Momentum-Space Skyrmions: Topological Textures

• Momentum-space skyrmions are topologically nontrivial spin or pseudo-spin textures in reciprocal space, defined by quantized winding numbers and Berry curvature effects.
• They are characterized using phase-space Berry curvature formalism, multi-momentum operator theory, and spin-orbit coupled models, revealing intricate band topology.
• Experimental realizations in topological insulators, magnonic lattices, and photonic systems demonstrate their impact on transport phenomena and advanced device functionalities.

Momentum-space skyrmions refer to topologically nontrivial spin or pseudo-spin textures distributed across momentum (k-) space, often associated with quantized winding numbers and arising from underlying band topology, mixed phase-space Berry curvature effects, or engineered optical field structures. The concept spans condensed matter, quantum magnetism, magnonics, photonics, and the dynamic evolution of wavepackets, with rich implications for transport phenomena, quasiparticle dynamics, and topological protection.

1. Fundamental Principles of Momentum-Space Skyrmions

Momentum-space skyrmions are defined as topological spin or pseudo-spin textures in the reciprocal space (Brillouin zone) of a quantum system. Mathematically, a normalized vector field $\hat{n}(\mathbf{k})$ assigns a unit vector at each momentum $\mathbf{k}$, and the skyrmion number is given by

$$N = \frac{1}{4\pi} \int_{BZ} \hat{n}(\mathbf{k}) \cdot (\partial_{k_x} \hat{n}(\mathbf{k}) \times \partial_{k_y} \hat{n}(\mathbf{k})) \, d^2k$$

This quantifies the topological winding of the vector field over the Brillouin zone. In topological insulators, superconductors, and magnonic systems, the momentum-space skyrmion number can indicate transitions between distinct topological phases (Mohanta et al., 2016, Loder et al., 2017, Ghader et al., 30 Apr 2024). Berry curvatures and phase-space gauge fields play a pivotal role, with mixed real/momentum-space Berry curvature ($\Omega^{Rk}$) driving phenomena such as the Dzyaloshinskii–Moriya (DM) interaction and the charge of real-space skyrmions in chiral magnets (Freimuth et al., 2013).

The mapping from the Brillouin zone (torus $T^2$) to the spin sphere $S^2$ may be incomplete (e.g., restricted to the hemisphere due to time-reversal or symmetry constraints) and can yield so-called "meron-like" (half-skyrmion) structures with quantized half-integer winding (Loder et al., 2017, Yessenov et al., 15 Mar 2025).

2. Theoretical Frameworks and Key Models

Diverse theoretical approaches underpin momentum-space skyrmion phenomena:

• Phase-space Berry curvature formalism: The Berry connection $$\boldsymbol{A}_n(\mathbf{x}) = \langle \mathbf{x}, n | i \partial/\partial x_j | \mathbf{x}, n \rangle$$, with $\mathbf{x}=(\mathbf{R},\mathbf{k})$, leads to a 6-component vector, capturing both real- and momentum-space winding as well as mixed couplings (Freimuth et al., 2013).
• Multi-momentum operator theory: In magnonic skyrmion lattices, fully transforming the magnon Hamiltonian into momentum space, with off-diagonal terms from Fourier expansion over skyrmion wave vectors, accurately incorporates umklapp scattering and complex band topology (Ghader et al., 30 Apr 2024).
• Spin-orbit coupled electronic models: Rashba or SU(2)-gauge-coupled Hubbard models link the DM interaction and magnetic anisotropies directly to spin-dependent hopping, producing momentum-space textures and quantized Chern numbers for Fermi pockets (Makuta et al., 2023).
• Quantum field theory and conservation laws: Dipole conservation of topological charge dictates collective coordinate dynamics. A direct consequence is the vanishing effective mass of an isolated quantum skyrmion, underpinned by the Girvin–MacDonald–Platzman (GMP) algebra for the topological density operator (Sorn et al., 20 Dec 2024).

In photonic systems, momentum-space skyrmionic and meronic textures are engineered by imprinting polarization winding onto spatial-frequency domains via digital holography and metasurfaces (Yessenov et al., 15 Mar 2025), or observed in the canonical momentum and Poynting vector fields generated by Mie scattering (Chen et al., 27 Nov 2024).

3. Mixed Berry Curvature, Dzyaloshinskii–Moriya Interaction, and Skyrmion Charge

The mixed Berry curvature tensor $$\Omega_{n,ij} = \partial A_{n,j}/\partial x_i - \partial A_{n,i}/\partial x_j$$ encodes real-space, momentum-space, and mixed phase-space curvatures. In chiral magnets, the DM interaction $D_{ij}$ is microscopically induced by phase-space Berry curvatures and is given by

$$D_{ij} = \sum_n \int \frac{d^3k}{(2\pi)^3} \left[ f_n A_{n,ij} + \frac{1}{\beta} \ln(1 + e^{-\beta (\varepsilon_n - \mu)}) B_{n,ij} \right]$$

where $A_{n,ij}$ and $B_{n,ij}$ stem from energy shifts and Berry curvatures, respectively (Freimuth et al., 2013). The DM interaction promotes twisted magnetic textures, such as skyrmion lattices, directly connecting phase-space topology to energetics.

Phase-space Berry curvature also modifies the density of states, yielding a redistribution of the electronic charge in skyrmions. In insulators, the total skyrmion charge is topologically quantized by the second Chern number,

$$\delta Q = -e \int \frac{d^2 R d^2 k}{(2\pi)^2} \frac{\epsilon_{ijkl}}{8} \operatorname{Tr}[\Omega_{ij} \Omega_{kl}]$$

As a result, momentum-space winding numbers not only determine transport properties (e.g., quantized Hall conductivity) but also reflect the underlying real-space topological textures (Freimuth et al., 2013).

4. Experimentally Observed Momentum-Space Skyrmions and Topological Textures

Momentum-space skyrmions have been realized or proposed in various experimental platforms:

• Topological insulators: On the surface of Bi$_2$Se$_3$, an applied exchange field can drive a transition from a hedgehog spin texture to a skyrmion texture in momentum space, with the skyrmion number changing discretely and Hall conductance remaining quantized at $e^2/(2h)$—a topological transition without gap closing (Mohanta et al., 2016).
• Magnonic skyrmion lattices: ARPES and magnon spectroscopy can resolve the momentum-space band topology and Berry curvature in large skyrmion lattices, exhibiting multiple magnetic-field-inducible topological phase transitions (Ghader et al., 30 Apr 2024).
• Ultrathin ferromagnets: In SrRuO$_3$/SrIrO$_3$ heterostructures, ARPES and Hall measurements confirm robust Weyl-point band topology and the simultaneous presence of skyrmions in real space, evidenced by a persistent anomalous Hall effect and topological Hall effect signatures (Zheng et al., 22 Nov 2024).
• Photonic systems: Deep-subwavelength photonic skyrmions are generated through spin–orbit coupling in focused and evanescent vector beams. The spin texture can be resolved with near-field microscopy to scales as small as 10 nm, and mapped to momentum-space topological features (Du et al., 2018). Multipole Mie scattering fields can generate skyrmion and meron textures in canonical momentum and Poynting fields, exhibiting topological robustness against geometric defects (Chen et al., 27 Nov 2024).
• Dynamic topological phases: In non-unitary quench dynamics of PT-symmetric quantum walks, momentum–time skyrmions emerge, protected by dynamic Chern numbers integrated over the (k, t) manifold, providing direct access to dynamic topology (Wang et al., 2018).
• Ultrafast space-time optical merons: 3D-localized optical skyrmionic (meron) structures are realized in momentum–energy space by imprinting half-skyrmion polarization textures onto spectral surfaces using digital holography and large-area metasurfaces. The polarization is modulated according to parameters $k_r$, $\phi_k$, and $\Omega$, and mapped back to real space via Fourier transformation (Yessenov et al., 15 Mar 2025).

5. Dynamics, Conservation Laws, and Analogies to Quantum Hall and Fractonic States

The dynamical behavior of quantum skyrmions in momentum space is governed by conservation laws rooted in field-theoretic and algebraic structure:

• Topological dipole conservation: The dipole moment $$D_i = \int x_i \rho_{top}(\mathbf{x}) d^2x = Q_{top} R_i$$ is strictly conserved for an isolated skyrmion, resulting in vanishing effective mass and intrinsic immobility (pinned dynamics), even after accounting for quantum spin-wave corrections (Sorn et al., 20 Dec 2024).
• Thiele equation and gyrocoupling: The Thiele equation $G \times \dot{\mathbf{R}} = 0$ reflects the balance between gyrocoupling and force. No inertial term arises for quantum skyrmions in the presence of dipole conservation.
• Girvin–MacDonald–Platzman algebra: The commutator for topological density $$[\hat{\rho}_{top}(\mathbf{x}), \hat{\rho}_{top}(\mathbf{x}')]$$ replicates the GMP algebra of the fractional quantum Hall effect, pointing to noncommutative geometry and robust collective modes (Sorn et al., 20 Dec 2024).
• Fractonic behavior: The conservation of both charge and dipole moment resembles fracton excitations—isolated skyrmions are immobile, while paired textures (dipoles) exhibit constrained mobility.

By analogy, skyrmion crystals or liquids may realize topological collective modes akin to quantum Hall edge states, and the vanishing kinetic energy of quantum skyrmions mirrors flat dispersions in Landau levels.

6. Extensions: CP$^2$ Skyrmions, Quantum Magnetic Solitons, and Multidimensional Textures

Beyond conventional $S^2$ (CP$^1$) skyrmions, spin-1 quantum magnets admit CP$^2$ skyrmion textures in both dipolar (magnetic) and quadrupolar (nematic) order parameters. These textures interpolate between magnetic and nonmagnetic states, and form skyrmion crystals with characteristic diffraction signatures in momentum space (Zhang et al., 2022). Rigorous SU(3) coherent-state formulations capture their complex topology,

$$C = -\frac{i}{32\pi} \int dx \, dy \, \epsilon_{jk} \text{Tr}[\mathfrak{n} (\partial_j \mathfrak{n} \partial_k \mathfrak{n} - \partial_k \mathfrak{n} \partial_j \mathfrak{n}) ]$$

Quantum skyrmions in frustrated ferromagnets constitute many-magnon bound states with exponential band-narrowing due to tunneling processes, and their bandstructure and transport are tightly locked to angular momentum and spin quantum numbers, resulting in distinct momentum-space dispersions for different configurations (Lohani et al., 2019).

In photonics, the use of spatiotemporal modulation and coordinate transformations (e.g., spectral-to-radial mapping by log–polar transformation) enables the synthesis and manipulation of 3D skyrmion or meron spin textures in momentum–energy (k, ω) spaces, opening new vistas in spin optics, imaging, and quantum technologies (Yessenov et al., 15 Mar 2025).

7. Implications, Applications, and Outlook

Momentum-space skyrmions have substantial impact in areas ranging from topological electronics, magnonics, and spintronics to photonics and quantum information:

• Transport phenomena: Berry curvature winding in momentum space drives anomalous and topological Hall effects, topological spin Hall currents, and quantized transport signatures in magnetic or topological materials (Freimuth et al., 2013, Yin et al., 2015, Mohanta et al., 2016, Zheng et al., 22 Nov 2024).
• Quantum and optical devices: Deep-subwavelength and robust skyrmion/meron spin textures enable high-fidelity data encoding, precision metrology, and advanced imaging (Du et al., 2018, Yessenov et al., 15 Mar 2025).
• Information processing: Skyrmion detection/control via Hall signals and engineered topology enables next-generation memory and logic devices.
• Fundamental physics: Skyrmions provide physical realizations of field-theoretic solitons, with rich connections to quantum Hall systems, fracton phases, and topological quantum computation (Sorn et al., 20 Dec 2024).
• Materials science: Momentum-space detection via diffraction, ARPES, and neutron or X-ray scattering facilitates identification of complex topological textures, including CP$^2$ skyrmion crystals (Zhang et al., 2022).
• Future directions: Tunable magnon band topology via field or lattice engineering, synthesis of multidimensional photonic skyrmions, and exploration of fractonic and quantum Hall–like liquid states in magnetic materials are prominent research avenues.

Momentum-space skyrmions thus unify spatial and reciprocal-space topological phenomena, providing a universal language for understanding and manipulating complex quantum and classical textures across physical platforms.