Quantum Valley Hall Topological Phases

• Quantum valley Hall-based topological phases are 2D systems where nontrivial topology is induced by valley degrees of freedom, resulting in valley-polarized edge channels and domain wall modes.
• They are realized in honeycomb and bilayer structures via symmetry-breaking mechanisms that generate distinct valley invariants such as valley Chern and Euler numbers.
• Tunable by external fields, twist engineering, and substrate effects, these phases hold promise for low-power electronics and advanced quantum device architectures.

Quantum Valley Hall-based Topological Phases

Quantum valley Hall-based topological phases constitute a broad class of two-dimensional systems in which nontrivial electronic topology is associated with the valley degree of freedom. In contrast to conventional quantum Hall or quantum spin Hall phases, these states derive their topological properties from the inequivalence of electronic valleys at distinct Brillouin zone points (commonly K and K′ in honeycomb or hexagonal lattices), and are characterized by valley-specific topological invariants. The fundamental distinguishing feature of quantum valley Hall (QVH) phases is the localization of Berry curvature or more general geometric quantities around valley points, and the appearance of protected boundary or domain wall modes whose existence is guaranteed by differences in valley-resolved topological indices. This valley selectivity underlies a diversity of phenomena, including charge-neutral edge transport, robust valley filtering, and tunable topological transitions, and has enabled a rich taxonomy of related topological states ranging from conventional QVH to zero–Berry-curvature valley phases and valley-polarized Chern/topological insulators.

1. Topological Invariants: From Valley Chern Number to Valley Euler Number

The archetype of quantum valley Hall phases is the standard QVH insulator, defined by a vanishing net Chern number $C = C_K + C_{K'} = 0$, but nonzero valley Chern number $C_v = C_K - C_{K'}$. Here $C_K$ ($C_{K'}$) is obtained by integrating the Berry curvature $\Omega(\mathbf{k})$ over a region in momentum space surrounding the K (K′) valley: $$C_K = \frac{1}{2\pi}\int_{V_K} \Omega(\mathbf{k})\,d^2k, \qquad C_{K'} = \frac{1}{2\pi}\int_{V_{K'}} \Omega(\mathbf{k})\,d^2k,$$ with the Berry curvature typically sharply concentrated at each valley in models with negligible intervalley scattering (Xue et al., 2024, Tahir et al., 2012, Xue et al., 2018). A valley Chern imbalance leads to counterpropagating valley-polarized edge channels.

A further generalization arises in the context of space-time inversion ($\mathcal{I}_{\mathrm{ST}} = \mathcal{P}\mathcal{T}$) symmetric systems, where the Berry curvature vanishes identically ($\Omega(\mathbf{k}) = 0$), but a nonzero Euler curvature $\Omega_E(\mathbf{k})$ can be defined using real gauge wavefunctions: $$\Omega_E(\mathbf{k}) = \left\langle \nabla_{\mathbf{k}} u^1(\mathbf{k}) | \times | \nabla_{\mathbf{k}} u^2(\mathbf{k}) \right\rangle.$$ The valley Euler number (VEN) is defined as

$$\chi_\eta = \frac{1}{2\pi}\int_{V_\eta} \Omega_E(\mathbf{k})\,d^2k,\qquad \eta = K,K',$$

supporting phases where domain wall states are protected by the difference in Euler class, even in the total absence of microscopic Berry curvature (so-called zero–Berry-curvature QVH: ZBC-QVHE) (Ghadimi et al., 2024).

These valley-resolved invariants can additionally coexist or compete with total Chern indices (producing valley-polarized quantum anomalous Hall phases) or with higher-genus geometric indices (as in fractional regimes or in the presence of multi-channel valley structures) (Pan et al., 2014, Saha et al., 2 Dec 2025, Das et al., 19 Sep 2025).

2. Minimal Models and Symmetry Protection

Prototypical minimal models for QVH phases are based on honeycomb lattices (graphene, silicene, germanene, or 2D transition-metal dichalcogenides) with symmetry-breaking mass terms: $$H_\eta(\mathbf{k}) = \hbar v_F (\eta k_x \sigma_x + k_y \sigma_y) + m_\eta \sigma_z,$$ where $m_\eta$ can arise from sublattice-staggered potentials (inversion breaking), spin-orbit coupling, or exchange fields, often yielding opposite mass terms ($m_K = -m_{K'}$) for the two valleys (Tahir et al., 2012, Xue et al., 2024, Yuan et al., 2021). The presence of time-reversal symmetry ensures $C = 0$, with the topological regime controlled by the sign and magnitude of the mass.

In ZBC-QVHE, AA′-stacked bilayer hexagonal lattices with space-time inversion symmetry enforce real Bloch wavefunctions and identically zero Berry curvature at all $\mathbf{k}$. Nonetheless, the antisymmetric real connection enables a nontrivial Euler class, with sharp locality of the Euler curvature near valleys. The domain wall Hamiltonian for a mass term $m(y)$ reads (Ghadimi et al., 2024): $$H_\eta^{\mathrm{DW}}(k_x,y) = v\eta k_x \sigma_x\tau_0 - i v \partial_y \sigma_y\tau_0 + m(y)\sigma_z\tau_z + t_\perp \sigma_0\tau_x,$$ supporting two counterpropagating helical modes protected by mirror or combined chiral + inversion symmetries, robust even when the Berry curvature is strictly zero.

Symmetry ensures the protection and quantization of these valley topological invariants as long as intervalley scattering is negligible; in particular, mirror symmetry about the domain wall and the presence of combined chiral and inversion symmetries are sufficient to enforce gapless crossing at the domain boundary in models with nontrivial VEN.

3. Realizations: Material Platforms and Band Engineering

QVH-based topological phases are realized or proposed in a variety of platforms:

• Buckled Honeycomb Monolayers and Bilayers: Germanene, silicene, and related materials, where inversion and time-reversal symmetry breaking can be controlled by substrates, external electric fields, or magnetic proximity (Xue et al., 2024, Pan et al., 2014, Ezawa, 2013, Tahir et al., 2012). Germanene-based van der Waals heterostructures support QVHE with valley Chern number $C_v = 2$ and gaps up to 70 meV, with transitions to QAHE or trivial phases driven by magnetic orientation.
• Layered and Twisted Systems: AA′-stacked h-BAs, h-BP, and large-angle twisted bilayer graphenes are calculated to realize ZBC-QVHE by DFT, supporting robust helical domain wall states protected by VEN (Ghadimi et al., 2024). In low-buckled bilayers, a QVH regime is induced via layer-resolved Rashba SOC—either extrinsically by design or emergently via moiré patterning and twist engineering (Shen et al., 2024).
• Transition Metal Dichalcogenide (TMD) Heterobilayers: MoTe$_2$/WSe$_2$ demonstrates a strong interaction-induced QVH insulating state (QVHI) at commensurate fillings ($v = 2$), where long-range Coulomb interactions mediate interlayer tunneling and generate nontrivial valley topology even without bare hopping (Saha et al., 2 Dec 2025). These QVHI states can transition to QAH insulators upon Zeeman splitting.
• Photonic and Classical-Wave Analogs: Spin- and valley-contrasting topological phases have been engineered in photonic crystals, where synthetic gauge fields for photonic pseudospin and valley are controlled through geometric structure, enabling direct tuning between QVH and QSH regimes (Xue et al., 2018).

4. Edge and Domain Wall Modes: Helical and Chiral Valley Channels

A defining signature of QVH-type phases is the presence of one-dimensional metallic channels at system edges or internal domain walls between regions of differing valley topological invariant. In conventional QVH phases, domain walls between regions of opposite valley Chern number host counterpropagating edge states, each associated with a specific valley and direction (Tahir et al., 2012, Xue et al., 2024). The boundary state solutions are of Jackiw-Rebbi type, with wavefunctions localized at the sign change of the mass term and dispersion $E_\eta \propto \eta k_x$ for valley $\eta$ (Ezawa, 2013).

In ZBC-QVHE, the edge states are 1D helical metallic modes, protected not by Berry curvature but by the change in Euler class across the DW and by symmetries such as mirror with respect to the DW or chiral + time-reversal (Ghadimi et al., 2024). At heterostructure DWs in h-BAs or twisted graphene, DFT and tight-binding calculations reveal pairs of modes per valley, each robust to disorder that preserves the protecting symmetry, and with the ability to annihilate if Euler curvature delocalizes (e.g., in h-BN under strong gap).

Interacting quantum Hall valley systems can exhibit Luttinger liquid behavior in boundary channels at domain walls, with the number and nature of metallic or gapped modes tuned by valley occupation (e.g., in Bi(111) QH valley ferromagnets) (Randeria et al., 2019). Coulomb interactions may open a gap for multi-channel DWs or preserve metallicity for single-channel cases, leading to rich 1D correlated physics.

In valley-polarized QAH or hybrid QVH–QAH phases (e.g., in silicene with Rashba SOC and exchange), the edge mode spectrum can become asymmetric, with a net chirality and an imbalance in valley-resolved propagating modes—realizing coexistence of Chern and valley Hall responses (Pan et al., 2013, Pan et al., 2014).

5. Competing and Coexisting Topological Orders

The valley Hall paradigm is frequently embedded within a wider context of competing or coexisting orders:

• QVH–QAH Transitions: In systems with both inversion and time-reversal breaking, transitions between QVH and QAHE (or intermediate valley-polarized QAH) phases are controlled by tuning exchange field direction, electric field, or spin–orbit coupling. In Bi-based, silicene, and germanene heterostructures, a sequence of QAHE–QVHE–trivial insulator phases is encountered as field parameters are swept, with domain structure leading to valley-polarized chiral interconnects (Xue et al., 2024, Liu et al., 2014, Pan et al., 2014, Pan et al., 2013).
• ZBC-QVHE versus Chern and Euler Class: The ZBC-QVHE regime demonstrates the independence and coexistence of new valley-resolved topological indices, such as the Euler class, with no reliance on net Chern or Berry curvature, opening new qualifications for topological protection (Ghadimi et al., 2024).
• Fractional and Multi-Channel QVH Phases: In systems with flat topological bands (e.g., in twisted MoTe$_2$), fractional quantum valley Hall phases and fractional topological insulators (both Abelian and non-Abelian) emerge through interaction physics in multi-valley Landau level models, controlled by the balance of inter- and intra-valley repulsion or tuning of external superlattice potentials (Das et al., 19 Sep 2025, Agarwal, 2022). These phases can admit exotic edge modes (e.g., parafermionic zero modes at defects).
• Correlated and Nematic QVH Insulators: In twisted multilayer graphene, interaction-driven nematic order at half filling can realize universal QVH phases, with transitions to QAH states under moderate magnetic fields (Zhang et al., 2021). Topological hexaborides show rich surface QVH ordering due to threefold Dirac valley degeneracy and interaction-induced symmetry breaking, leading to valley-polarized, valley-coherent, and nematic phases with unique quantum Hall plateau sequences (Li et al., 2015).

6. Control, Tunability, and Experimental Signatures

QVH-based topological phases are uniquely tunable by external fields, stacking geometry, strain, and interlayer bias:

• Electric and Magnetic Field Tuning: The energy gaps and phase boundaries in QVH insulators can be actively controlled by perpendicular electric fields (modulating inversion breaking), proximity-induced exchange fields, or field orientation, enabling direct switching between topological regimes (Tahir et al., 2012, Xue et al., 2024, Liu et al., 2014).
• Twist Engineering and Rashba SOC: Moiré patterns in twisted bilayer or hetero-bilayer structures naturally generate effective layer-resolved Rashba coupling and spatial variation in mass terms, allowing for twistronic control of the valley topology, edge state layout, and domain wall engineering (Shen et al., 2024, Ghadimi et al., 2024).
• Disorder and Edge Robustness: The topological edge states of QVH systems are protected against backscattering as long as intervalley mixing is weak, but they are not generally immune to short-range disorder. In valley-polarized QAH phases, disorder can even enhance valley polarization by selectively localizing counterpropagating modes (Pan et al., 2014).
• Observables and Device Applications: Hallmarks include nonlocal resistance in valley Hall transport, scanning tunneling spectroscopy of domain wall bound states (including symmetry-protected zero modes or Luttinger liquid behavior), sign reversals in valley Hall conductivity under field modulation, and realization of dissipationless and valley-polarized interconnects for low-power electronics—particularly in systems supporting large topological gaps (Bi, germanene, TMDs) (Soldatov, 2024, Xue et al., 2018, Randeria et al., 2019).

Experimental realization has been demonstrated or is anticipated in transition-metal dichalcogenide bilayers, germanene-based heterostructures, two-dimensional hexaborides, Bi-based systems under QH conditions, twisted multilayer graphene, and photonic platforms with synthetic gauge fields.

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References:

• Quantum Valley Hall effect without Berry curvature (Ghadimi et al., 2024)
• Valley-polarized quantum anomalous Hall phases and tunable topological phase transitions in half-hydrogenated Bi honeycomb monolayers (Liu et al., 2014)
• Emergent Quantum Valley Hall Insulator from Electron Interactions in Transition-Metal Dichalcogenide Heterobilayers (Saha et al., 2 Dec 2025)
• Valley-Polarized Quantum Anomalous-Hall Effects in Silicene (Pan et al., 2013)
• Valley-dependent Multiple Quantum States and Topological Transitions in Germanene-based Ferromagnetic van der Waals Heterostructures (Xue et al., 2024)
• Unification of valley and anomalous Hall effects in a strained lattice (Yuan et al., 2021)
• Spin-valley-controlled photonic topological insulator (Xue et al., 2018)
• Spin polarized nematic order, quantum valley Hall states, and field tunable topological transitions in twisted multilayer graphene systems (Zhang et al., 2021)
• Spontaneous symmetry breaking and quantum Hall valley ordering on the surface of topological hexaborides (Li et al., 2015)
• Fractional topological insulators at odd-integer filling: Phase diagram of two-valley quantum Hall model (Das et al., 19 Sep 2025)
• Valley-polarized quantum anomalous Hall phase and disorder induced valley-filtered chiral edge channels (Pan et al., 2014)
• Quantum valley Hall states in low-buckled counterparts of graphene bilayer (Shen et al., 2024)
• Interacting multi-channel topological boundary modes in a quantum Hall valley system (Randeria et al., 2019)
• Quantum spin/valley Hall effect and topological insulator phase transitions in silicene (Tahir et al., 2012)
• Quantum Hall valley Ferromagnets as a platform for topologically protected quantum memory (Agarwal, 2022)
• Spin-Valleytronics in Silicene: Quantum-Spin-Quantum-Anomalous Hall Insulators and Single-Valley Semimetals (Ezawa, 2013)