The magnetic monopole—a particle carrying isolated north or south magnetic charge—remains one of physics' most tantalizing absences. Despite decades of searching, no fundamental monopole has ever been detected in our universe. Yet in certain crystalline materials, electronic states emerge that behave exactly as if magnetic monopoles existed, not in real space, but in the abstract realm of momentum space where quantum mechanics describes electron motion.

Weyl semimetals represent a profound convergence of relativistic quantum mechanics and solid-state physics. Their electrons near specific points in momentum space obey the Weyl equation—the same mathematics describing massless fermions in particle physics. But here's the remarkable twist: these Weyl points act as sources and sinks of a quantum geometric quantity called Berry curvature, creating effective magnetic monopoles whose field lines thread through momentum space. This isn't mere mathematical analogy; it produces measurable electromagnetic responses unlike anything in conventional materials.

The discovery of Weyl semimetals in 2015 opened an entirely new chapter in quantum materials research. These crystals exhibit surface states that form open arcs rather than closed contours, transport phenomena that violate classical expectations, and electromagnetic responses that blur the boundary between electricity and magnetism. Understanding why these monopole-like structures emerge requires journeying into the topology of quantum mechanics itself—a realm where geometry becomes destiny and abstract mathematical structures manifest as observable physical phenomena.

In crystalline solids, electrons occupy energy bands—allowed energy ranges determined by the periodic atomic arrangement. When two bands approach each other and cross at isolated points in momentum space, something extraordinary can happen. If the crystal lacks either inversion symmetry or time-reversal symmetry, these crossing points become topologically protected and carry a mathematical charge that cannot be removed without fundamentally restructuring the electronic bands.

These crossing points are Weyl points, and their topological charge manifests as Berry curvature—a geometric quantity describing how quantum wavefunctions twist as electrons move through momentum space. Near a Weyl point, the Berry curvature radiates outward (or inward) in all directions, following the exact mathematical form of a magnetic monopole field. The total Berry flux through any surface surrounding the point equals a quantized value, precisely analogous to Gauss's law for magnetic monopoles.

The Nielsen-Ninomiya theorem guarantees that Weyl points always come in pairs with opposite topological charges—monopoles and antimonopoles in momentum space. If you integrate the Berry curvature over the entire Brillouin zone (the fundamental unit cell in momentum space), the total must vanish. This constraint means field lines emanating from one Weyl point must terminate at its partner, creating a geometric structure reminiscent of dipole fields but with each pole separately localized.

First-principles calculations played a crucial role in predicting and understanding Weyl semimetals. Density functional theory calculations by teams including those influenced by Marvin Cohen's approach to materials prediction identified candidates like TaAs and NbAs before their experimental confirmation. These computational methods solve the quantum mechanical equations governing electron behavior, revealing the band structure topology that creates Weyl points.

The topological protection of Weyl points makes them remarkably robust against perturbations. Unlike accidental band crossings that can be gapped by small changes in the crystal potential, Weyl points can only be destroyed by merging with their opposite-charge partners. This stability arises from the quantized nature of the topological charge—it cannot change continuously, only through discrete annihilation events.

Takeaway

Topological charges in momentum space behave like magnetic monopoles because Berry curvature follows the same mathematical laws as magnetic fields, but the Nielsen-Ninomiya theorem ensures these monopoles always come in opposite pairs that can only be destroyed through mutual annihilation.

Conventional metals possess Fermi surfaces—closed contours in momentum space where occupied and unoccupied electronic states meet. These closed surfaces determine nearly every metallic property, from electrical conductivity to heat capacity. Weyl semimetals shatter this paradigm by hosting surface states that form open arcs, connecting the projections of bulk Weyl points onto the surface Brillouin zone.

These Fermi arcs represent a topological necessity, not merely an interesting curiosity. Consider a surface perpendicular to the line connecting two Weyl points of opposite charge. The bulk Berry curvature flux passing through the surface must be absorbed somewhere, and this absorption occurs through the surface states. The arc connecting Weyl point projections carries a quantized Berry phase that exactly accounts for the monopole charges they connect.

Angle-resolved photoemission spectroscopy (ARPES) provided the smoking-gun evidence for Weyl semimetals by directly imaging these arc states. In TaAs, researchers observed surface states forming open contours—an impossible configuration for any conventional material. The arcs begin at one Weyl point projection and terminate at another, their length and curvature encoding information about the bulk electronic structure.

The open nature of Fermi arcs creates unprecedented quantum oscillation phenomena. In conventional metals, electrons traversing the Fermi surface under magnetic fields execute closed orbits, producing oscillations periodic in inverse magnetic field. Weyl semimetal surfaces host incomplete orbits; electrons reaching an arc endpoint tunnel through the bulk to the opposite surface, completing their journey on another arc. This creates quantum oscillations with unusual characteristics that fingerprint the Weyl state.

Computational prediction of Fermi arc configurations requires sophisticated surface Green's function methods that project bulk electronic structure onto semi-infinite crystal terminations. These calculations reveal how arc connectivity depends on surface orientation, termination chemistry, and the relative positions of bulk Weyl points—enabling targeted materials design for specific surface-state properties.

Takeaway

Fermi arcs are not merely exotic surface features but topological requirements that absorb the Berry flux from bulk monopoles, and their open configuration enables quantum oscillation experiments where electrons complete orbits by tunneling between opposite crystal surfaces.

Each Weyl point carries a chirality—a handedness determined by whether Berry curvature radiates outward or inward. Electrons near a given Weyl point have their spin locked to their momentum direction, either parallel or antiparallel depending on chirality. Under ordinary circumstances, the number of electrons near each Weyl point remains separately conserved. But apply parallel electric and magnetic fields, and something extraordinary happens: charge pumps from one chirality to the other.

This chiral anomaly was first discovered in particle physics, where it explains certain decay processes of neutral pions. In Weyl semimetals, the same mathematics manifests as a measurable transport phenomenon. Parallel electric and magnetic fields create a spectral flow that transfers electrons from one Weyl point to its opposite-chirality partner, temporarily violating what appeared to be a conservation law.

The experimental signature is negative longitudinal magnetoresistance—electrical resistance that decreases when magnetic field aligns with current flow. This contradicts classical expectations where magnetic fields typically increase resistance through Lorentz force deflection. The chiral anomaly provides an additional conduction channel: electrons pumped to one chirality can relax back to the other through inter-valley scattering, creating a steady-state current contribution that grows with magnetic field.

Observation of the chiral anomaly requires careful experimental design to separate intrinsic effects from artifacts. Current jetting, inhomogeneous current distribution, and contact effects can all mimic negative magnetoresistance. The most convincing demonstrations use carefully engineered sample geometries and demonstrate the angular dependence predicted by theory—maximum effect when field and current align, vanishing when perpendicular.

The chiral anomaly also produces the chiral magnetic effect: a current flowing parallel to magnetic field even without applied electric field, driven by any imbalance in chemical potential between Weyl points of opposite chirality. While controversial in equilibrium systems due to fundamental constraints, nonequilibrium realizations using optical pumping or contacts with different chemical potentials have been proposed and partially demonstrated.

Takeaway

The chiral anomaly transforms parallel electric and magnetic fields into a charge pump between Weyl points, producing experimentally observable negative magnetoresistance that serves as a transport signature of the momentum-space monopole structure.

Weyl semimetals exemplify how abstract mathematical structures—topology, Berry curvature, chirality—crystallize into tangible material properties. The momentum-space monopoles emerging from band topology aren't convenient fictions but physical entities with measurable consequences: arc-shaped surface states, anomalous transport, and electromagnetic responses that challenge classical intuition.

The predictive power of computational materials science proved essential in this discovery. First-principles calculations identified candidate materials, predicted their surface-state configurations, and guided experimental searches. This represents the modern paradigm of materials research: theory and computation leading experiment rather than merely explaining it.

Looking forward, Weyl semimetals offer platforms for exploring relativistic quantum mechanics in tabletop experiments, potential applications in spintronics and quantum computing, and templates for discovering yet-stranger topological phases. The magnetic monopoles that nature denied us in fundamental physics have emerged, transmuted, in the quantum geometry of crystalline matter—a reminder that the universe's mathematical elegance often reveals itself in unexpected domains.