# Exploring Weyl Semimetals: Emergence of Exotic Electrons and Topological Order

Mehmet Keçeci

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Mehmet Keçeci. Exploring Weyl Semimetals: Emergence of Exotic Electrons and Topological Order. 2025. hal-05146435

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# Exploring Weyl Semimetals: Emergence of Exotic Electrons and Topological Order

Mehmet Keçeci¹

¹ORCID ID: https://orcid.org/0000-0001-9937-9839, İstanbul, Türkiye

**Received:** 05.07.2025

**Abstract:**

Weyl semimetals (WSMs) have emerged as one of the most groundbreaking discoveries in condensed matter physics in recent years, deepening our fundamental scientific understanding and opening new horizons for future technologies. These materials are named after Hermann Weyl, who in 1929 predicted Weyl fermions – massless, chiral relativistic particles. WSMs, as low-energy excitations of these fermions in solid-state systems, possess special points in their electronic band structure where bands cross linearly, known as "Weyl points." These points act as sources or sinks (monopoles) of Berry curvature in momentum space and carry a definite chirality (right- or left-handed). For a material to host a WSM phase, either time-reversal symmetry or inversion symmetry must be broken. One of the most distinctive features of Weyl semimetals is the existence of unique topological surface states known as "Fermi arcs" on their surfaces. These arcs connect Weyl points of opposite chirality within the bulk and form open Fermi surfaces, which are absent in ordinary metals. The presence of Fermi arcs is strong evidence of the topological nature of WSMs and has been experimentally observed using techniques like angle-resolved photoemission spectroscopy (ARPES). Another crucial phenomenon is the "chiral anomaly." This is a quantum field theory effect where chiral charge is not conserved in the presence of parallel electric and magnetic fields, resulting in the pumping of Weyl fermions from one chirality to the other. In solid-state WSMs, the chiral anomaly leads to unusual transport properties, such as a characteristic negative longitudinal magnetoresistance and a giant anomalous Hall effect. These features are important experimental signatures for confirming the existence of Weyl fermions. WSM phases have been theoretically predicted and experimentally realized in various materials, including the TaAs family (TaAs, TaP, NbAs, NbP), WTe₂, MoTe₂, and magnetic WSMs. These materials exhibit a wide range of exotic physical properties, such as anomalous Hall conductivity, giant nonlinear optical responses, thermoelectric effects, and potential catalytic activities. The unique electronic and topological properties of Weyl semimetals make them highly promising candidates for potential applications such as low-energy-consumption electronics, spintronic devices, topological quantum computing, and even high-efficiency catalysts. Intensive research in material synthesis, characterization, and theoretical modeling is ongoing to unlock the full potential of these materials. The discovery of Weyl semimetals has opened up a new and exciting research area in condensed matter physics, highlighting the richness and complexity of topological matter.

**Keywords:**

Weyl Semimetals, Topological Phases, Weyl Fermions, Fermi Arcs, Chiral Anomaly, Electronic Band Structure, Berry Curvature, Anomalous Transport, Exotic Electronic States, Condensed Matter Physics.
# I. The Historical Development of Weyl Fermions and Weyl Semimetals: From Theoretical Prediction to Experimental Discovery

Condensed matter physics continually renews itself as an interdisciplinary field of discovery where theoretical concepts from fundamental particle physics can be realized in tangible materials. In this context, Weyl fermions, envisioned by Hermann Weyl in 1929 as massless, chiral, and relativistic fermions [1], remained a concept of significant theoretical importance for many years, albeit without a direct counterpart among fundamental particles. This situation changed radically with the discovery of Weyl semimetals (WSMs) [58, 59, 89–91], where low-energy excitations behaving as Weyl fermions can emerge in certain crystalline structures. This historical review details the chronological and conceptual journey from the theoretical origins of Weyl fermions, through their prediction in solid-state systems, to their experimental verification and current vibrant research frontiers, citing key relevant publications.

## 1. Theoretical Foundations and Early Concepts (1929 – Early 2000s)

The story of Weyl fermions begins in 1929 when Hermann Weyl proposed these particles as a two-component, massless solution to the Dirac equation [1]. The Weyl equation describes spin-1/2 massless particles in relativistic quantum mechanics, characterized by a definite chirality (right-handed or left-handed), with these chiralities being independent. At the time, neutrinos were considered potential candidates for Weyl fermions, but the later discovery of neutrino oscillations revealed that neutrinos possess a small mass and are therefore not pure Weyl fermions [2]. Consequently, Weyl fermions remained "missing" as fundamental particles for a long time.

In condensed matter physics, early hints of the potential existence of Weyl-like excitations emerged from studies of the topological properties of systems, notably with Gregory Volovik's work in the 1980s and 1990s. He showed that fermionic excitations in the A-phase of superfluid Helium-3 (³He-A) possess point-like Fermi surfaces in momentum space, analogous to Weyl points [3]. These studies theoretically established the possibility of topologically protected point-like band crossings that act as monopoles of Berry curvature and could lead to interesting quantum phenomena such as the chiral anomaly. However, these works did not offer a direct prediction for a specific solid-state material.

## 2. Prediction of Weyl Fermions in Solids (2007 – 2011)
The first significant steps towards the concrete realization of Weyl semimetals [58, 59, 89–91] in solid-state systems emerged as a byproduct of intensive research on topological insulators. In 2007, Shuichi Murakami theoretically demonstrated that in certain three-dimensional systems with broken inversion symmetry, a topological insulator could undergo a phase transition to a semimetal by tuning a critical parameter, and at this transition point, bands could cross linearly to form Weyl points [4]. This work emphasized that the stability of Weyl points relies on topological protection and that these points act as sources or sinks (monopoles) of Berry curvature in momentum space.

However, the specific prediction of a Weyl semimetal phase in real materials had to wait until 2011. This turning point occurred when Xiangang Wan, Ari Turner, Ashvin Vishwanath, and Sergey Savrasov, based on first-principles calculations, theoretically demonstrated the possibility of a Weyl semimetal phase in magnetic pyrochlore iridates (e.g., Y₂Ir₂O₇), where time-reversal symmetry is broken [5]. This groundbreaking work clearly established that Weyl points must appear in pairs (of opposite chirality) and that to separate these points in momentum space, either time-reversal symmetry (as in magnetic systems) or inversion (spatial parity) symmetry must be broken. The pyrochlore iridates were seen as promising candidates due to the combination of strong spin-orbit coupling and magnetic ordering. This theoretical work provided a concrete roadmap for experimental searches and gave significant impetus to the field.

### 3. Experimental Discoveries: Triumph Through Unexpected Routes (2012 – 2015)

Following the theoretical prediction by Wan et al. [5], intensive experimental efforts commenced on pyrochlore iridates. However, the synthesis of these materials, their growth in single-crystal form, and the complexity of their intrinsic magnetic ordering made the experimental observation of clear Weyl signatures quite challenging.

The actual experimental breakthrough came from systems where inversion symmetry is broken, but time-reversal symmetry is preserved (i.e., non-magnetic systems). The year 2015 became an "annus mirabilis" for the experimental discovery of Weyl semimetals [58, 59, 89–91]. Several independent research groups theoretically predicted and subsequently experimentally confirmed the Weyl semimetal phase in transition metal monopnictides known as the Tantalum Arsenide (TaAs) family (TaAs, TaP, NbAs, NbP). Notably, theoretical work by a group led by Hongming Weng, Chen Fang, Zhong Fang, and Xi Dai [6], and by groups led by Su-Yang Xu, B. Andrei Bernevig, and M. Zahid Hasan [7], showed that the crystal structure of these materials (space group I4₁md) naturally breaks inversion symmetry, and strong spin-orbit coupling
leads to the formation of Weyl points. Twelve pairs of Weyl points (24 total Weyl points) were predicted in TaAs.

Following these theoretical predictions, a group led by M. Zahid Hasan (Princeton University) [8] and a group led by Hong Ding (Chinese Academy of Sciences) [9] almost simultaneously reported the existence of Weyl points and their associated topological Fermi arcs in TaAs using Angle-Resolved Photoemission Spectroscopy (ARPES). The ARPES experiments clearly showed linearly dispersing conical bands (Weyl cones) in the bulk electronic band structure and, in the surface states, the characteristic Fermi arcs – disconnected segments of Fermi surface that connect bulk Weyl points of opposite chirality. The existence of Fermi arcs is one of the most striking and directly observable topological signatures of Weyl semimetals, and these discoveries paved the way for the field.

Concurrently, anomalous transport properties attributed to the chiral anomaly, such as negative longitudinal magnetoresistance, began to be observed in TaAs and similar materials. Notably, a group led by Jun Xiong and N. Phuan Ong observed negative magnetoresistance in Na₃Bi (a Dirac semimetal) [10], and subsequently, a group led by Cheng Zhang and Shuang Jia observed it in TaAs [11] when parallel electric and magnetic fields were applied. This is a solid-state manifestation of the chiral anomaly, a phenomenon where Weyl fermions of one chirality are pumped to the other, resulting in a non-conservation of chiral current. These transport experiments complemented the ARPES findings, strongly supporting the existence of Weyl fermions. These simultaneous discoveries generated a massive wave of excitement in condensed matter physics, leading to a research explosion that could be termed the "Weyl rush."

## 4. Diversification and New Discoveries in Weyl Semimetals (2015 – Present)

After the discovery of the TaAs family, researchers focused on searching for new Weyl semimetal materials and understanding different types and properties of these materials. An important distinction emerged: Type-I and Type-II Weyl semimetals.

* **Type-I Weyl Semimetals:** Materials like TaAs fall into this category. The Weyl points form point-like Fermi surfaces at the Fermi level, and the electronic bands cross with a symmetric, upright cone-like dispersion at the Weyl point.

* **Type-II Weyl Semimetals:** Theoretically predicted in 2015 by Alexei Soluyanov and colleagues [12] and soon after experimentally confirmed in transition metal dichalcogenides like WTe₂ and
$MoTe_2$, in this type, Weyl points emerge at the contact points of electron and hole pockets. Consequently, they possess a tilted and heavily "over-tilted" conical dispersion at the Fermi level, coexisting with electron and hole pockets, rather than a point-like Fermi surface. This leads Type-II WSMs to exhibit strongly anisotropic and layer-dependent properties distinct from Type-I. The discovery of Type-II Weyl fermions in $WTe_2$ [13, 14] was of particular interest, especially due to its connection with the material's already known extremely large and non-saturating magnetoresistance.

The investigation of magnetic Weyl semimetals also gained momentum. These systems, where time-reversal symmetry is broken, can host unique magnetoelectric effects such as a giant anomalous Hall effect. Materials like $Co_3Sn_2S_2$ [15], $Mn_3Sn$ [16], and $GdPtBi$ are being intensively investigated as candidates for magnetic Weyl semimetals, with some exhibiting remarkable properties like record-breaking anomalous Hall conductivity.

## 5. Emerging Phenomena, Exotic Properties, and Future Perspectives (Present Day)

The discovery of Weyl semimetals has not only deepened our understanding of fundamental physics but has also opened potential avenues for novel technological applications. Current research focuses include:

* **New Material Discovery and Synthesis:** The search and synthesis of new materials with cleaner and larger single crystals, exhibiting Weyl phases with different symmetry-breaking mechanisms (e.g., systems where both inversion and time-reversal symmetry are broken) or stability at higher temperatures, continues.

* **Exotic Transport and Optical Properties:** Phenomena such as the chiral anomaly and its associated quantum oscillations, anomalous Hall and Nernst effects, giant and tuneable nonlinear optical responses (e.g., second-harmonic generation, circular photogalvanic effect [17]), thermoelectric effects, and even potential catalytic activities are being actively investigated.

* **Topological Quantum Computing:** The topologically protected nature of Weyl fermions, and particularly their Majorana-like boundary states, could make them potential candidates for fault-tolerant quantum computing platforms, although research in this area is still in conceptual and early experimental stages.

* **Spintronic and Valleytronic Applications:** The spin-momentum locking, chirality-dependent spin textures, and valley degrees of freedom of Weyl semimetals could be harnessed for next-generation spintronic and valleytronic devices.
* **Axion Electrodynamics and Higher-Order Topology:** There are theoretical proposals and experimental hints that condensed matter analogues of axion electrodynamics might be realized in some Weyl semimetals [18]. Furthermore, beyond Weyl semimetals, new phases with more complex topological features, such as "higher-order" Weyl semimetals or other exotic topological semimetals, are being explored.

In conclusion, the journey from Hermann Weyl's abstract mathematical concept of Weyl fermions in 1929 [1] to their concrete realization in solid-state systems approximately 85 years later, in the early 2010s, strikingly demonstrates the power of fundamental scientific curiosity, theoretical foresight, and advanced experimental techniques coming together. Weyl semimetals have opened an exciting new chapter in the rich and complex world of topological matter, continuing to expand both our understanding of fundamental physics and our horizons for potential technological applications. This field remains one of the most dynamic, productive, and interdisciplinary research areas in condensed matter physics.

## II. Potential Application Frontiers of Weyl Semimetals and Their Groundbreaking Technological Promises

The discovery of Weyl semimetals (WSMs) has not only deepened our fundamental understanding of condensed matter physics but has also opened doors to groundbreaking technological applications, thanks to the extraordinary electronic, optical, and topological properties inherent in this exotic class of materials. Characteristics such as the chiral anomaly, topologically protected Fermi arcs, giant nonlinear optical responses, and unique magnetoelectric interactions position WSMs as a promising platform for next-generation devices, surpassing conventional semiconductors and metals. This section will interpret the potential application areas of WSMs and the revolutionary contributions they could offer in these fields, in light of current research.

### 1. Low-Energy-Consumption Electronics and Spintronic Devices

Energy efficiency is one of the biggest challenges for modern electronic devices. WSMs possess the potential to offer significant solutions in this regard.

* **Chiral Anomaly and Dissipationless Transport:** The chiral anomaly [88] in WSMs leads to the efficient pumping of Weyl fermions from one chirality to another under specific conditions (parallel
electric and magnetic fields), significantly reducing resistance in the process. This is notably
observed as negative longitudinal magnetoresistance [10, 11]. In idealized scenarios, these chiral
currents are thought to be nearly dissipationless. If this property can be harnessed in practical devices,
it could lead to the development of ultra-low-power transistors and logic circuits with vastly reduced
energy loss due to Joule heating. This could be revolutionary, especially for mobile devices and high-
performance computing.

*   **Topological Fermi Arcs and Spintronics:** The Fermi arcs present on the surfaces of WSMs are topologically protected conducting channels with spin-momentum locking [8, 9]. This means that electrons moving in a particular direction also have their spins aligned in a specific direction. This property is extremely valuable for spintronic devices. Fermi arcs can be used for highly efficient spin injection and detection, enabling the development of spintronic components like faster, smaller, and more energy-efficient magnetic random-access memory (MRAM) cells, spin-based transistors, and spin logic gates. The topological protection of WSMs could make these spin currents more robust against defects and impurities, enhancing device reliability [19].

## 2. Topological Quantum Computation

Quantum computing promises to solve complex problems intractable for classical computers, but the sensitivity of quantum bits (qubits) to environmental noise (decoherence) is a major hurdle. Topological quantum computation aims to overcome this issue by encoding information in the topology of the system.

*   **Majorana Fermions and Braiding:** WSMs are predicted to host Majorana fermions – particles that are their own antiparticles – under certain conditions, particularly when interfaced with a superconductor. Specifically, it has been theoretically shown that surface Fermi arcs of Type-II WSMs in proximity to a superconductor can give rise to Majorana boundary states (Majorana zero modes) [20]. These Majorana zero modes can be used as topologically protected qubits. Information is processed by "braiding" these Majorana modes in momentum or real space, and these operations are highly resistant to errors due to topological protection. WSM-based Majorana qubits could significantly reduce decoherence, one of the biggest obstacles in quantum computing, paving the way for scalable and fault-tolerant quantum computers. This field is still in intensive theoretical and early experimental research, but its potential is immense.
### 3. Advanced Optoelectronics and Photonic Applications

WSMs also exhibit unique properties in their interaction with light, making them attractive for next-generation optoelectronic and photonic devices.

* **Giant Nonlinear Optical Responses:** WSMs, especially those with broken inversion symmetry, have been shown to exhibit nonlinear optical (NLO) responses orders of magnitude larger than other known materials. For instance, very strong second-harmonic generation (SHG) has been observed in WSMs like TaAs [21]. Furthermore, the circular photogalvanic effect (CPGE) – the generation of a DC current by circularly polarized light – is quite prominent and chirality-dependent in WSMs [17]. These giant NLO effects can be used to develop more efficient frequency converters (e.g., for converting infrared light to visible light or vice versa), optical switches, modulators, and even terahertz (THz) radiation sources and detectors. The potential of WSMs in this area could lead to compact and high-performance photonic integrated circuits.

* **Broadband and Tuneable Photodetectors:** The linear band dispersion of WSMs allows them to absorb light over a broad spectral range (from infrared to visible). This makes them suitable candidates for broadband photodetectors. Moreover, since the position and number of Weyl points can be tuned by external parameters (e.g., strain, magnetic field), the spectral response and polarization sensitivity of these photodetectors can also be dynamically controlled.

### 4. Thermoelectric Energy Conversion

Converting waste heat into electricity (or vice versa, cooling with electricity) is critical for energy efficiency and sustainability. WSMs also show promise in this area.

* **Anomalous Nernst and Ettingshausen Effects:** Magnetic WSMs, in particular, can exhibit a large anomalous Nernst effect (ANE) [22]. ANE is the generation of an electric field (or current) transverse to both a temperature gradient and the magnetization in a material. Similarly, the anomalous Ettingshausen effect (generation of a temperature gradient in the presence of an electric current and magnetization) can also be observed. These effects could be used to develop thermoelectric generators and coolers that are potentially more efficient or can operate in different geometries than
conventional devices based on the Seebeck or Peltier effects. The topological band structure and
Berry curvature in WSMs can enhance these anomalous thermoelectric effects.

**5. Catalysis**

Heterogeneous catalysis is widely used in industrial processes to increase the speed and selectivity of chemical reactions.

*   **Topological Surface States and Active Sites:** Topological surface states like Fermi arcs in WSMs can possess unique electronic densities and reactive sites. These surface states could function as highly active catalytic centres for specific chemical reactions [23]. Furthermore, the chiral structure of some WSMs might be beneficial for the enantioselective (preferring one enantiomer over the other) synthesis of chiral molecules, which is particularly important for the pharmaceutical industry. The catalytic potential of WSMs is a newly emerging field, but initial results are promising.

**6. Exotic Electromagnetism and High-Sensitivity Sensors**

WSMs have been theoretically proposed to host more exotic electromagnetic phenomena, such as axion electrodynamics [18]. This could lead to the discovery of new interactions and particles beyond our current understanding of fundamental physics. In terms of practical applications, the giant magnetoresistance properties of WSMs (both negative longitudinal magnetoresistance and the positive giant magnetoresistance in Type-II WSMs) can be utilized to develop extremely sensitive magnetic field sensors.

**Challenges and Future Outlook**

Despite the groundbreaking potential of Weyl semimetals, several challenges still lie ahead for practical applications. Synthesizing high-quality, large-scale single crystals, developing suitable fabrication techniques for device manufacturing, finding materials with stable and prominent Weyl features at room temperature, and fully understanding the complex physical properties of these materials are some of these challenges. However, intensive ongoing research worldwide offers hope for overcoming these obstacles and realizing the technological revolutions promised by Weyl semimetals. WSMs remain a class of materials that pushes the boundaries of materials science and holds the potential to shape future technologies.
### III. Fundamental Concepts in Weyl Semimetals: Detailed Explanations and Connections

The study of Weyl semimetals (WSMs) delves into the depths of condensed matter physics and necessitates the reinterpretation of many fundamental concepts. In this section, key concepts crucial for understanding WSMs and associated phenomena will be explained in detail, continuing the previous reference numbering and providing both Turkish and English counterparts.

#### 1. Weyl Fermion

* **Description:** A fundamental type of fermion with spin-1/2 and a definite chirality (right-handed or left-handed), theoretically predicted by Hermann Weyl in 1929 as a massless solution to the Dirac equation [1]. Unlike Dirac fermions (e.g., electrons are massive Dirac fermions), ideal Weyl fermions are massless, and their chirality, i.e., the orientation of their spin relative to their momentum, is a well-defined quantum number. Although not yet definitively observed as free fundamental particles in particle physics (neutrinos were initially considered candidates but were found to have mass [2]), they effectively emerge as low-energy collective excitations (quasiparticles) around linear crossing points in the electronic band structure of Weyl semimetals.

* **Importance:** Their chiral nature leads to unique quantum phenomena such as the chiral anomaly [10, 11]. Their existence in solid-state systems demonstrates that fundamental particle concepts from high-energy physics can be realized as condensed matter analogues, bridging disciplines.

#### 2. Chirality

* **Description:** The property of an object that cannot be superimposed on its mirror image; the most familiar examples are our right and left hands. In particle physics and the context of Weyl fermions, chirality refers to the projection of a particle's spin onto its momentum vector. A right-handed (R) Weyl fermion has its spin aligned in the same direction as its momentum (positive helicity), while a left-handed (L) Weyl fermion has its spin aligned opposite to its momentum (negative helicity). For massless particles, chirality and helicity are equivalent.
* **Importance:** Weyl points possess a definite chirality (e.g., R or L), and this chirality directly determines the topological properties of the material, the connectivity of Fermi arcs, and chirality-dependent transport phenomena like the chiral anomaly.

### 3. Weyl Point / Weyl Node

* **Description:** Special points in the electronic band structure of a Weyl semimetal in three-dimensional momentum space where the conduction and valence bands cross linearly in a non-degenerate manner. These points are topologically protected singularities. Each Weyl point carries a definite chirality (right- or left-handed) and a topological charge (typically a Chern number of ±1). The energy dispersion around these points is conical (Weyl cone).

* **Importance:** They act as monopoles (sources or sinks) of Berry curvature in momentum space. Their stability is topological, meaning they cannot be gapped out or eliminated by small perturbations that preserve the symmetries of the material. Weyl points always appear in pairs of opposite chirality and opposite topological charge, and due to the Nielsen-Ninomiya theorem, the total chiral charge (or total Chern number) within a Brillouin zone must be zero [24].

### 4. Berry Curvature and Berry Phase

* **Description:**

* **Berry Phase:** A geometric phase acquired by the wavefunction of a quantum system when its parameters (e.g., the momentum vector k of a Bloch electron) are slowly (adiabatically) varied along a closed loop, in addition to the dynamical phase.

* **Berry Curvature ($\Omega(k)$):** A vector quantity in momentum space that describes how the Berry phase "curls" or changes. It can be thought of as an "effective magnetic field" or "pseudo-magnetic field" in momentum space that influences the motion of electrons.

* **Importance:** Weyl points function as singular sources (positive chirality) or sinks (negative chirality) of Berry curvature in momentum space. Berry curvature underlies many anomalous transport and response phenomena, such as the anomalous Hall effect, anomalous Nernst effect, and
orbital magnetization [25]. The flux of Berry curvature through a closed surface enclosing a Weyl point gives its topological charge (Chern number).

## 5. Topological Charge / Chern Number

* **Description:** A quantized topological invariant that characterizes Weyl points and topological band structures in general. For a Weyl point, it is calculated as the flux of the Berry curvature through a two-dimensional closed surface (e.g., a sphere) enclosing the point, divided by $2\pi$, and is always an integer (typically $\pm 1$ for Weyl points). This charge is directly related to the chirality of the Weyl point.

* **Importance:** The integer value of the topological charge signifies that Weyl points are topologically stable and can only be annihilated by merging with another Weyl point of opposite total charge. This ensures the robustness of the Weyl phase.

## 6. Fermi Arc

* **Description:** Unique, open-ended topological surface states that appear on the surfaces or boundaries of Weyl semimetals, connecting the momentum-space projections of bulk Weyl points with opposite chirality. Unlike the closed Fermi surfaces of ordinary metals, Fermi arcs do not form complete loops; they start at one Weyl point projection and terminate at another.

* **Importance:** They are one of the most striking and directly observable experimental signatures of Weyl semimetals and are a consequence of the bulk-boundary correspondence. Their existence confirms that the material is in a topologically non-trivial Weyl semimetal phase. They are directly observed by Angle-Resolved Photoemission Spectroscopy (ARPES) [8, 9].

## 7. Chiral Anomaly

* **Description:** A quantum field theory phenomenon where a classically conserved symmetry (e.g., the conservation of chiral current) is broken due to quantum effects (e.g., via a triangle Feynman diagram). In Weyl semimetals, this manifests as a net change in the number of Weyl fermions of one chirality relative to the other (a "pumping" from one chirality to the other) in the presence of parallel electric (E) and magnetic (B) fields. This means that while the total number of fermions is conserved,
the individual numbers of left-handed and right-handed fermions (and thus the chiral charge) are not
conserved.

* **Importance:** In Weyl semimetals, it leads to characteristic unusual transport properties such as a negative longitudinal magnetoresistance (resistance decrease when the magnetic field is parallel to the electric field) [10, 11, 26] and a giant anomalous Hall effect [15, 16]. It is one of the strongest experimental signatures confirming the existence of Weyl fermions in the solid state.

**8. Symmetry Breaking**

* **Description:** For a Weyl semimetal phase to form and for Weyl points to be separated in momentum space and exist stably, the Hamiltonian of the system must lack certain symmetries. Specifically, either time-reversal symmetry (TRS) or inversion symmetry (IS, spatial parity symmetry) must be broken. If both are preserved, the system can be a Dirac semimetal or an ordinary metal/insulator, but not a Weyl semimetal.

* **Time-Reversal Symmetry Breaking:** Typically occurs in materials with intrinsic magnetic order (ferromagnetism, antiferromagnetism, etc.). Examples: Magnetic WSMs $Co_3Sn_2S_2$ [15], $Mn_3Sn$ [16].

* **Inversion Symmetry Breaking:** Occurs in materials whose crystal structure lacks an inversion center (non-centrosymmetric). Examples: TaAs-class WSMs [6, 7, 8, 9].

* **Importance:** Which symmetry is broken greatly influences the specific properties of the Weyl semimetal (e.g., anomalous Hall effect in magnetic WSMs, nonlinear optical effects in non-centrosymmetric WSMs) and its potential applications.

**9. Spin-Orbit Coupling (SOC)**

* **Description:** A relativistic interaction between an electron's spin and its orbital motion around a nucleus. This interaction can be thought of as the spin interacting with an effective magnetic field generated by the nucleus appearing to orbit the electron in its own reference frame. The strength of
the effect is generally proportional to the fourth power of the atomic number ($Z^4$), making it much more pronounced in heavy elements (e.g., Ta, W, Bi, Ir).

* **Importance:** It plays a critical role in the formation of Weyl semimetals (and topological materials in general). Strong SOC can lead to the splitting of energy bands, band inversion, and consequently, the formation of Weyl points or other topological states. Without SOC, the Weyl phase would not emerge in many materials.

## 10. Linear Dispersion / Weyl Cone

* **Description:** The linear relationship between energy and momentum (dispersion relation) in the immediate vicinity of a Weyl point, meaning energy (E) is directly proportional to the magnitude of momentum (k) ($E \approx \hbar vF|k-k_{Weyl}|$, where vF is the Fermi velocity and $k_{Weyl}$ is the momentum of the Weyl point). The Hamiltonian for the continuum around the Weyl point is given by $\mathcal{H}(q) = v_x q_x \sigma_x + v_y q_y \sigma_y + v_z q_z \sigma_z$, where $\sigma_{x,y,z}$ are Pauli matrices. The momentum $q(q_x, q_y, q_z) = k - k_{Weyl}$ defines the distance to the Weyl point, and $q = 0$ at the Weyl point [27]. This is analogous to the energy-momentum relation of massless relativistic particles (E = pc). In three-dimensional momentum space, this linear dispersion forms a structure of two cones (one valence band cone and one conduction band cone) touching at their apices at the Weyl point; this structure is called a "Weyl cone."

* **Importance:** It reflects the effective massless nature of Weyl fermions and determines the behavior of low-energy electronic excitations in these materials. This linear dispersion can lead to high carrier mobility, broadband optical absorption, and other unique electronic properties.

## 11. Bulk-Boundary Correspondence

* **Description:** A fundamental and universal principle of topological materials (topological insulators, Weyl semimetals, etc.). It states that there is a direct and precise relationship between the topological properties of the bulk of a material (e.g., topological invariants of the bulk band structure, Chern numbers, $Z_2$ invariants, the existence and chirality of Weyl points) and the special, protected electronic states that emerge at its boundaries (surfaces, edges, or corners).
* **Importance:** If the bulk topological properties are known, the existence and some characteristics of boundary states can be predicted, and vice versa, the observation of boundary states provides information about the bulk topology. In Weyl semimetals, the existence of Fermi arcs [8, 9] is strong evidence that the bulk is topologically non-trivial and hosts Weyl points.

## 12. Type-I Weyl Semimetal (Type-I WSM)

* **Description:** The standard type of Weyl semimetal where the Weyl points form point-like (or very small) Fermi surfaces at the Fermi level (EF). In this type, the Weyl cone can be upright or slightly tilted at the Fermi level, but its cross-section at EF is a point, and electron or hole pockets are either separate from or originate/terminate at the Weyl point. TaAs-class materials [6, 7, 8, 9] are generally considered Type-I WSMs.

* **Importance:** They exhibit behavior closer to Lorentz invariance (at low energies). They were the first type of Weyl semimetal discovered and are the most studied, providing prototype systems for understanding many fundamental Weyl phenomena.

## 13. Type-II Weyl Semimetal (Type-II WSM)

* **Description:** A type of Weyl semimetal theoretically predicted in 2015 by Alexei Soluyanov and colleagues [12] and soon after experimentally confirmed in materials like $WTe_2$ and $MoTe_2$ [13, 14]. In this type, Weyl points emerge at the contact points of electron and hole pockets in momentum space. Consequently, the Weyl cone has a severely "overtilted" structure at the Fermi level, such that instead of a point-like Fermi surface, Weyl points exist along a line of contact between electron and hole pockets at EF.

* **Importance:** They strongly violate Lorentz invariance. This can lead to interesting phenomena distinct from Type-I, such as anisotropic transport properties, layer-dependent characteristics, and the giant, non-saturating magnetoresistance observed in $WTe_2$. Furthermore, the manifestation of properties like the chiral anomaly and Fermi arcs can differ in Type-II WSMs compared to Type-I.

## 14. Dirac Semimetal
* **Description:** A type of semimetal with linearly dispersing bands that cross at four-fold degenerate Dirac points in its electronic band structure, occurring when both time-reversal symmetry and inversion symmetry are preserved. A Dirac point can be thought of as two Weyl points of opposite chirality and opposite topological charge overlapping at the same point in momentum space. This degeneracy is protected by crystal symmetry.

* **Importance:** It can be viewed as a "parent" or "precursor" phase to Weyl semimetals. If either time-reversal symmetry or inversion symmetry (or both) is broken in a Dirac semimetal, the Dirac point typically splits into pairs of Weyl points of opposite chirality, and the system transitions into a Weyl semimetal. Na₃Bi [10] and Cd₃As₂ are known three-dimensional Dirac semimetals.

## 15. Topological Insulator (TI)

* **Description:** Materials that behave as an insulator in their bulk (i.e., have an energy gap at the Fermi level) but possess topologically protected, gapless conducting states on their surfaces or edges. These surface states often exhibit unique properties like spin-momentum locking, meaning an electron's spin is coupled to the direction of its momentum.

* **Importance:** They are an important conceptual and experimental step on the path to the discovery of Weyl semimetals. According to some theories, a topological insulator can undergo a phase transition to a Weyl semimetal or Dirac semimetal through specific symmetry breaking or parameter tuning [4]. It is a key member of the family of topological materials.

## 16. Nielsen-Ninomiya Theorem

* **Description:** An important "no-go" theorem concerning the behavior of chiral fermions (like Weyl fermions) in a periodic lattice structure (e.g., a crystal). It essentially states that within a Brillouin zone (or more generally, on a compact manifold), the number of left-handed and right-handed Weyl points must be equal. This implies that the total chiral charge or total Chern number per unit cell must be zero.
* **Importance:** It guarantees that Weyl points always emerge in pairs of opposite chirality, or at least that the total chirality sums to zero. This is a fundamental constraint on the band structure of Weyl semimetals and is considered in the design of new Weyl materials.

# 17. Anomalous Hall Effect (AHE)

* **Description:** The generation of a Hall voltage (or Hall current) transverse to an electric current in ferromagnetic or other materials with intrinsic magnetization, even in the absence of an external magnetic field. Unlike the ordinary Hall effect, the AHE depends on the presence of magnetization. In Weyl semimetals, particularly in magnetic WSMs where time-reversal symmetry is broken, an intrinsic AHE mechanism arising from the momentum-space distribution of Berry curvature can be very dominant. The separation between Weyl points and their role as Berry curvature monopoles can lead to a giant AHE.

* **Importance:** It is a key transport signature of magnetic WSMs [15, 16, 22]. The magnitude of the AHE can provide information about the topological band structure of the material and the location of Weyl points. It holds potential for spintronic applications.

# IV. The Future of Weyl Semimetals and Interdisciplinary Synergies: Perspectives from New Mathematical Tools, Quantum Technologies, and Materials Science

The discovery of Weyl semimetals (WSMs) and the intensive research in this field have not only pushed the boundaries of condensed matter physics but have also raised exciting questions about how concepts and tools from different disciplines can contribute to understanding this new class of materials and developing their potential applications. In this context, the works of Mehmet Keçeci on theoretical physics, mathematical structures, quantum technologies, and nanomaterials, along with contributions from other researchers in magnetism, quantum simulations, materials science, and fundamental particle physics, offer potential intersection points that can enrich the future research directions and technological promises of WSMs.

# 1. New Mathematical Structures and Weyl Physics
Mehmet Keçeci's works such as "The Keçeci Binomial Square" [28, 52], "Kececisquares" (as a software application) [29, 51], "Scalable Complexity: Mathematical Analysis and Potential for Physical Applications of the Keçeci Circle Fractal" [30, 54], "Kececifractals" (as a software application) [31, 53], "Keçeci Numbers and the Keçeci Prime Number" [32, 56, 57], and "Kececinumbers" (as a software application) [33, 55] focus on the reinterpretation of fundamental mathematical structures and the exploration of their potential applications. Such fundamental mathematical tools and fractal geometries could offer new analytical or numerical approaches to understanding the complex electronic band structures, topological properties, and especially the symmetries in momentum space of WSMs. For instance:

*   **Fractal Geometries and Topological States:** The complex geometries of Fermi arcs or other topological surface states in WSMs could be investigated using fractal analysis methods. Keçeci's work on fractals [30, 31, 53, 54] might bring new perspectives on the scalability and self-similarity properties of such intricate structures. Particularly, whether topological states exhibit fractal behaviours in systems affected by disorder or boundary conditions could be explored.

*   **New Numerical Tools and Modeling:** Software tools like "Kececisquares" [29, 51] and "Kececinumbers" [33, 55], if adaptable to numerical methods for WSM band structure calculations or the determination of topological invariants, could contribute to modeling efforts in this field. New numerical schemes are always valuable, especially for large systems or complex interactions.

## 2. Weyl Semimetals in the Context of Quantum Technologies and Nanomaterials

Keçeci's publications "From Majorana Fermions to Quantum Devices: The Role of Nanomaterials in the Second Quantum Era" [34] and "Discourse on the Second Quantum Revolution and Nanotechnology Applications in the Midst of the COVID-19 Pandemic of Inequality" [35] establish a direct link with quantum technologies, one of the most exciting potential application areas for WSMs.

*   **Majorana Fermions and Topological Quantum Computation:** WSMs, especially when interfaced with superconductors, are predicted to host Majorana zero modes [20], which could form the basis for topological quantum computation. Keçeci's work focusing on Majorana fermions and the role of nanomaterials in quantum devices [34] provides a general framework for how nanotechnology can be utilized to overcome challenges in this area, such as material selection, device design, and manipulation
of Majorana modes. WSM-based nanostructures (nanowires, thin films) could be promising platforms for realizing Majorana qubits.

*   **The Second Quantum Revolution and WSMs:** The "Second Quantum Revolution" emphasized by Keçeci [35] refers to the translation of fundamental principles of quantum mechanics into new technologies. WSMs, with applications like low-energy electronics based on the chiral anomaly, spintronic devices, and quantum sensors, have the potential to be significant players in this revolution.

### 3. Theoretical Physics and Fundamental Interactions

Keçeci's earlier work on "2n-dimensional at Fujii model instanton-like solutions and coupling constant's role between instantons with higher derivatives" [36] is in the context of fundamental particle physics and field theory. Although not directly related to WSMs, such a theoretical physics background could indirectly contribute to a deeper understanding of phenomena originating from quantum field theory observed in WSMs, like the chiral anomaly, or to the investigation of potential manifestations of more exotic theories like axion electrodynamics in WSMs. Similarly, the study of the thermal properties of the exotic X(3872) state via QCD sum rules by Veliev et al. (2018) [37] serves as an example of how fundamental interactions play a role in complex systems, and such analytical approaches might inspire modeling of many-body interactions in WSMs.

### 4. Materials Science, Magnetism, and Experimental Approaches

The realization and characterization of Weyl semimetals require advanced material synthesis and experimental techniques. In this context, the works of other researchers offer important parallels and complementary information:

*   **Magnetism and Spintronics:** The demonstration of the coexistence of perpendicular and in-plane exchange bias using a single ferromagnetic layer in Pt/Co/Cr/CoO thin film by Öztürk et al. (2016) [38] and the direct evidence of a nonorthogonal magnetization configuration in a single crystalline Fe₁-xCox/Rh/Fe/Rh(001) system by Yıldız et al. (2009) [39] highlight the importance of complex magnetic interactions and anisotropies, which are critical for understanding and developing magnetic WSMs and WSM-based spintronic devices. The position and number of Weyl points in magnetic WSMs are strongly dependent on the nature of the magnetic order.
* **Material Characterization and New Materials:** The investigation of crystallographic, structural, optical, and dielectric properties of aniline and aniline halide imprinted hydrogels for optoelectronic applications by Yalçın et al. (2023) [40] and the study of magnetodielectric effects in Co-implanted TiInS₂ and TlGaSe₂ crystals by Mikailzade et al. (2016) [41] exemplify the processes of synthesizing new materials, characterizing them, and exploring their potential applications. WSM research similarly relies on the quest for new materials and detailed investigation of their properties. In particular, magnetoelectric and magnetodielectric interactions can be significant in certain types of WSMs.

* **Quantum Simulations and Model Predictions:** The work by Domuschiev (2025) comparing quantum simulation versus model prediction in human medicine [42] underscores the power of simulations and theoretical models in understanding complex systems. Ab-initio calculations of WSMs and simulations based on model Hamiltonians are indispensable for explaining experimental observations and predicting new phenomena.

**5. Software and Computational Tools**

Keçeci's software packages like "Kececilayout" [43, 44, 61–70] and "Grikod/Grikod2" [45, 46, 71, 72], and his contribution to "SciencePlots" [47], demonstrate the growing importance of computational tools and data visualization in scientific research. Weyl semimetal (WSM) research also heavily relies on first-principles calculations, model simulations, and the analysis of large datasets. Such open-source software tools and visualization packages can accelerate research processes and help share results more effectively. Strikingly illustrating this trend and the power of computational methods is the research co-authored by Mustafa Özdemir, which investigates the effect of carbon substitution on the electronic and optical properties of the MgSiP₂ chalcopyrite compound using first-principles calculations based on Density Functional Theory (DFT) [48]. In this study, it was shown in detail, using the TB-mBJ approximation, how the band gap of MgSiP₂ can be tuned by carbon substitution and how its absorption coefficient in the visible light region can be improved. Although not directly studying WSMs, such *ab initio* calculations and band structure engineering approaches are precisely the fundamental methodologies used in the theoretical screening of new WSM candidates, prediction of their electronic properties, identification of topological phases, and optimization of material characteristics for potential applications. The effective use of DFT in investigating different material classes, such as chalcopyrites, provides a valuable foundation and methodological insight for the discovery and understanding of more exotic topological materials like WSMs.
# 6. Security Applications and Electromagnetic Techniques

The work by Bulat Rameev (2020) on magnetic resonance and microwave techniques for security applications [49] could have an indirect connection to some potential applications of WSMs. For example, the unique electromagnetic responses of WSMs (e.g., giant nonlinear responses in the THz region or magnetoelectric effects linked to the chiral anomaly) could form the basis for next-generation sensors or scanning technologies.

# v. Conclusion and Future Perspective

The works of Mehmet Keçeci on mathematical structures, quantum concepts, and software tools, along with the contributions of other researchers in materials science, magnetism, and theoretical physics, highlight the multidisciplinary nature and future potential of Weyl semimetals. The application of new mathematical approaches to model the complex physics of WSMs, the use of nanomaterial principles in developing WSM-based quantum devices, and the employment of advanced experimental techniques in discovering and characterizing new WSM materials will further expand the frontiers of this exciting field. Interdisciplinary collaboration and the integration of expertise from different areas will play a key role in realizing the scientific and technological revolutions promised by Weyl semimetals.

# The Convergence of Weyl and Dirac Semimetals with Majorana Modes: A Glimpse into the Quantum Future

The discovery of Weyl semimetals (WSMs) and their precursor phases, Dirac semimetals (DSMs), has not only unveiled exotic electronic states and topological phases but also provided profound insights into how fundamental concepts from particle physics can manifest in solid-state systems [1, 5, 6, 7, 8, 9, 10, 12]. One of the most captivating future promises of these materials is their potential to host Majorana zero modes (MZMs), which could be the building blocks for topologically protected quantum computation [20, 34]. Below are some possibilities and interpretations of what might happen in the future through the convergence and individual advancements of these fields:

# 1. Revolution in Topological Quantum Computation: Fault-Tolerant Qubits
* **Current Status and Potential:** Majorana zero modes are fermionic excitations that are their own antiparticles and are topologically protected. This protection makes MZM-based qubits highly resistant to local noise and decoherence, which is one of the biggest challenges for current quantum computing approaches. WSMs and DSMs, particularly when interfaced with superconductors (via the proximity effect), are theoretically predicted to host MZMs [20].

* **Future Possibilities:**

* **Experimental Realization and Manipulation:** One of the most crucial future steps will be the definitive and reproducible experimental demonstration of MZMs in WSM/DSM-superconductor heterostructures, followed by the controlled braiding of these MZMs. Braiding operations form topological quantum gates.

* **Materials Engineering:** The discovery of new WSM/DSM materials and superconductor combinations that can host MZMs at higher temperatures, with greater stability, and are easier to fabricate will be critical. Keçeci's emphasis on the role of nanomaterials in quantum devices [34] highlights the importance of nanostructures and interface engineering at this juncture.

* **Scalable Qubit Designs:** Beyond single MZM pairs, intensive research will continue on how to generate multi-qubit systems and design a scalable topological quantum computer architecture.

* **Unique Advantages of WSMs:** Due to their chiral nature and diverse Weyl point configurations, WSMs might offer potential advantages over conventional semiconductor nanowires for creating and manipulating MZMs. For instance, distinct Fermi arcs could provide natural pathways for transporting MZMs.

## 2. Next-Generation Low-Energy Electronics and Spintronic Devices

* **Chiral Currents and Dissipationless Transport:** The chiral anomaly in WSMs [10, 11, 26] can theoretically lead to dissipationless or very low-loss chiral currents.

* **Future Possibilities:**
* **Chiraltronics:** A completely new electronics paradigm could emerge where information is carried and processed by chiral currents. This could mean a massive leap in energy efficiency compared to current CMOS technology.

* **Topological Spintronics:** The spin-momentum locked surface and bulk states of WSMs and DSMs are ideal for ultra-fast and low-power spintronic devices (e.g., spin transistors, MRAMs) [19]. The giant anomalous Hall effect in magnetic WSMs [15, 16, 22, 49] could be used for magnetic field sensors or magnetic memory read heads.

* **Valleytronics:** In some WSMs and DSMs, an additional degree of freedom (valley index), defined by electrons residing in different momentum-space "valleys," can emerge. This valley degree of freedom could be used to encode information, leading to devices with new functionalities.

### 3. Advanced Photonic and Optoelectronic Applications

* **Giant Nonlinear Responses:** WSMs, particularly those with broken inversion symmetry, exhibit exceptionally large nonlinear optical (NLO) responses [17, 21].

* **Future Possibilities:**

* **Compact and Efficient Frequency Converters:** Compact devices capable of efficiently changing the frequency of laser light (e.g., from infrared to visible or THz) could be developed.

* **THz Technologies:** WSMs could offer new platforms for the generation, detection, and manipulation of terahertz (THz) radiation. THz technologies are important in many fields, including imaging, communication, and spectroscopy.

* **Optically Controlled Chirality and Topology:** It might become possible to dynamically control the position, number, or chirality of Weyl points in WSMs using light, leading to tuneable optical and electronic devices.

### 4. Exotic Fundamental Physics and New Material Discoveries
* **Axion Electrodynamics and Higher-Order Topology:** Some WSMs have been proposed to host condensed matter analogues of more exotic fundamental physics theories, such as axion electrodynamics [18].

* **Future Possibilities:**

* **Tabletop High-Energy Physics:** WSMs could offer a platform for "tabletop experiments" where the properties of fundamental particles and interactions can be studied in a laboratory setting, without the need for large particle accelerators.

* **Discovery of New Topological Phases:** Research into WSMs and DSMs will continue to inspire the discovery of more complex topological phases, such as "higher-order topological insulators/semimetals", "nodal-line semimetals" [60] or other exotic topological states.

* **Materials Design and Artificial Intelligence:** The discovery of new WSM and DSM materials could be accelerated by computational materials science and artificial intelligence (e.g., machine learning) methods. New mathematical approaches, like Keçeci's fractal [30, 31, 53, 54] or numerical [33, 34, 55] works, could potentially be integrated into such search algorithms. First-principles studies, like those of Mustafa Özdemir using DFT calculations [48], will be critical in verifying candidate materials.

## 5. Interdisciplinary Synergies and Computational Tools

* **Integration of Theory, Computation, and Experiment:** Progress in the fields of WSMs, DSMs, and MZMs will rely on strong synergy between theoretical predictions, advanced computational simulations (as emphasized by Domuschiev [42]), and precise experimental measurements (e.g., techniques used by Öztürk et al. [38], Yıldız et al. [39], and Mikailzade et al. [41] for magnetic properties). Contemporary challenges in advanced technological applications often require interdisciplinary approaches and sophisticated experimental methodologies. In this context, the statistical methods employed [87], such as design of experiments (DOE), Taguchi optimization, and response surface methodology—used to achieve minimum clad size in a laser-assisted metal deposition process—bear significant methodological resemblance to studies aiming to model and
optimize highly sensitive physical systems, such as quantum computing architectures or gravitational wave observatories. In both domains, the systematic analysis of multiple variables is crucial for achieving optimal outcomes and enabling automation in real-world applications. Therefore, the successful adaptation of such methodologies across different engineering fields highlights the importance of interdisciplinary scientific synthesis [78, 79].

* **Future Possibilities:**

* **Open Science and Data Sharing:** The trend towards open-source software contributions, like those by Keçeci [29, 31, 33, 43, 44, 45, 46, 47, 51], and the general movement in the scientific community will encourage more open sharing of large datasets, computational codes, and experimental results. This can accelerate the discovery process.

* **New Analytical and Numerical Methods:** New analytical theories and more powerful numerical simulation techniques (e.g., quantum Monte Carlo, tensor networks) will be needed to model the complex interactions and quantum dynamics of WSMs and MZMs.

**Challenges:**

Despite these exciting possibilities, significant challenges remain:

* **Material Quality and Fabrication:** Producing high-purity, large, and homogeneous WSM/DSM crystals and heterostructures is still difficult.

* **Room-Temperature Operation:** Many interesting phenomena are currently observed at low temperatures. Material discovery and engineering are needed for room-temperature devices.

* **Definitive Detection of MZMs:** While evidence for MZMs is growing, definitive and undisputed experimental signatures are still an active area of research.

One of the most remarkable types of these singularities is what we call the blow-up phenomenon. Blow-up is simply a form of spontaneous singularity that occurs when one or more of the dependent variables approach infinity as time approaches a certain finite value [50].
Singularities such as the blow-up phenomenon are important in quantum computing, especially in understanding the **stability of quantum dynamics, simulation accuracy, and algorithmic behavior** under extreme conditions.

* **Quantum Dynamics:** The dynamics of quantum states governed by differential equations like the Schrödinger equation can sometimes exhibit unstable or singular behaviours under certain initial or boundary conditions.

* **Simulation Accuracy:** When simulating quantum systems on classical or quantum computers, identifying and handling potential blow-ups helps improve the fidelity of the simulation.

* **Algorithm Design:** Understanding how solutions diverge or become unstable can lead to more robust quantum algorithms, especially in optimization and machine learning applications.

Modeling blow-up phenomena mathematically can therefore contribute to improving the **reliability, efficiency, and predictive power** of quantum computational models.

# VI. Conclusion: Horizons of Weyl Semimetals and the Ongoing Journey of Scientific Discovery

Weyl semimetals, Dirac semimetals, and Majorana zero modes are profoundly interconnected fields with the potential to transform both fundamental science and technology in the coming decades. They could revolutionize a spectrum of applications [73–86], from fault-tolerant quantum computation to ultra-efficient electronics and new discoveries in fundamental physics. Realizing this potential will require strong interdisciplinary collaboration [67–69, 87] among materials scientists, theoretical physicists, experimentalists, and engineers, along with continuous innovation.

The journey of Weyl semimetals (WSMs), from Hermann Weyl's theoretical prediction in 1929 [1] to their experimental verification in tangible materials in the early 2010s [5, 6, 7, 8, 9], constitutes one of the most captivating and productive narratives in modern condensed matter physics. These materials have not only forged an elegant bridge between fundamental particle physics and solid-state physics but have also vividly demonstrated how fundamental concepts like topology, symmetry, and quantum field theory can profoundly influence material properties. Weyl fermions, initially a theoretical curiosity, have transformed into real excitations observable in laboratories through WSMs, leading to a host of extraordinary physical
Exploring Weyl Semimetals: Emergence of Exotic Electrons and Topological Order, Mehmet Keçeci, 2025

phenomena such as the chiral anomaly [10, 11, 26], topological Fermi arcs [8, 9], and unique magnetoelectric

responses.

As discussed in this review, the discovery of WSMs has heralded a new era in materials science, enabling the investigation of different WSM types, including Type-I [8, 9] and Type-II [12, 13, 14], as well as magnetic WSMs [15, 16]. This diversity offers a rich platform of materials, each exhibiting unique electronic band structures, Berry curvature distributions [25], and symmetry-breaking mechanisms [4, 5]. Understanding these materials has underscored the critical role of advanced experimental techniques like Angle-Resolved Photoemission Spectroscopy (ARPES) and powerful computational methods such as Density Functional Theory (DFT) [48].

The potential technological applications [73–86] of WSMs are at least as exciting as their fundamental scientific importance. Low-energy "chiraltronic" devices, topologically protected spintronic components [19], advanced photonic applications based on giant nonlinear optical effects [17, 21], and high-efficiency thermoelectric energy converters [22] are just a few areas where WSMs could make an impact in the near future. Perhaps the most transformative potential lies in the possibility that WSMs, when interfaced with superconductors, could host Majorana zero modes [20, 33]. This could offer a long-sought path towards realizing fault-tolerant topological quantum computation and could be a significant driver of the "Second Quantum Revolution" [35].

However, this bright future vision is accompanied by significant challenges. The synthesis of high-quality, large-scale, and room-temperature stable WSM materials, the development of reliable fabrication techniques for integrating these materials into devices, and particularly the definitive detection and manipulation of Majorana modes, require further research and innovation.

It is at this juncture that the importance of contributions from diverse disciplines becomes apparent, such as Mehmet Keçeci's work on new mathematical structures [28, 30, 32, 52], software tools [29, 31, 33, 43, 44, 45, 46, 51], and his visionary work on quantum technologies [34, 35]. New analytical and numerical methods could potentially be used to model the complex behavior of WSMs, fractal geometries to understand topological states, or AI-assisted algorithms for materials discovery. Similarly, expertise in magnetism [38, 39], materials characterization [40, 41], quantum simulations [42], and fundamental particle physics [36, 37] will enrich WSM research and help overcome the challenges encountered. Specific expertise, such as
Mustafa Özdemir's work in computational materials science [48] and Bulat Rameev's focus on electromagnetic techniques [49], are tangible examples of this multidisciplinary endeavor.

In conclusion, Weyl semimetals, along with their associated Dirac semimetals and Majorana modes, represent a dynamic research area that pushes the frontiers of science, offers fertile ground for fundamental discoveries, and inspires future technologies. The future of this field depends on the collaboration of researchers from different disciplines, innovative ideas, and the unyielding flame of scientific curiosity. The story of WSMs is a striking example of the ongoing journey of scientific discovery, illustrating how knowledge accumulates, how theory guides experiment, and how unexpected discoveries open new horizons. As we further unravel the secrets of these materials and fully realize their potential in the coming years, we can anticipate many more revolutions in both our fundamental understanding of science and our technological capabilities.
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