Abstract
The Berry curvature (BC)—a quantity encoding the geometric properties of the electronic wavefunctions in a solid—is at the heart of different Halllike transport phenomena, including the anomalous Hall and the nonlinear Hall and Nernst effects. In nonmagnetic quantum materials with acentric crystalline arrangements, local concentrations of BC are generally linked to singleparticle wavefunctions that are a quantum superposition of electron and hole excitations. BCmediated effects are consequently observed in twodimensional systems with pairs of massive Dirac cones and threedimensional bulk crystals with quartets of Weyl cones. Here, we demonstrate that in materials equipped with orbital degrees of freedom local BC concentrations can arise even in the complete absence of hole excitations. In these solids, the crystals fields appearing in very lowsymmetric structures trigger BCs characterized by hotspots and singular pinch points. These characteristics naturally yield giant BC dipoles and large nonlinear transport responses in timereversal symmetric conditions.
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Introduction
Quantum materials can be generally defined as those solidstate structures hosting physical phenomena which, even at the macroscopic scale, cannot be captured by a purely classical description^{1}. Among such quantum phenomena, those related to the geometric properties of the electronic wavefunctions play undoubtedly a primary role. In an Nband crystalline system, the cellperiodic 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 FubiniStudy metric^{2} that measures the infinitesimal distance between Bloch states at different points of the BZ. The imaginary part of this quantum geometric tensor^{3,4} corresponds to the wellknown Berry curvature (BC), which, when integrated over the full BZ, gives the Chern number cataloging twodimensional insulators^{5}. In metallic systems with partially filled bands, the BC summed over all occupied states can result in a nonvanishing Berry phase if the system breaks timereversal symmetry. This Berry phase regulates the intrinsic part of the anomalous Hall conductivity of magnetic metals^{6,7,8,9}.
Materials with an acentric crystal structure can possess nonvanishing concentrations of BC even if magnetic order is absent. Probing the BC of these noncentrosymmetric and nonmagnetic materials via charge transport measurements usually requires externally applied magnetic fields. For instance, in timereversal invariant Weyl semimetals, such as TaAs^{10}, the strong BC arising from the Weyl nodes can be revealed using the planar Hall effect^{11}—a physical consequence of the negative longitudinal magnetoresistance associated with the chiral anomaly of Weyl fermions^{12}. Recently, it has been also shown that the planar Hall effect can display an anomalous antisymmetric response^{13,14}, which, at least in twodimensional materials, is entirely due to an unbalance in the BC distribution triggered by the Zeemaninduced spin splitting of the electronic bands.
In the absence of external magnetic fields, a BC charge transport diagnostic for nonmagnetic materials requires to go beyond the linear response regime^{15,16,17,18,19}. Halllike currents appearing as a nonlinear (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 threedimensional systems, nonvanishing BCDs have been linked to the presence of tilted Weyl cones, and have been shown to exist both in typeI and in typeII Weyl semimetals^{20} such as MoTe_{2}^{21} and the ternary compound TaIrTe_{4}^{22,23}. Furthermore, the Rashba semiconductor BiTeI has been predicted to host a BCD that is strongly enhanced across its pressureinduced topological phase transitions^{24}.
In twodimensional 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 nonlinear Hall currents can exist in symmetry groups containing also rotational symmetries. In these cases nonlinear skew and sidejump scatterings are the origin of the phenomenon^{18,19,25}. The concomitant presence of spin–orbit coupled massive Dirac cones with substantial BC and unusually lowsymmetry crystalline environments have suggested the surface states of SnTe^{26} in the lowtemperature ferroelectric phase^{27}, monolayer transition metal dichalcogenides in the socalled 1T_{d} phase^{28,29,30}, and bilayer WTe_{2} as material structures hosting sizable BCDs^{31,32}. Spin–orbit free twodimensional materials, including monolayer and bilayer graphene, have been also put forward as materials with relatively large BCDs^{33}. 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 curvatures^{34}.
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 threedimensional strong topological insulators^{35} as well as conventional twodimensional electron gases (2DEG) with Rashba spin–orbit coupling^{36} 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 measurements^{36}, provides a new avenue for investigations. It shows in fact that Berry curvaturemediated 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 nonmagnetic 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 manybody 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 outofplane component of the spin textures^{37,38}. Consequently, the BC does not possess the characteristic ‘hotspots’ appearing in close proximity to near degeneracy between two bands where the Bloch wavefunctions are rapidly changing. The absence of such BC hotspots 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 timereversal 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 pseudospin one orbital degree of freedom can display BC hotspots 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 singlevalley twolevel 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 longrange magnetic order, the two Fermi surfaces must originate from one of the four timereversal invariant point of the Brillouin zone (BZ) (n_{1}b_{1} + n_{2}b_{2})/2 with b_{1,2} the two primitive reciprocal lattice vectors of the BZ and n_{1,2} = 0, 1. We emphasize that our arguments hold also for systems with multiple pairs of pockets centered at timereversal invariant momenta. Timereversal symmetry guarantees that the two bands will be Kramers’ degenerate at the timereversal 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 lowenergy 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 outofplane axis and \({{{\mathcal{T}}}}\) timereversal, and thus allows for local BC concentrations^{13}. 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 iσ_{x}^{39}. Momentum and spin transform under \({{{{\mathcal{C}}}}}_{3}\) and \({{{{\mathcal{M}}}}}_{x}\) as follows
where k_{±} = k_{x} ± ik_{y} and σ_{±} = σ_{x} ± iσ_{y}. Furthermore, the Hamiltonian must satisfy the timereversal symmetry constraint \({{{\mathcal{H}}}}({{{\bf{k}}}})={{{\mathcal{T}}}}{{{\mathcal{H}}}}({{{\bf{k}}}}){{{{\mathcal{T}}}}}^{1}\), with the timereversal 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 theorem^{40} and hence can occur only on the isolated surfaces of threedimensional strong topological insulators^{41}. And indeed \({{{\mathcal{H}}}}({{{\bf{k}}}})\) coincides with the effective Hamiltonian for the surface states of the topological insulators in the Bi_{2}Se_{3} material class^{39,42,43}. In a genuine twodimensional 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. Timereversal symmetry implies that terms quadratic in momentum are coupled to the identity matrix. Therefore, we arrive at the wellknown Hamiltonian of a twodimensional electron gas with Rashbalike spin–orbit coupling that reads
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 twodimensional plane at all momenta.
There are two different ways to lift the d vector outofplane and thus trigger a nonvanishing BC^{44}. The first one consists in introducing a constant mass Δσ_{z}. This term removes the Kramers’ degeneracy at the TRIM [see Fig. 1b] and therefore breaks timereversal invariance. It can be realized by externally applying an outofplane magnetic field or by inducing longrange magnetic order. The BC then generally displays an hotspot located at the TRIM and a circular symmetric distribution [see Fig. 1c]. Moreover, timereversal symmetry breaking implies that the Berry phase accumulated by electrons on the Fermi surface is nonvanishing^{6}. The second route explicitly takes into account trigonal warping terms which are cubic in momentum and couple to the Pauli matrix σ_{z}. Such terms preserve timereversal invariance, and thus create a BC distribution with an angular dependence such that the Berry phase accumulated over any symmetryallowed Fermi line cancels out^{35}. Perhaps more importantly, the BC triggered by crystalline anisotropy terms^{36} does not display a hotspot, thus suggesting that in systems with conventional quasiparticles and a single internal degree of freedom timereversal 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 hotspots is entirely allowed even in timereversal 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 coupling^{45,46,47}, are able to induce BC hot spots with timereversal 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 p_{z}, p_{y}, p_{x} orbital basis, the generators of the point group are represented by
The two p_{x,y} orbitals form a twodimensional irreducible representation (IRREP) whereas the p_{z} orbital represents a onedimensional 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 matrices^{48} Λ_{i} [see “Methods”] as
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 Λ_{i}. This means that they should belong to the same representation of the crystal point group^{49}. From the representation of the Λ_{i}’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
Here the parameter Δ quantifies the energetic splitting between the p_{x,y} doublet and the p_{z} singlet. The second term in the Hamiltonian corresponds instead to the pseudospin one massless Dirac Hamiltonian^{50,51} predicted to occur for instance in the kagome lattice with a staggered magnetic π flux^{50}. Pseudospin one Dirac fermions are not subject to any fermion multiplication theorem^{52}. 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 ℏ^{2}k^{2}Λ_{0}/(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 timereversal broken 2DEG, a direct computation [see Methods] shows that the BC associated to the Hamiltonian above is vanishing for all momenta. Breaking timereversal symmetry introducing a constant mass term ∝Λ_{7} 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
Nothing prevents to have the interorbital mixing terms ∝Λ_{2,5} with different amplitudes. Without loss of generality, in the remainder we will consider a single parameter α_{R}. In the Hamiltonian above, we have also neglected a constant term ∝Λ_{1}. For materials with an hightemperature trigonal structure, its amplitude Δ_{1} is expected to be of the same order of magnitude as Δ_{m}. In this regime [see the Supplementary Note 1], a term ∝Λ_{1} 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 p_{x,y} implying that all levels at the Γ point of the BZ are singly degenerate. Second, the two p_{x,y} orbitals have band degeneracies along the mirror symmetric k_{x} = 0 line of the BZ. Such mirrorsymmetry 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 nontrivial geometry of the electronic wavefunctions associated with the porbital manifold. Note that the BC also displays hotspots [see Fig. 1f] with BC sources and sinks averaging to zero on any mirror symmetric Fermi surface as mandated by timereversal invariance.
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 twodimensional d electron systems of t_{2g} orbital character such as SrTiO_{3}^{53,54}, KTaO_{3}^{55}, and SrVO_{3}based heterostructures. When compared to conventional semiconductor heterostructures, complex oxide interfaces consist of d electrons with different symmetries, a key element in determining their manybody ground states that include, notably, unconventional superconductivity^{56}. In the hightemperature cubic phases of these materials, the octahedral crystal field pins the lowenergy physics to a degenerate t_{2g} manifold, which spans an effective angular momentum one subspace, precisely as the p orbitals discussed above. We note that for the t_{2g} orbitals the spin–orbit coupling has a minus sign due to the effective orbital moment projection from L = 2 to the L_{eff} = 1 in the t_{2g} manifold. This overall sign however is not altering the results. The reduced symmetry at interfaces lift the t_{2g} 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 \(\left\vert {a}_{1g}\right\rangle =\left(\left\vert xy\right\rangle +\left\vert xz\right\rangle +\left\vert yz\right\rangle \right)/\sqrt{3}\) onedimensional 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 = e^{2πi/3}, form the twodimensional IRREP.
The energetic ordering of the levels depends on the microscopic details of the interface. For example, at the (111)LaAlO_{3}/SrTiO_{3} interface, xray absorption spectroscopy^{57} 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, lowtemperature 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 SrTiO_{3}. A structural transition occurring at around 105 K, from the cubic phase to a tetragonal structure^{58} [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 onedimensional IRREP [see Fig. 2e]. SrTiO_{3}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 TiO_{6} oxygenoctahedron antiferrodistortive rotations^{59}. In addition, below about 50 K, SrTiO_{3} and KTaO_{3} approach a ferroelectric instability that is accompanied by strong polar quantum fluctuations. This regime is characterized by a soft transverse phonon mode that involves offcenter displacement of the Ti ions with respect to the surrounding octahedron of oxygen ions^{60}, 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 twolevel spin system the local Berry curvature of the spinsplit bands, if nonvanishing, is opposite. Due to the concomitant presence of both spinbands 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) vectors^{61}. 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 b_{k}⋆b_{k} precludes the appearance of BC with timereversal 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 Λ_{2,5,7}, together with the “mass” terms Λ_{3,8}, is a sufficient condition to obtain timereversal 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.
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 k_{y} = 0 line. These sources and sinks of BC are equidistant from the mirror symmetric k_{x} = 0 line since the BC, as any genuine pseudoscalar, must be odd under vertical mirror symmetry operations, i.e., Ω(k_{x}, k_{y}) = − Ω( − k_{x}, k_{y}). Note that the combination of timereversal symmetry and vertical mirror implies that the BC will be even sending k_{y} → − k_{y}, thus guaranteeing that, taken by themselves, the BC hotspots will be centered around the k_{y} = 0 line. Their finite k_{x} 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 k_{x}. 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 k_{x} = 0 line. The BC also displays a nodal ring around the pinch point, and thus possesses a characteristic dwave character around the singular point. This can be understood by constructing a k ⋅ p theory around each of the two timereversal 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 k_{x} ≡ 0 of the BZ. Close to the degeneracies, \({{{{\mathcal{M}}}}}_{x}\) can be therefore represented as σ_{z}. Under \({{{{\mathcal{M}}}}}_{x}\), k_{x} → − k_{x}, whereas k_{y} → k_{y}. Moreover, the Pauli matrices σ_{x,y} → − σ_{x,y}. An effective twoband model close to the degeneracies must then have the following form at the leading order:
where δk_{y} is the momentum measured relatively to the mirror symmetryprotected degeneracy and we have neglected the quadratic term coupling to the identity k^{2}σ_{0} that does not affect the BC. Using the usual formulation of the BC for a twoband model [see the “Methods” section], it is possible to show that the Hamiltonian above is characterized by a zeromomentum pinchpoint with two nodal lines [see the Supplementary Note 2] and dwave character. It is interesting to note that this also implies that the effective timereversal symmetry inverting the sign of k around the pinch point is broken. Perhaps even more importantly, the dwave 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 pinchpoint has an opposite angular dependence [see Fig. 3e] and consequently the BCD density has opposite sign [c.f., Fig. 3h].
Having the bandresolved 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} = ∫d^{2}k/(2π)^{2} and f_{0} being the equilibrium FermiDirac 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 2DEG^{36}. 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 hotspots 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 hotspots. By further increasing the chemical potential, a second Lifshitz transition signals the occupation of the first e_{g} band with two pockets centered around the k_{y} = 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 α_{R}, α_{m} but retains similar properties [see Fig. 4 and the Supplementary Note 3]. The strength of BCmediated 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.
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 λ_{so} is the spin–orbit coupling strength, the L = 1 angular momentum matrices correspond to the Gell–Mann matrices Λ_{2}, Λ_{5}, Λ_{7}, and the Pauli matrices τ_{x,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 offdiagonal terms in orbital space Λ_{2,5,7} cannot give any correction at first order in λ_{so}. 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 Rashbalike 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 threecomponent 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
Here, the momentum dependence of the phase ϕ and the angle θ is a byproduct of the orbital Rashba coupling: the effect of spin–orbit coupling, which is offdiagonal in orbital space, is modulated by the momentumdependent orbital content of the eigenstates \(\left\vert {\psi }_{0}^{\uparrow ,\downarrow }({{{\bf{k}}}})\right\rangle\). The abelian Berry connection of the two spinsplit 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 spinindependent 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 twolevel spin system^{44}. This also implies that the Berry curvature of a Kramers’ pair of bands Ω^{+,−}(k) = Ω(k) ± Ω_{so}(k). The contribution of the Berry curvature Ω_{so} is opposite for the timereversed 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 timereversal 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 narrowgap 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 timereversal symmetric spin–orbit mechanism^{35,36}, which strongly relies on crystalline anisotropy terms. We have shown in fact that the Berry curvature triggered by orbital degrees of freedom features both hotspots and singular pinchpoints. Furthermore, due to the crystalline symmetry constraints the Berry curvature is naturally equipped with a nonvanishing Berry curvature dipole. These characteristics yield a boost of three orders of magnitude in the quantum nonlinear Hall effect. In (111) LaAlO_{3}SrTiO_{3} heterointerfaces where the characteristic Fermi wavevector \({k}_{F}^{0}\simeq 1\) nm^{−1}, the Berry curvature dipole D_{x} ≃ 1nm. The corresponding nonlinear Hall voltage can be evaluated using the relation^{31,33}\({V}_{yxx}={e}^{3}\,\tau \,{D}_{x}\, {I}_{x}{ }^{2}/(2{\hslash }^{2}{\sigma }_{xx}^{2}W)\), with the characteristic relaxation time τ ≃ 1 pS and the longitudinal conductance σ_{xx} ≃ 5 mS. In a typical Hall bar of width W ≃ 10 μm sourced with a current I_{x} ≃ 100 μA, the non linear Hall voltage V_{yxx} ≃ 2 μV, which is compatible with the strong nonlinear Hall signal experimentally detected^{36}.
The findings of our study carry a dramatic impact on the developing area of condensed matter physics dubbed orbitronics^{62}. 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 porbitals 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 electronelectron and electronphonon interactions effects in the control of Berry curvaturemediated effects. For instance, orbital selective metalinsulator 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 Λ_{0}, the eight Gell–Mann matrices can be defined as
Let us now check the properties of these eight Gell–Mann matrices under timereversal symmetry. Since we are considering electrons that are effectively spinless due to the SU(2) spin symmetry, the timereversal operator can be represented as \({{{\mathcal{K}}}}\). Hence, the three Gell–Mann matrices Λ_{2}, Λ_{5}, Λ_{7} are odd under timereversal, i.e., \({{{{\mathcal{T}}}}}^{1}{{{\Lambda }}}_{2,5,7}{{{\mathcal{T}}}}={{{\Lambda }}}_{2,5,7}\), whereas the remaining matrices are even under timereversal. Similarly, Λ_{1,2,3,8} are even under the vertical mirror symmetry whereas Λ_{4,5,6,7} 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:
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 twodimensional 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 σ_{i} as \({{{\mathcal{H}}}}({{{\bf{k}}}})={d}_{0}({{{\bf{k}}}}){\sigma }_{0}+{{{\bf{d}}}}({{{\bf{k}}}})\cdot {{{\boldsymbol{\sigma }}}}\), where σ_{0} is the 2 × 2 identity matrix and the Pauli matrix vector σ = (σ_{x}, σ_{y}σ_{z}). The Berry curvature can be expressed in terms of d vector
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 b_{0}(k) is a scalar and b(k) is an eightdimensional vector. The Gell–Mann matrices satisfy an algebra that is a generalization of the SU(2). In particular, we have that
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
From these, one defines three bilinear operations of SU(3) vectors: the dot (scalar) product v ⋅ w = v_{a}w_{a}, the cross product (v × w)_{a} = f_{abc}v_{b}w_{c}, and the star product (v ⋆ w)_{a} = d_{abc}v_{b}w_{c}. The star product is a symmetric vector product which does not play any role for SU(2) since d_{abc} = 0. Moreover, the bandresolved Berry curvature is given by^{48,61}:
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 b_{k} 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 (γ_{k,n}b_{k} + b_{k}⋆b_{k}) is always orthogonal to the vector in the curly braces in the expression of the Berry curvature Eq. (9). Hence Ω_{n}(k) = 0. On the contrary, assuming a \({{{{\mathcal{C}}}}}_{s}\) pointgroup 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, b_{k}⋆b_{k} are those that yield the nonzero 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 nonvanishing BC can be thus obtained even in the absence of the Gell–Mann matrix Λ_{8}. This, on the other hand, would correspond to values of the crystal field splitting Δ_{m} = 2Δ/3 implying a very strong distortion of the crystal from the trigonal arrangement. The presence of the constant term ∝Λ_{8} is thus essential to describe systems with a parent hightemperature 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 f_{0}(k) is the equilibrium Fermi–Dirac distribution function. At zero temperature, this expression can be rewritten as a line integral over the Fermi line
where E_{n} = E_{n}(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 \({D}_{x}=\mathop{\sum }\nolimits_{n = 1}^{3}{D}_{x,n}\).
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
All numerical codes in this paper are available from the corresponding authors upon reasonable request.
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Acknowledgements
A.D.C., M.C., and M.T.M. acknowledge support by the EU’s Horizon 2020 research and innovation program under Grant Agreement No. 964398 (SUPERGATE). A.D.C. acknowledges support by the Swiss State Secretariat for Education, Research and Innovation (SERI) under contract number MB22.00071, by the Gordon and Betty Moore Foundation (Grant No. 332 GBMF10451 to A.D.C.) and by the Netherlands Organisation for Scientific Research (NWO/OCW) as part of the VIDI [project 016.Vidi.189.061] and ENWGROOT [project TOPCORE] programs. We thank M. Gabay for valuable discussions.
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C.O. conceived and supervised the project. M.T.M. performed the computations with help from C.N. and M.C. M.T.M., C.N., A.C., M,C. and C. O. analysed the results and wrote the manuscript.
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Mercaldo, M.T., Noce, C., Caviglia, A.D. et al. Orbital design of Berry curvature: pinch points and giant dipoles induced by crystal fields. npj Quantum Mater. 8, 12 (2023). https://doi.org/10.1038/s4153502300545y
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DOI: https://doi.org/10.1038/s4153502300545y
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