy quanta of the field. Along the Stark-shifted resonance line, the minimum quasienergy gap opens as ∣A∣m|A|^m∣A∣m, and the charge is nontrivial only for odd mmm. We analytically derive both results from a perturbative reduction to a spin-1/21/21/2 in an effective two-dimensional magnetic field and confirm them numerically through the Bargmann invariant. Moreover, we propose a chirality-based protocol that cancels the dynamical phase to isolate the geometric one, and an ancilla-based Ramsey readout that renders the topological charge directly observable.
We investigate the Floquet spectrum of a detuned, driven two-level system and show that it exhibits exact quasienergy crossings when the detuning is an integer multiple of the energy quantum of the driving field. This behavior can be explained by a hidden time-nonlocal parity, which allows the Floquet modes to be classified as even or odd. Then a generic feature is the emergence of exact crossings between quasienergies of different parity. A constructive proof of the existence of the symmetry is based on a scalar recurrence relation. Moreover, we present a general scheme for its numerical computation, which can be applied to models beyond the two-level system. Analytical results are illustrated with numerical data.
We explore the physics of a Chern insulator subjected to a two-step Floquet drive. We analytically obtain the phase diagram and show that the system can exhibit different topological phases characterized by the presence and chirality of edge modes in the two bulk gaps of the Floquet quasienergy spectrum, around 0 and $\ensuremath{\pi}$. We find that the phase of the system depends on the mean but not on the amplitude of the drive. The bulk topological invariants characterizing the phases can be extracted by mapping the unitary evolution within a time period to an energetically trivial but topologically nontrivial time evolution. An extensive numerical study of the bulk topological invariants in the presence of quenched disorder reveals transitions induced by strong disorder (i) from the different topological to trivial insulator phases and (ii) from a trivial to a topological Anderson insulator phase at intermediate disorder strengths. Careful analysis of level statistics of the quasienergy spectrum indicates a ``levitation-annihilation'' mechanism near these transitions.
Few-level quantum systems driven by nf{n}_{\mathrm{f}}nf​ incommensurate fundamental frequencies exhibit temporal analogs of noninteracting phenomena in nf{n}_{\mathrm{f}}nf​ spatial dimensions, a consequence of the generalization of Floquet theory in frequency space. We organize the fundamental solutions of the frequency lattice model for nf=2{n}_{\mathrm{f}}=2nf​=2 into a quasienergy band structure and show that every band is classified by an integer Chern number. In the trivial class, all bands have zero Chern number and the quasiperiodic dynamics is qualitatively similar to Floquet dynamics. The topological class with nonzero Chern bands has dramatic dynamical signatures, including the pumping of energy from one drive to the other, chaotic sensitivity to initial conditions, and aperiodic time dynamics of expectation values. The topological class is however unstable to generic perturbations due to exact level crossings in the quasienergy spectrum. Nevertheless, using the case study of a spin in a quasiperiodically varying magnetic field, we show that topological class can be realized at low frequencies as a prethermal phase, and at finite frequencies using counterdiabatic tools.
M\"obius topological insulators have dispersive edge bands with M\"obius twists in momentum space, which are protected by the combination of chiral and Z2{\mathbb{Z}}_{2}Z2​-projective translational symmetries. In this work, we reveal a unique type of M\"obius topological insulator, whose edge bands can twist around the quasienergy $\ensuremath{\pi}$ of a periodically driven system and are thus of Floquet origin. By applying time-periodic quenches to an experimentally realized M\"obius insulator model, we obtain interconnected M\"obius edge bands around zero and $\ensuremath{\pi}$ quasienergies, which can coexist with a gapped or gapless bulk. These M\"obius bands are topologically characterized by a pair of generalized winding numbers, which are integer-quantized as a result of an emergent chiral symmetry at a high-symmetry point in momentum space. Numerical investigations of the quasienergy and entanglement spectra provide consistent evidence for the presence of such M\"obius topological phases. A protocol based on the adiabatic switching of edge-band populations is further introduced to dynamically characterize the topology of Floquet M\"obius edge bands. Our findings thus extend the scope of M\"obius topological phases to nonequilibrium settings and unveil a unique class of M\"obius-twisted topological edge states without static counterparts.
The dynamic effect of particles in a periodically driven two-level system is studied with the help of SU(2) symmetry. General closed-form solutions for the quasienergy and the Floquet states are obtained, from which it is found that the avoided quasienergy crossing is confined in the range (−ω/2, ω/2), where ω is the frequency of the driving fields. Of particular interest are high-frequency driving fields, of which it is demonstrated that the exact quasienergy crossings occur at certain values of amplitude and frequency of driving fields, and this phenomenon of dynamic localization should be determined by experiments.
Periodically driven quantum systems have emerged as one of the most versatile tools for quantum simulation, enabling the engineering of effective Hamiltonians, topological phases, and inherently nonequilibrium phenomena such as discrete time crystals. Yet fundamental challenges persist: from a spectral ambiguity that precludes a unique Floquet ground state to an inability to manipulate Floquet eigenstates beyond the adiabatic regime. This thesis demonstrates that counterdiabatic driving—a technique from adiabatic quantum control—provides a unifying framework to address these challenges. We first show that the periodic lab-frame Hamiltonian generates counterdiabatic driving for the Floquet eigenstates, revealing Floquet theory as an inverse counterdiabatic driving problem. Exploiting this equivalence, we develop geometric Floquet theory: a reformulation based on the parallel-transport gauge, which uniquely decomposes Floquet dynamics into a geometric contribution, encoded in Berry phases, and a dynamical contribution, generated by the average-energy operator. The average-energy operator provides an unambiguous sorting of the quasienergy spectrum, identifies a unique Floquet ground state at all drive frequencies, and serves as a sensitive diagnostic for heating and symmetry-breaking transitions. Inherently nonequilibrium phenomena—the 𝜋-quasienergy splitting of discrete time crystals and the anomalous edge modes of Floquet topological insulators—are traced to geometric phases. On the practical side, we exploit the intrinsic robustness of prethermal discrete time crystals for a highly frequency-selective DTC-based AC magnetic field sensing scheme and demonstrate it experimentally on a nuclear spin quantum simulator. We then generalize counterdiabatic driving to periodically driven systems. In particular, we derive a non-perturbative variational principle for the Floquet adiabatic gauge potential; this yields local, experimentally feasible control protocols for transitionless manipulation of Floquet states. We demonstrate this formalism on Floquet state preparation in an ultracold atom quantum simulator and many-body state transfer across a quantum phase transition, with significant improvements over unassisted protocols in both cases.