Valley-Contrasted Optical Selection Rules

• Valley-contrasted optical selection rules are symmetry-governed principles where inequivalent valleys couple to opposite circular polarizations, enabling control over valley and spin states.
• They are demonstrated in systems like monolayer WSe2, silicene, and twisted heterobilayers, where broken inversion symmetry and quantum geometry dictate distinct optical responses.
• Generalizations of these rules extend to excitonic and nonlinear regimes, influencing interlayer, moiré excitons, and even core-level transitions in advanced spectroscopies.

Valley-contrasted optical selection rules are symmetry-governed relations between light polarization and inequivalent extrema of the electronic structure in momentum space. In their canonical form, opposite valleys couple to opposite circular polarizations, so that optical absorption or emission directly addresses a valley index as an internal quantum number. In monolayer WSe$_2$, this appears as $K\leftrightarrow \sigma^+$ and $K'\leftrightarrow \sigma^-$, enabling optical preparation of both valley polarization and SU(2) valley coherence (Jones et al., 2013). Later work showed that the same concept can become spin-valley selective in silicene and other honeycomb systems, stacking dependent in bilayers and heterobilayers, channel dependent in moiré excitons, and even linear-polarization selective through the quantum metric rather than Berry curvature (Ezawa, 2012, Schneider et al., 2019, Urano et al., 26 Jun 2026, Li et al., 12 Jul 2025).

1. Symmetry basis and canonical form

At the most basic level, valley-contrasted optical selection rules require at least two symmetry-related but inequivalent valleys, typically $K$ and $K'$, together with broken inversion symmetry or an equivalent mechanism that makes their dipole response nonidentical. In monolayer transition-metal dichalcogenides, the valleys are degenerate by time-reversal symmetry but separated by a large crystal momentum $\sim 2K$, so intervalley mixing is weak and valley is a good quantum number near the band edge. The band-edge Bloch states carry opposite orbital or angular-momentum character in the two valleys, and the interband dipole matrix elements transform differently under the crystal’s threefold rotational symmetry. The result is the standard rule that one circular helicity is allowed in one valley and the opposite helicity in the other (Jones et al., 2013).

A low-energy Dirac description makes the same structure explicit. In silicene, the optical operators satisfy $\mathcal P_\pm^K \propto \tau_\pm$ and $\mathcal P_\pm^{K'} \propto -\tau_\mp$, so opposite helicities couple to the two valleys, with time reversal enforcing $P_+^K(k)=P_-^{K'}(k)$ and $P_-^K(k)=P_+^{K'}(k)$. At $K\leftrightarrow \sigma^+$0, the optical polarization reaches $K\leftrightarrow \sigma^+$1, meaning perfect helicity discrimination at the valley edge (Ezawa, 2012). This supports a general interpretation: valley-contrasted optical selection rules are most naturally viewed as representation-theoretic statements about velocity or dipole operators at symmetry-related valleys, rather than as a material-specific phenomenology of circular dichroism alone.

2. Monolayer transition-metal dichalcogenides: polarization and coherence

Monolayer WSe$K\leftrightarrow \sigma^+$2 provides the benchmark realization. The neutral exciton $K\leftrightarrow \sigma^+$3, positive trion $K\leftrightarrow \sigma^+$4, negative trion $K\leftrightarrow \sigma^+$5, and fine-structure feature $K\leftrightarrow \sigma^+$6 all exhibit strongly helicity-preserving photoluminescence under circular excitation. The circular polarization is defined as

$K\leftrightarrow \sigma^+$7

Under $K\leftrightarrow \sigma^+$8 excitation the co-circular PL dominates, under $K\leftrightarrow \sigma^+$9 excitation the helicity reverses, and the effect persists even when the excitation energy is moved from $K'\leftrightarrow \sigma^-$0 eV to $K'\leftrightarrow \sigma^-$1 eV. These observations establish that the many-body excitonic complexes inherit the single-particle valley selection rules (Jones et al., 2013).

The same system also established that valley-contrasted selection rules can be used not only for valley polarization but for valley coherence. Because linear polarization is a coherent superposition of $K'\leftrightarrow \sigma^-$2 and $K'\leftrightarrow \sigma^-$3, linearly polarized excitation prepares a coherent superposition of $K'\leftrightarrow \sigma^-$4- and $K'\leftrightarrow \sigma^-$5-valley excitons. For neutral excitons, the emitted photoluminescence is itself linearly polarized, and its axis tracks the excitation axis one-to-one. The angular dependence is fitted by

$K'\leftrightarrow \sigma^-$6

with $K'\leftrightarrow \sigma^-$7, where $K'\leftrightarrow \sigma^-$8 is the excitation polarization angle. Since a single-valley exciton emits circularly polarized light, linearly polarized $K'\leftrightarrow \sigma^-$9 photoluminescence is direct evidence of a nonzero off-diagonal valley density-matrix element and therefore of excitonic valley coherence (Jones et al., 2013).

The trion case clarifies the distinction between polarization and coherence. Circular pumping still initializes valley polarization in $K$0 and $K$1, but linearly polarized excitation does not yield linearly polarized trion PL. For $K$2, the emitted photon is entangled with the remaining hole state, so the reduced photon state is mixed. For $K$3, exchange interactions split the relevant configurations by an amount estimated to be on the order of $K$4 meV, which destroys the transferred coherence. Valley polarization is therefore available for both neutral and charged excitonic complexes, whereas optically visible valley coherence was demonstrated only for the neutral exciton in this setting (Jones et al., 2013).

3. Interlayer, twisted, and moiré excitons

In twisted or lattice-mismatched heterobilayers, valley-contrasted selection rules acquire a finite-momentum structure. For interlayer excitons in MoX$K$5/WX$K$6 systems, the electron and hole valleys are displaced by twist or mismatch, so direct photon emission occurs only at finite center-of-mass momentum

$K$7

The bright states form six symmetry-related light cones, and the emitted light at each cone is generally elliptically polarized, with the major axis locked to the exciton velocity direction and the dominant helicity determined by the electron and hole valley indices. Valley contrast is thus preserved, but no longer as the monolayer rule of purely circular $K$8 emission at $K$9 (Yu et al., 2015).

A related registry dependence appears in lattice-matched heterobilayers. In MoSe$K'$0/WSe$K'$1 and related systems, broken out-of-plane mirror symmetry removes the monolayer restriction that spin-conserving transitions couple only to in-plane light and spin-flip transitions only to out-of-plane light. The interlayer exciton dipole takes the mixed form $K'$2, and as the local atomic registry varies, the dominant component crosses between in-plane dipoles of opposite circular polarization and the out-of-plane one. A central consequence is that the spin-triplet interlayer exciton can be in the same order of magnitude as the spin-singlet exciton in optical strength, and that at a given registry the spin-singlet and spin-triplet excitons obey distinct valley polarization selection rules (Yu et al., 2018).

In moiré interlayer excitons, the notion of a single valley-helicity channel can break into several emissive channels with opposite rules. In near-$K'$3, R-type WSe$K'$4/WS$K'$5, the A-like and B-like moiré channels emit opposite helicities. At gate conditions where the time-integrated circular polarization is nearly zero, the helicity-resolved photoluminescence traces cross in time, and the helicity-difference signal changes sign. This is captured by the two-channel form

$K'$6

with $K'$7. The result shows that vanishing steady-state circular polarization can be a false negative for valley polarization in multichannel emitters: opposite-selection channels can coexist dynamically and cancel only after time integration (Urano et al., 26 Jun 2026).

4. Material and phase dependence beyond monolayers

Outside monolayer TMDCs, valley-contrasted selection rules are often intertwined with additional discrete degrees of freedom. In silicene, strong spin-orbit coupling and electric-field-controlled band inversion transform the valley rule into a spin-valley optical selection rule. For the fundamental transition $K'$8, right-handed circularly polarized light excites only up-spin electrons in the $K'$9 valley in the topological-insulator phase, whereas in the band-insulator phase it excites only down-spin electrons in the same valley; the $\sim 2K$0 point carries the opposite handedness by time reversal. The rule flips at the critical field $\sim 2K$1, where one gap closes and the topological transition occurs (Ezawa, 2012).

In bilayer WS$\sim 2K$2, stacking symmetry determines whether the monolayer valley rule survives. The non-centrosymmetric AB stacking preserves monolayer-like valley-selective excitation and spin-valley locking, whereas inversion-symmetric AA' stacking does not retain straightforward valley-helicity selectivity for the bilayer as a whole. Circular dichroism can still occur in AA', but its origin is linked to spin-layer locking rather than to valley polarization. The contrast between AB and AA' therefore shows that optical helicity in a bilayer is not, by itself, sufficient evidence for valley contrast; the stacking symmetry is decisive (Schneider et al., 2019).

Monolayer $\sim 2K$3 compounds provide an infrared version of the same logic. Six materials—$\sim 2K$4, $\sim 2K$5, $\sim 2K$6, $\sim 2K$7, $\sim 2K$8, and $\sim 2K$9—were identified as direct-gap semiconductors with band edges at $\mathcal P_\pm^K \propto \tau_\pm$0, strong spin-valley coupling in the valence band, and valley-selective interband transitions in the infrared. In the representative $\mathcal P_\pm^K \propto \tau_\pm$1 device simulation, with $\mathcal P_\pm^K \propto \tau_\pm$2, right-handed circularly polarized light excites the $\mathcal P_\pm^K \propto \tau_\pm$3 valley and yields a valley and spin polarization close to $\mathcal P_\pm^K \propto \tau_\pm$4 at zero bias (Yuan et al., 2021).

More abstract honeycomb-lattice models show that broken symmetry and topology can abruptly alter the rule itself. In the charge-ordered and $\mathcal P_\pm^K \propto \tau_\pm$5-axis spin-ordered insulating states of a spin-orbit-coupled honeycomb model, circularly polarized light excites definite spin-valley sectors, but the assignment changes discontinuously at the topological transition $\mathcal P_\pm^K \propto \tau_\pm$6. For $\mathcal P_\pm^K \propto \tau_\pm$7, the charge-ordered state obeys $\mathcal P_\pm^K \propto \tau_\pm$8 and $\mathcal P_\pm^K \propto \tau_\pm$9; for $\mathcal P_\pm^{K'} \propto -\tau_\mp$0 this reverses. The optical selection rule is therefore not fixed by lattice geometry alone, but by the ordered state and the topological character of the bands (Yanagi et al., 2017).

5. Geometric, excitonic, and nonlinear generalizations

One major generalization replaces the simple valley-helicity rule by a combined rule involving the internal angular momentum of the exciton. In monolayer WS$\mathcal P_\pm^{K'} \propto -\tau_\mp$1, the valley-exciton locking relation is

$\mathcal P_\pm^{K'} \propto -\tau_\mp$2

where $\mathcal P_\pm^{K'} \propto -\tau_\mp$3 is the valley angular momentum change, $\mathcal P_\pm^{K'} \propto -\tau_\mp$4 the exciton angular momentum change, and $\mathcal P_\pm^{K'} \propto -\tau_\mp$5 the lattice contribution allowed by $\mathcal P_\pm^{K'} \propto -\tau_\mp$6 symmetry. This explains why $\mathcal P_\pm^{K'} \propto -\tau_\mp$7-resonant second-harmonic generation and $\mathcal P_\pm^{K'} \propto -\tau_\mp$8-resonant two-photon luminescence exhibit opposite helicity relations to the pump: the SHG helicity at resonance was measured as $\mathcal P_\pm^{K'} \propto -\tau_\mp$9 at $P_+^K(k)=P_-^{K'}(k)$0 eV, whereas the TPL helicity at the $P_+^K(k)=P_-^{K'}(k)$1 resonance was $P_+^K(k)=P_-^{K'}(k)$2 at $P_+^K(k)=P_-^{K'}(k)$3 eV (Xiao et al., 2015).

A second generalization arises from the pseudospin texture of massive Dirac cones. In this case the one-photon optical matrix element itself carries angular winding, so the bright exciton series is not limited to the $P_+^K(k)=P_-^{K'}(k)$4-like envelope. In the ideal massive Dirac model, $P_+^K(k)=P_-^{K'}(k)$5 at $P_+^K(k)=P_-^{K'}(k)$6 excites the $P_+^K(k)=P_-^{K'}(k)$7-orbital exciton while $P_+^K(k)=P_-^{K'}(k)$8 excites the $P_+^K(k)=P_-^{K'}(k)$9 exciton. Once trigonal warping is included, $P_-^K(k)=P_+^{K'}(k)$0-orbital excitons also become one-photon bright, again with helicity opposite to the $P_-^K(k)=P_+^{K'}(k)$1 series in the same valley. The rule is therefore set by the combined angular structure of the envelope and the Bloch-state optical matrix element, rather than by the envelope alone (Gong et al., 2017).

Recent geometric reformulations separate the circular and linear versions of valley contrast. Quantum metric-based optical selection rules identify the off-diagonal metric $P_-^K(k)=P_+^{K'}(k)$2 as the quantity controlling the difference between orthogonal linear polarizations. For a two-band transition,

$P_-^K(k)=P_+^{K'}(k)$3

In the altermagnetic and Kane-Mele models, and in monolayer $P_-^K(k)=P_+^{K'}(k)$4, this yields valley-contrasted linear selection rules in which orthogonal linear polarizations are locked to distinct valleys; in the cleanest case, the $P_-^K(k)=P_+^{K'}(k)$5 valley couples only to $P_-^K(k)=P_+^{K'}(k)$6 and the $P_-^K(k)=P_+^{K'}(k)$7 valley only to $P_-^K(k)=P_+^{K'}(k)$8, with $P_-^K(k)=P_+^{K'}(k)$9 (Li et al., 12 Jul 2025).

Nonlinear and subcycle regimes introduce yet another layer. In 2H-MoTe$K\leftrightarrow \sigma^+$00, a doubly resonant cascaded pathway

$K\leftrightarrow \sigma^+$01

obeys the same helicity preference at each step in a given valley, so the effective nonlinear rule is determined by the product of two valley-selective dipole couplings rather than by a single virtual two-photon transition. The experimentally observed valley asymmetry in CB+2 was approximately three times larger than in the CB (Courtade et al., 5 May 2026). At even shorter timescales, an instantaneous optical valley selection rule was formulated in terms of the product of the field’s instantaneous optical chirality and the valley chirality,

$K\leftrightarrow \sigma^+$02

and a chirality-separated optical field was shown to produce both $K\leftrightarrow \sigma^+$03-purity valley-polarized currents and pure valley current with zero net charge flow (He et al., 11 Aug 2025).

6. Spectroscopic extensions, transport signatures, and interpretive cautions

Valley-contrasted selection rules can survive in regimes where ordinary band-edge intuition would suggest that they disappear. In monolayer MoS$K\leftrightarrow \sigma^+$04 under Landau quantization, the generic inter-Landau-level rule $K\leftrightarrow \sigma^+$05 does not erase valley contrast. Instead, the allowed interband transitions are

$K\leftrightarrow \sigma^+$06

$K\leftrightarrow \sigma^+$07

for $K\leftrightarrow \sigma^+$08, and reversing the sign of $K\leftrightarrow \sigma^+$09 swaps the valley assignment. The Landau levels also exhibit a systematic valley splitting linear in magnetic field and comparable to the Landau-level spacing (Chu et al., 2014).

Localized impurity states introduce orbital-selective valley optics. For shallow impurities in a gapped Dirac material such as monolayer MoS$K\leftrightarrow \sigma^+$10, the $K\leftrightarrow \sigma^+$11 impurity state is threshold-suppressed, $K\leftrightarrow \sigma^+$12, but the orbital impurity states $K\leftrightarrow \sigma^+$13, with $K\leftrightarrow \sigma^+$14, obey a threshold rule of the form $K\leftrightarrow \sigma^+$15. Valley selectivity therefore survives for impurity-band transitions, but chiefly through the impurity states with nonzero orbital structure rather than through the $K\leftrightarrow \sigma^+$16 state (Ko et al., 2020).

The same selection rules can be converted into photonic directionality. In monolayer WS$K\leftrightarrow \sigma^+$17 on a hyperbolic metamaterial, the standard correspondence $K\leftrightarrow \sigma^+$18, $K\leftrightarrow \sigma^+$19 is mapped into opposite momentum-space directions because opposite chiral dipoles couple to opposite surface-plasmon and high-$K\leftrightarrow \sigma^+$20 channels. The measured directional contrast between $K\leftrightarrow \sigma^+$21 and $K\leftrightarrow \sigma^+$22 exceeded $K\leftrightarrow \sigma^+$23, realizing an optical valley Hall effect without nanostructuring the emitter itself (Guddala et al., 2020).

Soft-x-ray spectroscopy extends the concept to core levels. Group-theoretical and all-electron DFT analysis showed that in monolayer TMDs, metal $K\leftrightarrow \sigma^+$24, $K\leftrightarrow \sigma^+$25, and $K\leftrightarrow \sigma^+$26 core edges can exhibit valley selectivity for transitions involving empty valence-band-edge states under circularly polarized radiation, whereas analogous transitions from those core levels into the conduction-band minimum are not valley selective, and metal $K\leftrightarrow \sigma^+$27-core edges are not valley selective either (Geondzhian et al., 2021). A practical caution follows from the moiré literature: low or vanishing time-integrated circular polarization does not, by itself, imply rapid valley depolarization or absence of valley polarization. In multichannel emitters, opposite-selection channels can coexist and cancel only after integration over time (Urano et al., 26 Jun 2026).

Valley-contrasted optical selection rules are therefore best understood not as a single monolayer circular-dichroism rule, but as a hierarchy of symmetry constraints on light-matter coupling. In the simplest case they lock $K\leftrightarrow \sigma^+$28 and $K\leftrightarrow \sigma^+$29 to opposite helicities; in more elaborate settings they encode spin, layer, exciton angular momentum, moiré registry, quantum geometry, nonlinear resonance structure, or subcycle chirality. Across these regimes, the common principle is that inequivalent valleys are optically distinguishable because the relevant dipole, velocity, or excitonic matrix elements transform differently at symmetry-related points in momentum space.