Berry Curvature Research Articles

Topological magnon in exchange frustration driven incommensurate spin spiral of kagome-lattice YMn6Sn6

YMn6Sn6 consists of two types of Mn-based kagome planes stacked along the c-axis having a complex magnetic interaction. We report a spin reconstruction in YMn6Sn6 from ferromagnetic (FM) into a combination of two incommensurate spin spirals (SSs) originating from two different types of Mn kagome planes driven by frustrated magnetic exchanges along the c-axis with the inclusion of the Hubbard U. The pitch angle and wave vector of the incommensurate SSs are ∼89.3∘ and ∼ (0 0 0.248), respectively, which are in excellent agreement with the experiment. We employ an effective model Hamiltonian constructed out of exchange interactions to capture the experimentally observed nonequivalent nature of the two incommensurate SSs which also explain the FM-SS crossover due to antiferromagntic spin exchange with correlation. We further report the existence of a topological magnon with spin-orbit coupling in the incommensurate SS phase of YMn6Sn6 by calculating the topological invariants and Berry curvature profile. The location of the Dirac magnon in the energy landscape at 73 meV matches with another experimental report. We demonstrate the accuracy of our results by highlighting the experimental features in YMn6Sn6. Published by the American Physical Society 2024

Nonlinear anomalous transverse responses induced by the Berry curvature quadrupole in systems with broken time-reversal symmetry

Nonlinear anomalous transverse responses induced by the Berry curvature quadrupole in systems with broken time-reversal symmetry

Spinor Boltzmann equation with Berry curvature

Spinor Boltzmann equation with Berry curvature

Unusual Anomalous Hall Effect in Two-Dimensional Ferromagnetic Cr7Te8

Two-dimensional (2D) materials with inherent magnetism have attracted considerable attention in the fields of spintronics and condensed matter physics. The anomalous Hall effect (AHE) offers a theoretical foundation for understanding the origins of 2D ferromagnetism (2D-FM) and offers a valuable opportunity for applications in topological electronics. Here, we present uniform and large-size 2D Cr7Te8 nanosheets with varying thicknesses grown using the chemical vapor deposition (CVD) method. The 2D Cr7Te8 nanosheets with robust perpendicular magnetic anisotropy, even a few layers deep, exhibit a Curie temperature (TC) ranging from 180 to 270 K according to the varying thickness of Cr7Te8. Moreover, we observed a temperature-induced reversal in the sign of the anomalous Hall resistance, correlating with changes in the intrinsic Berry curvature. Additionally, the topological Hall effect (THE) observed at low temperatures suggests the presence of non-trivial spin chirality. Our findings about topologically non-trivial magnetic spin states in 2D ferromagnets provide a promising opportunity for new designs in magnetic memory spintronics.

Atomic disorder and intrinsic anomalous Hall effect in a half-metallic ferromagnet Co2VAl

Half-metallic ferromagnets, conducting for one spin channel while insulating for the other, are highly desirable for spintronic applications due to 100 % spin polarization around the Fermi level. Cobalt-based half-metallic Heusler compounds have attracted enormous attention due to their large spin polarization and a high magnetic transition temperature. In the present study, we report the experimental and theoretical investigation of crystal structure and anomalous Hall effect (AHE) in half-metallic ferromagnet Co2VAl. The structural investigation of high-resolution synchrotron x-ray diffraction data reveals 10 % antisite disorder between V and Al atoms within the L21 ordered crystal structure. The scaling analysis of anomalous Hall data shows that the AHE in our system is mainly driven by the Berry curvature in the momentum space. The magnitude of experimental intrinsic anomalous Hall conductivity (AHC) due to the momentum space Berry curvature is about 44.67 ± 0.02 S/cm at 5 K, which is less than the theoretically calculated AHC for the ordered structure. Our theoretical calculations suggest that the lower AHC obtained for the present system is due to the reduced Berry curvature in the disordered case with negligible impact on half-metallicity of the system.

Coexistence of anomalous Hall effect and weak magnetization in a nominally collinear antiferromagnet MnTe

In various material systems, an antiferromagnetic phase was found to coexist with a weak ferromagneticlike signal, while symmetry-based theoretical predictions indicate a possibility of a nonzero anomalous Hall effect (AHE) even in the absence of sample magnetization. This is the case of nominally collinear antiferromagnets, in particular, hexagonal MnTe, where the AHE and no detectable magnetization have been recently reported. To clarify the role of magnetization, we present a study of bulk MnTe samples, combining experiment and theory. We demonstrate that the existence of the AHE in the hexagonal MnTe is accompanied by the presence of a weak but detectable ferromagneticlike signal, vanishing at the Néel temperature. In contrast to thin layer samples, we find that the AHE hysteresis loop shows an opposite sign and Barkhausen-like jumps. We introduce a macrospin model involving the Dzyaloshinskii–Moriya type interaction, which explains the existence of a nonzero magnetic moment in the absence of external field and reproduces well hysteretic behavior of the AHE. Using analysis of Néel-vector-dependent Berry curvature, we show that the intrinsic AHE in hexagonal MnTe can be nonzero even when the magnetization vanishes and, also, that it changes sign depending on the Fermi energy position. Published by the American Physical Society 2024

Multiferroic properties and the layer-stacking quantum anomalous Hall effect in two-dimensional RuOHX (X = F, Cl, Br)

Exploring the physics coupled with layer degrees of freedom in materials has become a hot topic in quantum layertronics. We propose a robust second-order topological insulator monolayer RuOHX (X = F, Cl, and Br), a two-dimensional ferromagnetic semiconductor with large valley polarization, capable of undergoing topological phase transition induced by strain effect. In the bilayer RuOHX, we achieve layer-polarized anomalous Hall effect through interlayer sliding, originating from layer-stacking Berry curvature. Moreover, it can be controlled and reversed by the direction of ferroelectric polarization. Under appropriate biaxial strain, the bilayer RuOHX exhibits quantum layer spin Hall effect in which the helical edge states are manifested as spin-chirality-locking, due to the degeneracy of layer-polarized quantum anomalous Hall effect. Our work explores the potential application via layer-stacking topological properties for future quantum device applications.

Circular photocurrents in centrosymmetric semiconductors with hidden spin polarization

Centrosymmetric materials with site inversion asymmetries possess hidden spin polarization, which remains challenging to be converted into spin currents because the global inversion symmetry is still conserved. This study demonstrates the spin-polarized circular photocurrents in centrosymmetric transition metal dichalcogenide semiconductors at normal incidence without applying electric bias. The global inversion symmetry is broken by using a spatially-varying circularly polarized light beam, which could generate spin gradient owing to the hidden spin polarization. The dependence of the circular photocurrents on electrode configuration, illumination position, and beam spot size indicates an emergence of circulating electric current under spatially inhomogeneous light, which is associated with the deflection of spin-polarized current through the inverse spin Hall effect. The circular photocurrents is subsequently utilized to probe the spin polarization and the inverse spin Hall effect under different excitation wavelengths and temperatures. The results of this study demonstrate the feasibility of using centrosymmetric materials with hidden spin polarization and non-vanishing Berry curvature for spintronic device applications.

Non-Abelian Fractionalization in Topological Minibands.

Motivated by the recent discovery of fractional quantum anomalous Hall states in moiré systems, we consider the possibility of realizing non-Abelian phases in topological minibands. We study a family of moiré systems, skyrmion Chern band models, which can be realized in two-dimensional semiconductor-magnet heterostructures and also capture the essence of twisted transition metal dichalcogenide homobilayers. We show using many-body exact diagonalization that, in spite of strong Berry curvature variations in momentum space, the non-Abelian Moore-Read state can be realized at half filling of the second miniband. These results demonstrate the feasibility of non-Abelian fractionalization in moiré systems without Landau levels and shed light on the desirable conditions for their realization. In particular, we highlight the prospect of realizing the Moore-Read state in twisted semiconductor bilayers.

Dirac fermions and spin transport in the SrVO3/SrTiO3 quantum well

The type-II Dirac fermions are characterized by a tilted Dirac cone and sustain exotic quantum transport phenomena. Here, we show the emergence of a type-II Dirac nodal loop in the SrVO3/SrTiO3 quantum well structure by density functional theory calculations and symmetry arguments. When the spin–orbit coupling (SOC) is neglected, the unavoidable crossing between two t2g bands gives rise to a nodal loop in the Brillouin zone, which is protected by the mirror symmetry. Such a nodal loop is fully gapped by including SOC, which results in a large spin Hall effect due to large spin Berry curvatures. Our findings add the unexplored functionality to oxide quantum wells and offer a practical platform to explore the interplay between topological fermions and spin transport phenomena.

Ferroelectric Semimetals with α-Bi/SnSe van der Waals Heterostructures and Their Topological Currents.

We show that proximity effects can be utilized to engineer van der Waals heterostructures (vdWHs) displaying semimetallic spin-ferroelectricity locking, where ferroelectricity and semimetallic spin states are confined to different layers, but are correlated by means of proximity effects. Our findings are supported by first principles calculations involving α-Bi/SnSe bilayers. We show that such systems support ferroelectrically switchable nonlinear anomalous Hall effect originating from large Berry curvature dipoles as well as direct and inverse spin Hall effects with giant bulk spin-charge interconversion efficiencies. The giant efficiencies are consequences of the proximity-induced semimetallic nature of low energy electron states, which are shown to behave as two-dimensional pseudo-Weyl fermions by means of symmetry analysis and first principles calculations as well as direct angle-resolved photoemission spectroscopy measurements.

Theory of Classical Electrodynamics with Topologically Quantized Singularities as Electric Charges

AbstractA theory of classical electrodynamics, where the only admissible electric charges are topological singularities in the electromagnetic field, is formulated. Charge quantization is accounted by the Chern theorem, such that Dirac magnetic monopoles are not needed. The theory allows positive and negative charges of equal magnitude, where the sign of the charge corresponds to the chirality of the topological singularity. Given the trajectory of the singularity, one can calculate electric and magnetic fields identical to those produced by Maxwell's equations for a moving point charge, apart from a multiplicative constant factor related to electron charge and vacuum permittivity. The theory is based on the relativistic Weyl equation in frequency‐wavevector space, with eigenstates comprising the position, velocity, and acceleration of the singularity, and eigenvalues defining the retarded position of the charge. From the eigenstates, one calculates the Berry connection and the Berry curvatures, and identifies the curvatures as electric and magnetic fields.

Quantum geometry in condensed matter

Abstract One of the most celebrated accomplishments of modern physics is the description of fundamental principles of nature in the language of geometry. As the motion of celestial bodies is governed by the geometry of spacetime, the motion of electrons in condensed matter can be characterized by the geometry of the Hilbert space of their wave functions. Such quantum geometry, comprising of Berry curvature and quantum metric, can thus exert profound influences on various properties of materials. The dipoles of both Berry curvature and quantum metric produce nonlinear transport. The quantum metric plays an important role in flat-band superconductors by enhancing the transition temperature. The uniformly distributed momentum-space quantum geometry stabilizes the fractional Chern insulators and results in the fractional quantum anomalous Hall effect. We here review in detail quantum geometry in condensed matter, paying close attention to its effects on nonlinear transport, superconductivity, and topological properties. Possible future research directions in this field are also envisaged.

Designer spin-orbit superlattices: symmetry-protected Dirac cones and spin Berry curvature in two-dimensional van der Waals metamaterials

The emergence of strong relativistic spin-orbit effects in low-dimensional systems provides a rich opportunity for exploring unconventional states of matter. Here, we present a route to realise tunable relativistic band structures based on the lateral patterning of proximity-induced spin-orbit coupling. The concept is illustrated on a patterned graphene–transition metal dichalcogenide heterostructure, where the spatially periodic spin-orbit coupling induces a rich mini-band structure featuring massless and massive Dirac bands carrying large spin Berry curvature. The envisaged systems support robust and gate-tunable spin Hall responses driven by the quantum geometry of mini-bands, which can be tailored through metasurface fabrication methods and twisting effects. These findings open pathways to two-dimensional quantum material design and low-power spintronic applications.

Janus monolayers Fe2SSeX2(X = Ga, In, and Tl): Robust nontrivial topology with high Chern-number

Abstract High-performance quantum anomalous Hall (QAH) systems are the crucial materials to explore emerging quantum physics and magnetic topological phenomena. Inspired by layered FeSe materials with excellent superconducting properties, the Janus monolayers Fe2SSeX2 (X = Ga, In, and Tl) are built by the decoration of Ga, In, and Tl atoms in monolayer Fe2SSe. Using the first-principles calculations, the Fe2SSeX2 have stable structures and prefer ferromagnetic (FM) orders, which belong to the Weyl semimetal without spin-orbit coupling (SOC). For the out-of-plane (OOP) magnetic anisotropy, the large nontrivial gaps are opened, and the Fe2SSeX2 are predicted to be large-gap QAH insulators with high Chern-number C = 2, proved by two chiral edge states and Berry curvatures. When the magnetization is flipped, two chiral edge states can be simultaneously changed and C = -2 can be obtained, revealing one fascinating behavior of chiral spin-edge-state locking. It is found that the QAH properties of the Fe2SSeX2 are robust against the strain. Especially, the nontrivial topological quantum states can spontaneously appear for the Fe2SSeGa2 and Fe2SSeIn2, because the easy magnetic axis orientations are adjusted from in-plane to OOP by the biaxial strain. Our studies provide excellent candidate systems to realize the QAH properties with high Chern-number, and advocate more experimental explorations in the combination of superconductivity and topology.

Electric, thermal, and thermoelectric magnetoconductivity for Weyl/multi-Weyl semimetals in planar Hall set-ups induced by the combined effects of topology and strain

We continue our investigation of the response tensors in planar Hall (or planar thermal Hall) configurations where a three-dimensional Weyl/multi-Weyl semimetal is subjected to the combined influence of an electric field E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{E}}$$\\end{document} (and/or temperature gradient ∇rT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla _{{\ extbf{r}} } T$$\\end{document}) and an effective magnetic field Bχ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{B}}_\\chi$$\\end{document}, generalizing the considerations of Phys. Rev. B 108 (2023) 155132 and Physica E 159 (2024) 115914. The electromagnetic fields are oriented at a generic angle with respect to each other, thus leading to the possibility of having collinear components, which do not arise in a Hall set-up. The net effective magnetic field Bχ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{B}}_\\chi$$\\end{document} consists of two parts—(a) an actual/physical magnetic field B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{B}}$$\\end{document} applied externally; and (b) an emergent magnetic field B5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{B}}_5$$\\end{document} which quantifies the elastic deformations of the sample. B5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{B}}_5$$\\end{document} is an axial pseudomagnetic field because it couples to conjugate nodal points with opposite chiralities with opposite signs. Using a semiclassical Boltzmann formalism, we derive the generic expressions for the response tensors, including the effects of the Berry curvature (BC) and the orbital magnetic moment (OMM), which arise due to a nontrivial topology of the bandstructures. We elucidate the interplay of the BC-only and the OMM-dependent parts in the longitudinal and transverse (or Hall) components of the electric, thermal, and thermoelectric response tensors. Especially, for the co-planar transverse components of the response tensors, the OMM part acts exclusively in opposition (sync) with the BC-only part for the Weyl (multi-Weyl) semimetals.

Three-dimensional valley-contrasting sound.

Spin and valley are two fundamental properties of electrons in crystals. The similarity between them is well understood in valley-contrasting physics established decades ago in two-dimensional (2D) materials like graphene-with broken inversion symmetry, the two valleys in graphene exhibit opposite orbital magnetic moments, similar to the spin-1/2 behaviors of electrons, and opposite Berry curvature that leads to a half topological charge. However, valley-contrasting physics has never been explored in 3D crystals. Here, we develop a 3D acoustic crystal exhibiting 3D valley-contrasting physics. Unlike spin that is fundamentally binary, valley in 3D can take six different values, each carrying a vortex in a distinct direction. The topological valley transport is generalized from the edge states of 2D materials to the surface states of 3D materials, with interesting features including robust propagation, topological refraction, and valley-cavity localization. Our results open a new route for wave manipulation in 3D space.

Prediction of Intriguing Valley Properties in Two-Dimensional Hf2TeIX (X = I, Br) Monolayers

The valley degree of freedom, as a new information carrier, is important for basic physical research and the development of advanced devices. Herein, using first-principle calculations, we predict that two-dimensional Hf2TeIX (X = I, Br) monolayers harbor intriguing valley properties. Without considering spin–orbit coupling (SOC), the Hf2TeI2 monolayer has a semi-metallic nature, with Dirac cones located at the high-symmetry point K, and feature, with considerable Fermi velocity. When the SOC is taken into account, a band gap opening of 271 meV can be observed at the Dirac cones. More interestingly, the Hf2TeIBr monolayer exhibits intrinsic spatial inversion symmetry breaking, which leads to the emergence of valley-contrasting physics under SOC. This is demonstrated by the presence of spin–valley splitting and opposite Berry curvature at adjacent K points. Besides, the spin–valley splitting, the band gap and magnitude of the Berry curvature of the Hf2TeIBr monolayer can be effectively tuned by strain engineering. These findings contribute significantly to the design of valleytronic devices and extend opportunities for exploring two-dimensional valley materials.

Quantum geometry quadrupole-induced third-order nonlinear transport in antiferromagnetic topological insulator MnBi2Te4

The study of quantum geometry effects in materials has been one of the most important research directions in recent decades. The quantum geometry of a material is characterized by the quantum geometric tensor of the Bloch states. The imaginary part of the quantum geometry tensor gives rise to the Berry curvature while the real part gives rise to the quantum metric. While Berry curvature has been well studied in the past decades, the experimental investigation on the quantum metric effects is only at its infancy stage. In this work, we measure the nonlinear transport of bulk MnBi2Te4, which is a topological anti-ferromagnet. We found that the second order nonlinear responses are negligible as required by inversion symmetry, the third-order nonlinear responses are finite. The measured third-harmonic longitudinal (Vxx3ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${V}_{{xx}}^{3\\omega }$$\\end{document}) and transverse (Vxy3ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${V}_{{xy}}^{3\\omega }$$\\end{document}) voltages with frequency 3ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega$$\\end{document}, driven by an a.c. current with frequency ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega$$\\end{document}, show an intimate connection with magnetic transitions of MnBi2Te4 flakes. Their magnitudes change abruptly as MnBi2Te4 flakes go through magnetic transitions from an antiferromagnetic state to a canted antiferromagnetic state and to a ferromagnetic state. In addition, the measured Vxx3ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${V}_{{xx}}^{3\\omega }$$\\end{document} is an even function of the applied magnetic field B while Vxy3ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${V}_{{xy}}^{3\\omega }$$\\end{document} is odd in B. Amazingly, the field dependence of the third-order responses as a function of the magnetic field suggests that Vxx3ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${V}_{{xx}}^{3\\omega }$$\\end{document} is induced by the quantum metric quadrupole and Vxy3ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${V}_{{xy}}^{3\\omega }$$\\end{document} is induced by the Berry curvature quadrupole. Therefore, the quadrupoles of both the real and the imaginary part of the quantum geometry tensor of bulk MnBi2Te4 are revealed through the third order nonlinear transport measurements. This work greatly advanced our understanding on the connections between the higher order moments of quantum geometry and nonlinear transport.

Gate Modulation of Dissipationless Nonlinear Quantum Geometric Current in 2D Te.

Two-dimensional trigonal tellurium (2D Te), a narrow-bandgap semiconductor with a bandgap of approximately 0.3 eV, hosts Weyl points near the band edge and exhibits a narrow, strong Berry curvature dipole (BCD). By applying a back-gate bias to align the Fermi level with the BCD, a sharp increase in the dissipationless transverse nonlinear Hall response is observed in 2D Te. Gate modulation of the BCD demonstrates an on/off ratio of 104 and a responsivity of nearly 106 V/W, while the longitudinal current induced by band modulation reaches an on/off ratio of about 10. This current is sustained up to 200 K, exhibiting a change of 3 orders of magnitude. The inclusion of both transistor action and rectification enhances the temperature sensitivity of the dissipationless Hall current, offering potential applications in electrothermal detectors and sensors and highlighting the significance of topological properties in advancing electronic applications.