## Main

When moving along closed paths, electrons can accumulate a geometric Berry phase related to the flux of a field, called the Berry curvature (BC), encoding the geometric properties of the electronic wavefunctions. In magnetic materials, the adiabatic motion of electrons around the Fermi surface provides such a Berry phase. It is directly observable since it governs the intrinsic part of the anomalous Hall conductivity1,2. Anomalous Hall effect measurements, therefore, represent a charge transport footprint of the intrinsic geometric structure of electronic wavefunctions. In non-magnetic materials, the BC field is forced to vanish by symmetry when summed over the occupied electronic states. However, local concentrations of positive and negative BC in momentum space are allowed by acentric crystalline arrangements3. This segregation of BC in different regions of momentum space appears whenever electronic states with different internal quantum numbers are coupled to each other by terms that linearly depend on crystalline momentum k. In these regions, the electronic bands typically resemble the dispersion relations of relativistic Dirac or Weyl fermions. The spin–orbit linear-in-k coupling between different spin states shapes the Dirac cones at the surfaces of three-dimensional topological insulators4,5 as well as the Weyl cones of topological semimetals6. Couplings between different atomic orbital and sublattice states, instead, give rise to the (gapped) Dirac cones of transitional metal dichalcogenides and graphene. Conceptually speaking, the appearance of BC beyond this Dirac/Weyl paradigm is entirely allowed. The fundamental conditions for the occurrence of BC only involve the crystalline geometry of a material, with no restrictions on the specific properties of its low-energy electronic excitations. Achieving this challenge is of great interest. First, it could, in principle, result in the coexistence of different mechanisms of BC generation. This could be used, in turn, to endow a single-material system with different BC-mediated effects, for instance, spin and orbital Hall effects. Second, searching for BCs without Dirac or Weyl cones might allow the design of materials with interplay of correlated and topological physics—an unexplored frontier in condensed-matter physics.

Here we reach these two milestones in the two-dimensional electron system (2DES) confined at (111)-oriented oxide interfaces, with a high-temperature trigonal crystalline structure. This model system satisfies the crystalline symmetry properties for a non-vanishing BC. The combination of spin–orbit coupling, orbital degrees of freedom associated with the low-energy t2g electrons, and crystal fields leads to the coexistence of a spin-sourced and orbital-sourced BC. The two sources are independently probed using two different charge transport diagnostic tools. The observation of the BC-mediated anomalous planar Hall effect (APHE)7,8 grants direct access to the spin-sourced BC, whereas nonlinear Hall transport measurements in time-reversal symmetric conditions9,10 detect an orbital-mediated Berry curvature dipole (BCD)—a quantity measured so far only in gapped Dirac systems9,10,11,12,13,14,15,16,17,18,19 and three-dimensional topological semimetals20,21,22,23,24,25. We identify (111)-oriented LaAlO3/SrTiO3 heterointerfaces as an ideal material system because their 2DES features many-body correlations and a two-dimensional superconducting ground state26,27,28,29,30.

We synthesize (111)-oriented LaAlO3/SrTiO3 heterostructures by pulsed laser deposition (Methods). The samples are lithographically patterned into Hall bars oriented along the two orthogonal principal in-plane crystallographic directions: the $$[\bar{1}10]$$ and $$[\bar{1}\bar{1}2]$$ axis (Fig. 1a). The sheet conductance and carrier density of the 2DES are controlled by electrostatic-field effects in a back-gate geometry (Fig. 1b). We source an oscillating current (Iω) with frequency ω/2π along each Hall bar, and concomitantly measure the longitudinal response as well as the first- or second-harmonic transverse voltages in a conventional lock-in detection scheme (Fig. 1a).

The non-trivial geometric properties of the electronic waves in the 2DES derive entirely from the triangular arrangement of the titanium atoms at the (111)-oriented LaAlO3/SrTiO3 interface (Fig. 1c). Together with the $${{{{\mathcal{M}}}}}_{\bar{1}10}$$ mirror-line symmetry, this yields a $${{{{\mathcal{C}}}}}_{3v}$$ crystallographic point group symmetry. As a result of this trigonal crystal field and the concomitant presence of spin–orbit coupling, the entire d-orbital manifold of Ti atoms located at the centre of the surface Brillouin zone (BZ) is split into five distinct Kramers’ pairs (Supplementary Note I). The energy bands of the pairs are shifted in momentum due to spin–orbit coupling. In their simplest form, they acquire a parabolic dispersion reminiscent of a Rashba 2DES (Fig. 1d). However, the trigonal crystal field brings about a specific hexagonal warping31,32 that has a twofold effect. First, for each time-reversal related pair of bands, the Fermi lines acquire a hexagonal ‘snowflake’ shape33. Second, and the most important, the spin texture in momentum space acquires a characteristic out-of-plane component34,35, with alternating meron and antimeron wedges respecting the symmetry properties of the crystal (Fig. 1e). This unique spin–momentum locking enables a non-vanishing local BC entirely generated by spin–orbit coupling (Supplementary Note I). The local BC of the spin-split bands of each pair cancel each other at the same crystal momentum. However, there is a region of crystal momenta populated by a single spin band. In this region (namely, the annulus between the two Fermi lines of the system), alternating positive and negative regions of non-vanishing BC are present (Fig. 1f).

Apart from the spin channel, an inherently different source of BC exists. In systems with orbital degrees of freedom, the lack of crystal centrosymmetry yields coupling that are linear in k, and mix different atomic orbital states. These orbital Rashba couplings36 are independent of the presence of spin–orbit coupling. Precisely as its spin counterpart, the orbital Rashba coupling can generate a finite BC37, but only when all the rotational symmetries are broken (Methods and Supplementary Note I). With a reduced $${{{{\mathcal{C}}}}}_{\mathrm{s}}$$ symmetry, low-lying t2g orbitals are split into three non-degenerate levels. The corresponding orbital bands then realize a gapped Rashba-like spectrum with protected crossings along the mirror-symmetric lines of the two-dimensional BZ (Fig. 1g). These characteristics result in the appearance of dipolar BC hotspots and singular pinch points (Fig. 1h). Such orbital sources of BC are fully active at the (111) oxide interfaces owing to the reduced low-temperature symmetries. The cubic-to-tetragonal structural phase transition38,39 occurring at 110 K breaks the three-fold rotational symmetry along the [111] direction. In addition, the tetragonal to locally triclinic structural distortions at temperatures below ~70 K together with the ferroelectric instability40 below 50 K are expected to strongly enhance the orbital Rashba strength.

The orbital-sourced BC is expected to be very stiff in response to externally applied in-plane magnetic fields due to the absence of symmetry-protected orbital degeneracies. In contrast, the spin-sourced BC is substantially more susceptible to planar magnetic fields. As shown in Fig. 2a,b, an in-plane magnetic field is capable of generating a BC hotspot within the Fermi surface annulus. This BC hotspot corresponds to a field-induced avoided level crossing between the two spin-split bands that occurs whenever the applied magnetic field breaks the residual crystalline mirror symmetry. The momentum-integrated net BC is then non-zero (Supplementary Note II), and yields a transverse Hall conductance satisfying the antisymmetric property σxyρyx = −1, even in the absence of any Lorentz force. This effect, theoretically predicted elsewhere7,8 and known as the APHE, is different in nature with respect to the conventional planar Hall effect, which is instead related to the anisotropy in the longitudinal magnetoresistance and thus characterized by a symmetric response, namely, σxy(B) = σxy(–B).

Figure 2c shows the transverse (Hall) resistance measured with a current applied along the $$[\bar{1}\bar{1}2]$$ crystal direction and with collinear current and magnetic field. This ensures a vanishing symmetric planar Hall effect7. At fields well below 4 T, a small signal increasing linearly with the field strength is detected. This feature can be attributed to an out-of-plane misalignment of the magnetic field smaller than 1.5° (Supplementary Note III). Above a magnetic-field threshold instead, a large transverse Hall signal sharply emerges (Extended Data Fig. 3). At even larger fields, this response saturates. Electrostatic gating is found to decrease the magnetic-field threshold and promotes a non-monotonic evolution of the response amplitude (Fig. 2d,e). The experimental features of this Hall response can be captured by considering a single pair of spin-split bands coupled to the external field by the Zeeman interaction. In this picture, the sudden onset of the transverse response is associated with the appearance of the BC hotspot inside the Fermi surface annulus occurring at a critical magnetic-field strength (Supplementary Note II). Magnetoconductance measurements in the weak antilocalization regime (Extended Data Fig. 4) show that the onset of the transverse Hall signal precisely coincides with the appearance of the spin-sourced BC hotspot (Extended Data Fig. 5). The non-monotonic behaviour of the transverse response as a function of electrostatic gating and magnetic-field strength can also be ascribed to the BC origin of the Hall response. The angular dependence of the transverse resistance (Fig. 2f) indicates a vanishing transverse linear conductivity when the planar magnetic field is along the $$[\bar{1}10]$$ direction, due to mirror symmetry $${{{{\mathcal{M}}}}}_{[\bar{1}10]}$$. This is independent of whether the driving current is along the $$[\bar{1}10]$$ or $$[\bar{1}\bar{1}2]$$ direction. Note that the two angular dependencies are related to each other by a 180° shift, in agreement with the Onsager reciprocity relations41.

The absence of linear conductivity makes this configuration the ideal regime to investigate the presence of nonlinear transverse responses, which are symmetry-allowed when the driving current is collinear with the magnetic field (Supplementary Note II). We have, therefore, performed systematic measurements of the second-harmonic (2ω) transverse responses (Fig. 3a,b) by sourcing the a.c. current along the $$[\bar{1}10]$$ direction. We have subsequently disentangled the field-antisymmetric $${R}_{{{{{y}}}},{{{\rm{as}}}}}^{2\omega }=\left[{R}_{{{{{y}}}}}^{2\omega }(B)\,-\,{R}_{{{{{y}}}}}^{2\omega }(-B)\right]/2$$ and field-symmetric $${R}_{{{{{y,\mathrm{sym}}}}}}^{2\omega }=\left[{R}_{{{{{y}}}}}^{2\omega }(B)\,+\,{R}_{{{{{y}}}}}^{2\omega }(-B)\right]/2$$ contributions, since they originate from distinct physical effects. In particular, the antisymmetric part contains a semiclassical contribution that only depends on the conventional group velocity of the carriers at the Fermi level (Supplementary Note II). Conversely, the symmetric part originates from the anomalous velocity term of the carriers. It is a purely quantum contribution and can be expressed in terms of a BCD. We observe the following features in Fig. 3a,b. The semiclassical antisymmetric contribution has a sudden onset above a characteristic magnetic field (Fig. 3a) that is sensitive to gating (Fig. 3c). The gate dependence displays a monotonic growth consistent with its physical origin. On the contrary, the symmetric contribution displays the typical non-monotonous gate and field-amplitude dependence (Fig. 3b,d) of BC-mediated effects. The gate dependence of the nonlinear symmetric contribution obtained by sourcing the current along the $$[\bar{1}\bar{1}2]$$ direction is instead strongly suppressed and featureless (Fig. 3e). This is consistent with a $$[\bar{1}10]$$-oriented BCD, which gives a vanishing response in this configuration. We note that the symmetric nonlinear transverse resistance has a characteristic quadratic current–voltage (IωV2ω), which—combined with the response at double the driving frequency—establishes its second-order nature (Fig. 3f).

The fact that only the symmetric contribution persists even in the zero-field limit (Fig. 3a,b) indicates the presence of a finite BCD in the absence of externally applied magnetic fields and thus of a nonlinear Hall effect in time-reversal symmetric conditions. To support the existence of a finite BCD with time-reversal symmetry, we have individually evaluated the dipole originating from the spin-sourced BC and the dipole related to the orbital-sourced BC (Methods). Figure 4a shows that in the entire parameter space of our low-energy theory model, the spin-sourced BCD is two orders of magnitude smaller than the orbital-sourced BCD. The latter exceeds the inverse characteristic Fermi wavenumber $$k_\mathrm{F}^{-1}$$ ≈ 0.5 nm. Besides the intrinsic contribution to the BCD, the nonlinear Hall response with time-reversal symmetry also contains disorder-induced contributions10,42 due to nonlinear skew and side-jump scattering. We experimentally access such contributions by measuring the longitudinal signal $${V}_{{{{{yyy}}}}}^{2\omega }$$ that is symmetry-allowed but does not possess any intrinsic BCD contribution. As displayed in Fig. 4b, the strong difference in amplitude between the longitudinal signal and transverse $${V}_{{{{{yxx}}}}}^{2\omega }$$ signal over a large driving-current range proves the absence of three-fold rotation symmetry as well as a nonlinear Hall effect completely dominated by the intrinsic BCD. The anisotropy between longitudinal and transverse nonlinear signals also allows us to exclude a leading role played by thermoelectric effects due to Joule heating (Fig. 4b, inset). We further observe that both longitudinal $${V}_{{{{{xxx}}}}}^{2\omega }$$ and transverse $${V}_{{{{{xyy}}}}}^{2\omega }$$ responses have an amplitude comparable with the longitudinal signal $${V}_{{{{{yyy}}}}}^{2\omega }$$, thus suggesting their disorder-induced nature. We point out that the finite amplitudes of $${V}_{{{{{xxx}}}}}^{2\omega }$$ and $${V}_{{{{{xyy}}}}}^{2\omega }$$ imply $${{{{\mathcal{M}}}}}_{\bar{1}10}$$ symmetry breaking (Supplementary Note II). This can be related to the mirror-breaking arrangements of the oxygen atoms caused by the antiferrodistortive octahedron rotations.43 It might also be due to the presence of structural domain patterns appearing at the cubic-to-tetragonal structural transition.

We have systematically verified the occurrence of a sizable nonlinear transverse response over the full range of sheet conductances and concomitantly observed a large difference between the two nonlinear transverse-conductivity tensor component χyxx and χxyy (Fig. 4c). This further proves the main intrinsic BCD contribution to the nonlinear Hall response. By further evaluating the momentum relaxation time τ (Supplementary Note II), we can estimate the size of the BCD (Methods):

$${D}_{{{{{x}}}}}=\frac{2{\hslash }^{2}}{{e}^{3}\tau }\,{\chi }_{{{{{yxx}}}}}\,.$$
(1)

The resulting BCD (Fig. 4d) is two orders of magnitude larger than the dipole observed in systems with massive Dirac fermions, such as bilayer WTe2 (refs. 11,12) and—over a finite density range—a factor of two larger than the dipole observed in corrugated bilayer graphene13. We attribute the large magnitude of this effect to the fact that the orbital-sourced BC is naturally equipped with a large dipolar density due to the presence of singular pinch points and hotspots with dipolar arrangements. We also monitored the temperature dependence of transverse-conductivity tensor components χyxx and χxyy (Fig. 4e) and the corresponding behaviour of BCD Dx (Fig. 4f). All these quantities rapidly drop approaching 30 K, that is, the temperature above which the strong polar quantum fluctuations of SrTiO3 vanish. This further establishes the orbital Rashba coupling as the physical mechanism behind the orbital-sourced BC.

The pure orbital-based mechanism of BCD featured here paves the way to the atomic-scale design of quantum sources of nonlinear electrodynamics persisting up to room temperature. Oxide-based 2DES could be, for instance, combined with a room-temperature polar ferroelectric layer, triggering symmetry lowering and thus inducing orbital Rashba coupling by interfacial design. This and other alternative platforms combining a low-symmetry crystal with orbital degrees of freedom and polar modes, including room-temperature polar metals44 and conducting ferroelectric domain walls, are candidate oxide architectures to perform operations such as rectification45 and frequency mixing. Moreover, multiple sources of BC can be implemented for combined optoelectronic and spintronic functionalities in a single-material system: photogalvanic currents due to the orbital-sourced BC can be employed to create spin Hall voltages exploiting the spin-sourced BC. Our study also establishes a general approach to generate topological charge distributions in strongly correlated materials, opening a vast space for exploration at the intersection between topology and correlations.

## Methods

### Sample growth

The nine-unit-cell-thick LaAlO3 crystalline layer is grown on the TiO-rich surface of a (111)-oriented SrTiO3 substrate, from the ablation of a high-purity (>99.9%) LaAlO3 sintered target by pulsed laser deposition using a KrF excimer laser (wavelength, 248 nm). We perform the real-time monitoring of growth by following intensity oscillations, in a layer-by-layer growth mode, of the first diffraction spot using reflection high-energy electron diffraction (Extended Data Fig. 7a). This allows us to stop the growth at precisely the critical thickness of nine unit cells of LaAlO3 (ref. 46) necessary for the (111)-oriented LaAlO3/SrTiO3 2DES to form. The SrTiO3(111) substrate was first heated to 700 °C in an oxygen partial pressure of 6 × 10−5 mbar. The LaAlO3 layer was grown in those conditions at a laser fluence of 1.2 J cm−2 and laser repetition rate of 1 Hz. Following the growth of the LaAlO3 layer, the temperature is ramped down to 500 °C before performing one-hour-long in situ annealing in a static background pressure of 300 mbar of pure oxygen, to recover the oxygen stoichiometry of the reduced heterostructure. Finally, the sample is cooled down at –20 °C min−1, and kept in the same oxygen environment at zero heating power for at least 45 min.

### Device fabrication

The (111)-oriented LaAlO3/SrTiO3 blanket films were lithographically patterned into two Hall bars (width W = 40 μm; length L = 180 μm), oriented along the two orthogonal crystal-axis directions of $$[\bar{1}10]$$ and $$[\bar{1}\bar{1}2]$$. The Hall bars are defined by electron-beam lithography into a poly(methyl methacrylate) resist, which is used as a hard mask for argon-ion milling (Extended Data Fig. 7c). The dry-etching duration is calibrated and timed to be precisely stopped when the LaAlO3 layer is fully removed to avoid the creation of an oxygen-deficient conducting SrTiO3−δ surface. This leaves an insulating SrTiO3 matrix surrounding the protected LaAlO3/SrTiO3 areas, which host a geometrically confined 2DES.

### Electrical transport measurements

The Hall bars are connected to a chip carrier by an ultrasonic wedge-bonding technique in which the aluminium wires form ohmic contacts with the 2DES through the LaAlO3 overlayer. The sample is anchored to the chip carrier by homogeneously coating the backside of the SrTiO3 substrate with silver paint. A d.c. voltage Vg is sourced between the silver back-electrode and the desired Hall-bar device to enable electrostatic-field-effect gating of the 2DES, leveraging the large dielectric permittivity of strontium titanate at low T (~2 × 104 below 10 K)47,48. Non-hysteretic dependence of σxx (σyy) on Vg is achieved following an initial gate-forming procedure49.

Standard four-terminal electrical (magneto-)transport measurements were performed at 1.5 K in a liquid helium-4 flow cryostat, equipped with a superconducting magnet (maximum magnetic field, B = ±12 T). An a.c. excitation current Iω |Iω|sin(ωt), of frequency ω/(2π) = 17.77 Hz, is sourced along the desired crystallographic direction. The sheet resistance, $${R}_{{{{\rm{s}}}}}={\sigma }_{{{{{xx}}}}}^{-1}$$, of a Hall-bar device is related to the first-harmonic longitudinal voltage drop Vxx according to Rs = (Vxx/Ix)(W/L). When the a.c. current is sourced along $$\hat{{{{\bf{x}}}}}\parallel [\bar{1}10]$$ ($$\hat{{{{\bf{y}}}}}\parallel [\bar{1}\bar{1}2]$$), we make use of a standard lock-in detection technique to concomitantly measure the first-harmonic longitudinal response Vxx (Vyy), and either the in-phase first-harmonic $${V}_{{{{{xy}}}}}^{\omega }$$ ($${V}_{{{{{yx}}}}}^{\omega }$$) or out-of-phase second-harmonic $${V}_{{{{{yxx}}}}}^{2\omega }$$ ($${V}_{{{{{xyy}}}}}^{2\omega }$$) transverse voltages (Fig. 1a). We define the first- and second-harmonic transverse resistances as $${R}_{{{{{xy}}}}}^{\omega }={V}_{{{{{xy}}}}}^{\omega }/| {I}_{{{{{x}}}}}^{\omega }|$$ and $${R}_{{{{{y}}}}}^{2\omega }={V}_{{{{{yxx}}}}}^{2\omega }/| {I}_{{{{{x}}}}}^{\omega }{| }^{2}$$, respectively. First- and second-harmonic measurements are performed at 10 and 50 μA, respectively.

We systematically decompose both first- and second-harmonic magneto-responses into their field-symmetric $${R}_{{{{\rm{sym}}}}}^{(2)\omega }$$ and field-antisymmetric $${R}_{{{{\rm{as}}}}}^{(2)\omega }$$ contributions according to

$${R}_{{{{\rm{sym}}}}}^{(2)\omega }=\left[{R}^{(2)\omega }(B)+{R}^{(2)\omega }(-B)\right]/2\,,$$
(2a)
$${R}_{{{{\rm{as}}}}}^{(2)\omega }=\left[{R}^{(2)\omega }(B)-{R}^{(2)\omega }(-B)\right]/2\,.$$
(2b)

In particular, the first-harmonic transverse resistance is purely field antisymmetric, and hence, we chose the simplified notation of $${R}_{{{{{xy}}}}}\equiv {R}_{{{{{xy}}}},{{{\rm{as}}}}}^{\omega }$$.

### Estimation of the Rashba spin–orbit energy from magnetoconductance measurements in the weak antilocalization regime

In a 2DES, in the presence of a spin relaxation mechanism induced by an additional spin–orbit interaction, the conductance is subject to weak localization corrections at lower temperatures. Extended Data Fig. 4a shows the gate-modulated magnetoconductance curves of the 2DES, which exhibit a characteristic low-field weak antilocalization behaviour. The magnetoconductance curves, normalized to the quantum of conductance GQ = e2/(πħ), are fitted using a Hikami–Larkin–Nagaoka model that expresses the change in conductivity Δσ(B) = σ(B) – σ(0) of the 2DES under an external out-of-plane magnetic field B, in the diffusive regime (with negligible Zeeman splitting), as follows50,51:

$$\begin{array}{ll}\frac{{{\Delta }}\sigma ({B}_{\perp })}{{G}_{{{{\rm{Q}}}}}}&=-\frac{1}{2}{{\varPsi }}\left(\frac{1}{2}+\frac{{B}_{{{{\rm{i}}}}}}{{B}_{\perp }}\right)+\frac{1}{2}\ln \left(\frac{{B}_{{{{\rm{i}}}}}}{B}\right)\\ &+{{\varPsi }}\left(\frac{1}{2}+\frac{{B}_{{{{\rm{i}}}}}+{B}_{{{{\rm{so}}}}}}{{B}_{\perp }}\right)-\ln \left(\frac{{B}_{{{{\rm{i}}}}}+{B}_{{{{\rm{so}}}}}}{{B}_{\perp }}\right)\\ &+\frac{1}{2}{{\varPsi }}\left(\frac{1}{2}+\frac{{B}_{{{{\rm{i}}}}}+2{B}_{{{{\rm{so}}}}}}{{B}_{\perp }}\right)-\frac{1}{2}\ln \left(\frac{{B}_{{{{\rm{i}}}}}+2{B}_{{{{\rm{so}}}}}}{{B}_{\perp }}\right)\\ &-{A}_{{{{{\rm{K}}}}}}\frac{\sigma (0)}{{G}_{{{{\rm{Q}}}}}}{B}_{\perp }^{2}\,\end{array},$$
(3)

where Ψ is the digamma function; ħ = h/(2π) is the reduced Planck constant; $${B}_{{{{\rm{i}}}},{{{\rm{so}}}}}=\hslash /\left(4eD{\tau }_{{{{\rm{i}}}},{{{\rm{so}}}}}\right)$$ are the effective fields related to the inelastic and spin–orbit relaxation times (τi and τso, respectively); and D = πħ2σ(0)/(e2m*) is the diffusion constant. The last term in equation (3), proportional to $${B}_{\perp }^{2}$$, contains AK, the so-called Kohler coefficient, which accounts for orbital magnetoconductance.

Hence, from the fit to the weak antilocalization magnetoconductance curves, the effective Rashba spin–orbit coupling αR can be calculated as

$${\alpha }_{{{{\rm{R}}}}}={\hslash }^{2}/\left[2{m}^{* }\sqrt{\left(D{\tau }_{{{{\rm{so}}}}}\right)}\right]\,,$$
(4)

based on a D’yakonov–Perel’ spin relaxation mechanism51. A summary of the dependence of the extracted parameters on the 2DES’ sheet conductance is plotted in Extended Data Fig. 5b. The spin–orbit energy Δso can then be estimated according to

$${{{\varDelta }}}_{{{{\rm{so}}}}}=2{\alpha }_{{{{\rm{R}}}}}{k}_{{{{\rm{F}}}}}\,,$$
(5)

where, in two dimensions, the Fermi wavevector is given by $${k}_{{{{\rm{F}}}}}=\sqrt{2\uppi {n}_{2{{{\rm{D}}}}}}$$, assuming a circular Fermi surface. The sheet carrier density n2D is experimentally obtained for each doping value from the (ordinary) Hall effect (Supplementary Note III), measured concomitantly with the magnetoconductance traces.

### Spin-sourced and orbital-sourced BCD calculations

We first estimate the BCD due to spin sources in time-reversal symmetry condition as a function of carrier density considering the low-energy Hamiltonian for a single Kramers’-related pair of bands (Supplementary Note I):

$${{{\mathcal{H}}}}=\frac{{{{{\bf{k}}}}}^{2}}{2m({{{\bf{k}}}})}-{\alpha }_{{{{\rm{R}}}}}\,{{{\bf{\sigma }}}}\cdot {{{\bf{k}}}}\times \hat{{{{\bf{z}}}}}+\frac{\lambda }{2}({k}_{+}^{3}+{k}_{-}^{3}){\sigma }_{{{{{z}}}}},$$
(6)

where the momentum-dependent mass can be negative close to the Γ point (Supplementary Note I). Although this model Hamiltonian is equipped with a finite BC, its dipole is forced to vanish by the three-fold rotation symmetry (Supplementary Note I). We capture the rotation symmetry breaking of the low-temperature structure at the leading order by assuming inequivalent coefficients for the spin–orbit coupling terms linear in momentum. In other words, we make the substitution αR(σxky – σykx)→vykyσx – vxkxσy. Since the dipole is a pseudo-vector, the residual mirror symmetry $${{{{\mathcal{M}}}}}_{x}$$ forces it to be directed along the $$\hat{{{{\bf{x}}}}}$$ direction. In the relaxation-time approximation, it is given by

$${D}_{x}={\int}_{{{{\bf{k}}}}}{\partial }_{{k}_{x}}{{{\varOmega }}}_{{{{{z}}}}}({{{\bf{k}}}}),$$
(7)

where Ωz is the BC of our two-band model that we write in a dimensionless form by measuring energies in units of $${k}_{{{{\rm{F}}}}}^{2}/2m({k}_{{{{\rm{F}}}}})$$, lengths in units of 1/kF and densities in units of $${n}_{0}={k}_{{{{\rm{F}}}}}^{2}/2\uppi$$. Here kF is a reference Fermi wavevector. For simplicity, we have considered a positive momentum-independent effective mass. For the BCD shown in Fig. 4a, the remaining parameters have been chosen as vx = 0.4, vy = (1.2, 1.4, 1.6) × vx and λ = 0.1. Moreover, we account for orbital degeneracy by tripling the dipole of a single Kramers’ pair. This gives an upper bound for the spin-sourced BCD.

We have also evaluated the BCD due to orbital sources considering the low-energy Hamiltonian for spin–orbit-free t2g electrons derived from symmetry principles (Supplementary Note I) and reading

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

where we introduced the Gell–Mann matrices as

$$\begin{array}{l}{{{\varLambda }}}_{2}=\left(\begin{array}{rcl}0&-i&0\\ i&0&0\\ 0&0&0\end{array}\right){{{\varLambda }}}_{3}=\left(\begin{array}{rcl}1&0&0\\ 0&-1&0\\ 0&0&0\end{array}\right)\\ {{{\varLambda }}}_{5}=\left(\begin{array}{rcl}0&0&-i\\ 0&0&0\\ i&0&0\end{array}\right){{{\varLambda }}}_{7}=\left(\begin{array}{rcl}0&0&0\\ 0&0&-i\\ 0&i&0\end{array}\right)\\ {{{\varLambda }}}_{8}=\left(\begin{array}{lll}\frac{1}{\sqrt{3}}&0&0\\ 0&\frac{1}{\sqrt{3}}&0\\ 0&0&\frac{-2}{\sqrt{3}}\end{array}\right)\end{array},$$

and Λ0 is the identity matrix. In the Hamiltonian above, Δ is the splitting between the a1g singlet and $${e}_{\mathrm{g}}^{{\prime} }$$ doublet resulting from the t2g orbitals in a trigonal crystal field. Here Δm is the additional splitting between the doublet caused by rotational symmetry breaking. Finally, αOR and αm are the strengths of the orbital Rashba coupling. Note that in the presence of three-fold rotation symmetry, αm ≡ 0, in which case the BC is forced to vanish. For simplicity, we have evaluated the BC for the $${{{{\mathcal{C}}}}}_{\mathrm{s}}$$ point group-symmetric case assuming αm ≡ αOR. In our continuum SU(3) model, the BC can be computed using the method outlined elsewhere52. We have subsequently computed the corresponding dipole measuring, as before, energies in units of $${k}_{{{{\rm{F}}}}}^{2}/2m$$, lengths in units of 1/kF and densities in units of $${n}_{0}={k}_{{{{\rm{F}}}}}^{2}/2\uppi$$. The dimensionless orbital Rashba coupling has been varied between αOR = 1 and αOR = 2, whereas we have fixed Δ = –0.1 and Δm = 0.005. The value of the crystal field splitting Δ is consistent with the amplitude determined by X-ray absorption spectroscopy53 of the order 8 meV, and therefore, it is almost one order of magnitude smaller than our energy unit of ~40 meV for a reference $${k}_{\mathrm{F}}^{-1}$$ 0.5 nm and effective mass m 3me (Supplementary Note III). The calculated dipole (Fig. 4a) has been finally multiplied by two to account for spin degeneracy. As shown in Supplementary Note I, we remark that the model Hamiltonian for the spin sources of BC (equation (6)) and the model Hamiltonian for the orbital sources (equation (8)) derive from a single six-band model where orbital and spin degrees of freedom are treated on an equal footing.

### Estimation of BCD magnitude from nonlinear Hall measurements

The nonlinear current density is mathematically given by $${j}_{\alpha }^{2\omega }={\chi }_{\alpha \beta \gamma }\,{E}_{\beta }\,{E}_{\gamma }$$, where χαβγ is the nonlinear transverse-conductivity tensor. When an a.c. current density $${I}_{{{{{x}}}}}^{\omega }/W={\sigma }_{{{{{xx}}}}}{E}_{{{{{x}}}}}^{\omega }$$ is sourced along $$\hat{{{{\bf{x}}}}}$$, the second-harmonic transverse current density developing along $$\hat{{{{\bf{y}}}}}$$ is related to the BCD D according to9

$${{{{\bf{j}}}}}_{{{{{y}}}}}^{2\omega }\,=\,\frac{{e}^{3}\tau }{2{\hslash }^{2}(1+\mathrm{i}\omega \tau )}\left(\hat{{{{\bf{z}}}}}\times {{{{\bf{E}}}}}_{{{{{x}}}}}^{\omega }\right)\left({{{\bf{D}}}}\cdot {{{{\bf{E}}}}}_{{{{{x}}}}}^{\omega }\right)\,,$$
(9)

where τ is the momentum relaxation time and e is the elementary charge. Due to the mirror symmetry $${{{{\mathcal{M}}}}}_{{{{{x}}}}}\equiv {{{{\mathcal{M}}}}}_{[\bar{1}10]}$$, the dipole is found to point along $$\hat{{{{\bf{x}}}}}$$; in the quasi-d.c. limit, that is, (ωτ)  1, the BCD expression reduces to

$${D}_{{{{{x}}}}}=\frac{2{\hslash }^{2}}{{e}^{3}\tau }\frac{{j}_{{{{{y}}}}}^{2\omega }}{{\left({E}_{{{{{x}}}}}^{\omega }\right)}^{2}}=\frac{2{\hslash }^{2}}{{e}^{3}\tau }\frac{{V}_{{{{{yxx}}}}}^{2\omega }\,{\sigma }_{xx}^{3}\,W}{| {I}_{{{{{x}}}}}^{\omega }{| }^{2}},$$
(10)

which is the explicit expression for equation (1), in terms of experimentally measurable quantities only, and where

$${\chi }_{{{{{yxx}}}}}=\frac{{j}_{{{{{y}}}}}^{2\omega }}{{\left({E}_{{{{{x}}}}}^{\omega }\right)}^{2}},$$
(11a)
$${\chi }_{{{{{xyy}}}}}=\frac{{j}_{{{{{x}}}}}^{2\omega }}{{\left({E}_{{{{{y}}}}}^{\omega }\right)}^{2}},$$
(11b)

are the measured nonlinear transverse-conductivity tensor elements shown in Fig. 4c,e.