Quantum Valley Hall Insulator Phase

• Quantum Valley Hall Insulating phase is a two-dimensional topological state characterized by a bulk band gap, valley-contrasting Chern numbers, and robust counterpropagating edge modes.
• It is realized in systems like Bernal bilayer graphene and TMD heterobilayers where inversion symmetry breaking, external fields, and lattice relaxations open a gap and induce quantized valley Hall conductivity.
• Electrical control enables topological switching and valleytronics applications, with experiments showing precise resistance plateaus and strong disorder tolerance.

A quantum valley Hall insulator (QVHI) is a two-dimensional topological phase characterized by a bulk band gap and counterpropagating edge or domain-wall modes that are protected by valley-contrasting Chern numbers. Unlike the quantum spin Hall insulator, whose edge states are protected by time-reversal symmetry and spin topology, the QVHI relies on inversion symmetry breaking in multivalley band structures such as those of Bernal-stacked bilayer graphene, transition metal dichalcogenide (TMD) heterobilayers, silicene, and engineered Dirac materials. When a gapped system hosts valleys with opposite Chern numbers, externally induced fields, lattice relaxation, electron interactions, or spin-orbit coupling can stabilize quantized valley Hall conductivity accompanied by robust, valley-polarized edge states.

1. Microscopic Origin and Hamiltonian Construction

In Bernal bilayer graphene (BLG), applying a perpendicular displacement field $D$ breaks inversion symmetry, opening a bulk gap $\Delta$ at the valleys $K$ and $K'$ (Huang et al., 14 Aug 2024). The low-energy Hamiltonian near each valley is described by Dirac-type models with mass terms that take opposite signs in different valleys, thus producing valley Chern numbers of $\nu_K = +2$ and $\nu_{K'} = -2$ for $\Delta > 0$. At domain walls where $D$ changes sign, the valley Chern number difference induces multiple 1D kink (domain-wall) modes. In silicene, similar physics occurs when the sublattice potential $|\ell E_z|$ exceeds the intrinsic spin-orbit gap $\Delta_{SO}$, flipping the sign of the Dirac mass and leading to a QVHI phase (Tahir et al., 2012). In moiré TMD heterobilayers such as MoTe$_2$/WSe$_2$, both lattice relaxation-induced pseudo-magnetic fields and interlayer tunneling can create bands with opposite valley Chern numbers, even when interlayer tunneling is mediated purely by electron-electron interactions (Saha et al., 2 Dec 2025).

The generic minimal model is a two-valley Dirac Hamiltonian:

$$H_\tau(\mathbf{k}) = \hbar v_F (\tau k_x \sigma_x + k_y \sigma_y) + m_\tau \sigma_z$$

where $\tau = \pm1$ labels the valleys ($K$, $K'$), and $m_\tau$ is the valley-dependent mass set by the inversion-breaking field. Interlayer tunneling, interactions, SOC, and exchange fields can further split or invert the masses.

2. Topological Invariants: Valley Chern Numbers and Conductivity

The core topological invariant of the QVHI phase is the valley-resolved Chern number:

$$C_\tau = \frac{1}{2\pi} \int_{\text{BZ}} \Omega_\tau(\mathbf{k}) d^2k ,\quad \Omega_\tau(\mathbf{k}) = -\tau \frac{m_\tau v_F^2}{2[(v_F k)^2 + m_\tau^2]^{3/2}}$$

Opposite signs of $m_\tau$ in the two valleys yield $C_K = +1$, $C_{K'} = -1$ (or $[\pm 2,\mp 2]$ in BLG). The quantized valley Hall conductivity is (Huang et al., 14 Aug 2024, Saha et al., 2 Dec 2025):

$$\sigma_{xy}^v = \frac{e^2}{h}\, \frac{C_K - C_{K'}}{2}$$

This conductivity does not lead to net charge Hall response, but electric fields can drive valley-polarized transverse currents.

Edge or domain-wall states have their own correspondence: a kink or domain-wall where the mass term changes sign hosts $|\Delta\nu_K|/2$ counterpropagating modes per valley (Huang et al., 14 Aug 2024, Qian et al., 2018). In the presence of spin degeneracy (e.g. BLG), this doubles the number of kink modes.

3. Experimental Realizations and Transport Signatures

Empirical transport studies in BLG show resistance plateaus at $R_{\text{kink}} = h/(4e^2) \approx 6.45$ k$\Omega$, with deviations below 1% over temperature ranges up to 50 K and a wide DC bias window (Huang et al., 14 Aug 2024). Device architectures use hBN/graphite encapsulation and split gate arrangements to create tunable domain-wall junctions, with topological (kink-on) and trivial (kink-off) configurations selectable by gate voltages. Moiré TMD heterobilayers (especially MoTe$_2$/WSe$_2$) realize QVHI states at full moiré filling ($\nu=2$ holes per cell), exhibiting quantized nonlocal valley transport without net Hall voltage and robust to disorder (Saha et al., 2 Dec 2025, Xie et al., 2021).

A table summarizing platform-specific signatures follows:

Platform	Valley Chern ($C_K$)	Valley Hall Conductivity ($\sigma_{xy}^v$)
BLG (domain wall)	+2, –2	$4e^2/h$
MoTe$_2$/WSe$_2$ moiré bilayer	+1, –1	$e^2/h$
Silicene (above $E_z^c$)	+½, –½	$e^2/h$

4. Role of Interactions, Symmetry, and Band Topology

Interactions play a central role in the emergence and stability of QVHI phases, especially in moiré TMD bilayers. In MoTe$_2$/WSe$_2$, long-range interlayer repulsion $V$ can mediate interlayer tunneling and induce topologically nontrivial bands even when single-particle hopping is negligible (Saha et al., 2 Dec 2025). Distinct irreducible representations arise for the valley-mixing order parameter: "s-wave" (A$_1$) QVHI with uniform real tunneling and "p ± ip-wave" QVHI when spin-dependent complex hopping patterns dominate. The QVHI phase exists robustly for $V$ exceeding a threshold ($V_c \sim18$–19 meV) and displacement fields $D$ in the regime of band overlap.

Symmetry classifications further refine the topological response. Time-reversal symmetry leads to pairs of bands with opposite valley Chern numbers, preserving $C_{\text{total}}=0$ but enabling nonlocal valley transport. Zeeman fields can selectively lift the topological gap in one valley, producing a quantum anomalous Hall insulator (QAHI) state with net Chern number in the remaining valley (Saha et al., 2 Dec 2025).

5. Robustness and Disorder Tolerance

QVHI edge/domain-wall modes are protected against smooth (long-wavelength) disorder due to the requirement of large momentum transfer for intervalley scattering (Huang et al., 14 Aug 2024, Qian et al., 2018). In BLG, phonon-assisted intervalley backscattering is suppressed up to $\sim$100 K (ZA phonon $\hbar\omega_{\text{min}} \sim 10–68$ meV). Experimental mechanical analogues (magnetically coupled spinner lattices) confirm that domain-wall modes retain exponential localization and quantized conduction over wide disorder strengths (Qian et al., 2018).

Disorder-tolerance is further quantified via valley-symmetry commutators, with localization length $\Lambda\sim R^{-\nu}$ for off-diagonal valley-breaking terms (exponent $\nu\sim2.6$). To maximize robustness, device geometries align domain-walls along momentum directions connecting valleys (e.g. zigzag edges).

6. Topological Switching and Device Applications

Electrically controlled QVHI phases facilitate topological switching. In BLG, toggling the displacement field $D_L$ between opposite signs switches the junction from trivial to topological, yielding an on/off ratio $\sim$200, gate-limited rise/fall times $\sim$6 ms, and negligible operational hysteresis (Huang et al., 14 Aug 2024). Exotic device functionalities, such as valley valves, electron quantum-optics beam splitters, flying-qubit circuits, and proximity-induced topological superconducting channels are realized due to ballistic, valley-polarized domain-wall modes with large quantum efficiency, temperature robustness, and tunable conductance.

Moiré superlattice engineering in TMDs adds further avenues: the phase diagram can be navigated via twist angle, moiré potential depth, and strain-induced pseudo-magnetic field strength (Xie et al., 2021, Saha et al., 2 Dec 2025); transitions between QVHI, QAHI, and trivial phases are possible in a single device by varying filling factor, gate fields, and small Zeeman perturbations.

7. Outlook and Future Research

QVHI phases embody the intersection of crystallographic symmetry, band topology, electron correlations, and device engineering. High-temperature robustness, quantized valley Hall conductance, disorder-tolerant edge modes, and electrical controllability position the QVHI as a central platform for valleytronics, quantum information transfer, and topological logic. Ongoing research explores new material families (e.g. anisotropic Dirac compounds (Dong et al., 2022)), correlation-induced band mixing (e.g. competition between $s$- and $p ± ip$-wave QVHI (Saha et al., 2 Dec 2025)), and connections to fractionalized excitations, electron optics, and valley-polarized superconductivity. Theoretical tools leveraging spin–Chern insulator mappings provide precise definitions of topological invariants even in the presence of disorder (Qian et al., 2018), ensuring a rigorous framework for engineering and characterizing QVHI devices in next-generation quantum technologies.