Introduction

Quantum materials can be generally defined as those solid-state structures hosting physical phenomena which, even at the macroscopic scale, cannot be captured by a purely classical description1. Among such quantum phenomena, those related to the geometric properties of the electronic wavefunctions play undoubtedly a primary role. In an N-band crystalline system, the cell-periodic part of the electronic Bloch waves defines a mapping from the Brillouin zone (BZ) to a complex space naturally equipped with a geometric structure—its tangent space defines a Fubini-Study metric2 that measures the infinitesimal distance between Bloch states at different points of the BZ. The imaginary part of this quantum geometric tensor3,4 corresponds to the well-known Berry curvature (BC), which, when integrated over the full BZ, gives the Chern number cataloging two-dimensional insulators5. In metallic systems with partially filled bands, the BC summed over all occupied states can result in a non-vanishing Berry phase if the system breaks time-reversal symmetry. This Berry phase regulates the intrinsic part of the anomalous Hall conductivity of magnetic metals6,7,8,9.

Materials with an acentric crystal structure can possess non-vanishing concentrations of BC even if magnetic order is absent. Probing the BC of these non-centrosymmetric and non-magnetic materials via charge transport measurements usually requires externally applied magnetic fields. For instance, in time-reversal invariant Weyl semimetals, such as TaAs10, the strong BC arising from the Weyl nodes can be revealed using the planar Hall effect11—a physical consequence of the negative longitudinal magnetoresistance associated with the chiral anomaly of Weyl fermions12. Recently, it has been also shown that the planar Hall effect can display an anomalous antisymmetric response13,14, which, at least in two-dimensional materials, is entirely due to an unbalance in the BC distribution triggered by the Zeeman-induced spin splitting of the electronic bands.

In the absence of external magnetic fields, a BC charge transport diagnostic for non-magnetic materials requires to go beyond the linear response regime15,16,17,18,19. Hall-like currents appearing as a non-linear (quadratic) response to a driving electric field can have an intrinsic contribution governed by the Berry curvature dipole (BCD), which is essentially the first moment of the Berry curvature in momentum space. In three-dimensional systems, non-vanishing BCDs have been linked to the presence of tilted Weyl cones, and have been shown to exist both in type-I and in type-II Weyl semimetals20 such as MoTe221 and the ternary compound TaIrTe422,23. Furthermore, the Rashba semiconductor BiTeI has been predicted to host a BCD that is strongly enhanced across its pressure-induced topological phase transitions24.

In two-dimensional materials, the appearance of BCDs is subject to stringent symmetry constraints: the largest symmetry group is \({{{{\mathcal{C}}}}}_{s}\), which is composed by the identity and a single vertical mirror line. Note that non-linear Hall currents can exist in symmetry groups containing also rotational symmetries. In these cases non-linear skew and side-jump scatterings are the origin of the phenomenon18,19,25. The concomitant presence of spin–orbit coupled massive Dirac cones with substantial BC and unusually low-symmetry crystalline environments have suggested the surface states of SnTe26 in the low-temperature ferroelectric phase27, monolayer transition metal dichalcogenides in the so-called 1Td phase28,29,30, and bilayer WTe2 as material structures hosting sizable BCDs31,32. Spin–orbit free two-dimensional materials, including monolayer and bilayer graphene, have been also put forward as materials with relatively large BCDs33. In these systems, it is the interplay between the trigonal warping of the Fermi surface and the presence of massive Dirac cones due to inversion symmetry breaking that triggers dipolar concentration of Berry curvatures34.

Finite concentrations of BC and BCDs are symmetry allowed also in systems that do not feature quartets of Weyl cones and pairs of massive Dirac cones. The anomalous massless Dirac cones at the surface of three-dimensional strong topological insulators35 as well as conventional two-dimensional electron gases (2DEG) with Rashba spin–orbit coupling36 are generally characterized by finite local BC concentrations when subject to trigonal crystal fields. The existence of BC in 2DEGs, which has been experimentally probed through anomalous planar Hall effect measurements36, provides a new avenue for investigations. It shows in fact that Berry curvature-mediated effects can be generated entirely from conduction electrons. This overcomes the requirement of materials with narrow gaps in which the electronic wavefunctions at the Fermi level are a quantum superposition of electron and hole excitations, and extends the palette of non-magnetic materials displaying BC effects to, for instance, doped semiconductors with gaps in the eV range. It also proves that it is possible to trigger BC effects in conventional electron liquids with competing instabilities towards other many-body quantum phases.

In a spin–orbit coupled 2DEG, the BC is however triggered by crystalline anisotropy terms, which are cubic in momentum and linked to the out-of-plane component of the spin textures37,38. Consequently, the BC does not possess the characteristic ‘hot-spots’ appearing in close proximity to near degeneracy between two bands where the Bloch wavefunctions are rapidly changing. The absence of such BC hot-spots forbids, in turn, large enhancements of the BCD, which is a central quest for material design. This motivates the fundamental question on whether and how an electron system can develop strong local BC concentrations in time-reversal symmetric conditions even in the complete absence of hole excitations. Here, we provide a positive answer to this question by showing that spin–orbit free metallic systems with an effective pseudo-spin one orbital degree of freedom can display BC hot-spots and characteristic BC singular pinch points that yield dipoles order of magnitudes larger than those triggered by spin–orbit coupling in a 2DEG.

Results

Model Hamiltonian from symmetry principles

Let us first consider a generic single-valley two-level system in two dimensions with spin degree of freedom only. The corresponding energy spectrum is assumed to accurately represent the electronic bands close to the Fermi level of the metal in question. As long as we consider materials without long-range magnetic order, the two Fermi surfaces must originate from one of the four time-reversal invariant point of the Brillouin zone (BZ) (n1b1 + n2b2)/2 with b1,2 the two primitive reciprocal lattice vectors of the BZ and n1,2 = 0, 1. We emphasize that our arguments hold also for systems with multiple pairs of pockets centered at time-reversal invariant momenta. Time-reversal symmetry guarantees that the two bands will be Kramers’ degenerate at the time-reversal invariant momenta (TRIM). The effective Hamiltonian in the vicinity of the TRIM can be captured using a conventional k ⋅ p theory that keeps track of the point group symmetries of the crystal. To make things concrete, let us assume that the low-energy conduction bands are centered around the Γ point of the BZ and we are dealing with an acentric crystal with \({{{{\mathcal{C}}}}}_{3v}\) point group symmetry. This is the largest acentric symmetry group without \({{{{\mathcal{C}}}}}_{2}{{{\mathcal{T}}}}\) symmetry, \({{{{\mathcal{C}}}}}_{2}\) indicating a twofold rotation symmetry with out-of-plane axis and \({{{\mathcal{T}}}}\) time-reversal, and thus allows for local BC concentrations13. The generators of \({{{{\mathcal{C}}}}}_{3v}\) are the threefold rotation symmetry \({{{{\mathcal{C}}}}}_{3}\) and a vertical mirror symmetry, which, without loss of generality, we take as \({{{{\mathcal{M}}}}}_{x}\) sending x → − x. The threefold rotation symmetry can be represented as \({{{{\rm{e}}}}}^{-i\pi {\sigma }_{z}/3}\) while the mirror symmetry as iσx39. Momentum and spin transform under \({{{{\mathcal{C}}}}}_{3}\) and \({{{{\mathcal{M}}}}}_{x}\) as follows

$$\begin{array}{rcl}{{{{\mathcal{C}}}}}_{3}:{k}_{\pm }\to {{{{\rm{e}}}}}^{\pm 2\pi i/3}{k}_{\pm };&{\sigma }_{\pm }\to {{{{\rm{e}}}}}^{\pm 2\pi i/3}{\sigma }_{\pm }&{\sigma }_{z}\to {\sigma }_{z}\\ {{{{\mathcal{M}}}}}_{x}:{k}_{\pm }\to -{k}_{\pm };&{\sigma }_{y,z}\to -{\sigma }_{y,z}&{\sigma }_{x}\to {\sigma }_{x}\end{array}$$
(1)

where k± = kx ± iky and σ± = σx ± iσy. Furthermore, the Hamiltonian must satisfy the time-reversal symmetry constraint \({{{\mathcal{H}}}}({{{\bf{k}}}})={{{\mathcal{T}}}}{{{\mathcal{H}}}}(-{{{\bf{k}}}}){{{{\mathcal{T}}}}}^{-1}\), with the time-reversal operator that, as usual, can be represented as \({{{\mathcal{T}}}}=i{\sigma }_{y}{{{\mathcal{K}}}}\) and \({{{\mathcal{K}}}}\) the complex conjugation. When expanded up to linear order in k, the form of the Hamiltonian reads as \({{{\mathcal{H}}}}({{{\bf{k}}}})={\alpha }_{R}\left({k}_{x}{\sigma }_{y}-{k}_{y}{\sigma }_{x}\right)\). The Dirac cone energy spectrum predicted by this Hamiltonian violates the fermion doubling theorem40 and hence can occur only on the isolated surfaces of three-dimensional strong topological insulators41. And indeed \({{{\mathcal{H}}}}({{{\bf{k}}}})\) coincides with the effective Hamiltonian for the surface states of the topological insulators in the Bi2Se3 material class39,42,43. In a genuine two-dimensional system such anomalous states cannot be present, and an even number of Kramers’ related pair of bands must exist at each Fermi energy. Consequently, the effective Hamiltonian must be equipped with an additional term that is quadratic in momentum and such that it doubles the number of states at each energy. Time-reversal symmetry implies that terms quadratic in momentum are coupled to the identity matrix. Therefore, we arrive at the well-known Hamiltonian of a two-dimensional electron gas with Rashba-like spin–orbit coupling that reads

$${{{\mathcal{H}}}}({{{\bf{k}}}})=\frac{{\hslash }^{2}{k}^{2}}{2m}{\sigma }_{0}+{\alpha }_{R}\left({k}_{x}{\sigma }_{y}-{k}_{y}{\sigma }_{x}\right).$$
(2)

The corresponding energy spectrum consisting of two shifted parabolas is schematically shown in Fig. 1a. Although the crystalline symmetry requirements are fulfilled, the Hamiltonian above does not predict any finite BC local concentration. This is because the d vector associated to the Hamiltonian \({{{\bf{d}}}}=\left\{-{\alpha }_{R}{k}_{y},{\alpha }_{R}{k}_{x},0\right\}\) is confined to a two-dimensional plane at all momenta.

Fig. 1: Spin and orbital mechanisms of Berry curvature.
figure 1

a Schematic band structure of a two-dimensional electron gas with Rashba spin–orbit coupling. b An out-of-plane magnetic field breaks the Kramers' degeneracy at k = 0 and triggers a finite BC. c The local BC has a circular profile with an hot spot at the Γ point of the BZ. d Schematic band structure of a two-dimensional electron system characterized by an L = 1 orbital multiplet in a trigonal crystalline environment. e An additional crystalline symmetry lowering splits completely the energy levels at the Γ point of the BZ even if time-reversal symmetry is preserved. The presence of mirror symmetry protects crossing at finite momenta. f A characteristic time-reversal symmetric BC profile with the presence of hot-spots and singular pinch points. The BC has been obtained using the model Hamiltonian Eq. (5) with \({{\Delta }}=-0.2{{{{\mathcal{E}}}}}_{0}\), \({{{\Delta }}}_{m}=0.12{{{{\mathcal{E}}}}}_{0}\), \({\alpha }_{R}=1.0{{{{\mathcal{E}}}}}_{0}/{k}_{F}^{0}\) and \({\alpha }_{m}=0.5{{{{\mathcal{E}}}}}_{0}/{k}_{F}^{0}\) with \({{{{\mathcal{E}}}}}_{0}={\hslash }^{2}{({k}_{F}^{0})}^{2}/(2m)\) and \({k}_{F}^{0}\) a characteristic Fermi wavevector.

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There are two different ways to lift the d vector out-of-plane and thus trigger a non-vanishing BC44. The first one consists in introducing a constant mass Δσz. This term removes the Kramers’ degeneracy at the TRIM [see Fig. 1b] and therefore breaks time-reversal invariance. It can be realized by externally applying an out-of-plane magnetic field or by inducing long-range magnetic order. The BC then generally displays an hot-spot located at the TRIM and a circular symmetric distribution [see Fig. 1c]. Moreover, time-reversal symmetry breaking implies that the Berry phase accumulated by electrons on the Fermi surface is non-vanishing6. The second route explicitly takes into account trigonal warping terms which are cubic in momentum and couple to the Pauli matrix σz. Such terms preserve time-reversal invariance, and thus create a BC distribution with an angular dependence such that the Berry phase accumulated over any symmetry-allowed Fermi line cancels out35. Perhaps more importantly, the BC triggered by crystalline anisotropy terms36 does not display a hot-spot, thus suggesting that in systems with conventional quasiparticles and a single internal degree of freedom time-reversal symmetry breaking is a prerequisite for large local BC enhancements.

We now refute this assertion by showing that in systems with orbital degrees of freedom the formation of BC hot-spots is entirely allowed even in time-reversal symmetric conditions. Consider for instance a system of p orbitals. In a generic centrosymmetric crystal, interorbital hybridization away from the TRIM can only occur with terms that are quadratic in momentum. However, and this is key, in an acentric crystal interorbital mixing terms linear in momentum are symmetry allowed. These mixing terms, often referred to as orbital Rashba coupling45,46,47, are able to induce BC hot spots with time-reversal symmetry, as we now show. We assume as before an acentric crystal with \({{{{\mathcal{C}}}}}_{3v}\) point group, and electrons that are effectively spinless due to SU(2) spin symmetry conservation: we are thus removing spin–orbit coupling all together. In the pz, py, px orbital basis, the generators of the point group are represented by

$${{{{\mathcal{M}}}}}_{x}=\left(\begin{array}{ccc}1&0&0\\ 0&1&0\\ 0&0&-1\end{array}\right);{{{{\mathcal{C}}}}}_{3}=\left(\begin{array}{ccc}1&0&0\\ 0&\cos \frac{2\pi }{3}&\sin \frac{2\pi }{3}\\ 0&-\sin \frac{2\pi }{3}&\cos \frac{2\pi }{3}\end{array}\right),$$

The two px,y orbitals form a two-dimensional irreducible representation (IRREP) whereas the pz orbital represents a one-dimensional IRREP. The form of the effective Hamiltonian away from the TRIM can be captured using symmetry constraints. Specifically, any generic 3 × 3 Hamiltonian can be expanded in terms of the nine Gell–Mann matrices48 Λi [see “Methods”] as

$${{{\mathcal{H}}}}({{{\bf{k}}}})=\mathop{\sum }\limits_{i=0}^{8}{b}_{i}({{{\bf{k}}}}){{{\Lambda }}}_{i}.$$
(3)

The invariance of the Hamiltonian requires that the components of the Hamiltonian vector b(k) should have the same behavior as the corresponding Gell–Mann matrices Λi. This means that they should belong to the same representation of the crystal point group49. From the representation of the Λi’s [see “Methods” and Table 1] and those of the polynomials of k [see Table 1], we find that the effective Hamiltonian up to linear order in momentum reads as

$${{{\mathcal{H}}}}({{{\bf{k}}}})={{\Delta }}\left({{{\Lambda }}}_{3}+\frac{1}{\sqrt{3}}{{{\Lambda }}}_{8}\right)-{\alpha }_{R}\left[{k}_{x}{{{\Lambda }}}_{5}+{k}_{y}{{{\Lambda }}}_{2}\right].$$
(4)

Here the parameter Δ quantifies the energetic splitting between the px,y doublet and the pz singlet. The second term in the Hamiltonian corresponds instead to the pseudo-spin one massless Dirac Hamiltonian50,51 predicted to occur for instance in the kagome lattice with a staggered magnetic π flux50. Pseudo-spin one Dirac fermions are not subject to any fermion multiplication theorem52. Therefore, a doubling of the number of states at each energy is not strictly required. However, since we are interested in systems without the concomitant presence of electrons and holes, we will introduce a term ℏ2k2Λ0/(2m) with an equal effective mass for all three bands. The ensuing Hamiltonian can be then seen as a generalization of the Rashba 2DEG to an SU(3) system with the effect of the trigonal crystal field that leads to a partial splitting of the energy levels at the TRIM, entirely allowed by the absence of Kramers’ theorem. Despite the spectral properties [c.f. Fig. 1d] have a strong resemblance to those obtained in a time-reversal broken 2DEG, a direct computation [see Methods] shows that the BC associated to the Hamiltonian above is vanishing for all momenta. Breaking time-reversal symmetry introducing a constant mass term ∝Λ7 or considering crystalline anisotropy terms that are cubic in momentum represent two possible routes to trigger a finite Berry curvature. The crux of the story is that in the present SU(3) system at hand, another possibility exists. It only relies on the crystal field effects that are generated by lowering the crystalline point group to \({{{{\mathcal{C}}}}}_{s}\). From the representations of the Gell–Mann matrices and the polynomials of k in this group, we find that the effective Hamiltonian reads

$$\begin{array}{ll}{{{\mathcal{H}}}}({{{\bf{k}}}})=&\frac{{\hslash }^{2}{k}^{2}}{2m}{{{\Lambda }}}_{0}+{{\Delta }}\left({{{\Lambda }}}_{3}+\frac{1}{\sqrt{3}}{{{\Lambda }}}_{8}\right)+{{{\Delta }}}_{m}\left(\frac{1}{2}{{{\Lambda }}}_{3}-\frac{\sqrt{3}}{2}{{{\Lambda }}}_{8}\right)\\ &-{\alpha }_{R}\left[{k}_{x}{{{\Lambda }}}_{5}+{k}_{y}{{{\Lambda }}}_{2}\right]-{\alpha }_{m}{k}_{x}{{{\Lambda }}}_{7}.\end{array}$$
(5)

Nothing prevents to have the interorbital mixing terms ∝Λ2,5 with different amplitudes. Without loss of generality, in the remainder we will consider a single parameter αR. In the Hamiltonian above, we have also neglected a constant term ∝Λ1. For materials with an high-temperature trigonal structure, its amplitude Δ1 is expected to be of the same order of magnitude as Δm. In this regime [see the Supplementary Note 1], a term ∝Λ1 has a very weak effect on the energy spectrum and BC properties, and can be thus disregarded. The energy spectrum reported in Fig. 1e shows that the effect of the crystal symmetry lowering is twofold. First, there is an additional energy splitting between the px,y implying that all levels at the Γ point of the BZ are singly degenerate. Second, the two px,y orbitals have band degeneracies along the mirror symmetric kx = 0 line of the BZ. Such mirror-symmetry protected crossings give rise to BC singular pinch points [see Fig. 1f and the Supplementary Note 2]. It is the presence of these pinch points that represents the hallmark of the non-trivial geometry of the electronic wavefunctions associated with the p-orbital manifold. Note that the BC also displays hot-spots [see Fig. 1f] with BC sources and sinks averaging to zero on any mirror symmetric Fermi surface as mandated by time-reversal invariance.

Table 1 Character tablesa for the point groups \({{{{\mathcal{C}}}}}_{3v}\) and \({{{{\mathcal{C}}}}}_{s}\).
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Material realizations

Before analyzing the origin and physical consequence of the BC and its characteristic pinch points, we now introduce a material platform naturally equipped with orbital degrees of freedom and the required low crystalline symmetry: [111] interfaces of transition metal oxides hosting two-dimensional d electron systems of t2g orbital character such as SrTiO3-53,54, KTaO3-55, and SrVO3-based heterostructures. When compared to conventional semiconductor heterostructures, complex oxide interfaces consist of d electrons with different symmetries, a key element in determining their many-body ground states that include, notably, unconventional superconductivity56. In the high-temperature cubic phases of these materials, the octahedral crystal field pins the low-energy physics to a degenerate t2g manifold, which spans an effective angular momentum one subspace, precisely as the p orbitals discussed above. We note that for the t2g orbitals the spin–orbit coupling has a minus sign due to the effective orbital moment projection from L = 2 to the Leff = 1 in the t2g manifold. This overall sign however is not altering the results. The reduced symmetry at interfaces lift the t2g energetic degeneracy and modify their orbital character. At the [111] interface the transition metal atoms form a stacked triangular lattice with three interlaced layers [see Fig. 2a, b]. This results in a triangular planar crystal field that hybridizes the \(\left\vert xy\right\rangle\), \(\left\vert xz\right\rangle\) and \(\left\vert yz\right\rangle\) orbitals to form an \(\left\vert {a}_{1g}\right\rangle =\left(\left\vert xy\right\rangle +\left\vert xz\right\rangle +\left\vert yz\right\rangle \right)/\sqrt{3}\) one-dimensional IRREP whereas the two states \(\vert {e}_{g\pm }^{{\prime} }\rangle =(\vert xy\rangle +{\omega }^{\pm 1}\vert xz\rangle +{\omega }^{\pm 2}\vert yz\rangle)/\sqrt{3}\), with ω = e2πi/3, form the two-dimensional IRREP.

Fig. 2: Materials, crystal symmetries and orbital splittings.
figure 2

a Schematic representation of an ABO3 pervoskite cubic unit cell displaying the three interlaced transition metal [111] planes. b Corresponding top view along the [111] crystallographic direction. We only show the B transition metal atoms. c, d show the effect of a tetragonal distortion with the [001] direction being the tetragonal axis. The distortion breaks the threefold rotation symmetry around the [111] axis but leaves a residual mirror symmetry. e Evolution of the orbital states at the Γ point of the BZ with quenched angular momentum.

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The energetic ordering of the levels depends on the microscopic details of the interface. For example, at the (111)LaAlO3/SrTiO3 interface, x-ray absorption spectroscopy57 sets the \(\left\vert {a}_{1g}\right\rangle\) state at lower energy [see Fig. 2e]. By further considering the structural inversion symmetry inherently present at the heterointerface, we thus formally reach the situation we discussed for the set of p orbitals, be it for the trigonal symmetry that excludes any local concentrations of BC. However, and this is key, low-temperature phase transitions in oxides lower the crystal symmetry, often realizing a tetragonal or orthorhombic phase with oxygen octahedra rotations and (anti)polar cation displacements. Let us consider the paradigmatic case of SrTiO3. A structural transition occurring at around 105 K, from the cubic phase to a tetragonal structure58 [see Fig. 2c], breaks the threefold rotational symmetry leaving a single residual mirror line. Assuming the tetragonal axis to be along the [001] direction, the surviving mirror symmetry at the [111] interface corresponds to \({{{{\mathcal{M}}}}}_{[\bar{1}10]}\) [see Fig. 2d]. This structural distortion lifts the degeneracy of the \({e}_{g}^{{\prime} }\) doublet. The bonding and antibonding states \(\left\vert {e}_{g+}^{{\prime} }\right\rangle \pm \left\vert {e}_{g-}^{{\prime} }\right\rangle\) have opposite mirror \({{{{\mathcal{M}}}}}_{[\bar{1}10]}\) eigenvalues and realize two distinct one-dimensional IRREP [see Fig. 2e]. SrTiO3-based heterointerfaces undergo additional tetragonal to locally triclinic structural distortions at temperatures below ≃70 K, which involves small displacements of the Sr atoms along the [111] directions convoluted with TiO6 oxygen-octahedron antiferrodistortive rotations59. In addition, below about 50 K, SrTiO3 and KTaO3 approach a ferroelectric instability that is accompanied by strong polar quantum fluctuations. This regime is characterized by a soft transverse phonon mode that involves off-center displacement of the Ti ions with respect to the surrounding octahedron of oxygen ions60, which, in the static limit, would correspond to a ferroelectric order parameter. This can potentially enhance the interorbital hybridization terms allowed in acentric crystalline environments, and thus boost the appearance of large BC concentrations.

Berry curvature dipole

Having identified (111)-oriented oxide heterointerfaces as ideal material platforms, we next analyze the specific properties of the BC and its first moment. We first notice that in the case of a two-level spin system the local Berry curvature of the spin-split bands, if non-vanishing, is opposite. Due to the concomitant presence of both spin-bands at each Fermi energy, the spin split bands cancel their respective local BC except for those momenta which are occupied by one spin band. In the SU(3) system at hand, there is a similar sum rule stating that at each momentum k the BC of the three bands [c.f. Fig. 3a] sum to zero. However, and as mentioned above, the orbital bands are not subject to fermion multiplication theorems. In certain energy ranges a single orbital band is occupied [c.f., Fig. 3a, b] and BC cancellations are not at work. There is also another essential difference between the BC associated to spin and orbital degrees of freedom. In general, the commutation and anticommutation relations of the SU(N) Lie algebra define symmetric and antisymmetric structure constants, which, in turn, define the star and cross products of generic SU(N) vectors61. Differently from an SU(3) system spanning an angular momentum one subspace, in SU(2) spin systems the symmetric structure constant vanishes identically. The ensuing absence of star products bk⋆bk precludes the appearance of BC with time-reversal symmetry as long as crystalline anisotropies are not taken into account [see “Methods”]. On the other hand, for SU(3) the presence of all three purely imaginary Gell–Mann matrices Λ2,5,7, together with the “mass” terms Λ3,8, is a sufficient condition to obtain time-reversal symmetric BC concentrations even when accounting only for terms that are linear in momentum [see “Methods”]. This, however, strictly requires that all rotation symmetries must be broken.

Fig. 3: Orbital design of Berry curvature hot-spots and pinch points.
figure 3

a Ordering of the crystal field split t2g (p) orbitals with their associated band index. b Energy spectrum of the model Eq. (5) obtained using the parameter set \({{\Delta }}=-0.2{{{{\mathcal{E}}}}}_{0}\), \({{{\Delta }}}_{m}=0.01{{{{\mathcal{E}}}}}_{0}\), \({\alpha }_{R}={\alpha }_{m}=1.0{{{{\mathcal{E}}}}}_{0}/{k}_{F}^{0}\). c–e show the ensuing band-resolved Berry curvature. f–h are the corresponding BC dipole densities \({\partial }_{{k}_{x}}{{\Omega }}\). Note that the presence of mirror symmetry guarantees that the orthogonal dipole density \({\partial }_{{k}_{y}}{{\Omega }}\) averages to zero.

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Next, we analyze the properties of the band resolved local BC starting from the lowest energy band, which corresponds to the \(\left(\left\vert xy\right\rangle +\left\vert xz\right\rangle +\left\vert yz\right\rangle \right)/\sqrt{3}\) state at (111) LAO/STO heterointerfaces. Figure 3c shows a characteristic BC profile. It displays two opposite poles centered on the ky = 0 line. These sources and sinks of BC are equidistant from the mirror symmetric kx = 0 line since the BC, as any genuine pseudoscalar, must be odd under vertical mirror symmetry operations, i.e., Ω(kx, ky) = − Ω( − kx, ky). Note that the combination of time-reversal symmetry and vertical mirror implies that the BC will be even sending ky → − ky, thus guaranteeing that, taken by themselves, the BC hot-spots will be centered around the ky = 0 line. Their finite kx values coincide with the points where the (direct) energy gap between the n = 1 and the n = 2 bands is minimized [see Fig. 3a, b and Supplementary Note 2], and thus the interorbital mixing is maximal. The properties of the BC are obviously reflected in the BCD local density \({\partial }_{{k}_{x}}{{\Omega }}({k}_{x},{k}_{y})\): it possesses [see Fig. 3f] a positive area strongly localized at the center of the BZ that is neutralized by two mirror symmetric negative regions present at finite kx. Let us next consider the Berry curvature profile arising from the two degenerate \({e}_{g}^{{\prime} }\) states that are split by the threefold rotation symmetry breaking. Fig. 3d shows the BC profile of the lowest energy band: it is entirely dominated by the BC pinch points induced by the mirror symmetry protected degeneracies on the kx = 0 line. The BC also displays a nodal ring around the pinch point, and thus possesses a characteristic d-wave character around the singular point. This can be understood by constructing a k ⋅ p theory around each of the two time-reversal related degeneracies. To do so, we first recall that the two bands deriving from the \({e}_{g}^{{\prime} }\) states have opposite \({{{{\mathcal{M}}}}}_{x}\) mirror eigenvalue along the full mirror line kx ≡ 0 of the BZ. Close to the degeneracies, \({{{{\mathcal{M}}}}}_{x}\) can be therefore represented as σz. Under \({{{{\mathcal{M}}}}}_{x}\), kx → − kx, whereas ky → ky. Moreover, the Pauli matrices σx,y → − σx,y. An effective two-band model close to the degeneracies must then have the following form at the leading order:

$${{{{\mathcal{H}}}}}_{eff}={v}_{x}{k}_{x}\,{\sigma }_{x}+\beta {k}_{x}\,\delta {k}_{y}\,{\sigma }_{y}+{v}_{y}\delta {k}_{y}\,{\sigma }_{z},$$
(6)

where δky is the momentum measured relatively to the mirror symmetry-protected degeneracy and we have neglected the quadratic term coupling to the identity k2σ0 that does not affect the BC. Using the usual formulation of the BC for a two-band model [see the “Methods” section], it is possible to show that the Hamiltonian above is characterized by a zero-momentum pinch-point with two nodal lines [see the Supplementary Note 2] and d-wave character. It is interesting to note that this also implies that the effective time-reversal symmetry inverting the sign of k around the pinch point is broken. Perhaps even more importantly, the d-wave character implies a very large BCD density in the immediate neighborhood of the pinch point [see Fig. 3g]. Similar properties are encountered when considering the highest energy band, with the difference that the pinch-point has an opposite angular dependence [see Fig. 3e] and consequently the BCD density has opposite sign [c.f., Fig. 3h].

Having the band-resolved BC and BCD density profiles in our hands, we finally discuss their characteristic fingerprints in the BCD defined by \({D}_{x}={\int}_{{{{\bf{k}}}}}{\partial }_{{k}_{x}}{{\Omega }}({{{\bf{k}}}}){f}_{0}\), with ∫k = ∫d2k/(2π)2 and f0 being the equilibrium Fermi-Dirac distribution function. By continuously sweeping the Fermi energy, we find that the BCD shows cusps and inflection points [see Fig. 4a], which, as we now discuss, are a direct consequence of Lifshitz transitions and their associated van Hove singularities [see the Supplementary Note 3]. Starting from the bottom of the first band, the magnitude of the BCD continuously increases until it reaches a maximum where the dipole is larger than the inverse of the Fermi momentum of a 2DEG \(1/{k}_{F}^{0}\) and thus gets an enhancement of three order of magnitudes with respect to a Rashba 2DEG36. In this region, there are two distinct Fermi lines encircling electronic pockets at finite values of k [c.f., Fig. 4b], which subsequently merge on two disconnected regions in momentum space [c.f. Fig. 4c]. Since the states in the immediate vicinity of the center of the BZ are not occupied, the BCD is entirely dominated by the two mirror symmetric negative hot-spots of Fig. 3f. By further increasing the chemical potential, the internal Fermi line collapses at the Γ point and therefore a first Lifshitz transition occurs [c.f. Fig. 4d]. In this regime, the BCD has exponentially small values due to the fact that the strong positive BCD density area around the center of the BZ counteracts the mirror symmetric negative hot-spots. By further increasing the chemical potential, a second Lifshitz transition signals the occupation of the first eg band with two pockets centered around the ky = 0 line [see Fig. 4e]. This Lifshitz transition coincides with a rapid increase of the BCD due to the contribution coming from the local BCD density regions external to the BC nodal ring of Fig. 3g. The subsequent sharp negative peak originates from a third Lifshitz transition in which the two electronic pockets of the second band merge, and almost concomitantly a tiny pocket of the third band centered around Γ arises [see Fig. 4f]. By computing the band resolved BCD [see the Supplementary Note 4] one finds that it is this small pocket the cause of the negative sharp peak. For large enough chemical potentials, the BCD develops an additional peak corresponding to the fermiology of Fig. 4g. This peak, which is again larger than \(1/{k}_{F}^{0}\), can be understood by noticing that due to the BC local sum rule the momenta close to the center of the BZ do not contribute to the BCD. On the other hand, the regions external to the BC nodal ring are unoccupied by the third band and consequently have a net positive BCD local density. Thermal smearing can affect the strongly localized peaks at lower chemical potential but will not alter the presence of this broader peak. Note that the BCD gets amplified by increasing the interorbital mixing parameters αR, αm but retains similar properties [see Fig. 4 and the Supplementary Note 3]. The strength of BC-mediated effects depends indeed on the ratio between the characteristic orbital Rashba energy \(2m{\alpha }_{R(m)}^{2}/{\hslash }^{2}\) and the crystal field splittings Δ(m). The BCD properties and values comparable to the Fermi wavelength are hence completely generic.

Fig. 4: Berry curvature dipole and topology of the Fermi lines.
figure 4

a Behavior of the Berry curvature dipole obtained by changing the chemical potential μ measured in units of \({{{{\mathcal{E}}}}}_{0}\). The different curves correspond to the different values of αR measured in units of \({{{{\mathcal{E}}}}}_{0}/{k}_{F}^{0}\). The other model parameters have been instead fixed as \({{\Delta }}=-0.2{{{{\mathcal{E}}}}}_{0}\), \({{{\Delta }}}_{m}=-0.01{{{{\mathcal{E}}}}}_{0}\), αm = αR. b–f display the Fermi lines and the band-resolved occupied regions in momentum space for \({\alpha }_{R}={\alpha }_{m}=1.5{{{{\mathcal{E}}}}}_{0}/{k}_{F}^{0}\).

Full size image

Let us finally discuss the role of spin–orbit coupling. It can be included in our model Hamiltonian Eq. (5) as \({{{{\mathcal{H}}}}}_{{\rm{so}}}={\lambda }_{{\rm{so}}}\left({L}_{x}\otimes {\tau }_{x}+{L}_{y}\otimes {\tau }_{y}+{L}_{z}\otimes {\tau }_{z}\right)\), where λso is the spin–orbit coupling strength, the L = 1 angular momentum matrices correspond to the Gell–Mann matrices Λ2, Λ5, Λ7, and the Pauli matrices τx,y,z act in spin space. Its effect can be analyzed using conventional (degenerate) perturbation theory. At the center of the Brillouin zone, \({{{{\mathcal{H}}}}}_{{\rm{so}}}\) is completely inactive—the eigenstates of the Hamiltonian Eq. (5) are orbital eigenstates and the off-diagonal terms in orbital space Λ2,5,7 cannot give any correction at first order in λso. The situation is different at finite values of momentum. The two spin–orbit free degenerate eigenstates are a superposition of the different orbitals (due to the orbital Rashba coupling). Therefore, the spin–orbit coupling term will lift their degeneracy resulting in a Rashba-like splitting of the bands.

In order to explore the consequence of this spin splitting on the Berry curvature, let us denote with \(\left\vert {\psi }_{0}^{\uparrow }({{{\bf{k}}}})\right\rangle\) and \(\left\vert {\psi }_{0}^{\downarrow }({{{\bf{k}}}})\right\rangle\) the two spin–orbit free degenerate eigenstates at each value of the momentum. Note that \(\left\vert {\psi }_{0}\right\rangle\) is a three-component spinor for the orbital degrees of freedom. When accounting perturbatively for spin–orbit coupling the eigenstates will be a superposition of the spin degenerate eigenstates and will generally read

$$\begin{array}{r}\vert {\psi }^{+}({{{\bf{k}}}})\rangle =\cos \theta ({{{\bf{k}}}}){{{{\rm{e}}}}}^{i\phi ({{{\bf{k}}}})}\vert {\psi }_{0}^{\uparrow }({{{\bf{k}}}})\rangle +\sin \theta ({{{\bf{k}}}})\vert {\psi }_{0}^{\downarrow }({{{\bf{k}}}})\rangle \\ \left\vert {\psi }^{-}({{{\bf{k}}}})\right\rangle =-\sin \theta ({{{\bf{k}}}}){{{{\rm{e}}}}}^{i\phi ({{{\bf{k}}}})}\vert {\psi }_{0}^{\uparrow }({{{\bf{k}}}})\rangle +\cos \theta ({{{\bf{k}}}})\vert {\psi }_{0}^{\downarrow }({{{\bf{k}}}})\rangle \end{array}$$

Here, the momentum dependence of the phase ϕ and the angle θ is a by-product of the orbital Rashba coupling: the effect of spin–orbit coupling, which is off-diagonal in orbital space, is modulated by the momentum-dependent orbital content of the eigenstates \(\left\vert {\psi }_{0}^{\uparrow ,\downarrow }({{{\bf{k}}}})\right\rangle\). The abelian Berry connection of the two spin-split states \({{{{\mathcal{A}}}}}_{{k}_{x},{k}_{y}}^{+,-}=\left\langle {\psi }^{+,-}({{{\bf{k}}}})| i{\partial }_{{k}_{x},{k}_{y}}{\psi }^{+,-}({{{\bf{k}}}})\right\rangle\) will therefore contain two terms: the first one is the spin-independent Berry connection \({{{{\mathcal{A}}}}}_{{k}_{x},{k}_{y}}^{0}=\left\langle {\psi }_{0}({{{\bf{k}}}})| i{\partial }_{{k}_{x},{k}_{y}}{\psi }_{0}({{{\bf{k}}}})\right\rangle\); the second term is instead related to the derivatives of the phase ϕ and angle θ. This Berry connection is opposite for the +,− states and coincides with the Berry connection of a two-level spin system44. This also implies that the Berry curvature of a Kramers’ pair of bands Ω+,−(k) = Ω(k) ± Ωso(k). The contribution of the Berry curvature Ωso is opposite for the time-reversed partners and the net effect only comes from the difference between the Fermi lines of two partner bands. However, the purely orbital Berry curvature Ω(k), which can be calculated directly from Eq. (5), sums up. The values of the BCD presented in Fig. 4 are thus simply doubled in the presence of a weak but finite spin–orbit coupling.

Discussion

In this study, we have shown an intrinsic pathway to design large concentrations of Berry curvature in time-reversal symmetric conditions making use only of the orbital angular momentum electrons acquire when bound to atomic nuclei. Such mechanism is different in nature with respect to that exploited in topological semimetals and narrow-gap semiconductors where the geometric properties of the electronic wavefunctions originate from the coupling between electron and hole excitations. The orbital design of Berry curvature is also inherently different from the time-reversal symmetric spin–orbit mechanism35,36, which strongly relies on crystalline anisotropy terms. We have shown in fact that the Berry curvature triggered by orbital degrees of freedom features both hot-spots and singular pinch-points. Furthermore, due to the crystalline symmetry constraints the Berry curvature is naturally equipped with a non-vanishing Berry curvature dipole. These characteristics yield a boost of three orders of magnitude in the quantum non-linear Hall effect. In (111) LaAlO3-SrTiO3 heterointerfaces where the characteristic Fermi wavevector \({k}_{F}^{0}\simeq 1\) nm−1, the Berry curvature dipole Dx ≃ 1nm. The corresponding non-linear Hall voltage can be evaluated using the relation31,33\({V}_{yxx}={e}^{3}\,\tau \,{D}_{x}\,| {I}_{x}{| }^{2}/(2{\hslash }^{2}{\sigma }_{xx}^{2}W)\), with the characteristic relaxation time τ ≃ 1 pS and the longitudinal conductance σxx ≃ 5 mS. In a typical Hall bar of width W ≃ 10 μm sourced with a current Ix ≃ 100 μA, the non linear Hall voltage Vyxx ≃ 2 μV, which is compatible with the strong non-linear Hall signal experimentally detected36.

The findings of our study carry a dramatic impact on the developing area of condensed matter physics dubbed orbitronics62. Electrons in solids can carry information by exploiting either their intrinsic spin or their orbital angular momentum. Generation, detection and manipulation of information using the electron spin is at the basis of spintronics. The Berry curvature distribution we have unveiled in our study is expected to trigger also an orbital Hall effect, whose origin is rooted in the geometric properties of the electronic wavefunctions, and can be manipulated using the orbital degrees of freedom. This opens a number of possibilities for orbitronic devices. This is even more relevant considering that our findings can be applied to a wide class of materials whose electronic properties can be described with an effective L = 1 orbital multiplet. These include other complex oxide heterointerfaces as well as spin–orbit free semiconductors where p-orbitals can be exploited. Since Dirac quasiparticles are not required in the orbital design of Berry curvature, it is possible to reach carrier densities large enough to potentially exploit electron-electron and electron-phonon interactions effects in the control of Berry curvature-mediated effects. For instance, orbital selective metal-insulator transitions can be used to switch on and off the electronic transport channels responsible for the Berry curvature and its dipole. We envision that this capability can be used to design orbitronic and electronic transistors relying on the geometry of the quantum wavefunctions.

Methods

Representation of the Gell–Mann matrices in the symmetry groups

Apart from the identity matrix Λ0, the eight Gell–Mann matrices can be defined as

$$\begin{array}{rc}{{{\Lambda }}}_{1}=\left(\begin{array}{rcl}0&1&0\\ 1&0&0\\ 0&0&0\end{array}\right),&{{{\Lambda }}}_{2}=\left(\begin{array}{rcl}0&-i&0\\ i&0&0\\ 0&0&0\end{array}\right),\\ {{{\Lambda }}}_{3}=\left(\begin{array}{rcl}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right),&{{{\Lambda }}}_{4}=\left(\begin{array}{rcl}0&0&1\\ 0&0&0\\ 1&0&0\end{array}\right),\\ {{{\Lambda }}}_{5}=\left(\begin{array}{rcl}0&0&-i\\ 0&0&0\\ i&0&0\end{array}\right),&{{{\Lambda }}}_{6}=\left(\begin{array}{rcl}0&0&0\\ 0&0&1\\ 0&1&0\end{array}\right),\\ {{{\Lambda }}}_{7}=\left(\begin{array}{rcl}0&0&0\\ 0&0&-i\\ 0&i&0\end{array}\right),&\qquad \,\,{{{\Lambda }}}_{8}=\left(\begin{array}{rcl}\frac{1}{\sqrt{3}}&0&0\\ 0&\frac{1}{\sqrt{3}}&0\\ 0&0&\frac{-2}{\sqrt{3}}\end{array}\right).\end{array}$$

Let us now check the properties of these eight Gell–Mann matrices under time-reversal symmetry. Since we are considering electrons that are effectively spinless due to the SU(2) spin symmetry, the time-reversal operator can be represented as \({{{\mathcal{K}}}}\). Hence, the three Gell–Mann matrices Λ2, Λ5, Λ7 are odd under time-reversal, i.e., \({{{{\mathcal{T}}}}}^{-1}{{{\Lambda }}}_{2,5,7}{{{\mathcal{T}}}}=-{{{\Lambda }}}_{2,5,7}\), whereas the remaining matrices are even under time-reversal. Similarly, Λ1,2,3,8 are even under the vertical mirror symmetry whereas Λ4,5,6,7 are odd. Let us finally talk about the threefold rotational symmetry. Since the rotation symmetry operator \({{{{\mathcal{C}}}}}_{3}=\exp \left[2\pi i{{{\Lambda }}}_{7}/3\right]\), the transformation properties of the Gell–Mann matrices are determined by the commutation relations \(\left[{{{\Lambda }}}_{7},{{{\Lambda }}}_{i}\right]\). The commutation relations are listed as follows:

$$\begin{array}{ll}\left[{{{\Lambda }}}_{7},{{{\Lambda }}}_{1}\right]=i{{{\Lambda }}}_{4}&\left[{{{\Lambda }}}_{7},{{{\Lambda }}}_{2}\right]=i{{{\Lambda }}}_{5}\\ \left[{{{\Lambda }}}_{7},{{{\Lambda }}}_{4}\right]=-i{{{\Lambda }}}_{1}&\left[{{{\Lambda }}}_{7},{{{\Lambda }}}_{5}\right]=-i{{{\Lambda }}}_{2}\\ \left[{{{\Lambda }}}_{7},{{{\Lambda }}}_{6}\right]=2i\left(\frac{{{{\Lambda }}}_{3}}{2}-\frac{\sqrt{3}}{2}{{{\Lambda }}}_{8}\right)&\left[{{{\Lambda }}}_{7},\frac{{{{\Lambda }}}_{3}}{2}-\frac{\sqrt{3}}{2}{{{\Lambda }}}_{8}\right]=-2i{{{\Lambda }}}_{6}\\ \left[{{{\Lambda }}}_{7},{{{\Lambda }}}_{3}+\frac{{{{\Lambda }}}_{8}}{\sqrt{3}}\right]=0&\end{array}$$

The results above indicate that the three pairs of operators \(\left\{{{{\Lambda }}}_{1},{{{\Lambda }}}_{4}\right\}\), \(\left\{{{{\Lambda }}}_{2},{{{\Lambda }}}_{5}\right\}\), and \(\left\{{{{\Lambda }}}_{6},\frac{{{{\Lambda }}}_{3}}{2}-\frac{\sqrt{3}}{2}{{{\Lambda }}}_{8}\right\}\) behave as vector under the threefold rotation symmetry and therefore form two-dimensional IRREPS.

Berry curvature of S U(2) and S U(3) systems

For SU(2) systems, a generic Hamiltonian can be written in terms of Pauli matrices σi as \({{{\mathcal{H}}}}({{{\bf{k}}}})={d}_{0}({{{\bf{k}}}}){\sigma }_{0}+{{{\bf{d}}}}({{{\bf{k}}}})\cdot {{{\boldsymbol{\sigma }}}}\), where σ0 is the 2 × 2 identity matrix and the Pauli matrix vector σ = (σx, σyσz). The Berry curvature can be expressed in terms of d vector

$${{{\Omega }}}_{\pm }({{{\bf{k}}}})=\mp \frac{1}{2| {{{\bf{d}}}}({{{\bf{k}}}}){| }^{3}}{{{\bf{d}}}}({{{\bf{k}}}})\cdot [{\partial }_{{k}_{x}}{{{\bf{d}}}}({{{\bf{k}}}})\times {\partial }_{{k}_{y}}{{{\bf{d}}}}({{{\bf{k}}}})]\ .$$
(7)

For SU(3) system, we can proceed analogously using the Gell–Mann matrices introduced above. The Hamiltonian of a system described by three electronic degrees of freedom in a 3 × 3 manifold can be written as \({{{\mathcal{H}}}}({{{\bf{k}}}})={b}_{0}({{{\bf{k}}}}){{{\Lambda }}}_{0}+{{{\bf{b}}}}({{{\bf{k}}}})\cdot {{{\boldsymbol{\Lambda }}}}\), where b0(k) is a scalar and b(k) is an eight-dimensional vector. The Gell–Mann matrices satisfy an algebra that is a generalization of the SU(2). In particular, we have that

$${{{\Lambda }}}_{a}{{{\Lambda }}}_{b}=\frac{2}{3}{\delta }_{ab}+\left({d}_{abc}+i{f}_{abc}\right){{{\Lambda }}}_{c},$$
(8)

where repeated indices are summed over. In the equation above, we have introduced the antisymmetric and symmetric structure factors of SU(3) that are defined, respectively, as

$${f}_{abc}=-\frac{i}{4}{{{\rm{Tr}}}}([{{{\Lambda }}}_{a},{{{\Lambda }}}_{b}]{{{\Lambda }}}_{c})\ ,\quad {d}_{abc}=\frac{1}{4}{{{\rm{Tr}}}}(\{{{{\Lambda }}}_{a},{{{\Lambda }}}_{b}\}{{{\Lambda }}}_{c}).$$

From these, one defines three bilinear operations of SU(3) vectors: the dot (scalar) product v ⋅ w = vawa, the cross product (v × w)a = fabcvbwc, and the star product (v ⋆ w)a = dabcvbwc. The star product is a symmetric vector product which does not play any role for SU(2) since dabc = 0. Moreover, the band-resolved Berry curvature is given by48,61:

$$\begin{array}{lll} {\mathrm{\Omega}}_n ({{\mathbf{k}}}) &=& -4\frac{\left(\gamma_{{{\mathbf{k}}},n} {\mathbf{b}}_{{{\mathbf{k}}}} + {\mathbf{b}}_{{{\mathbf{k}}}}\star{\mathbf{b}}_{{{\mathbf{k}}}}\right)} {\left(3\gamma^{2}_{{{\mathbf{k}}},n} -\left|{\mathbf{b}}_{{{\mathbf{k}}}}\right|^{2}\right)^{3}} \cdot \left\{ \left[\gamma_{{{\mathbf{k}}},n}\partial_{k_{x}}{\mathbf{b}}_{{{\mathbf{k}}}}\right.\right. \\ & +& \left.\left. \partial_{k_{x}} ({\mathbf{b}}_{{{\mathbf{k}}}} \star {\mathbf{b}}_{{{\mathbf{k}}}})\right] \times [\gamma_{{{\mathbf{k}}},n} \partial_{k_{y}}{\mathbf{b}}_{{{\mathbf{k}}}} +\partial_{k_y} ({\mathbf{b}}_{{{\mathbf{k}}}} \star {\mathbf{b}}_{{{\mathbf{k}}}})] \right\} \end{array}$$
(9)

where we introduced \({\gamma }_{{{{\bf{k}}}},n}=\frac{2}{\sqrt{3}}| {{{{\bf{b}}}}}_{{{{\bf{k}}}}}| \cos \left({\theta }_{{{{\bf{k}}}}}+\frac{2\pi }{3}n\right)\), \({\theta }_{{{{\bf{k}}}}}=\frac{1}{3}\arccos \left[\frac{\sqrt{3}{{{{\bf{b}}}}}_{{{{\bf{k}}}}}\cdot \left({{{{\bf{b}}}}}_{{{{\bf{k}}}}}\star {{{{\bf{b}}}}}_{{{{\bf{k}}}}}\right)}{| {{{{\bf{b}}}}}_{{{{\bf{k}}}}}{| }^{3}}\right]\), and bk is a shorthand for b(k). Generally speaking, the Berry curvature in Eq. (9) can be split in two contributions as \({{{\Omega }}}_{n}({{{\bf{k}}}})={{{\Omega }}}_{n}^{(0)}({{{\bf{k}}}})+{{{\Omega }}}_{n}^{(\star )}({{{\bf{k}}}})\), with \({{{\Omega }}}_{n}^{(0)}({{{\bf{k}}}})=-4\frac{{({\gamma }_{{{{\bf{k}}}},n})}^{3}}{{(3{\gamma }_{{{{\bf{k}}}},n}^{2}-| {{{{\bf{b}}}}}_{{{{\bf{k}}}}}{| }^{2})}^{3}}{{{{\bf{b}}}}}_{{{{\bf{k}}}}}\cdot [{\partial }_{{k}_{x}}{{{{\bf{b}}}}}_{{{{\bf{k}}}}}\times {\partial }_{{k}_{y}}{{{{\bf{b}}}}}_{{{{\bf{k}}}}}]\) that strongly resembles the BC expression for SU(2) systems of Eq. (7). In trigonal systems described by the effective Hamiltonian in Eq. (4), we have that for any momentum k and for any value of the parameters, the b vector associated with the Hamiltonian is such that (γk,nbk + bk⋆bk) is always orthogonal to the vector in the curly braces in the expression of the Berry curvature Eq. (9). Hence Ωn(k) = 0. On the contrary, assuming a \({{{{\mathcal{C}}}}}_{s}\) point-group symmetry the effective Hamiltonian of Eq. (5) defines \({{{{\bf{b}}}}}_{{{{\bf{k}}}}}=\left(0,{\alpha }_{R}{k}_{y},{{\Delta }}+\frac{1}{2}{{{\Delta }}}_{m},0,{\alpha }_{R}{k}_{x},0,{\alpha }_{m}{k}_{x},\frac{{{\Delta }}}{\sqrt{3}}-\frac{\sqrt{3}}{2}{{{\Delta }}}_{m}\right)\), for which we get that \({{{\Omega }}}_{n}^{(0)}({{{\bf{k}}}})=0\), and the BC is substantially given by \({{{\Omega }}}_{n}^{(\star )}({{{\bf{k}}}})\). In other words, the terms obtained by doing the star product, bk⋆bk are those that yield the non-zero BC. We point out that the BC is proportional to the combination of parameters \({\alpha }_{R}^{2}{\alpha }_{m}\,\left(2{{\Delta }}+{{{\Delta }}}_{m}\right)\). A non-vanishing BC can be thus obtained even in the absence of the Gell–Mann matrix Λ8. This, on the other hand, would correspond to values of the crystal field splitting Δm = 2Δ/3 implying a very strong distortion of the crystal from the trigonal arrangement. The presence of the constant term ∝Λ8 is thus essential to describe systems with a parent high-temperature trigonal crystal structure.

Calculation of the Berry curvature dipole

The first moment of the Berry curvature, the Berry curvature dipole, for each energy band n is given by \({D}_{x,n}=\int\frac{{d}^{2}k}{{(2\pi )}^{2}}{\partial }_{{k}_{x}}{{{\Omega }}}_{n}({{{\bf{k}}}}){f}_{0}({{{\bf{k}}}})\), where f0(k) is the equilibrium Fermi–Dirac distribution function. At zero temperature, this expression can be rewritten as a line integral over the Fermi line

$${D}_{x,n}=\int\frac{{d}^{2}k}{{(2\pi )}^{2}}{{{\Omega }}}_{n}({{{\bf{k}}}})\frac{\partial {E}_{n}}{\partial {k}_{x}}\delta ({E}_{n}-\mu )\ ,$$
(10)

where En = En(k) (n = 1, 2, 3) are the energy bands and μ is the chemical potential. We have used the latter expression (10) to evaluate the BCD, where \({D}_{x}=\mathop{\sum }\nolimits_{n = 1}^{3}{D}_{x,n}\).