Intervalley Floquet Band Crossing

• The paper illustrates how CPL-induced valley-specific Floquet mass splitting drives controlled band inversion in bilayer transition metal dichalcogenides.
• It employs a high-frequency Floquet Hamiltonian expansion truncated at O(1/Ω²) to reveal the critical condition where Floquet splitting equals the static inter-valley gap.
• The study links band inversion to many-body effects, such as Laughlin-type fractional Chern insulators and emergent multi-Weyl points, showcasing advanced quantum control.

Intervalley Floquet band crossing refers to the phenomenon whereby Floquet bands originating from distinct valleys in momentum space—such as the $K$ and $K'$ points in transition metal dichalcogenide bilayers—are tuned, via external periodic driving (typically circularly polarized light), into direct energetic overlap or inversion. This driven band inversion is not only of fundamental theoretical interest for its connection to topological and symmetry-breaking physics, but also provides a knob for controlling collective phases, such as fractional Chern insulators (FCIs) and multi-Weyl semimetals, via selective manipulation of valley degrees of freedom (Shi et al., 6 Jan 2026, Yan et al., 2017).

1. Floquet Engineering in Multivalley Systems

Floquet engineering describes the use of periodic driving to modify the properties of quantum materials, by effectively generating time-averaged Hamiltonians with symmetry properties and topological structure unattainable in equilibrium. In multivalley systems such as twisted bilayer MoTe$_2$, coupling the system to a circularly polarized light (CPL) drive is implemented via the Peierls substitution in the kinetic Hamiltonian, $$\boldsymbol{k} \rightarrow \boldsymbol{k} + e\boldsymbol{A}(t)/\hbar$$, with $$\boldsymbol{A}(t) = A_0(\cos\Omega t, -\sin\Omega t)$$. The high-frequency limit ($\hbar \Omega \gg$ bandwidth) justifies truncation of the Magnus (van Vleck) expansion at $\mathcal{O}(1/\Omega^2)$, yielding an effective static Floquet Hamiltonian (Shi et al., 6 Jan 2026).

When both conduction and valence bands are included in the microscopic model, the Floquet Hamiltonian exhibits explicit time-reversal symmetry (TRS) breaking, with valley-contrasting terms. Valley-resolved Floquet masses $m_{\mathrm{floq}}(\xi)$ (where $\xi$ distinguishes the valley) shift the bands in energy, facilitating direct control over the relative band alignment between valleys.

2. Microscopic Description and Band Inversion Mechanism

The single-particle Floquet Hamiltonian for twisted bilayer MoTe$_2$ is constructed from a basis combining band, layer, and valley degrees of freedom. Explicitly, to leading nontrivial orders in the inverse CPL frequency,

$$H_F = H_0 + \frac{1}{\hbar\Omega}[H_1, H_{-1}] + \frac{1}{2(\hbar\Omega)^2}[H_1,[H_0,H_{-1}]] + \mathrm{h.c.}$$

The dominant physical effect is a valley-contrasting shift in the band energies given by $\sim \xi\Delta\sigma_z$, where $\Delta = \hbar(v_F A_0)^2/\Omega$ is proportional to the CPL intensity. This Floquet mass breaks TRS, lowering the bands in valley $\xi=+$ by $-\Delta$ and raising them in $\xi=-$ by $+\Delta$. Each valley’s effective mass is also renormalized (Shi et al., 6 Jan 2026).

The critical condition for intervalley band crossing (band inversion) is reached when the energy splitting induced by Floquet engineering exceeds the static inter-valley gap, i.e.,

$$2\Delta = \delta E^0,\qquad \delta E^0 = E_{+,1}^0(\mathbf{k}_c) - E_{-,2}^0(\mathbf{k}_c)$$

where $E_{+,1}^0$ and $E_{-,2}^0$ are the quasienergies of the top valence band in valley $K$ and the second valence band in valley $K'$, respectively, at high-symmetry point $\mathbf{k}_c$. The critical intensity $\Delta_c$ for the driving field thus reads $\Delta_c = \frac{\delta E^0}{2}$, typically $\sim 10$ meV for $\hbar\Omega=3$ eV (Shi et al., 6 Jan 2026).

3. Intervalley Band Crossing: Many-Body Consequences and FCI Switching

Projection of interacting electrons (or holes) into the top two Floquet valence bands in each valley, at total hole filling $\nu_h=5/3$, reveals profound many-body effects. In the absence of drive ($\Delta=0$), the ground state is degenerate: it realizes a Laughlin-type $\nu=2/3$ FCI in one valley, with occupations $\nu_{h,+}=1$, $\nu_{h,-}=2/3$ or vice versa. This is characterized by a threefold ground-state degeneracy, a finite many-body gap, and a gapped particle-entanglement spectrum (Shi et al., 6 Jan 2026).

As $\Delta$ increases, valley pseudospin degeneracy is split by the single-particle Floquet mass. For $\Delta<\Delta_c$, holes preferentially occupy the energetically favorable band. At $\Delta = \Delta_c$, when the Floquet-induced splitting matches the bare inter-valley band gap, a sharp transition is enacted: all holes instantaneously redistribute to the now-lowest valence band, fully occupying the second band of one valley. The system’s ground state thus switches its valley polarization—transforming from a Laughlin-type FCI in valley $+$ to a Laughlin-type FCI in valley $-$. The transition is sharp because it is triggered by a direct single-particle level crossing, with the many-body FCI gap remaining open throughout (Shi et al., 6 Jan 2026).

4. Floquet Multi-Weyl Point Formation in Crossing-Nodal-Line Semimetals

In crossing-nodal-line semimetals, intervalley Floquet band crossing can also manifest as the merging or creation of multi-Weyl points under periodic driving. In such systems, the driven Hamiltonian acquires a term $\propto \gamma \eta F(\boldsymbol{k}) \tau_y$ (with drive helicity $\eta$ and function $F$ of momentum), which couples different valleys (nodal rings). The incidence angles of the drive determine the monopole charges and locations of emergent Weyl nodes. When tuning parameters so that two Weyl nodes of the same monopole charge merge at a crossing point, a double-Weyl node (monopole charge $\pm2$) is realized; by contrast, the merger of nodes with opposite charge results in annihilation (Yan et al., 2017).

The effective low-energy description near the merged point has the canonical double-Weyl form: $$H_{\mathrm{double}}(\mathbf{q}) = \alpha q_z \sigma_z + \beta(q_x^2 - q_y^2)\sigma_x + 2\beta q_x q_y \sigma_y$$ with coefficients $\alpha, \beta$ tunable by drive parameters (Yan et al., 2017).

5. Symmetry Breaking, Topology, and Experimental Signatures

Intervalley Floquet band crossing is fundamentally tied to explicit TRS breaking, induced by the CPL drive. This is evidenced by Floquet masses $\propto \xi\Delta$ and the splitting of valley bands by $\approx 2\Delta$, which act as valley-contrasting pseudospin Zeeman fields (Shi et al., 6 Jan 2026). Despite the Floquet band inversion, the high-frequency limit preserves the valley Chern number structure; each valley’s topological character persists under drive.

Experimentally, these transitions are observable spectroscopically and in collective response. In crossing-nodal-line semimetals, pump-probe angle-resolved photoemission spectroscopy (ARPES) detects Floquet side-bands and new crossing points, whose movement and merging indicate nodal manipulation. Fermi-arc surface states connected to multi-Weyl nodes, as well as chiral charge-flipping with drive helicity, provide further evidence (Yan et al., 2017). In moiré-engineered 2D materials, the valley-polarized FCI switch could be inferred via transport measurements sensitive to the Hall response and entanglement spectra (Shi et al., 6 Jan 2026).

6. Summary Table: Key Features of Intervalley Floquet Band Crossing

Phenomenon	Model System	Core Mechanism
Valley-polarized FCI switching	Floquet twisted bilayer MoTe$_2$	CPL-induced Floquet mass splitting, interaction-driven redistribution at filling $\nu_h=5/3$ (Shi et al., 6 Jan 2026)
Double-Weyl point formation and merging	Crossing-nodal-line semimetal	CPL Floquet terms mediate valley coupling, angle-driven Weyl node coalescence (Yan et al., 2017)

Inclusion of conduction and valence bands in Floquet expansion, retention of all $\mathcal O(1/\Omega^2)$ terms, and interactions in partially filled topological bands are necessary for capturing the full spectrum of physical behavior intrinsic to intervalley Floquet band crossing. The phenomenon provides a route to dynamically and reversibly engineer correlated, valley-contrasting quantum matter via light-matter interaction.