Beyond the Standard Model: Anisotropic Interactions in Weyl Semimetals

Author: Denis Avetisyan

New research reveals that Coulomb interactions within generalized Weyl semimetals create direction-dependent electronic behavior, challenging traditional Fermi liquid theory.

This study demonstrates an anisotropic marginal Fermi liquid state arising from long-range Coulomb interactions in generalized Weyl semimetals with higher monopole charge, as confirmed by renormalization group analysis and the Ward-Takahashi identity.

Conventional descriptions of interacting electron systems often struggle to reconcile Fermi liquid behavior with long-range Coulomb interactions and complex band structures. This is addressed in ‘Anisotropic marginal Fermi liquid for Coulomb interacting generalized Weyl fermions’, which investigates the emergent electronic properties of three-dimensional generalized Weyl semimetals. Through a Wilsonian renormalization group analysis, this work demonstrates that these materials, characterized by monopole charge $n \ge 2$ , exhibit an anisotropic marginal Fermi liquid state arising from direction-dependent Coulomb screening and a finite anomalous dimension. Could this anisotropic behavior offer a pathway to novel electronic devices and a deeper understanding of strongly correlated topological phases?

The Unfolding of Quantum Topology

A newly recognized phase of quantum matter, the Weyl semimetal, challenges conventional understandings of electronic behavior through its unique topological properties. Unlike traditional materials, these semimetals don’t simply conduct or insulate; they host special points in momentum space called Weyl nodes. These nodes aren’t merely features of the material’s band structure, but are topologically protected – meaning their existence is guaranteed by the fundamental laws of physics and robust against imperfections or disturbances. This protection arises from the material’s topology, a mathematical property describing its connectivity, which dictates the behavior of electrons near these nodes. Consequently, electrons behave as if possessing a peculiar “chirality,” leading to unusual transport properties like high mobility and the potential for dissipationless conduction, opening doors to next-generation electronic devices and fundamentally altering the landscape of condensed matter physics.

Conventional Weyl semimetals are characterized by band crossings – Weyl nodes – that behave as monopoles of Berry curvature in momentum space, giving rise to unique electronic properties. However, recent theoretical advancements demonstrate the possibility of ‘Generalized Weyl Semimetals’ where these nodes are not limited to a single monopole charge; instead, they can exhibit higher-order charge configurations. These higher-charge nodes arise from specific crystalline symmetries within the material, effectively creating more complex topological structures. This generalization significantly expands the landscape of possible quantum phenomena, offering a pathway to engineer materials with tailored electronic transport, novel surface states, and potentially even exotic phases of matter not found in standard Weyl semimetals. The ability to control and manipulate these higher-charge configurations presents a compelling frontier in the search for advanced quantum materials.

Generalized Weyl semimetals represent a significant advancement in the study of topological materials, offering a more complex landscape for exploring emergent quantum phenomena. Unlike conventional Weyl semimetals characterized by simple monopole charge nodes, these materials host nodes with higher charges, fundamentally altering their electronic and transport properties. Crucially, these higher-charge nodes aren’t simply accidental features; they are robustly protected by the material’s crystalline symmetries, ensuring their persistence even in the presence of imperfections. This symmetry protection unlocks the potential for novel quasiparticles and exotic electronic states, potentially leading to breakthroughs in areas like low-dissipation electronics and quantum computing. The richer topology afforded by these nodes provides a versatile platform for investigating phenomena such as the chiral anomaly and the emergence of unconventional superconductivity, promising a deeper understanding of quantum matter and its potential applications.

Renormalization and Topological Resilience

The Wilsonian Renormalization Group (WRG) is a method for studying the effective behavior of physical systems as the energy scale is varied. It achieves this by systematically eliminating high-energy degrees of freedom and their associated interactions, thereby simplifying the description of the system at lower energies. This process involves defining a momentum cutoff Λ and integrating out all modes with energies above this cutoff. The resulting low-energy effective action then describes the remaining degrees of freedom, but with modified couplings that incorporate the effects of the eliminated modes. By iteratively lowering Λ, the WRG generates a sequence of effective theories, each valid at a different energy scale, and reveals how interactions evolve and contribute to the system’s low-energy physics. This framework is particularly useful for identifying relevant and irrelevant operators, which dictate the long- and short-distance behavior of the system, respectively.

The Ward-Takahashi Identity is a fundamental equation in quantum field theory that establishes a relationship between the vector current and the generating functional of Green’s functions. Its application within the renormalization group framework guarantees that any calculated physical observables remain consistent with underlying symmetries, such as translational and rotational invariance, even after accounting for quantum fluctuations and divergences. Specifically, the identity ensures that the conservation of current, derived from symmetries, is maintained throughout the renormalization process, preventing the generation of anomalous terms that would violate these fundamental principles. This is achieved by relating changes in correlation functions to the symmetry transformations, effectively enforcing symmetry preservation at each energy scale during renormalization.

Application of the Wilsonian Renormalization Group to the generalized Weyl semimetal demonstrates that electron-electron interactions induce modifications to the single-particle electronic structure, specifically altering the band dispersion and quasiparticle weights. These interactions, treated perturbatively, generate momentum-dependent self-energies that renormalize the system’s parameters and can lead to the emergence of new topological phases or the suppression of existing ones. Importantly, the renormalization process can drive changes in the Fermi surface topology, potentially leading to the formation or annihilation of Weyl nodes and impacting the material’s transport properties, such as its anomalous Hall conductivity and chiral anomaly. The strength of these effects is determined by the specific form and magnitude of the interactions and the dimensionality of the system.

The Dance of Screening and Anomalous Behavior

The strength of the Coulomb interaction, normally described by $1/r$ behavior, is diminished in systems containing charge carriers due to their collective response. This reduction, termed screening, arises from the polarization of the charge carrier density around a given charge, effectively shielding its bare potential. The mobile charge carriers rearrange themselves to partially cancel the electric field, resulting in a screened potential that decays more rapidly with distance than the unscreened Coulomb potential. The degree of screening is dependent on the density and properties of the charge carriers, and is a fundamental factor influencing the electronic behavior of materials.

Random Phase Approximation (RPA) is a technique used to calculate the dielectric function $\epsilon(q,\omega)$ , which describes the response of a system to an external perturbation. In the context of screening, RPA considers the collective response of charge carriers to a test charge, effectively reducing the Coulomb interaction. The calculation involves summing over all possible excitations of the system, accounting for the polarization of the medium. This summation yields an expression for the screened Coulomb potential $V_{screened}(q) = \frac{V_{bare}(q)}{\epsilon(q,\omega)}$ , where $V_{bare}(q)$ is the bare Coulomb potential and $\epsilon(q,\omega)$ is the dielectric function calculated within RPA. The accuracy of the RPA calculation depends on the validity of the perturbative approach and neglects local-field effects and vertex corrections.

Anisotropic screening of the Coulomb interaction results in the formation of an Anisotropic Marginal Fermi Liquid (AMFL) state. This state is defined by a finite anomalous dimension, $\gamma_0$ , which quantifies the deviation from standard Fermi liquid behavior. The value of $\gamma_0$ is dependent on the number of flavors, denoted as N, and the spatial dimension, n. Specifically, calculations indicate a value of $\gamma_0 = 0.344/N$ for n=2 and $\gamma_0 = 0.252/N$ for n=3. This finite anomalous dimension is a key characteristic distinguishing the AMFL state and impacts its thermodynamic and transport properties.

Logarithmic corrections to scaling in thermodynamic and transport properties arise due to the screened Coulomb interaction and the resulting ‘Anisotropic Marginal Fermi Liquid’ state. These corrections manifest as deviations from standard power-law behavior in quantities like specific heat, susceptibility, and resistivity as temperature approaches zero. Specifically, the anomalous dimension $\gamma_0$ dictates the strength of these logarithmic terms; values of $\gamma_0 = 0.344/N$ for n=2 and $\gamma_0 = 0.252/N$ for n=3, where N represents the number of flavors, quantify the degree to which these properties deviate from the expected scaling behavior. The presence of these corrections indicates a breakdown of the simple mean-field theory and necessitates more sophisticated theoretical treatments to accurately describe the system’s behavior at low temperatures.

Probing the Vanishing Act: Experimental Signatures

Angle-Resolved Photoemission Spectroscopy, or ARPES, functions as a crucial experimental technique for directly observing the electronic structure of materials and, critically, quantifying the strength of quasiparticle interactions. By measuring the energy and momentum of emitted electrons, ARPES maps out the $E(k)$ dispersion relation – the relationship between energy and momentum – revealing the nature of electronic excitations. The intensity of the spectral function at a given energy and momentum is directly proportional to the quasiparticle residue, effectively providing a measure of how much an electron retains its original particle-like character despite interactions with the surrounding material. A diminished quasiparticle residue indicates stronger interactions and a greater departure from the simple, free-electron picture, offering valuable insights into the complex behavior of correlated electron systems and confirming theoretical predictions regarding the breakdown of the quasiparticle concept.

Theoretical models predicting non-Fermi liquid behavior rely heavily on the concept of logarithmic corrections to conventional scaling laws, and these predictions can be experimentally verified through careful analysis of material properties. The strength of interactions, particularly those arising from screening effects, directly influences the precise form of these corrections; deviations from expected power laws manifest as logarithmic dependencies. Specifically, specific heat and compressibility are predicted to scale with temperature as $T^{(1+2/n)} / [\ln(\Lambda/T)]^p$ , while optical conductivity exhibits scaling in frequency as $\omega / [\ln(\Lambda/\omega)]^x$ and $\omega^{(2/n-1)} / [\ln(\Lambda/\omega)]^z$ . By meticulously observing these logarithmic dependencies-and extracting the associated exponents-researchers can confirm the validity of the underlying theoretical framework and gain a deeper understanding of the complex interplay between interactions and emergent behavior in these materials.

Detailed investigations into the thermal and electromagnetic properties reveal a characteristic signature of the quasiparticle residue. Specifically, the specific heat and compressibility exhibit a scaling behavior described by $T^{(1+2/n)} / [\ln(\Lambda/T)]^{p_{th}}$ , indicating how these properties diminish with temperature and the logarithmic suppression of quasiparticles. Complementing this, the optical conductivity demonstrates a dependence on frequency ω, scaling as $\omega / [\ln(\Lambda/\omega)]^{p_x}$ at lower frequencies and transitioning to $\omega^{(2/n-1)} / [\ln(\Lambda/\omega)]^{p_z}$ at higher frequencies; these relationships collectively provide a precise measure of the quasiparticle residue’s influence on the material’s response to external stimuli, confirming the theoretical predictions regarding the system’s behavior near a quantum critical point.

The very existence of quasiparticles in strongly correlated materials is challenged by the system’s tendency towards collective behavior, and this manifests as a suppression of their spectral weight. Measurements reveal this suppression isn’t simply a vanishing of the quasiparticle peak, but rather a characteristic logarithmic decay of the quasiparticle residue-a measure of the fraction of electrons that actually behave as coherent quasiparticles. This residue scales as $\left[\ln\left(\Lambda/\omega\right)\right]^{-ηψ}$ , where Λ represents a high-energy cutoff, ω is the frequency, and $ηψ$ is a scaling exponent that directly reflects the strength of interactions. This logarithmic dependence indicates that even at relatively high energies, interactions continue to subtly erode the quasiparticle character, leaving behind only a diminished spectral signature and providing crucial insight into the collective quantum phenomena at play within the material.

Beyond the Horizon: Implications for Quantum Technologies

These novel materials exhibit topological properties-characteristics stemming from the material’s fundamental shape and connectivity at a quantum level-that are profoundly influenced by how electrons interact with each other and their surroundings. Specifically, the effects of ‘screening’ – where intervening electrons diminish the impact of Coulomb interactions – coupled with the direct Coulomb forces between electrons, give rise to unexpected, or emergent, behaviors. This interplay doesn’t simply modify existing properties; it can create entirely new functionalities, potentially enabling the development of devices that exploit these unique quantum states. Researchers anticipate that controlling these interactions will allow for the engineering of materials with tailored electronic and magnetic properties, opening doors to advancements beyond conventional semiconductor technology and potentially revolutionizing fields like quantum computing and spintronics.

The potential for creating remarkably stable quantum devices stems from a fascinating interplay between a material’s Berry curvature and its topological charge. Berry curvature, a geometric property arising from the quantum mechanical phase of electron wavefunctions, dictates how charged particles move within a material. When coupled with non-trivial topological charge – a characteristic describing the ‘twistedness’ of a material’s electronic band structure – this creates protected states resistant to scattering from imperfections or disorder. These topologically protected states act as robust pathways for quantum information, minimizing decoherence – a major obstacle in quantum computing. Researchers are actively investigating materials where strong spin-orbit coupling and tailored electronic structures maximize this effect, envisioning quantum bits, or qubits, that maintain their quantum state for significantly longer durations and are thus far more reliable than current designs. This approach promises to move beyond the limitations of traditional semiconductor-based quantum devices and usher in a new era of fault-tolerant quantum technologies.

Continued investigation into these novel materials holds the potential to unify the abstract realm of fundamental physics with concrete technological progress. Researchers anticipate that a deeper comprehension of their quantum properties-particularly those arising from topological effects-could unlock pathways to more stable and efficient quantum bits, or qubits. This enhanced qubit stability is crucial for building scalable quantum computers capable of tackling currently intractable computational problems. Beyond computing, the unique characteristics of these materials may also find applications in advanced sensing technologies, secure communication networks, and the development of entirely new classes of electronic devices, suggesting a future where fundamental scientific discovery directly fuels groundbreaking innovation.

The study of these generalized Weyl semimetals reveals a fascinating truth about complex systems-their inherent impermanence. Much like all architectures, these materials possess a finite lifespan dictated by interactions and external forces. As the research demonstrates with its findings on anisotropic marginal Fermi liquid behavior, even fundamental properties are subject to directional dependencies and modifications over time. This aligns with Emerson’s observation that “The only true grandeur consists in growth.” The observed alterations in electronic properties, stemming from Coulomb interactions, aren’t signs of decay, but rather evidence of the system’s continuous evolution and adaptation within its energetic landscape.

What Lies Ahead?

The demonstration of anisotropic marginal Fermi liquid behavior in these generalized Weyl semimetals isn’t a revelation of stability, but a precise charting of decay. The system doesn’t avoid the inevitable influence of Coulomb interactions; it organizes around them, exhibiting direction-dependent vulnerabilities. The long-range nature of these interactions suggests a broader susceptibility to external perturbations, a subtle reshaping of the electronic landscape that will, ultimately, lead to a different equilibrium. It is not a question of whether the topological protection will fail, but how and when.

Current theoretical frameworks, largely predicated on simplified models, struggle to account for the full complexity of these interacting systems. The Ward-Takahashi identity, while a powerful tool, offers only a static snapshot. The renormalization group calculations, though illuminating, remain approximations-maps of a territory never fully explored. Future work must address the interplay between Berry curvature, anisotropic screening, and the emergence of novel collective modes. A deeper understanding necessitates moving beyond perturbative approaches.

The pursuit of “robust” topological phases feels increasingly like a delaying tactic. Time is not a metric to be overcome, but the medium in which all systems exist. The true challenge lies not in preventing the system’s evolution, but in accurately predicting the form it will take as it ages. The field will progress not by seeking perpetual stability, but by accepting the inherent impermanence of the materials it studies.

Original article: https://arxiv.org/pdf/2602.17666.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/