This research explores the emergence of novel quantum phases due to geometrical constraints, drawing parallels with known phenomena like surface states in topological insulators and edge states in quantum Hall systems. The focus shifts to naturally occurring constraints imposed by mesoscale domain patterns or topological defects. Domain walls (DWs) in ferroelectrics have shown unusual functionalities like conductivity and rectification, and the sharp structural changes at DWs can significantly alter magnetic exchange interactions and spin ordering. The study focuses on lacunar spinels (AB₂X₄), a family of narrow-gap Mott insulators exhibiting various correlation and spin-orbit effects, including Néel-type skyrmion lattice (SkL) states in GaV₄S₈, GaV₄Se₈, and GaMo₄S₈. These materials, initially cubic, transform to a rhombohedral state upon a Jahn-Teller transition, creating polar axial symmetry necessary for the SkL state. The research aims to demonstrate the emergence of a novel magnetic state confined to polar DWs in GaV₄Se₈, likely due to the constraints of matching distinct spin cycloids in adjacent domains.

The literature review extensively discusses previous research on geometrical constraints and their influence on quantum phases. It highlights examples such as metallic surface states in topological insulators, superconducting vortex states, interface-induced 2D electron gas and superconductivity, and integer and fractional quantum Hall edge states. It then examines domain walls in ferroelectrics, noting their unusual conductivity, rectification, and super-switching properties in various materials (YMnO₃, ErMnO₃, BiFeO₃, LiNbO₃, PbₓSr₁₋ₓTiO₃). The literature also covers the influence of geometrical confinement on magnetic skyrmions, including their enhanced thermal stability and the proposal of exotic edge states. Finally, it introduces lacunar spinels and their diverse properties, such as pressure-induced superconductivity, metal-to-insulator transitions, and the discovery of Néel-type skyrmion lattices in some of these compounds.

The study employed a multi-faceted approach involving several techniques to image and manipulate polar domains in GaV₄Se₈ crystals. This included non-contact atomic force microscopy (nc-AFM), scanning dissipation microscopy (SDM), and frequency-modulated Kelvin-probe force microscopy (KPFM) to characterize the polar domains and estimate DW density. Magnetic force microscopy (MFM) was used to explore magnetic states potentially confined to DWs. The electric control of polar domains was achieved by applying electric fields along the (111)-type axes, selecting specific domains or combinations thereof. The magnetic state's response to field orientation was investigated via small-angle neutron scattering (SANS) at 12 K by rotating a magnetic field of 220 mT in the (110) plane. Magneto-current and magnetic torque measurements, with the magnetic field finely rotated in the (110) plane, were used to map the phase boundaries in both mono-domain and multi-domain states. Magnetization and ac-susceptibility measurements complemented the SANS data, offering a complete picture of the material's magnetic behavior. Sample synthesis involved chemical vapor transport, and crystallographic orientation was determined by X-ray Laue and neutron diffraction.

The key finding is the identification of a novel magnetic state in GaV₄Se₈ that exists exclusively in samples with multiple polar domains, and is absent in single-domain samples. This state manifests as a sharp anomaly in magnetic susceptibility, magnetic torque, and magneto-current measurements. This anomaly is distinct from other anomalies associated with the known magnetic transitions in GaV₄Se₈ (cycloidal, Néel-type skyrmion lattice, and ferromagnetic states). It is observed only in multi-domain samples, indicating that the presence of domain walls is a necessary condition for this state's existence. The anomaly's sharpness and reproducibility suggest a well-defined transition rather than effects from DW pinning, domain-domain interactions, or defects. The transition is linked to the formation of magnetic states confined to domain walls, which transition to the ferromagnetic state at lower fields than the bulk. Based on DW density and magnetization jump, the thickness of the DW-confined magnetic states is estimated to be 3-6 nm. The study shows that magnetic field applied along the [001] direction doesn't reveal the transition in mono-domain samples indicating that the new magnetic states do not form at DWs between domains with polar axes of [111] and [111] but rather between domains with axes that are not perpendicular to the field. This suggests that the new state is likely formed due to the mismatch between cycloids with different spin rotation planes.

The results demonstrate that the presence of domain walls is crucial for the observed novel magnetic state. The high DW density and the mismatch in magnetic interactions (Dzyaloshinskii-Moriya vectors) at adjacent domains are proposed as the driving forces for this new state. The confined magnetic textures at DWs become ferromagnetic at lower fields than the bulk material, accounting for the observed anomaly. This finding suggests that the magnetic frustration arising from changing Dzyaloshinskii-Moriya interactions near DWs reduces the stability of modulated states, leading to this transition. The estimated thickness of the DW-confined states is consistent with the macroscopic observations, supporting the proposed mechanism. The study highlights that the magnetic equivalence of DWs depends on the magnetic field direction, and only certain DW types contribute to the observed anomaly.

The study reveals a novel magnetic state confined to polar domain walls in GaV₄Se₈, observable through macroscopic magnetic and magnetoelectric measurements. This state, absent in mono-domain samples, emphasizes the critical role of domain walls in inducing new forms of magnetism. The findings open avenues for exploring novel magnetic phenomena by manipulating domain wall structures and interactions. Future research could focus on directly imaging the spin texture within the DW-confined states to validate the proposed mechanisms and explore the tunability of these states using external stimuli.

The study primarily relies on macroscopic measurements. While the findings strongly suggest a DW-confined magnetic state, direct real-space imaging of the spin texture within the domain walls would provide more definitive proof. The estimation of the DW-confined state thickness relies on assumptions regarding magnetization changes. Further refinement of this estimation could be achieved with improved techniques for measuring the magnetization change.