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Study of the Anomalous Hall effect by tuning the spin orientation in the Altermagnetic material CrSb

Sreedevi Chintalapudi,1 Upasana Agrawal,1 , 2 and Suvadip Das1 , ∗

arXiv:2607.29646v1 [cond-mat.str-el] 31 Jul 2026

1 Department of Physics, Birla Institute of Technology and Science Pilani Hyderabad, Telangan 500078, India

2 Department of Mathematics and Computing, Birla Institute of Technology and Science Pilani Hyderabad, Telangan 500078, India

Recent development in the field of altermagnetism, and increased demand for the search of
applications of anomalous hall effect have ushered in a new era for novel quantum phases in
materials. Quantum materials previously anticipated to be scientifically predictable have unfolded
novel properties that brought them into the spotlight. These manifestations have led us to rethink
our understanding of existing classification of magnetic materials and preexisting notions about
anomalous hall effect in the light of topologically nontrivial phases of matter. One such recent de-
velopment lies in the novel class of alter-magnetic materials with prospect for quantum computing.
In this article, we delineate the spin and orbital resolved electronic spectrum, mode-decomposed
phonon dispersion relations, geometrical berry curvature and topological surface states and their
implications on anomalous Hall conductivity in the promising altermagnetic compound CrSb.
We further utilize first principles calculations coupled with computationally efficient maximally
localized wannier states of numerous magnetic configurations of the altermagnet to simulate
the effect of external fields and elucidate the fact that the linear behaviour of anomalous hall
conductivity with magnetization does not necessarily hold true for all magnetic classes, such as
altermagnets.

I. INTRODUCTION The quantum manifestation of Quantum Hall effect[1– 3], an effect resulting from the quantization of the electrons in a two dimensional electron gas[4] in the presence of external magnetic fields at low temperatures, has revolutionized both worlds of quantum materials and modern day technology. With precise quantized values of hall resistances measured to integer multiples[5] of h/e2 and shortly after, fractional multiples h/f e2 leading to integer and fractional quantum hall effect in systems hosting two dimensional electron gas has ushered in novel concepts in quantum many-body physics such as composite fermions. The magnitude of hall resistance has been observed to be much larger in ferromagnets in external magnetic fields, and the additional effect of spin-orbit coupling[6] leading to spin accumulation at the edges results in anomalous hall effect[6–8] and quantum anomalous hall effect respectively.[9] The technological implications of quantum anomalous Hall (QAH) [10] effect has been immense. Majorana modes [11–13], crucial for quantum computing applica- tions, appears to predominantly occur in topological su- perconductors. Chiral topological superconductors have been observed in materials with QAH-superconductor hybrid structures. Further, proximity induced supercon- ductivity in QAH insulators have been found to induce chiral Majorana edge modes[14, 15]. Dissipation less chiral edge currents in QAH materials are promising for ∗ [email protected] potential chiral spintronic devices. The immense potential of Anomalous Hall effect (AHE) and Quantum anomalous Hall effect (QAHE) have led to the exploration of multiple classes of mate- rials and their interfaces over the years with promise in electrical, optoelectronic and topological devices. This includes, but is not limited to, the double exchange mediated monotonous temperature dependence of AHE in chiral spin texture material SrRuO3 [ 16], earlier studies of AHE measurements in prototypical transition metal oxides[17], spiral and helical magnetic compounds like MnSi[18, 19], Co-doped FeSi[20], Ca and Sr doped LaCoO3 [ 21, 22], heusler alloys such as Co2 C rAl[23], and Mn5 G e3 [ 24–26]. It has been noted that the Hall resistivity in most conventional compounds scale as ρx y ∼ ρβ [8] where β varies in between 1 and 2. One of the earlier theoretical descriptions of the microscopic theory of AHE were formulated by Karplus and Luttinger (1954)[27]. The concept of AHE was soon connected to the geometry and topology of the electronic energy spectrum, and the geometrical berry connection was observed to correlate with the microcopic theory of AHE. This led to the detailed study and thereafter the decomposition of the AHE conductivity into three distinct contributions: intrinsic, skew-scattering and side-jump contributions[8, 9]. It was noted that the Hall resistivity ρx y ∼ ρ2 (intrinsic) and ρx y ∼ ρ (extrinsic) respectively. With the exploration of topological materials, quan- tum anomalous hall effect gained momentum. QAHE is primarily observed in systems hosting non-trivial

2 topological bands resulting in chiral edge states. QAHE, as opposed to AHE, results from broken time reversal symmetry driven by internal magnetization or local moments leading to dissipation-less edge states and associates itself with the occurence of Chern bands[28]. Recent recipes for tracking altermagnets found anoma- lous hall effect as one of the key quantities for identifying altermagnetic compounds[29]. Inspite of the popularlity of the field of anomalous hall conductivity, the exact dependence of the anomalous hall resistivity is still under debate.[8] Recently, a novel class of magnetism, termed “altermagnets”[30–32], has gained significant momen- tum. The phenomenon of altermagnetism appears in systems with opposite spin sublattices linked by crystal rotational symmetry, thereby leading to compensated magnetic order in the direct lattice space. The above presents itself with unconventional ordering of spins in the reciprocal lattice reflecting the coexistence of rotational symmetries. The spin space rotations in altermagnets can be expressed in group theoretical no- tation as [30] [E ∥ H] + [C2 ∥ AH], where the first term provides the symmetry transformations that interchange the atoms between two spin sublattices while the next term represents real space rotational transformations during interchange of the atoms from two sublattices. Lately, motivated by the enormous ramifications of the field, a significant effort has been invested to identify altermagnetic compounds. Some of the potential al- termagnetic candidates being investigated include, but are not limited to the following materials: RuO2 [ 33], Mn5 S i3 [ 18], CrSb[34], FeSb2 [ 35], MnF2 , CuF2 , MnO2 , MnTe[36], LaMnO3 and CaVO3 [ 37]. The understanding of the associated symmetries of the altermagnet order parameters strengthens our understanding of the fields of magnetism and superconductivity. In this paper, we provide first-principles calculations combined with maximally localized wannier functions for mapping the topological signatures in the altermagnetic compound CrSb. Further, the effect of spin orientation and the dependence of quantum anomalous hall conduc- tivity on the magnetization for different non-collinear spin orientations have been investigated. In order to in- spect the possibility of hosting surface states, calculations of slabs oriented along the (001) direction have been con- structed and studied employing surface Green’s functions and tight binding models. In Section II we discuss the theoretical background, basis and methodology for our first principles calculations, Section III details the find- ings from our calculations of anomalous hall conductivity, whereas in Section III, we delineate the conclusions and discussions following our observations in this study. II. THEORETICAL DISCOURSE AND COMPUTATIONAL METHODOLOGY The Hall effect has been traditionally divided into the two main categories in metals - ordinary Hall ef- fect (OHE), resulting from the Lorenz force acting on electrons, and extraordinary Hall effect (EHE), which is a consequence of the fermionic property of elec- trons. Within the level of approximation for indepen- dent electrons, the OHE [2] is given in form of the following integral over the Fermi surface for the Hall conductivity[7, 8, 30, 38]: Jx = σ0 E x + σx y (Ey Hz − Ez Hy ) (1) σOx y = − 1 12 dSF |v(k)| τk v x ( k)[v(k) × ∇ k] z [τk v y (k)] (2) = − τ 2 12 dSF |v(k)| vx ( k)[vx ( k)∇ y − vy (k)∇ x] vy (k) (3) = − τ 2 12 dSF |v(k)| [v2x ( k)µy y − vx ( k)vy (k)µx y ], (4) where the other components are obtained by cyclic per- mutations. We have utilized the isotropic relaxation time approximation in our recipe. Here µ is the k-dependent invers mass tensor, defined as µ− 1 αβ ≡ ∂vα ∂kβ ≡ ∂2 ϵk ∂kα ∂ kβ , (5) and vα = ∂ϵk /dkα is the corresponding Fermi velocity and ϵk is the electron energy respectively. If the band energies are different for each spin direction, the sum- mation over the spin index should be incorporated. In all our discussions we will be addressing all physical quantities in the atomic units. From an observation of the above expression one can make following conclusions: (1) the OHE conductivity is intrinsically a property of the one-electron band struc- ture, and, as long as the latter does not depend on the magnetic field, OHC doesn’t either (except for a small correction proportional to H 2 for a nonmagnetic mate- rial and proportional to H for a ferromagnetic one, but in either case will be small and scales as the product HN (0), where N is the density of states) (2) as long as the OHC is small compared to the diagonal component of conductivity σ0 , one can invert this relation obtaining the Hall resistivity as ROH = aO H, (6) where a is the coefficient determined by the one-electron nonrelativistic band structure. It was discovered during the 1930s[39] that in the case of ferromagnets, there is another contribution to the Hall effect, which is independent of the applied magnetic field

3 FIG. 1 : Band structure of CrSb for the different spin orientations of the Cr spin moment. The results elucidate the symmetry leading to altermagnetism with separate spin polarizations of the band along particular brillouin zone directions connected by rotational symmetry. FIG. 2 : Phonon dispersion relations for the altermagnetic compound CrSb. (a) Phonon dispersion decomposed by Inverse participation ratio (IPR) as a function of the Brillouin zone (b) Atom decomposed phonon density of states (c) Sublattice participation fraction of phonon energy for the Cr and Sb atoms. in single-domain or saturated samples, and proportional to the net magnetization within the hysteresis term, i.e., it is a constant for a single domain, whose sign depend on the sign of the magnetization (so that both the total magnetization and the Hall conductivity are independently proportional to the imbalance of the domain population). This leads to two corollaries from the above obser- vation: (1) the effect has to be relativistic in nature, otherwise it cannot depend on the sign of magnetization (or on it direction), since spin-orbit coupling interaction decouples the spin and the spatial directions and (2) apart from the magnetic field dependence due to domain

4 wall dyanmics (i.e., extrinsic) there is no dependence on the applied external field. The latter appears natural since magnetic field is small compared to the typical electron energy scale and cannot affect the electron dynamics in a saturated ferromagnet. Unfortunately this extrinsic dependence of the Hall conductivity on the applied field led to the introduction of an unphysical concept of an “Anomalous Hall effect” (AHE), intrinsically proportional to the field-induced magnetization [39], RH = aO H + aA M, (7) where the coefficient aA was assumed to be a constant. Since this formula is unphysical and cannot be applied to a single domain sample, the deviation from Eq. 7 have been routinely observed experimentally and dubbed by authors as either “topological HE” (THE)[40], “antifer- romagnetic HE”, or “crystal HE”. While the above are very good and insightful research dissemination, they focus primarily on the violation of linear dependence on M (often to the extend that the HE appear at zero H and zero M ), and provide a picture where Eq. 7 is microscopically valid, although there are various esoteric exceptions. As a result experimentalists have worked out an intrinsically flawed procedure that is routinely applied to magnetotransport measurements, whereupon an attempt is made to fit the experimentally measured Hall resistivity to the modified Eq. 7, namely RH (H) = aO H + aA M (H) + RT (H), (8) where the coefficients a are assumed to be field- and moment-independent. The superscript T here stands for “topological”,[41] since this is the most common designation of this residual term. A microscopically consistent definition of AHE was provided in numerous publications where the AHE arises from the off-diagonal conductivity not resulting from the Lorenz force, and expressed, via the Kubo formalism[42, 43], as σAx y = n̸= m d3 k[f (ϵk n) − f (ϵk m) ] (9) × Im ⟨kn|ˆvx | km⟩ ⟨km|ˆvx | kn⟩ (ϵk n − ϵk m) (10) where the integration is properly normed, and the ve- locity operator is defined as ˆv =∂H(k)/∂k, the deriva- tive of the Hamiltonian. Here we omit dissipative terms and f (ϵ) is the Fermi distribution function. Nowadays a more elegant formulation in terms of the geometrical Berry curvature[44] is more often used, namely σAx y = − n d3 kf (ϵk n) Bzn ( k) (11) Bn ( k)=i ∂ ∂k × kn| ∂ ∂k |kn . (12) There are various mechanisms that render nonzero values for this expression, including, but not limited to, uncom- pensated collinear magnetism (“classical” AHE) [2], non- collinear co-planar antiferromagnetism[45], compensated collinear magnetism (so-called altermagnetism)[38], and non-coplanar spin-chiral magnets[46]. In this didacti- cal paper we will show that deviations from this Equa- tion may arise in absence of any esoterical mechanism, through trivial dependence of electronic structure on magnetic order. III. RESULTS AND DISCUSSIONS In this section, we display the spin and orbital resolved electronic structure, phonon dispersion, spin orientation and topological properties such as surface states and geo- metrical berry curvature of the altermagnetic compound CrSb from first principles calculations[47, 48] combined with maximally localized wannier functions[49]. We have investigated the electronic structure, phonon dispersion, magnetic properties and topological features using first principles Density functional theory (DFT). DFT calcu- lations were carried out using the projector-augmented wave method implemented in the Vienna ab initio Sim- ulation Package (VASP)[48] and Quantum espresso[47] codes. The exchange-correlation functional form of the generalized gradient approximation implemented within the PBE functional was leveraged for our first principles calculations. In our ground state calculations, plane wave basis sets were utilized with energy cutoff of 600 eV and convergence was achieved with a 8×8×8 k-point grid. The relaxation of the internal coordinates were performed while keeping the cell shape fixed using the VASP code until the ellmann–Feynman forces reached the threshold of 0.001 eV/˚A . The ion steps of the converged with a threshold of 10− 6 eVwhereas the self-consistent electronic steps were converged upto a threshold of 10− 8 . Fig. 1 illustrates the zero magnetization due to the alternating spin sub lattices as dictated by the crystal symmetry [E ∥ H] + [C2 ∥ AH] depicted along the L1 − Γ − L2 direction. CrSb crystallizes in the P63 / mmc- D46 h space group[30] with hexagonal symmetry. The non-trivial spin sublattice [E ∥ H] corresponds to sym- metry transformations responsible for the exchange of atoms belonging to one of two spin sublattices, while the other half of transformations in real space are dictated by the spin group symmetry [C2 ∥ AH][31]. The nature

5 FIG. 3 : Fermi surface of CrSb color coded by the momentum dependent berry curvature contributions from (a) lateral view (b) top view (c) quiver plot of berry curvature in the hexagonal ab plane within the Brillouin zone (d)-(f) band decomposed berry curvature contributions to the anomalous hall conductivity. FIG. 4 : Slab calculations of the projected energy spectrum along the (001) plane obtained from tight binding calculations showing (a) the bulk energy spectrum (c) surface density of states and energy spectrum (c) existence of topological surface states. of anisotropies of the sublattice spin densities and the individual spin resolved fermi surfaces are characterized by these symmetries. The spin splitting from the two sublattices alternates between up to down spin across the entire Brillouin zone (BZ) leading to a net zero magnetization. Fig. 1(a) and (b) displays the spin resolved (up and down respectively) bands along the L1 − Γ − L2 direction, whereas Fig. 1(c)-(f) displays the orbital and spin resolution of the electronic bands. Note that the Cr 3d orbitals contribute significantly towards both the valence and conduction states of the electronic spectrum, whereas the Sb 5p orbitals contribute signifi- cantly to the conduction states respectively. A detailed comparison reveal that our DFT calculations utilizing VASP and Quantum espresso captures the features of the electronic spectrum and spectral properties in reasonably good agreement with that of experimentally measured Angle-resolved Photoemission Spectroscopy (ARPES) findings[34, 50]. Figure 2(a) exhibits the phonon spectrum of the altermagnetic compound along with the density of states resolved by the atomic decom- position (Fig. 2 (b). Note that the phonon spectrum is painted by the inverse participation ratio (IPR)[51] of the phonon modes. The calculated IPR contributions hint towards the real-space localization of the phonon modes in the altermagnetic compound. We observe that the lowest energy (highly dispersive) acoustic phonon modes present the lowest IPR rates, similar to that

6 FIG. 5 : The spin resolved density of states projected spectrum from the (001) slab calculations of the tight binding model showing the projected Pauli spin matrices (a) Sx (b) Sx (c) Sx respectively. (a) (b) (c) (d) (e) (f) (g) (h) FIG. 6 : The four magnetic configurations with (a) coplanar (b) canted at 45◦ (c) canted at an angle of 135◦ and (d) collinear orientations of the spin moment on the Cr sublattices (e)-(h) Anomalous hall conductivity as a function of the shift in the fermi energy for the four configurations. of the (highly localized) high energy optical modes. Fig. 2(c) exhibits the fractional energy participation of the sublattice atoms (Sb and Cr atoms) revealing a significantly higher energy participation from the Cr (Sb) atoms for the optical (acoustic) modes compared to the acoustic (optical) modes respectively. This is further confirmed by the phonon density of states showing alternate participation of the Cr and Sb atoms at the high frequency limit (5-8 THz) and lower frequency range (0-4 THz) respectively of the phonon spectrum. Next, we have utilized DFT to investigate the mo- mentum and band resolved geometrical berry curvature contributions and possibility of surface states in CrSb. The formalism involves the generation of maximally localized wannier functions from first principles density functional calculations. In order to perform these cal- culations, we have carried out a unitary transformation from the Bloch states to obtain localized Wannier functions. Here, the degree of freedom provided by the indeterminacy of the phase of the Bloch wavefunctions have been utilized for the construction of “maximally localized”Wannier functions. An accurate convergence of the minimum spread of the wannier functions is crucial for the above step. Following our wannier interpolation scheme, a mixed Fourier transform of the regular grid of k points in addition with recursive adaptive mesh iter- ations have been performed utilizing the WannierBerri code further in order to obtain berry curvatures with fine k-meshe grids of 200×200×200 without significant

7 (a) (b) (c) (d) (e) (f) FIG. 7 : The three magnetic configurations with spin mo- ments on the Cr aotoms opposite to each other along (a) a axis (b) b axis (c) c axis respectively resulting in zero mag- netic net moment. (d)-(f) Anomalous hall conductivity as a function of the shift in the fermi energy for the three config- urations. FIG. 8 : Plot of the anomalous hall conductivity as a func- tion of the net magnetization for four different magnetic con- figurations for the altermagnetic compound CrSb. computational resources. The accuarte berry curvature contributions are obtained by a combination of the VASP interface of Wannier90[49] and WannierBerri codes[52]. Fig. 3 (a),(b) exhibit the fermi surface of CrSb colour- coded by the geometrical berry curvature contribution across the entire BZ (red and blue for positive and neg- ative berry curvature contributions respectively). Fig. 3(c) displays a quiver plot indicating the distribution of berry curvature in the material along the ab plane, providing computational estimates of the topological phases, anomalous hall coefficient and nontrivial band crossings. Note that the berry curvature contributions from the quiver plots in the ab plane mimics the berry curvature resolved fermi surfaces when onserved from the c axis depicted in Fig. 3(b). Fig. 3 (d)-(f) present contribution to the geometrical curvature from each of the individual electronic bands crossing the fermi surface and reflects the symmetry properties characteristic of altermagnetic compounds. During the course of this study, we came across recent experimental ARPES publications that identified topological surface states in the electronic properties of this materials[53]. Motivated by these experimental findings, we have utilized the maximally localized wannier functions (MLWF)[49] for the construction of a semi-emperical tight binding (TB) model for evaluating the topological calculations in CrSb. In an event, the topological bulk bands are non-trivial, topologically protected surface states are obtained. The WannierTools[54] code was utilized for the construction of tight binding model from the wannier functions for the estimation of the existence of surface states in the material. We have constructed a slab system for our calculations of surface states with 30 slabs (periodic along the two surfaces of the slab)[54]. The solution of the Hamiltonian of the slab system provides the solutions of the surface states in the material. Figure 4 and 5 provides the spectral features of the surface states from our first-principles and tight binding calculations. Fig 4(a) shows the computed surface density of states and electronic structure for bulk CrSb near the fermi level, whereas Fig. 4(b)-(c) provide the surface density of states and surface state (SS) spectrum respectively for the slab along the high symmetry k path[49, 54]. Note that, following our calculations fermi arcs were obtained from the ends of the six connected fermi surface segments at the periphery (similar to the six spokes of a wheel). Figure 5 depicts the surface spin resolved density of states (DOS) for the slab system, where Fig. 5 (a), (b) and (c) displays the Pauli matrices Sx , Sy and Sz respectively evaluated by computing the spectral functions of the surface Green’s functions. Note that the DOS as a function of energy (Figure 4) and spin resolved DOS (Figure 5) are shown as observed from the top of the slab geometry (c axis). The signatures of symmetry characteristic of altermagnets are observed from the plots of the spin resolved density of states in conformity with the electronic structure of Figure 1 and previously published results.[30, 31] In the final part of the dissemination, we address the controversy regarding the linear behavior of AHC and the justification for assigning the disgression from linearity to topological Hall effect. In order to validate the linearity of the dependence of AHC[8] with the net magnetization in the case of an altermagnet, we have simulated the effect of applying magnetic fields in the altermagnetic material CrSb[38]. In order to do so, the magnetization on the two different Cr sublattices, have been considered separate from each other (except one scenario) for different directions of magnetic moments in response to applied external fields. In Figure 6, we depict four orientations of the sublattice spin moments and the corresponding AHC for the test material

8 CrSb. Fig. (a)-(d) shows configurations where the spin orientations are coplanar (a), collinear and parallel (d), staggered at 45◦ (b) and staggered at 135◦ respectively. Note that the first principles noncollinear calculations are performed by constraining the spin moments along these directions in the Cr sub-lattices, followed by construction of maximally localized wannier functions, mixed fourier transforms and adaptive mesh iterations for computation of the berry curvature contributions for each of the configurations above. The berry curvature contributions from all the electronic bands over the entirety of BZ integrated upto the fermi level provides an accurate estimation of the intrinsic AHC contribution in the material[52]. Fig. 6 (e)-(h) shows the AHC contributions for the four different configurations of CrSb with small variations around the fermi level Ef . An intersection of the plot with E = Ef provides an estimate of the value of AHC as illustrated by the black cross in the figures. A similar exercise has been shown in Figure 7, where the spin moment on the Cr sublattice (blue) is oriented along the a, b and c directions respectively (Fig. (a)-(c), whereas the spin moment on the other sublattice is in the opposite direction i.e. −a, −b and −c directions leading to two coplanar and one collinear configuration. Note that in all three scenario, there is a zero net magnetic moment from the Cr atoms in the material. First principles DFT calculations aided by wannier interpolations and adaptive mesh iterations utilizing WannierBeeri code, result in different AHC contributions from all three cases (nonzero from two, zero from one). For the purpose of visual illustration, the dependence of the AHC on the net magnetic moment[26] from the four configurations in Fig. 5 have been displayed in Fig- ure 7. We note that, for the altermagnetic compound CrSb hosting topological features in the electronic spec- trum, the AHC doesn’t depend linearly on the magne- tization, suggestive of the fact that the previous theory of linear dependence of AHC is valid, at best, for ex- periments on a subset of magnetic materials with single magnetic domains. We find it to be unjustified to extrap- olate the linear dependence of AHC on magnetization to include all classes of collinear and non-collinear magnets, anti ferromagnets and altermagnetic compounds[32]. IV. SUMMARY AND CONCLUSIONS In summary, we have investigated the classification of Hall effect into the didactic classification of OHE, EHE and AHE, and the misnomer of generalization of devi- ations from the linearity of AHE with topological Hall effect (THE). A detailed first principles DFT calculation and tight binding model constructed from Wannier func- tions of the electronic spectrum, and topological prop- erties of CrSb has been presented. Our results indicate symmetry enforced momentum and band resolved geo- metrical berry curvatures and surface states originating from nontrivial topological signatures in the altermag- netic compound. The spin resolved density of states from construction of slabs along the (001) surface uti- lizing tight binding model has been discussed. Finally, quantum simulations of two different set of configura- tions, with zero and non-zero net magnetic moment have been investigated. First principles calculations of AHC in these configurations reveal the fact that the altermag- netic compound oriented in external fields fail to concur with the intrinsically flawed[26, 55] hypothesis of linear behaviour of AHC with net magnetic moment, inspite of being routinely applied experimentally to all classes of magnetic materials. 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