Altermagnetism and Altermagnets: A Brief Review

The Hall effect is one of the best known effects in (solid-state) physics. Conventionally, this phenomenon describes the occurrence of charge currents that are perpendicular to an externally applied electric field due to a time-reversal symmetry breaking magnetic field. Besides, in ferromagnetic systems, the net magnetization can break time-reversal symmetry even in the absence of a magnetic field which allows the so-called anomalous Hall effect. This effect originates from extrinsic and intrinsic contributions that are both related to the existence of spin-orbit coupling [1]. Moreover, another contribution to the Hall effect, which is known as the topological Hall effect, can exist even if spin-orbit coupling is negligible. It may occur in certain noncollinear noncoplanar magnetic textures with a nonzero scalar spin chirality like skyrmions [2, 3].However, recent works [4, 5] reported the occurrence of an anomalous Hall effect in several compensated kagome magnets (cf. Fig. 1). These materials are coplanar antiferromagnets with vanishing net magnetization, and still, a group theoretical analysis allows the existence of the effect. The large conductivities obtained via first-principle calculations have been confirmed in experiments for Mn3Sn [6] and Mn3Ge [7]. However, a straightforward microscopic picture for this phenomenon was still missing.In this talk, we present an explanation on a microscopic level based on tight-binding calculations and analytical considerations [8]. For coplanar kagome magnets, we show the equivalence of spin-orbit coupling and an out-of-plane tilting of the magnetic moments. The existence of spin-orbit interaction does not only break a combined time-reversal and mirror symmetry of the Hamiltonian but can be transformed to a magnetic texture that is virtually canted, whereas, the original texture remains coplanar [cf. Fig. 2(a)]. Consequently, the ‘new’ anomalous Hall effect can be interpreted as a combination of an effective anomalous and topological Hall effect due to the net magnetic moment and the net scalar spin chirality of this virtual magnetic texture, respectively.Furthermore, as we demonstrate, a noncoplanar kagome magnet with spin-orbit coupling is able to behave like a system that is virtually coplanar and with compensated spin-orbit coupling [cf. Fig. 2(b)]. In this case, the combination of mirror and time-reversal symmetry of the Hamiltonian that was broken before has been restored. A critical out-of-plane tilting angle of the real texture can be found, where the virtual texture is coplanar and the Hall effect is absent for all energies. As we show in detail, the electronic properties are determined by this virtual texture that is hidden in the Hamiltonian.In consequent investigations, the calculations have been repeated for other transport quantities like the spin Hall effect where charge currents are converted into spin currents. These results can again be related to the virtual spin texture which has, however, different consequences for the spin Hall effect. Besides, in order to simulate the experimental situation, the investigated model was extended from a two-dimensional kagome lattice, as considered here, to a more realistic model including d-orbitals and kagome planes that are stacked along the out-of-plane direction. **

The interplay between symmetry breaking and topological electronic structure is crucial to design anomalous transport properties in materials. Materials with strong or quantum electromagnetic responses have an extensive impact on the development of data storage, information processing, energy conversion, etc. In magnetic materials, the anomalous transport of anomalous Hall effect, anomalous Nernst effect, and magneto-optical effect et al. can be understood from the Berry curvature of the electronic band structures. Two typical band structures of Weyl points and nodal line band structures host strong local Berry curvature. Since the Berry curvature is time-reversal symmetry odd, such strong Berry curvatures can lead to strongly enhanced anomalous transport signals. With this guiding principle, we studied the anomalous Hall effect in magnetic Weyl semimetal Co3Sn2S2 [1-3] and magnetic nodal line semimetal in Heusler compound Co2Mn(Ga/Al) [4].With a mirror symmetry, the inverted band structure forms a nodal loop in the absence of spin-orbital coupling. This nodal line can be broken by spin-orbital coupling and a bandgap opens, which generates non-zero Berry curvature in the bandgap and forms a hot loop, see Figure 1a-b. Such strong Berry curvature in the magnetic system can lead to a strongly enhance or even quantized anomalous Hall effect. We applied this idea to real materials of magnetic Heusler compounds Co2Mn(Ga/Al). Protected by mirror symmetries the band inversion between the bands with opposite mirror eigenvalue forms three gapless nodal lines in the kx=0, ky=0, and kz=0 mirror planes, respectively. With spin-orbital coupling, the symmetry of the system is reduced. Taking magnetic along z, the mirror symmetries in kx=0 and ky=0 planes are broken, which leads to band anti-crossings with strong local Berry curvature locating in the opened bandgap around original nodal lines, see Figure 1d. Integral of the Berry curvature in the whole k-space gives a large intrinsic anomalous Hall conductivity reaching ~1500 to ~2000 S/cm [4].Weyl points is another typical band structure and present as the Berry curvature monopole, and therefore naturally results in a strong anomalous Hall effect. In ideal models with only one pair of Weyl points locating at the Fermi level, the intrinsic anomalous Hall conductivity can be presented as the combination distance of Weyl points and the quantized anomalous Hall conductance. Inspired by these excellent relations, we studied the anomalous Hall effect in Co3Sn2S2, and a new record of three-dimensional anomalous Hall angel (~20%) was observed, which offers the 1st three-dimensional material with both strong anomalous Hall conductivity and anomalous Hall angle [1]. It indeed shows as a Weyl semimetal from electronic band structure analysis. One crucial symmetry in Co3Sn2S2 is the three mirror planes parallel to the c direction, which results in three pairs of nodal lines connected by a c3z rotation symmetry. Because the magnetization is aligned along the z-direction, the mirror symmetries are broken by spin-orbital coupling. Meanwhile, one pair of Weyl points with opposite chirality remains along each of the former nodal lines, leading to the so large anomalous Hall effect.Though the strong anomalous Hall effect provides a promising signature for the existence of Weyl and nodal line band structure. Our transport work about Co3Sn2S2 and Co2MnGa/Al inspired the direct band structure detection by ARPES and STM [5-7], and they are in turn became the 1st experimentally verified magnetic Weyl semimetal and nodal line semimetal, respectively.Applying temperature gradient instead of the electrical field, the Weyl points and nodal lines induced Berry curvature can also lead to strongly enhanced anomalous Nernst effect. From our theoretical calculations and experimental measurements, the anomalous Nernst conductivity can reach around 3 and 6 A/(m-K) in Co3Sn2S2 [8] and Co2MnGa [9], respectively, with Co2MnGa keeping the record. Owing to the large anisotropy, Co3Sn2S2 is, so far, the only material with a large anomalous Nernst effect with zero magnetic fields. In addition, very recently, a giant magneto-optical response was observed in Co3Sn2S2 with the applied field from polarized light [10].Very recently, a strong interest in antiferromagnets is rising. In an antiferromagnet without such kind of joint TO symmetry to reverse Berry curvature, it allows the existence of anomalous Hall effect, anomalous Nernst effect, magneto-optical responses, and special spin current, etc. The nonzero anomalous Hall effect in antiferromagnets was proposed as early as 2001 in distorted non-linear magnetic structures [11]. However, its experimental realization was not successful until 2015 [12-16]. This understanding can be further expanded into collinear antiferromagnets. Different from non-linear antiferromagnets, the collinear antiferromagnetic structure can be usually understood from two sublattices connected by translation of inversion operation. Therefore, there are mainly two ways to break the joint symmetry, to replace the magnetic atoms connected by the joint TO symmetry, or change the the nonmagnetic sites. With this understanding, we predicted the anomalous Hall and Nernst effect in anti-Heusler Weyl semimetal Ti2MnAl [17-18]. **

A Busy March Meeting Is Brewing in Seattle

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.

A phase transition is associated with a change of certain symmetry. This symmetry change is captured by an order parameter which is zero above the transition temperature (or pressure) and non-zero below it. An important class of functional materials is that of ferroics which are characterized by two or more orientation states with the ability to switch between them via an applied field. In terms of broken spatial inversion and time reversal symmetry, there are four types of primary ferroics: ferroelectrics described by polarization (a polar vector with broken spatial inversion symmetry), ferromagnets described by magnetization (an axial vector with broken time reversal symmetry), ferrotoroidics described by torodization (an axio-polar vector with both spatial inversion and time reversal symmetries broken) and ferroelastics described by strain ε (a symmetric second rank polar tensor with neither spatial inversion nor time reversal symmetry broken but with broken rotational symmetry). Materials possessing two or more ferroic properties are called multiferroics. In particular, crystals exhibiting simultaneous ferroelectricity and magnetism are called magnetoelectrics. We consider the effect of disorder in these crystals which above the transition temperature may result in a tweed structure whereas it may lead to a glassy state below the transition. We also explore the properties of ferroics at nanoscale which are dominated by the surface/interface energy contribution. Finally, we describe in detail the magnetic symmetry of low-dimensional multiferroic materials and study a representative phase transition.

Though Weyl fermions have recently been observed in several materials with broken inversion symmetry, there are very few examples of such systems with broken time reversal symmetry. Various Co$_{2}$-based half-metallic ferromagnetic Heusler compounds are lately predicted to host Weyl type excitations in their band structure. These magnetic Heusler compounds with broken time reversal symmetry are expected to show a large momentum space Berry curvature, which introduces several exotic magneto-transport properties. In this report, we present systematic analysis of experimental results on anomalous Hall effect (AHE) in Co$_2$Ti$X$ ($X$=Si and Ge). This study is an attempt to understand the role of Berry curvature on AHE in Co$_2$Ti$X$ family of materials. The anomalous Hall resistivity is observed to scale quadratically with the longitudinal resistivity for both the compounds. The detailed analysis indicates that in anomalous Hall conductivity, the intrinsic Karplus-Luttinger Berry phase mechanism dominates over the extrinsic skew scattering and side-jump mechanism.

The anomalous Hall effect (AHE) is a transport phenomenon typically observed in ferromagnetic materials with broken time-reversal symmetry . Recently, the AHE has been observed in several archetype antiferromagnets (AFMs), including altermagnets, and AFMs with noncollinear, noncoplanar or canted Néel order, due to the breaking of joint symmetry of sublattice-transposing and time-reversal operation. However, the AHE is generally not allowed in collinear AFMs due to symmetry constraints. Here, we report the observation of the AHE in a collinear AFM L10-IrMn (001) film. Scanning transmission electron microscopy investigation shows the presence of (200)-oriented grains in the L10-IrMn (001)-oriented film due to the large lattice mismatch between the films and substrate. Consequently, the joint symmetry, with being the translation operation, may be locally broken in our samples, thus enabling the AHE, which is further supported by ab initio calculations. Our work provides a novel way to generate the AHE in AFMs by engineering the local symmetry.

The anomalous Hall effect (AHE) has emerged as a key indicator of time-reversal symmetry breaking (TRSB) and topological features in electronic band structures. Absent of a magnetic field, the AHE requires spontaneous TRSB but has proven hard to probe due to averaging over domains. The anomalous component of the Hall effect is thus frequently derived from extrapolating the magnetic field dependence of the Hall response. We show that discerning whether the AHE is an intrinsic property of the field-free system becomes intricate in the presence of strong magnetic fluctuations. As a study case, we use the Weyl semimetal PrAlGe, where TRSB can be toggled via a ferromagnetic transition, providing a transparent view of the AHE's topological origin. Through a combination of thermodynamic, transport, and muon spin relaxation measurements, we contrast the behavior below the ferromagnetic transition temperature to that of strong magnetic fluctuations above. Our results on PrAlGe provide general insights into the interpretation of anomalous Hall signals in systems where TRSB is debated, such as families of kagome metals or certain transition metal dichalcogenides.

The chirality associated with broken time reversal symmetry in magnetically doped topological insulators has important implications to the quantum transport phenomena. Here we report the anomalous Hall effect studies in Mn- and Cr-doped Bi$_2$Te$_3$ topological insulators with varied thickness and doping content. By tracing the chirality of the Hall loops, we find that the Mn-type anomalous Hall effect with clockwise chirality is strengthened by the reduction of film thickness, which is opposite to that of the Cr-type anomalous Hall effect with counterclockwise chirality. We provide a phenomenological model to explain the evolution of magnetic order and anomalous Hall effect chirality in magnetically doped topological insulators.

Inverse spin Hall effect (ISHE) allows the conversion of pure spin current into charge current in nonmagnetic materials (NM) due to spin-orbit interaction (SOI). In ferromagnetic materials (FM), SOI is known to contribute to anomalous Hall effect (AHE), anisotropic magnetoresistance (AMR), and other spin-dependent transport phenomena. However, SOI in FM has been ignored in ISHE studies in spintronic devices, and the possibility of "self-induced ISHE" in FM has never been explored until now. In this paper, we demonstrate the experimental verification of ISHE in FM. We found that the spin-pumping-induced spin current in permalloy (Py) film generates a transverse electromotive force (EMF) in the film itself, which results from the coupling of spin current and SOI in Py. The control experiments ruled out spin rectification effect and anomalous Nernst effect as the origin of the EMF.

Motived by time-reversal symmetry breaking and giant anomalous Hall effect in kagome superconductor $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ ($A$ = Cs, K, Rb), we carried out the thermal transport measurements on $\mathrm{Cs}{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$. In addition to the anomalous Hall effect, the anomalous Nernst effect and the anomalous thermal Hall effect emerge. Interestingly, the longitudinal thermal conductivity ${\ensuremath{\kappa}}_{xx}$ largely deviates from the electronic contribution obtained from the longitudinal conductivity ${\ensuremath{\sigma}}_{xx}$ by the Wiedemann-Franz law. In contrast, the thermal Hall conductivity ${\ensuremath{\kappa}}_{xy}$ is roughly consistent with the Wiedemann-Franz law from electronic contribution. All these results indicate the large phonon contribution in the longitudinal thermal conductivity. Moreover, the thermal Hall conductivity is also slightly greater than the theoretical electronic contribution, indicating other charge neutral contributions. More than that, the Nernst coefficient and Hall resistivity show the multiband behavior with possible additional contribution from Berry curvature at the low fields.

We consider a junction consisting of an extended one-dimensional Kitaev chain which incorporates both time-reversal symmetry (TRS) breaking and long-range interaction, sandwiched between two metallic leads from two sides. In this hybrid device, we study electrical transport under voltage bias for varying strengths of the TRS breaking phase. We compare the transport characteristics of the long-range type Kitaev chain with those of the short-range Kitaev chain as the strength of the TRS breaking phase varies. We find that the TRS breaking modifies the density of states and localisation/delocalisation property of the eigenstates, which in turn affect the transport characteristics. Moreover, we find that the impact of the TRS breaking is not identical for the long-range Kitaev chain and its short-range counterpart. Therefore, noticeable differences in the transport properties can be observed due to the interplay between the TRS breaking and the range of interaction.

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

The anomalous Hall effect (AHE), typically observed in ferromagnetic (FM) metals with broken time-reversal symmetry, depends on electronic and magnetic properties. In Co3Sn2-xInxS2, a giant AHE has been attributed to Berry curvature associated with the FM Weyl semimetal phase, yet recent studies report complicated magnetism. We use neutron scattering to determine the spin dynamics and structures as a function of x and provide a microscopic understanding of the AHE and magnetism interplay. Spin gap and stiffness indicate a contribution from Weyl fermions consistent with the AHE. The magnetic structure evolves from c-axis ferromagnetism at x = 0 to a canted antiferromagnetic (AFM) structure with reduced c-axis moment and in-plane AFM order at x = 0.12 and further reduced c-axis FM moment at x = 0.3. Since noncollinear spins can induce non-zero Berry curvature in real space acting as a fictitious magnetic field, our results revealed another AHE contribution, establishing the impact of magnetism on transport.

We study the spontaneous nonmagnetic time-reversal symmetry breaking in a two-dimensional Fermi liquid without breaking either the translation symmetry or the U(1) charge symmetry. Assuming the low-energy physics is described by fermionic quasiparticle excitations, we identified an ``emergent'' local $\text{U}{(1)}^{N}$ symmetry in momentum space for an $N$-band model. For a large class of models, including all one-band and two-band models, we found that the time-reversal and chiral symmetry breaking can be described by the $\text{U}{(1)}^{N}$ gauge theory associated with this emergent local $\text{U}{(1)}^{N}$ symmetry. This conclusion enables the classification of the time-reversal symmetry-breaking states as types I and II, depending on the type of accompanying spatial symmetry breaking. The properties of each class are studied. In particular, we show that the states breaking both time reversal and chiral symmetries are described by spontaneously generated Berry phases. We also show examples of the time-reversal symmetry-breaking phases in several different microscopically motivated models and calculate their associated Hall conductance within a mean-field approximation. The fermionic nematic phase with time-reversal symmetry breaking is also presented and the possible realizations in strongly correlated models such as the Emery model are discussed.