Valley Vortex in Photonics & Condensed Matter

• Valley vortex are topological defects that tie a valley index to a vortex field, displaying phase winding and chiral energy flow in various wave and quantum systems.
• They manifest in photonic, acoustic, and semiconductor setups by exploiting inversion symmetry breaking and Berry curvature to enable robust mode coupling and directional transport.
• Valley vortices serve as a cross-disciplinary template, linking optical Bloch modes, excitonic orders, and superconducting phenomena through valley-resolved vortex textures.

Searching arXiv for recent and foundational papers on “valley vortex” across photonics, acoustics, and condensed matter. “Valley vortex” denotes a family of phenomena in which a valley degree of freedom is tied to a vortex-like structure. In hexagonal wave systems, the term commonly refers to Bloch modes near inequivalent valleys $K$ and $K'$ whose real-space fields exhibit phase winding, orbital angular momentum, or circulating energy flow with chirality locked to the valley index. In semiconductor and moiré settings, the same phrase can denote topological defects of a complex valley-coupling or excitonic order parameter, where zeros of the valley field are accompanied by quantized phase winding. Across these usages, the common structure is a valley label encoded in a vortex texture of a complex field, whether that field is an electromagnetic mode, an acoustic pressure field, a quantum-dot valley coupling $\Delta(x,y)$, or an inter-valley coherent order parameter (Chen et al., 2018).

1. Valley degree of freedom and the meaning of “vortex”

In valley photonics and valley acoustics, the starting point is a hexagonal or honeycomb band structure with inequivalent extrema at $K$ and $K'$. These valleys are related by time-reversal symmetry but not by a reciprocal-lattice vector, so they define a binary pseudospin-like label. When inversion symmetry is broken, Dirac degeneracies are lifted, Berry curvature becomes localized near the valleys, and valley Chern numbers can be assigned locally. In this setting, “valley vortex” typically means that the Bloch eigenmode at a valley carries a real-space phase singularity or circulating energy flux whose chirality is locked to the valley index (Chen et al., 2016).

A representative formulation appears in valley photonic crystals where the out-of-plane electric field near the unit-cell center takes the form

$E_z(r,\phi)\propto e^{i l\phi},$

with $l=\pm 1$. In the microwave valley photonic crystal waveguide of Gao et al., the $K'$ valley is associated with a left-hand circular-polarization phase vortex with $l=-1$, while the $K$ valley is associated with a right-hand circular-polarization phase vortex with $K'$0. The same work states that the valley index is locked to the vortex chirality, and uses this locking to control directional beam splitting and edge-state transport (Chen et al., 2018).

In sonic crystals, the pressure eigenfield $K'$1 can likewise host singular points where $K'$2 and the phase winds by $K'$3 around a loop, with $K'$4. The associated time-averaged acoustic Poynting vector circulates around the core, so the valley state carries orbital angular momentum even though acoustics is scalar. This is the basis of the “acoustic valley vortex state,” in which chirality is locked to valley and can be detected by phase mapping and by chirality-dependent beam splitting (Lu et al., 2017).

A broader usage appears in condensed-matter settings where the relevant complex field is not a wave amplitude but a valley-coupling or valley-coherent order parameter. In Si/SiGe quantum dots, a valley vortex is a point where the complex valley coupling

$K'$5

vanishes and the phase winds by $K'$6 around the zero. In moiré Chern bands, the paper on the vortex spin liquid treats inter-valley coherent order as an excitonic condensate whose vortices proliferate into a liquid phase. These are not Bloch-mode vortices in the photonic sense, but they retain the same core topological feature: a phase singularity attached to a valley-resolved complex field (Woods et al., 7 Jul 2025).

2. Valley-locked vortices in photonic and sonic crystals

The most developed usage of “valley vortex” is in classical wave crystals. In the 2016 photonic valley crystal of Dong et al., each inequivalent valley carries its own vortex-like orbital angular momentum texture in real space. The system is a two-dimensional all-dielectric photonic crystal in TM polarization with two silicon rods of different diameters per unit cell, $K'$7 and $K'$8, which breaks inversion symmetry and opens a gap at $K'$9 and $\Delta(x,y)$0. Around each valley, the modes are described by a massive Dirac Hamiltonian, Berry curvature is sharply peaked, and the valley Chern numbers are $\Delta(x,y)$1 and $\Delta(x,y)$2 for one choice of mass sign. The out-of-plane electric field $\Delta(x,y)$3 exhibits opposite phase winding at the two valleys, giving what the paper calls valley-contrasting orbital angular momentum (Chen et al., 2016).

The 2017 sonic-crystal work of Lu et al. made the same structure directly visible in acoustics. In a two-dimensional hexagonal sonic crystal of rotated triangular steel rods in water, inversion breaking opens a band gap and the valley states $\Delta(x,y)$4 and $\Delta(x,y)$5 become acoustic vortex states. At high-symmetry points inside the unit cell, the pressure amplitude vanishes while the phase winds by $\Delta(x,y)$6, and the acoustic energy flux circulates around the core. The lower and upper valley states carry opposite topological charges $\Delta(x,y)$7, and the corresponding $\Delta(x,y)$8 states have opposite chirality by time reversal. The same paper establishes a $\Delta(x,y)$9-based azimuthal selection rule for exciting these states and shows a mimicked spin Hall effect of sound in which beams with opposite chirality split in different directions (Lu et al., 2017).

The 2018 microwave valley photonic crystal waveguide of Gao et al. provided an experimentally tunable version of the same idea. The structure consists of a honeycomb lattice of ceramic rods with $K$0, diameters $K$1 and $K$2, lattice constant $K$3, and zero-order TM$K$4 modes between parallel metal plates. The valley extrema occur at $K$5. The $K$6 valley state exhibits a circular phase vortex in $K$7 that decreases by $K$8 upon a counterclockwise loop, while the $K$9 valley exhibits the opposite winding. By exciting the structure with a three-monopole antenna array whose phases are tuned to synthesize a left-handed or right-handed source vortex, the authors selectively excite $K'$0 or $K'$1 bulk modes and tune the left/right splitting ratio smoothly from $K'$2 to $K'$3. The same valley topology supports edge states at VPC1/VPC2 domain walls and at a VPC–PEC boundary, where the edge dispersion evolves from gapless valley-dependent modes to almost flat bands as the boundary-rod diameter is varied (Chen et al., 2018).

A related but distinct mechanism appears in the 2018 photonic-graphene paper of Noh et al. There, inversion symmetry is not broken. Instead, two equivalent valleys are excited in a photonic graphene lattice, Bragg reflection transfers amplitude to a third equivalent valley, and the resulting field carries an optical vortex whose chirality is determined by whether the excitation is in the $K'$4 or $K'$5 sector. The paper reports single vortices and vortex–antivortex pairs, and argues that a three-band effective Hamiltonian is necessary because the commonly used two-band model cannot explain the observed vortex degeneracy lifting (Song et al., 2018).

3. Topology, selection rules, and transport

A recurring feature of valley-vortex systems is that the real-space vortex texture is tied to momentum-space topology. In valley photonic and phononic crystals, inversion breaking turns the Dirac crossing into a massive Dirac point, producing Berry curvature localized near $K'$6 and $K'$7. Integrating the curvature over a single valley yields half-integer valley Chern numbers, while the total Chern number remains zero because time-reversal symmetry is preserved. Interfaces between domains with opposite mass signs therefore support valley-polarized edge states whose counterpropagating branches reside at different valleys, so intervalley backscattering requires a large momentum transfer (Chen et al., 2018).

In the all-dielectric photonic valley crystal of Dong et al., a domain wall between two structures with opposite inversion asymmetry has $K'$8 and $K'$9, yielding edge modes whose transport is robust along zigzag boundaries. A Z-shaped waveguide exhibits nearly unit transmission over the normalized band $E_z(r,\phi)\propto e^{i l\phi},$0–$E_z(r,\phi)\propto e^{i l\phi},$1, and the authors interpret the edge states as topological channels built from valley modes whose unit-cell fields carry opposite orbital angular momentum at $E_z(r,\phi)\propto e^{i l\phi},$2 and $E_z(r,\phi)\propto e^{i l\phi},$3 (Chen et al., 2016).

The same coupling between chirality and valley also generates selection rules. In the microwave VPC waveguide, a three-antenna source with phases $E_z(r,\phi)\propto e^{i l\phi},$4 produces a left-hand circular-polarization phase vortex and preferentially couples to the $E_z(r,\phi)\propto e^{i l\phi},$5 valley, whereas $E_z(r,\phi)\propto e^{i l\phi},$6 produces the opposite chirality and excites $E_z(r,\phi)\propto e^{i l\phi},$7. Continuous phase tuning changes the source from an LCP vortex to an RCP vortex and continuously changes the relative excitation strength of the two valleys, which directly tunes the output splitting ratio (Chen et al., 2018).

In the sonic crystal of Lu et al., the selection rule is formulated in terms of the overlap $E_z(r,\phi)\propto e^{i l\phi},$8 between a valley eigenmode of charge $E_z(r,\phi)\propto e^{i l\phi},$9 and a chiral source of charge $l=\pm 1$0. Because the vortex core sits at a $l=\pm 1$1-symmetric point, rotational invariance requires $l=\pm 1$2 with $l=\pm 1$3. This allows one to excite a chosen valley state by matching the source chirality to the valley-vortex charge. The same system exhibits chirality-dependent beam splitting under trigonal warping, which the paper describes as a mimicked spin Hall effect of sound (Lu et al., 2017).

These examples motivate a general interpretation: valley-vortex locking is a bulk property of Bloch eigenmodes, while selection rules and protected transport are interface- and source-level consequences of that locking. This suggests a unifying picture in which valley pseudospin is encoded either as local orbital angular momentum or as a phase-vortex sign, and external beams couple efficiently only when their angular structure matches that encoded valley texture.

4. Supercells, higher-order modes, and near-field vortex engineering

Later work extended valley-vortex concepts beyond primitive-cell valley Hall systems to supercells and higher-order topological phases. In the 2021 photonic-crystal study of higher-order valley vortices, synchronized rotation of six dielectric cylinders in a $l=\pm 1$4-symmetric triangular lattice opens valley gaps and simultaneously changes the bulk polarization from $l=\pm 1$5 to $l=\pm 1$6 as the rotation angle changes sign. The resulting higher-order topological insulator supports both one-dimensional edge states and zero-dimensional corner states, and the corner states are reported to carry chiral orbital angular momenta locked by valleys, with phase vortices located at maximal Wyckoff points (Zhou et al., 2021).

A related 2022 photonic Su–Schrieffer–Heeger variant uses six dielectric triangular pillars per unit cell, with geometry tuned by $l=\pm 1$7. At $l=\pm 1$8, the structure has Dirac degeneracies near $l=\pm 1$9; for $K'$0, symmetry is reduced to $K'$1, a valley gap opens, Berry curvature localizes near $K'$2 and $K'$3, and the valley Chern number flips between $K'$4 and $K'$5. The valley band-edge modes possess phase singularities and circulating Poynting vectors at maximal Wyckoff positions, and chiral sources obey a valley-chirality selection rule. The same topology supports edge states in a rhombus-shaped beam splitter waveguide that remains robust at sharp corners (Yu et al., 2022).

The 2024 supercell-photonic-crystal work on nearfield vortex dynamics shifted emphasis from edge transport to controlled rearrangement of vortex singularities. A $K'$6 supercell built from triangular holes on silicon-on-insulator folds primitive-cell valley modes from $K'$7 to $K'$8. Paired rotations of the triangles drive a sequence from evanescent valley modes to quasi-BICs, frustrated modes, and quasi-valleys. Throughout this sequence, pseudospin vortices remain pinned at triangle centers while “free vortices” move, merge, and create asymmetric near-field patterns. At a critical rotation, three free vortices merge with the pseudospin at the top triangle to form a frustrated mode whose strong asymmetry enhances second-harmonic generation via a larger nonlinear overlap factor $K'$9. This work treats valley vortices as movable near-field objects whose spatial arrangement directly tunes $l=-1$0 factor and nonlinear response (Ye et al., 2024).

A recent extension in valley photonic crystal waveguides further hybridizes valley topology with spin-texture topology. In 2026, optical spin skyrmions were shown to arise as eigenstates of topological valley edge states in a VPC waveguide. The edge modes at $l=-1$1 and $l=-1$2 possess opposite phase-vortex charges in $l=-1$3 and $l=-1$4, and the evanescent spin angular momentum density forms Néel-type skyrmions with $l=-1$5. The sign of the skyrmion number is locked to the valley index, so the work effectively turns valley vortices into valley-locked spin skyrmions that propagate robustly around bends and defects (He et al., 4 May 2026).

5. Valley vortices in semiconductor and moiré condensed-matter systems

In semiconductor quantum dots, “valley vortex” has a more literal defect-theoretic meaning. The 2025 Si/SiGe work defines the complex valley coupling as

$l=-1$6

A valley vortex is a point where $l=-1$7, equivalently $l=-1$8, and the valley phase winds by $l=-1$9 around the point: $K$0 Because $K$1 is modeled as a complex Gaussian random field, its zeros are generically vortices. The paper derives a vortex density

$K$2

where $K$3 is the dot radius. In the disorder-dominated regime $K$4, vortices are dense; in the deterministic regime they are exponentially suppressed. The same paper proposes to calibrate a $K$5-factor mapping of the valley phase by moving a quantum dot around a loop enclosing a single valley vortex, because along such a loop the effective $K$6-factor must attain both $K$7 and $K$8, allowing direct extraction of $K$9 and $K'$00 (Woods et al., 7 Jul 2025).

In moiré Chern bands, the phrase appears in a different dual description. The 2024 vortex-spin-liquid paper considers two half-filled Chern bands with $K'$01 in the two valleys of twisted MoTe$K'$02. Strong inter-valley repulsion favors neutral excitons, but because opposite Chern numbers generate an effective flux for the exciton sector, a simple inter-valley-coherent condensate is frustrated. The proposed phase is a vortex liquid of excitons, and the paper interprets the neutral fermions $K'$03 as fermionic vortices of the nearby inter-valley-coherent order. These valley vortices form neutral Fermi surfaces, while the phase retains a fractional quantum spin Hall response and helical charged edge modes (Zhang, 2024).

A still more direct superconducting realization appears in rhombohedral graphene. The 2025 work on the spontaneous vortex–antivortex lattice considers a spin- and valley-polarized quarter metal, so that only one valley participates in pairing. Trigonal warping and broken time-reversal symmetry allow a cubic gradient term in the Ginzburg–Landau functional,

$K'$04

which favors finite-momentum pairing at three symmetry-related wavevectors. The resulting triple-$K'$05 state,

$K'$06

forms a spontaneous honeycomb vortex–antivortex lattice at zero applied field. The underlying mechanism is not the same as in photonic valley vortices, but the vortices are rooted in a valley-chiral superconducting background whose time-reversal breaking derives from valley polarization (Gaggioli et al., 20 Mar 2025).

6. Valley polarization, orbital magnetization, and spontaneous vortex matter

Another condensed-matter usage links valleys to actual superconducting vortices through orbital magnetization rather than phase texture of a Bloch or coupling field. In a 2025 theory of two-dimensional valley-polarized superconductors, orbital magnetization generated by valley polarization couples to the magnetic field of a superconducting vortex through

$K'$07

Because screening is weak in a thin film, a Pearl vortex has broad magnetic flux distribution, and the Zeeman energy gain $K'$08 can overcome the vortex self-energy. The criterion for spontaneous vortex formation is

$K'$09

with $K'$10. When this is satisfied, a vortex lattice forms even at zero applied field; in a macroscopic film the globally favorable state becomes an alternating stripe pattern of vortex-lattice domains with opposite polarization, because a single-domain net flux would generate a large magnetic self-energy (Jahin et al., 28 May 2025).

This use of valley and vortex is conceptually adjacent to the rhombohedral-graphene vortex–antivortex lattice but physically distinct. In the rhombohedral case, vortices arise from finite-momentum pairing in a valley-chiral superconductor (Gaggioli et al., 20 Mar 2025). In the orbital-magnetization case, vortices arise because valley polarization produces a magnetization that lowers the energy of ordinary superconducting vortices (Jahin et al., 28 May 2025). A plausible implication is that “valley vortex” in condensed-matter superconductivity can refer either to a topological defect in a valley-resolved order parameter or to ordinary flux vortices whose stability is controlled by valley-derived orbital magnetization.

7. Conceptual synthesis and scope of the term

The literature does not use “valley vortex” with a single universal definition. Instead, at least four technically distinct meanings appear.

Usage	Complex field carrying the vortex	Valley role
Valley photonics / acoustics	Bloch-mode field such as $K'$11 or $K'$12	Valley index locked to vortex chirality
Quantum dots	Valley coupling $K'$13	Valley phase winds around zeros of $K'$14
Moiré excitonic systems	Inter-valley coherent order parameter	Valley vortices are vortices of exciton order
Valley-polarized superconductors	Superconducting order or magnetic-flux sector	Valley polarization stabilizes vortex matter

In photonic and sonic crystals, the defining feature is real-space phase winding of valley Bloch modes, often accompanied by circulating Poynting or acoustic energy flow and by selection rules for chiral excitation (Lu et al., 2017). In semiconductor valley qubits, the defining feature is a topological defect of the complex valley coupling field, useful for reconstructing the valley phase landscape and for improving characterization of valley disorder (Woods et al., 7 Jul 2025). In moiré and superconducting systems, the phrase broadens to encompass vortices of inter-valley order or flux structures stabilized by valley polarization (Zhang, 2024).

What unifies these cases is the intertwining of a valley label with a vortex invariant. The invariant may be a real-space winding number $K'$15, a circulation of energy flow, a phase singularity in $K'$16, or a vortex of an inter-valley order parameter. This suggests that “valley vortex” is best understood as a cross-disciplinary template rather than a single object: a valley-resolved topological defect or vortex texture whose chirality, winding, or energetic stability is controlled by valley physics.