Wavefront Dislocations in Wavefields

Updated 9 February 2026

Wavefront dislocations are topological defects where the wave amplitude vanishes and the phase becomes indeterminate, resulting in quantized phase winding.
In systems like graphene and photonic lattices, WDs serve as real-space fingerprints of band topology and pseudospin texture, evidenced by distinct LDOS patterns.
Experimental techniques such as STM/STS and cavity array measurements enable the controlled detection and manipulation of WDs, aiding topological diagnostics.

A wavefront dislocation (WD) is a fundamental topological defect that arises in any complex scalar or vector wavefield at points (or lines) where the local amplitude vanishes and the phase becomes indeterminate. At such singularities, the phase winds by a quantized multiple of $2\pi$ upon encircling the defect, endowing the WD with an integer-valued topological "charge" or "index." WDs have been observed and analyzed across a diverse range of physical systems—including optics, electron waves, surface water waves, magnetohydrodynamic (MHD) waves, quasi-1D photonic lattices, and low-dimensional quantum materials such as graphene and bilayer graphene—where they serve as robust real-space signatures of the underlying wavefunction topology, pseudospin texture, and associated band invariants.

1. Mathematical and Physical Foundation of Wavefront Dislocations

A general wavefield can be locally expressed as $\psi(\mathbf{r}) = A(\mathbf{r})\,e^{i\phi(\mathbf{r})}$ , with $A(\mathbf{r})$ the real amplitude and $\phi(\mathbf{r})$ the phase. A WD occurs at $\mathbf{r}_0$ where $A(\mathbf{r}_0) = 0$ and $\phi(\mathbf{r}_0)$ is undefined. The phase winding around a small loop $C$ enclosing $\mathbf{r}_0$ is quantized: $m = \frac{1}{2\pi} \oint_{C} \nabla \phi \cdot d\mathbf{\ell} \in \mathbb{Z}$ Here $m$ is the topological charge of the dislocation, physically counting the number of extra (or missing if $m<0$ ) wavefronts that end or emerge from the singularity. The essence of the WD is thus a phase singularity reflecting nontrivial topology in the wave’s phase field (Dutreix et al., 2020, Dutreix et al., 2019, Zhuang et al., 2024).

2. WDs in Electronic and Photonic Quantum Materials

In condensed matter systems, especially low-dimensional and topological materials, WDs provide direct real-space access to topological invariants:

Graphene and Related Materials: Friedel oscillations in the LDOS near point defects manifest prominent WDs. For example, in monolayer graphene, intervalley backscattering yields an LDOS pattern whose phase winds by $\Delta\phi=4\pi$ (i.e., $m=2$ ) around the defect—a direct measurement of the pseudospin winding and Berry phase ( $\pi$ ) (Dutreix et al., 2019, Zhang et al., 2021). The number of dislocations $N_d$ corresponds to the difference of winding numbers (pseudospin winding) between valleys or bands, generalized as $N_d=|n|$ , with $n$ the integer winding (Abulafia et al., 2023, Ghadimi et al., 6 Feb 2026). In bilayer graphene, WDs with $m=4$ reflect doubled pseudospin winding and the $2\pi$ Berry phase at quadratic band touchings (Zhuang et al., 2024, Ghadimi et al., 6 Feb 2026).
Photonic Insulators: In quasi-1D photonic SSH lattices, WDs in LDOS interference patterns (e.g., in a chain of coupled resonators) reflect changes in the topological winding number at a band transition. The number of LDOS fringes on certain sublattices changes by precisely the winding number $w$ extracted from the Bloch Hamiltonian. The emergence or annihilation of WDs provides a direct, local measure of band topology (e.g., the Zak phase in 1D) (Dutreix et al., 2020).
Orbital Angular Momentum and Symmetry Breaking: Recent STM/STS studies demonstrate that the number and orientation of WDs in graphene can be controlled by breaking rotational symmetry locally (tip-induced, substrate-induced) or by engineering OAM-states, producing robust single (or multiple) WDs and directly probing orbital degrees of freedom in quantum interference (Liu et al., 2024).

3. The Role of Berry Phase, Pseudospin Texture, and Topological Invariants

WDs emerge generically at phase singularities and serve as real-space fingerprints of abstract band invariants:

Berry Phase and Pseudospin Winding: The integer $m$ extracted by integrating the phase gradient around a dislocation core is proportional to the Berry phase acquired by the wavefunction along a closed loop in $k$ -space. In monolayer (bilayer) graphene, $m=2$ ( $m=4$ ) recapitulates the Bloch-state winding and Berry phase ( $\gamma = \pi$ , $2\pi$ ). However, as shown in (Ghadimi et al., 6 Feb 2026), WD number responds exclusively to pseudospin winding $W$ (the in-plane rotation of the Bloch off-diagonal components), not to the total vorticity or patch Euler class of band touchings, thus highlighting the subtle distinction between Berry phase and other topological charges.
Chern and Winding Numbers in Defective Graphene: In graphene with vacancies or Kekulé distortions, the number of WDs in LDOS maps directly counts the winding (Chern) number of the defect-induced band structure, providing a local, quantized diagnostic of topological phases as formalized by the index theorem (Abulafia et al., 2023).
Robustness and Generality: The correspondence between WD count and topological invariants is robust against perturbations that preserve valley and chiral structure, and extends to gapped Dirac materials and two-dimensional semiconductors (e.g., MoS $_2$ , WS $_2$ ) (Zhang et al., 2021).

4. Experimental Detection and Measurement Protocols

WDs are observed experimentally via multiple techniques:

Scanning Tunneling Microscopy/Spectroscopy (STM/STS): Spatial conductance maps reveal fork-like dislocation patterns at defects. Fourier filtering at characteristic intervalley wavevectors $\Delta K$ isolates the oscillatory components, and inverse-FFT transforms yield real-space LDOS maps with clear WD signatures. The winding number is extracted by tracking the phase around the singular core (Dutreix et al., 2019, Abulafia et al., 2023, Liu et al., 2024).
Photonic and Microwave Cavity Arrays: LDOS interference patterns in coupled-resonator arrays are measured by network analyzers and moveable antennas. Topological transitions in SSH chains are seen as abrupt changes (birth or annihilation) of fringes on specific sublattices (Dutreix et al., 2020).
Transport and Dual-Probe STM: In monolayer and bilayer graphene, mapping the two-terminal differential conductance $G(\mathbf{r})$ as a function of tip position reveals WDs due to phase singularities in transmission amplitudes. This allows for transport-based diagnostics of valley-resolved topology (Zhuang et al., 2024).

5. WDs in Classical and Astrophysical Wave Systems

WDs are not exclusive to quantum systems; they appear in diverse classical wave contexts:

Surface Water Waves: Analytical signal representations of nonlinear dispersive surface waves (e.g., solutions to the NLS equation) demonstrate that vanishing amplitude points with unbounded Chu-Mei quotient $Q=(\partial_t^2 a)/a$ generically yield phase singularities and WDs. These manifest as merging or splitting of wave crests and are a generic feature of modulated wavegroups (Karjanto et al., 2019).
Magnetohydrodynamic and Coronal Waves: In MHD (Alfvén, magneto-acoustic) waves, especially in stellar atmospheres, WDs are identified as phase singularities (vortices and edges) in Doppler shift maps or line-of-sight velocity time series. Their observed charge $m$ permits decomposition into kink and sausage modes, providing modal analysis of MHD turbulence and insights into energy and torque transfer in stellar coronas (Ariste et al., 2013, Ariste et al., 2015).

Typology: WDs can be classified into edge (fork, $m=1$ ), screw (vortex, $m\neq 0$ ), and glide (mixed) dislocations, depending on the spatial and temporal structure of phase winding (Ariste et al., 2013, Ariste et al., 2015).
Modal Analysis: Modal content—the coexistence of kink ( $m=1$ ) and sausage ( $m=0$ ) waves, or higher-order modes—can be quantitatively extracted from WD charge and spatial distribution in both laboratory and astrophysical contexts.
Engineering and Control: Control over local potentials, stacking geometry (e.g., BA $\to$ AA stacking in bilayer graphene), OAM, and interference conditions enables the targeted creation, manipulation, and annihilation of WDs for pseudospin texture engineering and valleytronic applications (Liu et al., 2024, Ghadimi et al., 6 Feb 2026).

7. Implications, Generalizations, and Open Directions

Wavefront dislocations are universal, arising wherever complex wavefields support singularities of vanishing amplitude. In quantum materials, their detection provides an unambiguous, real-space window into topological invariants, pseudospin textures, and the local geometry of Bloch wavefunctions—without recourse to bulk boundary states, magnetic field sweeps, or momentum-space imaging. Their robustness, manipulability, and direct relationship to invariants such as winding or Chern numbers position WDs as essential observables for topological metrology, band engineering, and the study of emergent quantum phases (Dutreix et al., 2019, Dutreix et al., 2020, Abulafia et al., 2023, Ghadimi et al., 6 Feb 2026, Zhuang et al., 2024).

Challenges persist in extending these techniques to more complex, multi-band or non-Abelian systems, identifying the precise role of crystal symmetry, and developing high-throughput detection algorithms for WDs in large real-space datasets. Nevertheless, WDs provide a powerful, quantized diagnostic tool, linking wave-phase topology across the entire spectrum of physical wave phenomena.