Signatures of the Quantum Geometric Dipole of Interlayer Excitons in Counterflow Conductivity

Abstract: Collective excitations of many-body electron systems can carry internal structure, supporting novel quantum geometric and topological properties. Among these are a quantum geometric dipole (QGD), which for excitons have direct significance as an internal polarization. For interlayer excitons of a bilayer system, this represents an in-plane dipole moment, which can be used to drive them with in-plane electric fields. In this work, we consider counterflow electric currents associated with driven excitons in such a bilayer system as a probe of their QGD structure. As a simple but non-trivial example, we analyze a structure with a one-dimensional periodic potential in a strong perpendicular magnetic field. The resulting magnetoexciton bands host QGD structure that distinguishes it from the exciton QGD of a uniform system. To model exciton transport we adopt a Boltzmann approach that includes inter-band tunneling, allowing us to consider non-equilibrium momentum distributions that result from strong layer-antisymmetric driving fields. We show how linear response to a layer-symmetric component of the driving fields provide information about the QGD, and that the broad QGD structure of the exciton bands can be probed by the varying the layer-antisymmetric field. Our results demonstrate that counterflow conductivity serves as a tunable probe of the internal quantum geometric structure carried by the interlayer excitons, connecting transport to the quantum geometry of many-body excitations.

• The paper establishes a theoretical framework linking the quantum geometric dipole to counterflow conductivity in interlayer excitons.
• It uses a semiclassical Boltzmann approach with Landau-Zener tunneling to model exciton band dynamics under periodic potentials.
• Numerical results reveal that band structure modifications control the transverse conductivity, offering direct experimental probes of quantum geometry.

Signatures of the Quantum Geometric Dipole of Interlayer Excitons in Counterflow Conductivity

Introduction and Context

This work develops a comprehensive theoretical framework to probe the quantum geometric dipole (QGD) of interlayer excitons in bilayer systems via counterflow conductivity measurements. The QGD, which emerges from differences in the Berry connections associated with the electron and hole constituents of the exciton, acts as an internal, momentum-dependent in-plane polarization. Interlayer excitons—wherein electrons and holes reside in different layers—are of particular current interest due to their extended lifetimes, tunability via interlayer bias, and direct relevance in van der Waals materials and quantum Hall bilayers.

Theoretical formulations and numerical analyses are specialized to the case of interlayer magnetoexcitons under a strong out-of-plane magnetic field and a unidirectional periodic potential, which generates a nontrivial exciton band structure with momentum-dependent QGD profiles. The central focus is the linkage between transport observables, particularly the transverse counterflow conductivity, and quantum geometric aspects of the excitonic bands.

Theoretical Framework

Quantum Geometric Dipole

The QGD $D(\mathbf{K})$ is defined as the gauge-invariant difference between the Berry connections associated with the hole and electron components of an exciton wavefunction with total momentum $\mathbf{K}$: $$D(\mathbf{K}) = A^{(h)}(\mathbf{K}) - A^{(e)}(\mathbf{K})$$. This formalism generalizes the familiar momentum-dipole relation for magnetoexcitons to a broad family of collective modes characterized by quantum geometry, and provides a direct, microscopic route to the internal excitonic dipole moment.

Within the model system, the periodic potential induces zone folding and opens avoided crossings in the band structure. The band-projected QGD is determined by a weighted sum over folded momenta, with each contribution linked to the expansion coefficients of the exciton band eigenstates.

Semiclassical Dynamics and Boltzmann Transport

The exciton dynamics are formulated in a band-projected semiclassical framework, yielding coupled equations for the center-of-mass coordinate and momentum of the driven exciton wavepackets. The electric fields applied to the electron and hole layers are decomposed into symmetric ($\mathbf{E}_+$) and antisymmetric ($\mathbf{E}_-$) components. Whereas $\mathbf{E}_+$ couples linearly to the QGD, generating a drift velocity sensitive to quantum geometry, $\mathbf{E}_-$ drives interlayer antisymmetric momentum and can control the occupation profile over the band landscape.

To capture non-equilibrium phenomena in transport, particularly the role of interband tunneling near avoided crossings, a steady-state Boltzmann equation is solved using the relaxation-time approximation, with explicit modeling of Landau-Zener tunneling. The occupation function is thus allowed band discontinuities at the avoided crossings, a critical aspect for correctly modeling transport in the presence of strong driving.

Numerical Results and Major Findings

Band Structure and Quantum Geometric Profiles

Numerical diagonalization of the multiband exciton Hamiltonian reveals that the application of a periodic potential progressively opens gaps at zone boundaries and flattens low-lying bands. The QGD demonstrates helical structure near the Brillouin zone center and undergoes vanishing at Brillouin zone boundaries due to the periodicity constraints. As the periodic potential strength $W$ increases, the QGD for the lowest band is suppressed, whereas higher bands at large $|K|$ acquire enhanced QGD due to admixture with states of higher momentum.

Counterflow Conductivity as a Probe

The counterflow conductivity tensor $\sigma^{(\mathrm{CF})}$ is shown to be sensitive to the quantum geometry of the exciton bands. Analytically, the off-diagonal component $\mathbf{K}$0 is an averaged derivative of the QGD with respect to momentum and is directly related to the internal polarization structure of the bands.

Key numerical findings:

• Suppression of $\mathbf{K}$1 in the flat-band regime: A large periodic potential confines excitons to the flat, low-lying bands where the QGD gradient is suppressed, resulting in reduced transverse counterflow conductivity.
• Enhancement of $\mathbf{K}$2 with driving field: Increasing the layer-antisymmetric electric field $\mathbf{K}$3 increases the probability of interband tunneling, populates higher bands with steeper QGD profiles, and thus amplifies counterflow conductivity. This strong dependence on band occupation reflects the underlying quantum geometry.
• The impact of dielectric environment, encoded via the dielectric constant $\mathbf{K}$4, is twofold: increasing $\mathbf{K}$5 suppresses the low-$\mathbf{K}$6 QGD but also facilitates excitation into higher bands, which can increase overall counterflow response.

An important analytical result is the exact vanishing of the diagonal component $\mathbf{K}$7, reflecting symmetry constraints and the purely transverse QGD response.

Boltzmann Distribution and Tunneling

The structure of the nonequilibrium Boltzmann distribution function is highly sensitive to both the magnitude of the periodic potential and the antisymmetric field. At low $\mathbf{K}$8, exciton populations reflect the unperturbed (uniform) band structure, but increasing $\mathbf{K}$9 or $$D(\mathbf{K}) = A^{(h)}(\mathbf{K}) - A^{(e)}(\mathbf{K})$$0 leads to population redistribution across avoided crossings. The model explicitly incorporates the interplay of occupation discontinuities (due to Landau-Zener transitions) and overall distribution asymmetry driven by strong antisymmetric fields.

Implications and Future Directions

This work firmly establishes counterflow conductivity as a sensitive probe of the internal quantum geometric structure of interlayer excitons. The results demonstrate that the QGD, a fundamentally quantum mechanical and gauge-invariant measure of internal polarization, has clear transport signatures accessible in experimental counterflow measurements.

Future theoretical and experimental extensions suggested by the analysis include:

• Temporal and noise probes: Frequency-dependent response and noise, as well as co-flow conductivity, may provide enhanced sensitivity to sharp spikes and high-gradient features of the QGD, especially near avoided crossings.
• Real-space inhomogeneity and pairing: Incorporation of exciton generation/recombination, spatially varying densities, and possible Bose condensates would enable a more complete picture of multicomponent excitonic transport.
• Topological phenomena: Potential exists to explore the link between QGD profiles and topological invariants in exciton band structures—especially relevant for moiré superlattices and two-dimensional vdW heterostructures.
• Experimental platforms: The framework is adaptable to other bilayer systems, including those based on graphene, TMDs, or engineered quantum Hall bilayers, providing an avenue for direct comparison with transport and optical probes of excitonic systems in high magnetic fields.

Conclusion

This work provides a rigorous theoretical foundation for understanding and probing the quantum geometric dipole of interlayer excitons via counterflow transport. By linking bulk transport observables to internal band geometry, it opens pathways for both systematic experimental characterization of excitonic bands and the exploration of interaction-driven quantum geometric phenomena in multilayer materials. The semiclassical Boltzmann approach, combined with explicit multiband and tunneling modeling, demonstrates both the tractability and richness of quantum geometric effects in collective excitation transport. The implications extend to a broad array of contemporary quantum materials where band geometry and many-body effects intersect.