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Carmine Autieri

Publications and source records attributed to Carmine Autieri.

At least 19 recordsLinked to original sources
arXiv2026

Rank-Selective Optical Tomography of Higher-Wave Altermagnetism

Identifying the spatial rank of higher-wave altermagnetic order optically is challenging because local electric-dipole response does not uniquely resolve distinct continuum harmonics. We show that finite photon momentum turns one-photon spin-resolved absorption into a rank-selective tomography. For the planar $|m|=\ell$ sector of an even-parity $\ell$-wave component, a joint Fourier projection in polarization and momentum angle isolates $\mathcal T_\ell\propto\eta_\ell q^{\ell-2}$, yielding the hierarchy $d:q^0$, $g:q^2$, and $i:q^4$; phase changes track rotations of the selected magnetic harmonic. A Ward-consistent finite-$q$ microscopic calculation reproduces these powers without imposing them. Although discrete crystal symmetry can generate lower-order local aliases, they are orthogonal to the selected momentum harmonic and cannot contaminate it below $q^{\ell-2}$. In MnTe, whose nonrelativistic parent order is three-dimensional $g$ wave while spin--orbit coupling lowers the exact relativistic spin-momentum-locking symmetry, first-principles calculations show that more than $99.9\%$ of the Fourier power of the N\'eel-projected $A$-region spin-energy contrast remains in the parent $g$-wave-derived $m=3$ harmonic. Structured near fields place the required momentum window within experimental reach.

cond-mat.mes-hall↗
arXiv2026

Revisiting the topological properties of XMg2Bi2 (X = Ca, Sr, Ba, Yb and Eu)

Density functional theory is known to underestimate band gaps in semiconductors and to over- estimate inverted band gaps, frequently exaggerating the predicted size of the topological phase diagram of materials. Employing hybrid functionals and calculating the topological invariants, we revisit the topological properties of compounds crystallizing in the CaAl2Si2-type and demonstrate that the overestimation of the inverted band gaps is particularly pronounced in compounds with this crystal structure. Among these, the class of XMg2Bi2 materials (X = Ca, Sr, Ba, Yb, and Eu) is topologically trivial for all considered cations. Our calculations show that these materi- als are narrow-gap semiconductors with direct band gaps of 0.24-0.34 eV, slightly decreasing with increasing the atomic weight of the element X. We confirm this by applying uniaxial strain and hydrostatic pressure, confirming these results. We emphasize that the experimental observation of surface states alone is insufficient to establish nontrivial topology, as trivial semiconductors may host surface states without a surface Dirac point. Consequently, since these materials are intrinsi- cally topologically trivial, any experimentally observed topological signatures should be attributed to extrinsic effects such as doping or surface reconstruction. Our results underscore the importance of accurately treating electronic correlations when assessing topological character, even in materials containing heavy elements with strong spin-orbit coupling, such as bismuth.

cond-mat.mtrl-sci↗
arXiv2026

Defect Geometry Selects Polar and Anomalous Hall Phases in Two-Dimensional Altermagnets

Point defects in altermagnets can create phases absent in the pristine host by selectively breaking crystal symmetries. Combining symmetry analysis, first-principles calculations, and Hamiltonian modeling, we identify how point impurities modify the altermagnetic phase. Using the pristine d- wave altermagnetic monolayer V2Se2O as a testbed, we identify three distinct classes of impurities: those that preserve spin-momentum locking, those that induce a hybrid-parity state associated with Edelstein spin conversion, and those that produce a metallic ferrimagnetic state with an anomalous Hall effect. We further discuss the robustness of two-dimensional altermagnets against point impurities. Results for other two-dimensional systems, such as Mn4N2 and 2H-FeBr3, reveal the same symmetry-based control across distinct lattices and parent spin harmonics, establishing defect geometry as a general route for engineering spin textures and transport properties.

cond-mat.mtrl-sci↗
arXiv2026

Symmetry-Dependent Mechanical and Vibrational Response of Formamidinium Lead Halide Perovskites: A DFT Study

Formamidinium-based hybrid halide perovskites (FAPbX3, X = Cl, Br, and I) have attracted considerable attention for optoelectronic applications owing to their outstanding optical and electronic properties. However, the influence of crystal symmetry reduction on their mechanical behavior and stability has not yet been comprehensively understood. In this work, density functional theory (DFT) calculations were performed to investigate the structural, elastic, dynamical, and nonlinear mechanical properties of the cubic and ps-cubic phases of FAPbX3. The elastic constants, bulk, shear, and Young's moduli, Poisson's ratio, sound velocities, and Debye temperature were evaluated and correlated with the second Piola-Kirchhoff stress-strain response under tensile and compressive loading. The results reveal that the effect of symmetry reduction is strongly dependent on the halide composition. For FAPbCl3 and FAPbBr3, the transition from the cubic to the ps-cubic phase reduces the lattice stiffness, decreases the acoustic phonon velocities, and lowers the Debye temperature, whereas the opposite trend is observed for FAPbI3. The stress-strain analysis further reveals pronounced nonlinear, anisotropic, and asymmetric mechanical behavior, demonstrating that symmetry reduction can either activate or suppress strain-accommodation mechanisms depending on the halide species, thereby governing the mechanical stability and the onset of structural softening. These findings provide microscopic insight into the relationship between crystal symmetry, lattice dynamics, and nonlinear mechanical response in formamidinium-based halide perovskites, offering useful guidance for the design of mechanically robust optoelectronic materials.

cond-mat.mtrl-sci↗
arXiv2026

Magnetically tunable symmetry-enforced nodal lines producing huge anomalous Hall conductivity in altermagnetic $\alpha$-MnTe

Altermagnetic $\alpha$-MnTe exhibits huge anomalous Hall conductivity (AHC) up to room-temperature together with weak ferromagnetism arising from spin and orbital polarizations. We clarify the origin of the large value of the AHC by identifying two sets of distinct symmetry-enforced nodal lines in the valence bands with Mn character, located at $k_z=0$ and $k_z=\frac{\pi}{c}$, protected by mirror symmetry $M_z$ and glide symmetry $G_z = \{M_z\,|\,0,0,\tfrac{c}{2}\}$, respectively. Both nodal lines are energy-dependent with an approximate C$_6$ symmetry, which is reduced to an exact C$_2$ symmetry due to the presence of the N\'eel vector. The highest valence band exhibits a Mexican-hat dispersion, whereas the second-highest valence band exhibits an inverted Mexican-hat dispersion, with nodal lines at the crossing between the two bands. Within first-principles accuracy, we demonstrate that these nodal lines give rise to the large AHC observed experimentally and exhibit a strong interplay with the weak ferromagnetism. We further show that even a small spin canting strongly modifies the nodal lines and the AHC, making them both magnetically tunable. By disentangling the altermagnetic and ferromagnetic contributions to the AHC, the altermagnetic contribution dominates at small canting angles, while the ferromagnetic contribution becomes sizeable for larger values. Using linear dichroism in angle-resolved photoemission spectroscopy, we show a signature of the nodal line at the border of the Brillouin zone.

cond-mat.mtrl-sci↗
arXiv2026

Chiral Magnons and Cycloidal Phonons in Altermagnetic CuF$_{2}$ Monolayer

Altermagnetism establishes momentum-dependent spin splitting through non-symmorphic crystal symmetries, yet whether these same symmetries simultaneously govern spin and lattice collective excitations remains open. Here we show, using first-principles calculations and linear spin-wave theory, that monolayer CuF$_2$ hosts both chirality-split magnons and cycloidal phonons controlled by the same $P2_1/c$ symmetry operations. The altermagnetic order drives strongly anisotropic magnon chirality via symmetric anisotropic exchange, with Dzyaloshinskii--Moriya interactions acting as a weak secondary modulation. Crucially, the phonon and magnon chiral responses are directionally complementary: cycloidal phonon angular momentum emerges precisely where magnon chirality is symmetry-suppressed, and vice versa. The magnon bands further carry quantized Chern numbers $C^M = \pm 2$, confirming non-trivial altermagnetic topology. These results establish monolayer CuF$_2$ as a platform where a single symmetry framework engineers magnonic, phononic, and topological responses, providing a direct connection between altermagnetism and spin-lattice chirality in two-dimensional materials.

cond-mat.mtrl-sci↗
arXiv2026

Ferroelectric Altermagnetic Chern Insulator in magnetic field: electrical control of the Chern number

We investigate electrically controllable Chern topology in a two-dimensional compensated (d)-wave altermagnet described by a lattice-regularized Bernevig--Hughes--Zhang model. In the altermagnetic reference state, the two Kramers sectors acquire momentum-dependent spin splitting and opposite sector Chern numbers, while the combined $C_{4z}\mathcal T$ symmetry enforces a vanishing total charge Chern number. We show that this hidden topological structure can be activated by orbital-selective magnetic coupling and independently tuned by ferroelectric orbital hybridization. The magnetic coupling removes the sector cancellation and generates $C=\pm1$ and $\pm2$ phases, whereas the polar distortion shifts the Dirac gap closing away from high-symmetry momenta and enables electrical transitions such as $C=0\rightarrow-1$ and $C=1\rightarrow2$. The resulting phases exhibit the expected chiral edge-state multiplicity and quantized anomalous Hall conductivity. Their topology is further reflected in the orbital magnetization through the in-gap relation $\partial_\mu \widetilde M_z=-C$. Finally, we show that the Chern phases remain robust against transverse Kramers-sector mixing and symmetry-allowed inversion-asymmetric spin--orbit coupling. These results establish a symmetry-based route to electrically tunable Chern insulating phases in compensated altermagnets.

cond-mat.mes-hall↗
arXiv2026

Effect of polar distortions on the linear and nonlinear anomalous Hall conductivity of altermagnetic $\alpha$-MnTe

Altermagnetic $\alpha$-MnTe with N\'eel vector along the $y$-axis exhibits a finite anomalous Hall conductivity (AHC) and weak ferromagnetism along the $z$-axis. As already demonstrated in the bulk, there is the breaking of the C$_6$ symmetry by the in-plane N\'eel vector, leaving a C$_2$-type magnetic symmetry. The surface of $\alpha$-MnTe breaks the C$_2$, leaving only a time-reversed mirror symmetry with respect to the $x=0$ plane. Therefore, we demonstrate that on the surface, the interplay between breaking of the crystal symmetry and N\'eel vector orientation produces a reduction of the space group from hexagonal P6$_3$/mmc to orthorhombic Amm2. As a result, the surface exhibits not only a polar distortion along the $z$-axis, but also a polar distortion and a weak ferrimagnetism along the $y$-axis. To describe the surface of MnTe in an accessible way, we simplify the problem and examine the effect of the in-plane electric field in bulk MnTe. Moreover, as a doped ionic semiconductor, the properties of MnTe can be influenced by lattice polarization under an applied electric field. We investigate the interplay between the intrinsic anomalous Hall effect and lattice polarization, showing that polarization effects can substantially affect the AHC. Since the electric field breaks inversion symmetry, this contribution from the lattice polarization coexists with the non-linear anomalous Hall effect, highlighting the rich transport phenomenology of altermagnets.

cond-mat.mtrl-sci↗
arXiv2026

Berry-Curvature Activation by Orbital Flux in a Kagome Altermagnet

We investigate topological electronic responses in a kagome altermagnetic metal hosting a compensated coplanar $120^\circ$ magnetic texture. Using a minimal tight-binding model containing nearest-neighbor hopping, noncollinear exchange coupling, intrinsic spin--orbit coupling, and a time-reversal-odd loop-current order, we disentangle the magnetic, orbital, and relativistic mechanisms governing the electronic response. The exchange field produces pronounced momentum-dependent spin splitting and spin-polarized Fermi surfaces without generating a net magnetization. Nevertheless, in the absence of loop-current order, a hidden antiunitary symmetry $\mathcal{T}C_{2z}$ enforces vanishing Berry curvature and intrinsic anomalous Hall conductivity, even for finite spin--orbit coupling. A directed imaginary bond order breaks this protection and activates finite Berry curvature and a sizable, strongly filling-dependent Hall response already in the nonrelativistic limit. Spin--orbit coupling subsequently reconstructs the avoided crossings and redistributes the Berry curvature, enhancing or suppressing the Hall response depending on filling. For sufficiently strong loop-current order and spin--orbit coupling, a global gap opens at $n_e=3$, and the Hall conductivity approaches $2e^2/h$, consistent with an occupied-band Chern number of magnitude two. Parameter-space and filling-dependent calculations further demonstrate that the Hall-active regime extends over broad ranges of exchange coupling, spin--orbit coupling, and chemical potential and remains robust against symmetry-preserving longer-range hopping. These results identify orbital-current order as an independent route for converting a Hall-silent kagome altermagnet into an anomalous Hall metal or a gapped topological phase without net magnetization, noncoplanar spin order, or scalar spin chirality.

cond-mat.mtrl-sci↗
arXiv2026

Symmetry-protected nodal planes and accidental nodal surfaces in mixed odd-even wave spin-momentum locking of relativistic altermagnets

Non-relativistic spin--momentum locking in altermagnets exhibits an even number of nodal planes. In the relativistic limit, the number of nodal planes can be lowered by symmetry reduction due to the N\'eel vector and spin--orbit coupling in noncentrosymmetric systems. Therefore, an analysis of the evolution of the nodal planes in relativistic altermagnets is required. While $g$-wave spin--momentum locking is straightforward to realize in non-relativistic altermagnets, this $g$-wave does not necessarily survive in the relativistic case. In this work, we investigate the relativistic spin--momentum locking of the centrosymmetric CrSb and the noncentrosymmetric wurtzite MnTe. As a first result, we show that in both systems the dominant spin component retains its $g$-wave character in the relativistic regime only when the N\'eel vector is oriented along the $z$-axis, while the subdominant components exhibit $d$-wave symmetry in CrSb and $p$-wave symmetry in ferroelectric wurtzite MnTe. More generally, the $g$-wave character is preserved in the relativistic limit only when both the N\'eel vector and the electric field associated with inversion-symmetry breaking are oriented along the $z$-axis. As a second result, we show that relativistic spin--momentum locking of ferroelectric altermagnets can exhibit $p$-wave magnetism with one symmetry-protected nodal plane and an accidental nodal surface not protected by symmetry, or can have two accidental nodal surfaces. With the N\'eel vector aligned along the $x$-axis, selected bands of ferroelectric altermagnet wurtzite MnTe exhibit $p$-wave magnetism. Our results establish that altermagnets can host distinct spin components that realize a mixture of angular-momentum wave symmetries in momentum space in the relativistic limit.

cond-mat.mtrl-sci↗
arXiv2026

Quantum anomalous Hall conductivity in altermagnets under applied magnetic field

We investigate the emergence of quantum anomalous Hall conductivity in a two-dimensional $d$-wave altermagnet on a Lieb lattice under an external magnetic field. Altermagnetic order induces momentum-dependent spin splitting without net magnetization in the relativistic limit, producing distinct spin-resolved bands at the $X$ and $Y$ valleys. The phase diagram features a normal insulator and a spin Chern insulator separated by an accidental Dirac semimetal. The magnetic field breaks rotational symmetry between valleys while maintaining vanishing total magnetization, enabling independent valley contributions to topology. One valley supports Chern numbers $C=-1$ or $0$, while the other hosts $C=0$ or $+1$, governed by field strength and bandwidth. This competition yields valley-dependent topology. Berry curvature analysis reveals fully gapped phases with total Chern numbers $C=\pm1$, separated by valley-selective gap closings. We uncover a mechanism for rapid magnetic control of the quantum anomalous Hall effect near the semimetal phase and highlight key distinctions from ferro-valleytronic and quantum spin Hall systems.

cond-mat.mes-hall↗
arXiv2026

Contemporary Insights into Electronic Structure and Microscopic Transport in Nodal-Line Semimetals

Topological semimetals have emerged as an important class of quantum materials with novel electronic responses and unconventional transport phenomena. Among them, nodal-line semimetals are distinguished by band crossings that extend along one-dimensional lines in momentum space rather than occurring at discrete points, forming closed loops, chains, or extended lines. The stability of these nodal structures is governed by crystalline symmetries such as mirror, spin-rotation, and nonsymmorphic operations, which give rise to characteristic topological invariants and surface states, including drumhead-like bands. In this review, we present a comprehensive overview of the theoretical framework and experimental realization of nodal-line semimetals, with particular emphasis on symmetry protection and the consequences of symmetry breaking. We discuss the classification of nodal-line structures, their evolution into other topological phases, and their signatures in electronic structure measurements and transport phenomena. Special attention is given to insights obtained from angle-resolved photoemission spectroscopy and related probes. By bringing together symmetry analysis, band topology, and experimental observations, this review aims to clarify the relationship between topology, magnetism, and measurable electronic responses in nodal-line semimetals. These considerations highlight their potential as a versatile platform for next-generation topological electronic functionalities and emergent quantum phenomena beyond conventional paradigms.

cond-mat.mtrl-sci↗
arXiv2026

Long-Lived Interlayer Excitons and Type-II Band Alignment in Janus MoTe2/CrSBr van der Waals Heterostructures

Identifying two-dimensional heterostructures with exceptional electronic and optical properties remains an active area of research in advanced optoelectronics. Here, we present a comprehensive first-principles investigation of the electronic, optical, and excitonic properties of a MoTe2/CrSBr van der Waals heterostructure using density functional theory combined with fully relativistic GW and Bethe-Salpeter equation calculations. The close lattice matching between the two monolayers enables the formation of stable heterobilayers with two inequivalent interfaces (Te-S and Te-Br) arising from the Janus nature of CrSBr. Both interfaces are dynamically and thermally stable and exhibit type-II band alignment with a direct quasiparticle gap, promoting efficient spatial separation of electrons and holes. The heterostructure hosts interlayer excitons with lifetimes 18-45 ps significantly longer than those of the intralayer excitons in the isolated MoTe2, 3.6 ps, and CrSBr, 8.1 ps, monolayers. Moreover, the optical gap, exciton binding energy, and exciton lifetime of the heterostructure are strongly modulated by the built-in electric field associated with the Janus layer. These results establish the MoTe2/CrSBr heterostructure as a versatile platform for engineering long-lived interlayer excitons and highlight its potential for next-generation optoelectronic and light-harvesting applications.

cond-mat.mtrl-sci↗
arXiv2026

Interplay between Relativistic Spin-Momentum Locking and Breaking of Inversion Symmetry: conditions for p-wave magnetism

We investigate the interplay between relativistic spin-momentum locking arising from altermagnetism and various forms of inversion symmetry breaking. Depending on the symmetry breaking, this can give rise to Rashba-type spin-orbit coupling (SOC), Weyl-type SOC, or the coexistence of two distinct spin-momentum lockings. We focus on the altermagnetic Ca2RuO4 as a testbed material. Our results reproduce the experimentally observed ground state, which is an A-centered magnetic order with the Neel vector along the b-axis, hosting spin cantings along the a- and c-axes but without weak ferromagnetism. Ca2RuO4 exhibits relativistic spin-momentum locking, with different even-parity wave orders for the three spin components. We interpret the experimental results on doped samples as evidence for a transition from a pure altermagnetic phase to a weak ferromagnetic phase. Under ferroelectric- and antiferroelectric-like distortions, there are no qualitative changes in the non-relativistic spin-momentum locking and in the weak ferromagnetism. However, we observe the rise of the Rashba or Weyl-type SOC. Using numerical and analytical models, we investigate which nodal planes persist when inversion symmetry is broken in the relativistic case. The spin-momentum locking of the other components adopt a p-wave character in the case of Rashba; in contrast, Weyl-type SOC disrupts all nodal planes, leaving only nodal lines. Finally, to simulate a stripe phase with structural distortions along the z-axis, we studied a modulated electric field inducing atomic displacements within one Ca2RuO4 layer. This produces a magnetic phase transition to an exotic altermagnetic state with two non-relativistic spin-momentum lockings hosting weak ferromagnetism. Our research presents a comprehensive analysis of various possible scenarios in altermagnets with breaking of inversion symmetries under relativistic effects

cond-mat.mtrl-sci↗
arXiv2026

Relativistic spin-momentum locking in ferromagnets

The relativistic spin-momentum locking has been proven in time-reversal-breaking classes of materials with zero net magnetization in the non-relativistic limit, such as altermagnets and other non-collinear magnets. Using density functional theory calculations, we aim to show relativistic spin-momentum locking in ferromagnets, focusing on a broad class of ferromagnetic materials with magnetic sites connected by rotational symmetry, and compare with fcc Ni. In SrRuO3, the antisymmetric exchange interaction produces a spin canting orthogonal to the easy axis, while in all other cases, spin canting is forbidden. Even when the canted magnetic moment in real space is forbidden, relativistic spin-momentum locking shows sizable contributions in k-space. Using prototypical ferromagnets such as orthorhombic SrRuO3, hexagonal CrTe and CrAs with the NiAs crystal structure, half-Heusler MnPtSb, and fcc Ni, we demonstrate that relativistic spin-momentum locking can generate strong effects in ferromagnets. Subdominant components of centrosymmetric ferro-magnetic materials with magnetic sites connected by rotational symmetry host spin-momentum locking similar to altermagnets, while noncentrosymmetric MnPtSb hosts relativistic p-wave due to the spin-orbit coupling. Fcc Ni shows a more complex behavior with a combination of two spin-momentum locking patterns characteristic of altermagnets. Because ferromagnets typically have larger bandwidths than altermagnets, they provide a promising platform for observing even-wave relativistic spin-momentum locking and associated emergent phenomena. From an application standpoint, relativistic spin-momentum locking governs symmetry-allowed spin Hall currents, spin photocurrents, and other momentum-dependent spin responses in k-space.

cond-mat.mtrl-sci↗
arXiv2026

Staggered Dzyaloshinskii-Moriya and canting angle in centrosymmetric altermagnetic and ferromagnetic phases: influence on the anomalous Hall effect and Weyl points

We present a simple methodology to compute the anomalous Hall conductivity (AHC) as a function of the canting angles in ferromagnets and altermagnets, starting from a nonmagnetic Hamiltonian obtained from first-principles calculations that preserves the full symmetry of the crystal structure. Magnetism is introduced by including on-site spin splitting, spin-orbit coupling, and spin-canting angles. As a representative material, we study SrRuO$_3$, which supports spin canting and exhibits a sign change of the AHC. In the ferromagnetic phase, the low-energy AHC is found to be close to zero at the Fermi level, in agreement with experimental observations. We show that the dependence of the AHC on the relevant physical parameters is most pronounced in the central region of the electronic bandwidth. We determine the symmetry-allowed components of the AHC for different magnetic orders in the large family of transition-metal perovskite ABO$_3$ compounds with space group $62$, including the spontaneous in-plane anomalous Hall effect. Within density functional theory, we evaluate the range of spin-canting angles in SrRuO$_3$ and demonstrate that it is suppressed as electronic correlations increase. By analyzing the AHC as a function of the canting angle, we find that the collinear magnetic configurations contribute most to the AHC, while spin canting plays a secondary role in determining its magnitude in non-collinear ferromagnets and altermagnets. However, canting can become relevant and induce a sign change of the AHC when the collinear magnetic state exhibits an AHC close to zero. Finally, we investigate the locations of Weyl points in the Brillouin zone and their evolution as a function of the canting angle.

cond-mat.mtrl-sci↗
arXiv2026

Competing magnetic states in a non-coplanar Kagome magnet

Non-collinear Kagome antiferromagnets (AFMs) Mn3X (X = Sn, Ga, Ge, Ir, Pt) can generate an anomalous Hall effect (AHE) despite vanishing net magnetization, enabled by broken time-reversal and inversion symmetries. However, strong in-plane anisotropy has limited studies of the AFM-AHE and electronic applications to coplanar spin configurations. Non-coplanar spin textures in these systems have been realized only in low temperature spin-glass states or at interfaces with heavy metals. Here, we report an intrinsic non-coplanar spin configuration persisting up to 400 K in cubic-phase Mn3Ge, originating from coexisting symmetric and antisymmetric exchange interactions. Competing magnetic states associated with this non-coplanar spin configuration give rise to an unconventional AHE with a magnetic-field-induced sign reversal and a hump-like feature. Our findings establish a platform for non-coplanar magnetism in AFM spintronics.

cond-mat.mtrl-sci↗
arXiv2025

Dual topology and edge-reconstruction in $\alpha$-Sn

We formulate the tight-binding model for cubic $\alpha$-Sn based on the DFT calculations. In the model, we incorporate a variable bond angle, which allows us to simulate the effect of the in-plane strain. In the bulk, we demonstrate the presence of the $\mathbb{Z}_2$ topological invariant and a non-zero mirror Chern number, making $\alpha$-Sn one of the rare cases where dual topology can be observed. We calculate the topological phase diagram of multi-layer $\alpha$-Sn as a function of strain and number of layers. We find that a non-trivial quantum spin Hall state appears only for compressive strain above five layers of thickness. Quite surprisingly, both in the trivial and non-trivial phases, we find a plethora of edge-states with energies inside the bulk gap of the system. Some of these states are localized at the side surfaces of the slab, some of them prefer top/bottom surfaces and some are localized in the hinges. We trace the microscopic origin of these states back to a minimal model that supports chiral symmetry and multiple one-dimensional winding numbers that take different values in different directions in the Brillouin zone.

cond-mat.mes-hall↗