Berry curvature and semimetals

We review recent theoretical progress in the understanding and prediction of novel topological semimetals. Topological semimetals define a class of gapless electronic phases exhibiting topologically stable crossings of energy bands. Different types of topological semimetals can be distinguished on the basis of the degeneracy of the band crossings, their codimension (e.g., point or line nodes), and the crystal space group symmetries on which the protection of stable band crossings relies. The dispersion near the band crossing is a further discriminating characteristic. These properties give rise to a wide range of distinct semimetal phases such as Dirac or Weyl semimetals, point or line node semimetals, and type I or type II semimetals. In this review we give a general description of various families of topological semimetals, with an emphasis on proposed material realizations from first-principles calculations. The conceptual framework for studying topological gapless electronic phases is reviewed, with a particular focus on the symmetry requirements of energy band crossings, and the relation between the different families of topological semimetals is elucidated. In addition to the paradigmatic Dirac and Weyl semimetals, we pay particular attention to more recent examples of topological semimetals, which include nodal line semimetals, multifold fermion semimetals, and triple-point semimetals. Less emphasis is placed on their surface state properties, their responses to external probes, and recent experimental developments.

Weyl semimetals have attracted considerable research interest over the past decade, with a number of intriguing transport phenomena reported. Magnetic Weyl semimetals, which break time reversal symmetry, have been predicted and recently discovered. Co3Sn2S2 is a magnetic Weyl semimetal that exhibit a giant anomalous Hall effect (AHE) when the magnetic moments are aligned along the c axis. In this paper, we report the evolution of the AHE with an external magnetic field applied in the ab plane and current along the c axis, namely, the out-of-plane AHE of Co3Sn2S2. Density functional theory calculations predict a finite out-of-plane AHE when the spins are fully aligned in the ab plane. The evolution of the magnetic structure modifies the nodal line distribution and the Berry curvature, resulting in a weaker AHE with an amplitude of 200Scm−1 at the saturation field. To ensure the alignment of the magnetic field in the ab plane, a two-round scanning process was performed experimentally. After this, the out-of-plane AHE was measured at multiple temperatures. The observed AHE signals with applied fields of 1 and 2 T were in good agreement with theoretical predictions. These results suggest that engineering the anomalous Hall effect may be possible by designing the symmetry relation of the local Berry curvature. Published by the American Physical Society 2025

Ferromagnet (FM)\ifmmode\text{\textcelsius}\else\textcelsius\fi{}heavy-metal (HM) nanostructures can be used for magnetic state readout in the proposed magnetoelectric spin-orbit logic by locally injecting a spin-polarized current and measuring the spin-to-charge conversion via the spin Hall effect. However, this local configuration is prone to spurious signals. Here, we address spurious Hall effects that can contaminate the spin Hall signal in these FM/HM T-shaped nanostructures. The most pronounced Hall effects in our ${\mathrm{Co}}_{50}{\mathrm{Fe}}_{50}/\mathrm{Pt}$ nanostructures are the planar Hall effect and the anomalous Hall effect generated in the ferromagnetic nanowire. We find that the planar Hall effect, induced by misalignment between magnetization and current direction in the ferromagnetic wire, is manifested as a shift in the measured baseline resistance, but does not alter the spin Hall signal. In contrast, the anomalous Hall effect, arising from the charge-current distribution within the FM, adds to the spin Hall signal. However, the effect can be made insignificant by minimizing the shunting effect via proper design of the device. We conclude that local spin injection in FM/HM nanostructures is a suitable tool for measuring spin Hall signals and, therefore, a valid method for magnetic state readout in prospective spin-based logic.

The anomalous transverse transport properties, including the anomalous Hall and Nernst effects, are crucial for probing the topological band features in magnetic materials. In this study, we present a comprehensive investigation of the magnetic, electrical, and thermal transport, as well as the electronic band structure, of the noncollinear ferromagnet ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$. Our findings reveal that ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$ exhibits prominent anomalous Hall and Nernst effects. Scaling analysis suggests that the anomalous Hall effect is predominantly influenced by the intrinsic Berry curvature contribution. Additionally, the large anomalous Nernst signal exceeds the magnetization scaling relations observed in conventional ferromagnets, and the calculated anomalous Nernst conductivity exhibits a $T\text{ln}T$ relation. These observations align with the behavior seen in other magnetic topological materials, further suggesting a substantial intrinsic Berry curvature effect. Complemented by detailed theoretical calculations, we clarify the electronic band properties and confirm a substantial net Berry curvature near the Fermi level in this compound. Our work elucidates that the large anomalous Hall and Nernst effects of ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$ are dominated by an intrinsic mechanism and presents it as a probable magnetic topological material with a significant Berry phase effect in the momentum space.

Weyl semimetals exhibit unusual surface states and anomalous transport phenomena. It is hard to manipulate the band structure topology of specific Weyl materials. Topological transport phenomena usually appear at very low temperatures, which sets challenges for applications. In this work, we demonstrate the band topology modification via a weak magnetic field in a ferromagnetic Weyl semimetal candidate, Co2MnAl, at room temperature. We observe a tunable, giant anomalous Hall effect (AHE) induced by the transition involving Weyl points and nodal rings. The AHE conductivity is as large as that of a 3D quantum AHE, with the Hall angle (ΘH) reaching a record value (tan {Theta }^{H}=0.21) at the room temperature among magnetic conductors. Furthermore, we propose a material recipe to generate large AHE by gaping nodal rings without requiring Weyl points. Our work reveals an intrinsically magnetic platform to explore the interplay between magnetic dynamics and topological physics for developing spintronic devices.

We study the low-frequency properties of the bulk photovoltaic effect in\ntopological semimetals. The bulk photovoltaic effect is a nonlinear optical\neffect that generates DC photocurrents under uniform irradiation, allowed by\nnoncentrosymmetry. It is a promising mechanism for a terahertz photodetection\nbased on topological semimetals. Here, we systematically investigate the\nlow-frequency behavior of the second-order optical conductivity in point-node\nsemimetals. Through symmetry and power-counting analysis, we show that Dirac\nand Weyl points with tilted cones show the leading low-frequency divergence. In\nparticular, we find new divergent behaviors of the conductivity of Dirac and\nWeyl points under circularly polarized light, where the conductivity scales as\n$\\omega^{-2}$ and $\\omega^{-1}$ near the gap-closing point in two and three\ndimensions, respectively. We provide a further perspective on the low-frequency\nbulk photovoltaic effect by revealing the complete quantum geometric meaning of\nthe second-order optical conductivity tensor. The bulk photovoltaic effect has\ntwo origins, which are the transition of electron position and the transition\nof electron velocity during the optical excitation, and the resulting\nphotocurrents are respectively called the shift current and the injection\ncurrent. Based on an analysis of two-band models, we show that the injection\ncurrent is controlled by the quantum metric and Berry curvature, whereas the\nshift current is governed by the Christoffel symbols near the gap-closing\npoints in semimetals. Finally, for further demonstrations of our theory beyond\nsimple two-band models, we perform first-principles calculations on magnetic\nDirac semimetal MnGeO$_3$and Weyl semimetal PrGeAl. Our work brings out new\ninsights into the structure of nonlinear optical responses as well as for the\ndesign of semimetal-based terahertz photodetectors.\n

Symmetry protected Dirac semimetals can be transformed into Weyl semimetals by breaking the protecting symmetry, leading to many exotic quantum phenomena such as chiral anomaly and anomalous Hall effect. Here we show that, due to the large Zeeman g factor and small band width along b-axis in Dirac semimetal ZrTe5, a magnetic field of about 8 T along b-axis direction may annihilate the Weyl points and open up a two-dimensional (2D) Dirac mass gap, when the Zeeman splitting exceeds the band width along b-axis. This is manifested by a sharp drop of magnetoresistance (MR) above 8 T, which is probably due to additional carriers induced by the orbital splitting of the zeroth Landau level associated with the 2D Dirac point, which is descendant of the original Weyl points. Further evidence of the additional carriers is provided by the Hall effect and different anisotropic magnetoresistance (AMR) in low and high field regions. Our experiment reveals a probable topological quantum phase transition of field induced Weyl points annihilation in Dirac semimetal ZrTe5 and gives an alternative explanation for the drop of MR at high field.

Chiral edge states that propagate oppositely at two parallel strip edges are a hallmark feature of Chern insulators which were first proposed in the celebrated two-dimensional (2D) Haldane model. Subsequently, counterintuitive antichiral edge states that propagate in the same direction at two parallel strip edges were discovered in a 2D modified Haldane model. Recently, chiral surface states, the 2D extension of one-dimensional (1D) chiral edge states, have also been observed in a photonic analogue of a 3D Haldane model. However, despite many recent advances in antichiral edge states and chiral surface states, antichiral surface states, the 2D extension of 1D antichiral edge states, have never been realized in any physical system. Here, we report the experimental observation of antichiral surface states by constructing a 3D modified Haldane model in a magnetic Weyl photonic crystal with two pairs of frequency-shifted Weyl points (WPs). The 3D magnetic Weyl photonic crystal consists of gyromagnetic cylinders with opposite magnetization in different triangular sublattices of a 3D honeycomb lattice. Using microwave field-mapping measurements, unique properties of antichiral surface states have been observed directly, including the antichiral robust propagation, tilted surface dispersion, a single open Fermi arc connecting two projected WPs and a single Fermi loop winding around the surface Brillouin zone (BZ). These results extend the scope of antichiral topological states and enrich the family of magnetic Weyl semimetals.

Rare-earth hexaborides offer a rich tapestry for exploring the interplay of topological orders, magnetism, and electron-correlation effects associated with protected quantum phenomena. Using first-principles modeling, we identify in cerium hexaboride (CeB<sub>6</sub>) a topological state that harbors a pseudonodal surface (PNS) along with double-Weyl fermions. An analysis of the electronic states in the vicinity of the Fermi level reveals the presence of M<sub>z</sub>-mirror-symmetry-protected band crossings that involve itinerant (d) and localized (f) states of Ce. These band crossings are found to remain nearly gapless as one goes away from the mirror plane, driving the formation of PNS states. We also demonstrate the presence of coexisting double-Weyl fermions with chiral charges of ±2 on the C<sub>4z</sub> rotational axis. By analyzing the Berry curvature field associated with the topological states, we predict that CeB<sub>6</sub> supports a giant anomalous Hall conductivity of -1840Ω<sup>-1</sup> cm<sup>-1</sup> in the slightly electron-doped regime, which would be larger than the values reported to date in magnetic Weyl semimetals. Our study thus indicates that CeB<sub>6</sub>, which combines ferromagnetic order with topological physics in a heavy-fermion matrix, would provide a promising playground for investigating topological magnetic states with giant anomalous Hall responses.

A model based on the Dirac equation allows one to describe topological insulators. In this paper we extend this model by adding a Zeeman term to introduce magnetism. By this means we show that it can be used to describe the electronic properties of ferromagnetic Weyl and nodal line semimetals, which arise for distinct parameters of the model. We confirm the topological nontriviality of the nodal objects by calculating the topological invariants, as well as by demonstrating the existence of topological surface states in the spectral function of the semi-infinite systems. Furthermore, we calculate the anomalous and spin Hall conductivities for various model parameters, which allows us to identify typical signatures of Weyl points and nodal lines in electronic transport.

In this letter, we study the origin of the anomalous Hall effect (AHE) in ferrimagnetic insulator ${\mathrm{Tm}}_{3}{\mathrm{Fe}}_{5}{\mathrm{O}}_{12}$ (TmIG)/Pt heterostructures. A monotonic decrease of the anomalous Hall resistivity ($\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathrm{AHE}}$) with decreasing temperature is observed for TmIG/Pt, and a sign reversal of $\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathrm{AHE}}$ occurs at around 80 K. With the addition of a Cu interlayer, the $\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathrm{AHE}}$ similarly decreases as a function of temperature, but maintains the same sign across the full temperature range. This indicates that both the magnetic-proximity effect and spin Hall effect in the TmIG/Pt bilayer contribute to the AHE signal with opposing signs. The spin-Hall contribution to the AHE is dominant at room temperature but decreases with decreasing temperature. Meanwhile, the magnetic-proximity contribution to the AHE becomes dominant with decreasing temperatures, leading to a change of sign for $\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathrm{AHE}}$. We exclude a dominant influence of a ferrimagnetic compensation point in the temperature region by complementary magnetic hysteresis and neutron diffraction measurements. Our work, based on a simple method, sheds light on the origin of the AHE in magnetic insulator heterostructures, where the competition between the magnetic-proximity effect and spin Hall effect governs the sign and amplitude of the AHE.

Generally, the dissipationless Hall effect in solids requires time-reversal symmetry breaking (TRSB), where TRSB induced by external magnetic field results in the ordinary Hall effect, while TRSB caused by spontaneous magnetization gives rise to the anomalous Hall effect (AHE) which scales with the net magnetization. The AHE is therefore not expected in antiferromagnets with vanishing small magnetization. However, large AHE was recently observed in certain antiferromagnets with noncollinear spin structure and nonvanishing Berry curvature. Here, we report another origin of AHE in a layered antiferromagnet EuAl_{2}Si_{2}, namely, the domain wall (DW) skew scattering with Weyl points near the Fermi level, in experiments for the first time. Interestingly, the DWs form a unique periodic stripe structure with controllable periodicity by external magnetic field, which decreases nearly monotonically from 975nm at 0T to 232nm at 4T. Electrons incident on DW with topological bound states experience strong asymmetric scattering, leading to a giant AHE, with the DW Hall conductivity (DWHC) at 2K and 1.2T reaching a record value of ∼1.51×10^{4} Scm^{-1} among bulk systems and being 2 orders of magnitude larger than the intrinsic anomalous Hall conductivity. The observation not only sets a new paradigm for exploration of large anomalous Hall effect, but also provides potential applications in spintronic devices.

We examine a possible spin Hall effect for localized spin systems with no charge degrees of freedom. In this scenario, a longitudinal magnetic field gradient induces a transverse spin current carried by spin wave excitations with an anomalous velocity which is caused by a topological Berry-phase effect associated with spin chirality, in analogy with anomalous Hall effects in itinerant electron systems. Our argument is based on semiclassical equations of motion applicable to general spin systems. Also, a microscopic model of frustrated magnets which exhibits the anomalous spin Hall effect is presented.

Kondo-lattice systems are recognized for potentially hosting a variety of rich topological phases. Several pioneering studies have demonstrated significant anomalous Hall and anomalous Nernst effects in these systems, attributed to the Berry curvature of the hybridization bands. In this paper, we investigate UBiTe, a ferromagnetic Kondo-lattice system. Our findings reveal that the intrinsic contribution to the anomalous Hall conductivity is closely tied to the Kondo coherence temperature. Moreover, we demonstrate that slight shifts in the Fermi level across three different samples significantly influence this intrinsic contribution, aligning with the Berry curvature localized within the narrow hybridization bands. This provides a stark contrast to the less pronounced sensitivity observed in weakly correlated Weyl semimetals, underscoring the distinctive electronic properties of Kondo-lattice systems. The anomalous Hall conductivity of one sample ranks among the highest reported for topological magnetic materials.

The intrinsic anomalous Hall effect in ferromagnetic materials depends on the spin–orbit-coupling-induced effect in the electronic structure. The presence of anomalous Hall effect can be realized from the existence of the finite Berry curvature. Since the Heusler material Pt2MnGa is ferromagnetic in its collinear state and no report on its transport properties has yet been made, the anomalous Hall effect in this system is investigated using a combined approach of the density functional theory and maximally localized Wannier function. Analysis of the Berry curvature reveals positive and negative peaks along the high symmetry k-points. These peaks together lead to a significant negative anomalous Hall conductivity, which makes the material a promising candidate for applications in spintronics.