Valley Phases in 2D Materials

• Valley phases are regimes where inequivalent K and K' valleys drive symmetry breaking, topological order, and coherent dynamics.
• They encompass distinct orders such as valley Hall states with opposite Berry curvatures, valley-polarized states, and intervalley-coherent phases.
• Engineered systems from twisted bilayer graphene to silicon quantum dots and acoustic crystals enable practical control via valley-orbit coupling.

Searching arXiv for the cited valley-phase literature to ground the article in the relevant papers. {"query":"valley phases twisted bilayer graphene valley Hall intervalley coherence arXiv", "max_results": 10} {"query":"(Breiø et al., 2022) Chern-insulator phases and spontaneous spin and valley order in a moire lattice model for magic-angle twisted bilayer graphene", "max_results": 5} Valley phases are phases, textures, and control regimes in which the inequivalent valleys $K$ and $K'$, or $\tau=\pm1$, become the relevant organizing degree of freedom for symmetry breaking, topology, or coherent dynamics. In current usage, the term spans several non-equivalent notions: valley Hall states with opposite Berry curvature at opposite valleys; valley-polarized states with unequal occupation of the valleys; intervalley-coherent states that break a valley $U_v(1)$; spin-valley-entangled and layer-valley phases in multicomponent systems; Berry-phase effects tied to valley-dependent Bloch-band geometry; and device-level control based on the phase of the valley-orbit coupling in silicon quantum dots (Breiø et al., 2022, Murthy et al., 2017, Yong et al., 2018, Wu et al., 2012).

1. Definitions and conceptual scope

In many-body valley systems, several distinct order parameters coexist and should not be conflated. In the moiré-lattice Hartree–Fock treatment of magic-angle twisted bilayer graphene, valley polarization means an occupation imbalance between the two valleys, intervalley coherence means nonzero valley-off-diagonal expectation values, and quantum valley Hall order is instead defined by spontaneous imaginary next-nearest-neighbor bond expectation values, $$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$ (Breiø et al., 2022). This distinction is foundational: valley Hall order need not imply valley polarization, and intervalley coherence need not imply either a net Chern number or a valley Hall response.

In topological band problems, the standard valley-Hall construction relies on Berry curvature sharply concentrated near $K$ and $K'$, with opposite valley-resolved contributions and vanishing total Chern number under time-reversal symmetry. Bilayer sonic crystals make this distinction explicit by separating an acoustic valley Hall phase from an acoustic layer-valley Hall phase; the former is characterized by a nontrivial ordinary valley invariant and layer-mixed edge states, whereas the latter is characterized by a nontrivial layer-resolved valley invariant and layer-polarized edge states (Lu et al., 2018). Closely related logic appears in dual-gap valley photonic crystals, where a single structure may realize different valley topologies in different frequency windows, labelled by $(C_{v1},C_{v2})$ (Tang et al., 2020).

The word “phase” is also used in a different sense for geometric and complex phases. In monolayer MoSe$_2$, the relevant object is the Berry phase of Bloch states, whose sign reverses between $+K$ and $K'$0 and thereby imprints itself on exciton fine structure (Yong et al., 2018). In silicon quantum dots, by contrast, the valley phase is the phase of a complex valley-orbit coupling, $K'$1, whose interdot difference becomes a directly measurable control parameter in double-dot tunneling (Wu et al., 2012). This suggests that “valley phases” is best treated as a family of valley-structured phenomena rather than a single universal invariant.

2. Valley-polarized topological phases in electronic band structures

A central class of valley phases consists of valley-polarized quantum anomalous Hall states, in which a nonzero total Chern number coexists with unequal valley-resolved topological charges. In a low-buckled honeycomb lattice with exchange field and competing intrinsic and extrinsic Rashba couplings, a conventional quantum anomalous Hall state with $K'$2 and $K'$3 can be driven into a valley-polarized quantum anomalous Hall state with $K'$4, $K'$5, $K'$6, and $K'$7; the transition occurs through a one-valley gap closing and is interpreted in terms of additional skyrmions in the real-spin texture at $K'$8 (Pan et al., 2014). In that setting, short-range disorder can destroy counterpropagating valley pairs yet leave a quantized $K'$9 plateau with fully valley-polarized chiral transport.

First-principles work on half-hydrogenated Bi honeycomb monolayers established a closely related but materially distinct realization. There, two valley-polarized quantum anomalous Hall phases appear, with the chiral edge state near $\tau=\pm1$0 for one magnetization orientation and near $\tau=\pm1$1 after magnetization reversal; the associated Chern number changes from $\tau=\pm1$2 to $\tau=\pm1$3, and the topological gap reaches about $\tau=\pm1$4 at an appropriate buckled angle (Liu et al., 2014). The mechanism combines strong Bi spin-orbit coupling, spontaneous magnetization of the dehydrogenated $\tau=\pm1$5 band, sublattice asymmetry from half-hydrogenation, and intrinsic Rashba coupling.

A broader electrically tunable class emerges when Kane–Mele spin-orbit coupling is combined with inequivalent exchange fields on the two sublattices. In that model, intrinsic spin-orbit coupling plus inequivalent exchange fields already generates valley asymmetry, while Rashba coupling and a perpendicular electric field enlarge the phase space to valley-polarized quantum anomalous Hall states with $\tau=\pm1$6 and valley-contrasting insulating states with $\tau=\pm1$7 (Pan et al., 2024). The electric field changes which valley undergoes gap closing and reopening, so a fixed system can be tuned continuously between different valley-polarized topological states by varying only the field strength and direction.

Another route uses the orbital effect of an in-plane magnetic field in a buckled honeycomb lattice. There, a Haldane-like Peierls phase in next-nearest-neighbor hopping combines with a perpendicular electric field to create valley-dependent masses $\tau=\pm1$8 and $\tau=\pm1$9, enabling field-controlled transitions among quantum anomalous Hall, semimetal, and quantum valley Hall regimes (Nualpijit et al., 2023). In the semimetal phase one valley can be gapless while the other remains gapped, which yields perfect valley filtering in transverse conductivity.

3. Interaction-driven valley phases in graphene-based systems

In moiré and quantum Hall graphene systems, valley phases are often many-body ground states rather than single-particle band labels. In the Wannier-based moiré lattice model for magic-angle twisted bilayer graphene, unrestricted real-space Hartree–Fock finds that a quantum valley Hall phase exists at all integer fillings for sufficiently large interaction strength, but its flavor content depends strongly on filling (Breiø et al., 2022). At charge neutrality, only quantum valley Hall order appears. At even fillings $U_v(1)$0, the quantum valley Hall state coexists with full spin polarization. At odd fillings $U_v(1)$1, it coexists with both spin and valley polarization, so the cancellation between opposite-valley Chern sectors no longer occurs and the odd-integer states become quantum anomalous Hall or Chern-insulator phases.

The same work sharply distinguishes strong- and weak-coupling valley physics. At strong coupling, the integer-filling phases are homogeneous and insulating, with the quantum valley Hall order generated by the assisted-hopping term inherited from Wannier obstruction. At weaker interactions, the nonzero integer fillings become metallic and translation-breaking, and the valley order is often intervalley coherent rather than valley polarized. Representative examples include a metallic intervalley-coherent state at $U_v(1)$2, $U_v(1)$3, with ordering vectors of magnitude $U_v(1)$4 and enlarged unit cell $U_v(1)$5, and half-integer intervalley-coherent spirals with $U_v(1)$6 and $U_v(1)$7 (Breiø et al., 2022).

Bilayer graphene at $U_v(1)$8 in the quantum Hall regime supports a different hierarchy of valley phases. Hartree–Fock analysis with nonperturbative trigonal warping and short-range anisotropies yields the familiar fully layer-polarized and Kekulé states, but also a spin-valley-entangled phase and a new broken $U_v(1)$9 phase in which spin and valley sectors break independent continuous symmetries (Murthy et al., 2017). In this language, the Kekulé phase is a genuine intervalley-coherent state, the fully layer-polarized state is valley polarized without spontaneous valley $$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$0 breaking, the spin-valley-entangled state mixes spin and valley coherently, and the BU$$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$1 phase realizes simultaneous spin and valley coherence.

In Bernal bilayer graphene aligned to hBN, valley-layer locking becomes an active ingredient of miniband topology and many-body energetics. Magnetotransport on high-quality BBG-hBN heterostructures shows that the moiré-induced isolated bands have opposite Chern numbers in opposite valleys, that displacement field $$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$2 controls the valley-selective reconstruction of Landau levels, and that the resulting Hofstadter minibands host field-induced correlated insulators, helical edge signatures, and symmetry-broken Chern insulators (Jeong et al., 2023). The helical $$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$3 state near $$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$4 is interpreted as a valley-Hall-like phase with opposite valley chiralities and strong nonlocal transport, although the role of spin cannot be excluded.

4. Berry-phase and valley-orbit phases

Not all valley phases are thermodynamic phases. In monolayer MoSe$$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$5, the Berry phase of Bloch states produces an effective valley-orbital coupling for excitons, lifting the $$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$6/$$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$7 degeneracy with opposite sign in opposite valleys (Yong et al., 2018). The measured splitting is $$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$8 between $$\mathrm{Im}\,\langle c^\dagger_{p\tau\sigma} c_{p+2\,\tau\sigma}\rangle\neq 0$$9 and $K$0 in the $K$1 valley and $K$2 in the $K$3 valley, a time-reversal-symmetric analogue of the orbital Zeeman effect. Because the $K$4 and $K$5 intraexciton transitions are helicity selective, this Berry-phase-imprinted fine structure also produces valley-dependent optical Stark shifts and Autler–Townes doublets under resonant mid-infrared driving.

In silicon double quantum dots, the valley phase is instead the phase of the complex valley-orbit coupling, $K$6, induced by the interface and the vertical electric field (Wu et al., 2012). A single dot reveals only $K$7, but in a double dot the phase difference $K$8 cannot be gauged away simultaneously on both dots. As a result, the valley-eigenstate tunnel matrix elements become

$K$9

so a gate-controlled phase mismatch generates inter-valley tunneling and opens anticrossings between two-electron valley states. The proposed coherent rotation times are in the $K'$0–$K'$1 range for realistic structures (Wu et al., 2012).

Configuration-interaction work on exchange gates generalized this picture from coherent rotations to two-electron exchange. In a symmetric silicon double quantum dot, the exchange splitting is suppressed at finite valley phase difference and reaches its minimum value, approximately $K'$2, when the phase difference is $K'$3; in the minimal interpretation, $K'$4 (Tariq et al., 2021). The mechanism is the valley-phase-dependent dressing of the low-energy singlet and triplet by doubly occupied states, while higher orbital states provide substantial quantitative corrections. This makes the interdot valley phase difference a central parameter for scalable exchange-gate design.

5. Classical-wave realizations

Classical-wave systems provide unusually transparent realizations of valley phases because Berry curvature, edge fields, and mode localization can all be imaged directly. A bilayer sonic crystal with rotatable triangular scatterers realizes two distinct valley-projected topological acoustic insulators: the acoustic valley Hall phase, with nonzero ordinary valley invariant and layer-mixed edge states, and the acoustic layer-valley Hall phase, with nonzero layer-resolved valley invariant and layer-polarized edge states (Lu et al., 2018). Their low-energy description,

$K'$5

shows explicitly how layer-independent and layer-contrasting valley masses compete with interlayer coupling.

Photonic implementations extend this logic to vertically coupled designer-surface-plasmon crystals. There, layer pseudospin becomes a measurable degree of freedom, and the system supports both conventional valley-Hall phases with layer-mixed states and layer-polarized valley-Hall phases with layer-chiral edge transport (Wu et al., 2018). Devices built from LVH and CVH interfaces include a layer convertor with working bandwidth of about $K'$6 and transmission above $K'$7, and a layer-selected delay line with a measured dwell-time offset of about $K'$8 (Wu et al., 2018).

Other photonic and phononic constructions broaden the notion of valley topology. In metallic valley photonic crystals with dual band gaps, the same crystal can realize four distinct topological combinations $K'$9, so topology becomes frequency dependent rather than fixed by structure alone (Tang et al., 2020). In Kagome phononic lattices, valley Hall phases arise not only from inversion breaking but also from mirror-symmetry breaking, and both mechanisms lead to the same effective massive Dirac structure near $(C_{v1},C_{v2})$0 and $(C_{v1},C_{v2})$1 (Lera et al., 2018).

The standard valley-Hall picture has also been challenged and generalized. A $(C_{v1},C_{v2})$2-symmetric triangular photonic crystal with zero Berry curvature at the valleys can nevertheless support valley-polarized edge states through a local valley Hall effect built from spatially varying phase vortices; here valley polarization is location defined rather than fixed throughout the bulk (Bisharat et al., 2023). At the opposite extreme, hybrid Chern/valley photonic crystals can realize chiral valley edge states that are simultaneously unidirectional and valley polarized, by engineering the valley Dirac masses so that the interface mode exists around only one valley (Liu et al., 20 May 2025). That construction addresses the usual depolarization problem of ordinary valley-Hall transport and enables valley multiplexers, demultiplexers, and valley-locked waveguide crossings.

6. Diagnostics, robustness, and recurring distinctions

Across the literature, valley phases are diagnosed by a recurring set of observables: valley-localized Berry curvature, gap closing and reopening at a specific valley, valley-projected or layer-projected Chern indices, chiral or helical edge spectra, nonlocal transport, valley-selective optical selection rules, and tunneling or exchange anticrossings. In magic-angle twisted bilayer graphene, the distinction between quantum valley Hall order, valley polarization, and intervalley coherence is made directly at the level of one-body expectation values (Breiø et al., 2022). In MoSe$(C_{v1},C_{v2})$3, helicity-resolved intraexciton Stark spectroscopy isolates valley-dependent Berry-phase splittings (Yong et al., 2018). In silicon double dots, resonant tunneling and exchange spectroscopy make the relative valley phase physically observable through $(C_{v1},C_{v2})$4 and $(C_{v1},C_{v2})$5 (Wu et al., 2012, Tariq et al., 2021).

A recurring misconception is that all valley phases are equally robust. Ordinary valley-Hall edge states are protected only in the valley-projected sense: smooth interfaces and moderate disorder suppress $(C_{v1},C_{v2})$6 scattering, but strong intervalley mixing destroys the protection (Lu et al., 2018). The same caveat applies to local-valley-Hall edge states beyond inversion breaking (Bisharat et al., 2023). By contrast, chiral valley edge states in hybrid Chern/valley systems inherit one-way transport from the Chern side while remaining confined to a single valley, so backscattering and valley depolarization are strongly suppressed (Liu et al., 20 May 2025). Valley-polarized QAH phases occupy an intermediate position: their net chirality is protected by the total Chern number, but their valley content can still be altered by disorder or edge orientation, as shown explicitly in disorder-driven valley-filtered transport (Pan et al., 2014).

Another recurring misconception is to identify valley Hall order with valley polarization. The moiré Hartree–Fock results make the separation explicit: at charge neutrality a homogeneous quantum valley Hall insulator exists without spin or valley polarization, whereas odd fillings require both quantum valley Hall order and additional flavor selection (Breiø et al., 2022). Likewise, $(C_{v1},C_{v2})$7 does not imply valley triviality: electrically tunable Kane–Mele systems host $(C_{v1},C_{v2})$8 states with opposite nonzero Berry curvature near $(C_{v1},C_{v2})$9 and $_2$0, and BBG-hBN supports $_2$1 states interpreted as opposite-valley helical edge transport (Pan et al., 2024, Jeong et al., 2023). A plausible implication is that “valley phase” is most precise when the operative valley structure—polarization, coherence, local Berry curvature, layer locking, or complex phase difference—is specified explicitly rather than assumed from the word alone.