Magnetic injection photocurrents in valley polarized states of twisted bilayer graphene

Magic-angle twisted bilayer graphene displays a complex phase diagram as a function of flat band filling, featuring compressibility cascade transitions and a variety of competing ground states with broken spin, valley and point group symmetries. Recent THz photocurrent spectroscopy experiments have shown a dependence on the filling which is not consistent with the simplest cascade picture of sequential filling of equivalent flat bands. In this work, we show that when time-reversal symmetry is broken due to valley polarization, a magnetic injection photocurrent develops which can be used to distinguish different spin-valley polarization scenarios. Using the topological heavy fermion model we compute both shift and injection currents as a function of filling and argue that current experiments can be used to determine the spontaneous valley polarization.

💡 Research Summary

In this paper the authors address a puzzling aspect of the phase diagram of magic‑angle twisted bilayer graphene (TBG): recent terahertz (THz) photocurrent measurements reveal filling‑dependent features that cannot be reconciled with the simplest “cascade” picture of sequentially filling four equivalent flat bands. They propose that when time‑reversal symmetry (TRS) is broken by spontaneous valley polarization, a magnetic injection photocurrent (MIP) appears. This current is odd under TRS, even under the combined PT operation, and its sign flips with the direction of the valley magnetization. Consequently, the MIP can serve as a direct, non‑contact probe of the valley‑polarized ground state.

To quantify the effect the authors employ the Topological Heavy Fermion Model (THFM), a lattice formulation that faithfully reproduces the continuum description of TBG near the magic angle. In THFM the low‑energy sector is split into localized, strongly interacting f‑orbitals (two per moiré unit cell, carrying spin and valley indices) and itinerant c‑electrons (four bands). The non‑interacting part yields two ultra‑flat bands separated from dispersive bands by a hybridization gap set by the parameters M and γ. Interactions are introduced via an on‑site Hubbard term U for the f‑electrons and a ferromagnetic exchange J that couples f‑ and c‑electrons, producing an Anderson‑like Hamiltonian. A Hartree‑Fock decoupling provides self‑consistent mean‑field solutions as a function of the filling factor ν.

The authors also incorporate substrate effects arising from alignment with hexagonal boron nitride (hBN). The hBN potential breaks the C₂z rotational symmetry while preserving C₃z, and can be expressed as three constant sub‑lattice potentials Δ₁, Δ₂, and Δ₃ with distinct irreducible‑representation symmetries (B₂, B₁, and A₂ respectively). By symmetry analysis and explicit projection onto the THFM basis they derive the corresponding operators, notably a non‑local “z‑position” operator ˆr_z that captures the opposite energy shifts at the K and K′ valleys. Δ₁ couples to the local f‑orbital polarization σ_z (B₂), whereas Δ₂ and Δ₃ are non‑local and involve sin(k·a_n) factors.

Photocurrent generation is described within the standard second‑order response formalism. The linear photogalvanic effect (LPGE) consists of a shift contribution σ_sh (P‑odd, T‑even) and an injection contribution σ_inj (T‑odd, PT‑even). The shift term depends on Berry‑connection matrix elements and is insensitive to the scattering time τ; it is therefore governed primarily by the substrate potentials Δ₁ and Δ₂. The injection term, however, contains the derivative of the transition frequency ∂_k ω_nm and scales linearly with τ, making it dominant in clean samples where τ is large. Importantly, in TBG the two valleys are related by TRS; when TRS is intact the injection currents from K and K′ cancel. Valley polarization (a B₁, time‑odd perturbation) lifts this cancellation, allowing a net σ_inj_yyy component that flips sign with the valley magnetization.

Hartree‑Fock calculations reveal a cascade of mean‑field ground states as ν is varied: at ν≈1 and ν≈3 the system prefers a spin‑valley‑polarized Chern insulator (|C|=1), at ν≈2 a near‑degenerate competition between a valley‑polarized quantum anomalous Hall (QAH) state (C=2) and trivial valley‑Hall (VH) or spin‑valley‑Hall (SVH) states (C=0), and at ν≈4 a fully filled insulating state. The valley polarization of the f‑electrons, ⟨τ_z⟩, tracks these transitions and directly determines the sign of the magnetic injection current.

The computed photocurrent spectra show that the injection current exhibits pronounced peaks at photon energies resonant with flat‑to‑flat (FF) and flat‑to‑dispersive (FD) interband transitions (∼10 meV). For odd fillings the σ_inj_yyy peak changes sign, reproducing the experimentally observed sign reversal. The shift current, by contrast, is largely temperature‑independent and persists up to ∼60 K, reflecting its origin in the static sub‑lattice potentials. The authors argue that the relative magnitude of shift versus injection contributions can be tuned by temperature (through τ) and by engineering the substrate potential, providing a practical route to isolate the magnetic injection component.

In summary, the paper demonstrates that magnetic injection photocurrents are a robust, symmetry‑sensitive probe of spontaneous valley polarization in TBG. By combining the THFM framework with realistic Hartree‑Fock solutions and a careful symmetry analysis of substrate effects, the authors predict filling‑dependent sign changes and temperature trends that match existing THz photocurrent data. They suggest that future experiments, possibly with controlled magnetic fields or gate‑tunable τ, can unambiguously distinguish between QAH, VH, and SVH phases, offering a powerful complement to transport and magnetometry measurements in the ongoing exploration of correlated topological states in twisted graphene systems.