Berry Curvature and Quantum Metric in N-band systems -- an Eigenprojector Approach
EEigenprojectors, Bloch vectors and quantum geometry of N -band systems Ansgar Graf * , Fr´ed´eric Pi´echon † Universit´e Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France * [email protected] † [email protected] 22, 2021 Abstract
The eigenvalues of a parameter-dependent N × N Hamiltonian matrix form aband structure in parameter space. Quantum geometric properties (Berry curva-ture, quantum metric, etc.) of such N -band systems are usually computed fromparameter-dependent eigenstates. This approach faces several difficulties, includ-ing gauge ambiguities and singularities in the multicomponent eigenfunctions.In order to circumvent this problem, this work exposes an alternative approachbased on eigenprojectors and (generalized) Bloch vectors. First, an expansion ofeach eigenprojector as a matrix polynomial in the Hamiltonian is deduced, andusing SU( N ) Gell-Mann matrices an equivalent expansion of each Bloch vector isfound. In a second step, expressions for the N -band Berry curvature and quan-tum metric in terms of Bloch vectors are obtained. This leads to new explicitBerry curvature formulas in terms of the Hamiltonian vector, generalizing thewell-known two-band formula to arbitrary N . Moreover, a detailed treatment isgiven for the case of a particle-hole symmetric energy spectrum, which occurs insystems with a chiral or charge conjugation symmetry. For illustrating the for-malism, several model Hamiltonians featuring a multifold linear band crossing arediscussed; they have identical energy spectra but completely different geometricand topological properties. The methodology used in this work is more broadlyapplicable to compute any physical quantity, or to study the quantum dynamicsof any observable without the explicit construction of energy eigenstates. Contents N -band systems 10 P α ( H, E α ) . . . . . . . . . 103.2 Generalized Bloch vectors: b α ( h , E α ) . . . . . . . . . . . . . . . . . . . . . . . . 121 a r X i v : . [ c ond - m a t . o t h e r] F e b Quantum geometric tensor without eigenstates 16 N -band Berry curvature from the Hamiltonian vector . . . . . . . . . . . . . . 18 N -band systems with particle-hole symmetric energy spectrum 19 S multifold fermions and beyond . . . . . . . . . . . . . . . . . . . 246.2.1 Pseudospin- S multifold fermions . . . . . . . . . . . . . . . . . . . . . . 256.2.2 Beyond spin- S multifold fermions . . . . . . . . . . . . . . . . . . . . . . 27 In a quantum mechanical system characterized by an N -dimensional Hilbert space, a partic-ularly frequent scenario is the one where the N × N Hamiltonian matrix H ( x ) depends ona set of parameters x , say the crystal momentum in solid state physics, the intensity of anexternal field, an applied strain or any mean-field order parameter. A Hamiltonian of thiskind has a set of eigenvalues E α ( x ) that form a band structure, and may therefore be referredto as an N -band system , or equivalently a (parametric) SU( N ) system – these designationswill be used synonymously during this work.In this paper, we will be mainly concerned with the (quantum) geometric properties of N -band systems. The concept of quantum geometry can be viewed as originating from theoverlap of energy eigenstates | ψ α ( x ) (cid:105) (assumed to be non-degenerate and labeled by α ∈{ , ..., N } ) that are infinitesimally close in parameter space: (cid:104) ψ α ( x ) | ψ α ( x + d x ) (cid:105) . Obviously, aTaylor expansion of this overlap will produce terms determined by derivatives ∂/∂x i acting onenergy eigenstates | ψ α ( x ) (cid:105) . Hence arises the conventional point of view on quantum geometry,which regards quantum geometric information as being(i) encoded in rates of changes of energy eigenstates | ψ α ( x ) (cid:105) upon variation of x .This perspective is of considerable conceptual value, but leads to several issues in practicalcomputations (see Section 2). An equivalent point of view, which is adopted more rarely,consists in saying that quantum geometry is(ii) encoded in rates of changes of eigenprojectors P α ( x ) = | ψ α ( x ) (cid:105) (cid:104) ψ α ( x ) | . This is due to the mathematical fact that the Hermitian matrix H ( x ) is an element of the Lie algebra su ( N ),closely related to the Lie group SU( N ), and therefore H ( x ) may be called a (parametric) SU( N ) Hamiltonian . H ( x ) = (cid:80) Nα =1 E α ( x ) P α ( x ), a third point of view may be predicted: If we caninvert the spectral decomposition to find a unique function P α ( x ) = P α ( E α ( x ) , H ( x )), it isclear from point of view (ii) that quantum geometry can also be regarded as being(iii) encoded in rates of changes of the Hamiltonian H ( x ) and its eigenvalues E α ( x ).In fact, for N = 2, the relation P α = (1 + H/E α ), with 1 the 2 × N .It should be stressed that no information is lost when replacing point of view (i) by (ii)or (iii); when necessary, any eigenstate can easily be recovered as | ψ α ( x ) (cid:105) = P α ( x ) | ψ (cid:105) fromthe projectors, where | ψ (cid:105) can be chosen at convenience for each α and x . In other words, theoverlap (cid:104) ψ α ( x ) | ψ α ( x + d x ) (cid:105) and more generally the full quantum geometric information abouta given SU( N ) Hamiltonian is encoded in the eigenprojectors P α ( x ). In contrast to point ofview (i), however, point of view (iii) is extremely convenient for practical computations, sincethe only ingredients required are the Hamiltonian H ( x ) (known at the very outset) and itsset { E α ( x ) , α = 1 , ..., N } of eigenvalues (easy to find in many cases).Let us state more precisely what we have in mind, quantitatively, when speaking of quan-tum geometric properties . Historically, their fundamental importance was first revealed in thecontext of geometric phases which can have striking effects on physical observables. Whileapparently distinct notions of geometric phase were initially introduced, such as the Aharonov-Bohm [1, 2], Pancharatnam [3] and Stone-Longuet-Higgins [4] phases, the discovery of Berry’sphase [5, 6] then formalized the notion of a geometric phase in a very general way. Berryphases and their extensions [7–11] are now omnipresent in the physics literature.More fundamental than Berry’s phase is the underlying gauge field, known as the (Abelian)Berry connection A α,i ( x ) ≡ i (cid:104) ψ α ( x ) | ∂ i ψ α ( x ) (cid:105) , where ∂ i ≡ ∂/∂x i . Before its formalization [5,6], this concept popped up in some noteworthy early works, most explicitly in Blount’s For-malisms of Band Theory [12]. The origin of the Berry connection lies in the above-mentionedoverlap (cid:104) ψ α ( x ) | ψ α ( x + d x ) (cid:105) ; the phase of this complex number gives rise to the Berry connec-tion gauge field whose curvature field Ω α,ij ( x ) = ∂ i A α,j ( x ) − ∂ j A α,i ( x ), the Berry curvature ,plays the role of a pseudo-magnetic field in parameter space. However, also the modulus is ofimportance; it defines a metric tensor g α,ij ( x ) of Riemannian type on the manifold of quantumstates [13], known as quantum metric . Both quantum metric and Berry curvature Ω α,ij ( x )can be summarized in terms of a single complex tensor, called the quantum geometric tensor(QGT) [14–16]: T α,ij ( x ) ≡ g α,ij ( x ) − i α,ij ( x ) . (1)This object vanishes identically in the case of a one-dimensional Hilbert space, but is poten-tially non-zero for any N -band system with N ≥ entire QGT , not only the Berry curvature part, are essential for understanding the geometriccontributions to observables in N -band systems. While higher-order geometric quantities3xist, we will, in this paper, typically have in mind the QGT when speaking about quantumgeometric properties .In accordance with point of view (i) on quantum geometry, the QGT is formulated interms of energy eigenstates | ψ α ( x ) (cid:105) in the large majority of the afore-mentioned references.However, as we mentioned above, practical computations of the QGT via the | ψ α ( x ) (cid:105) faceseveral problems. First, closed-form expressions for energy eigenstates, i.e. expressions thatcan be used independently of the Hamiltonian of interest, are very hard to construct. Second,the gauge arbitrariness in the parameter-dependent global phase is problematic, especially ifderivatives with respect to these parameters are to be carried out. Third, it may be thatsingularities in some of the components of multicomponent states occur at certain points inparameter space, which cannot be excluded a priori . These issues are of particular relevancewhen addressing geometric information encoded in the eigenstates. Already for two-bandsystems, the task of finding the QGT from energy eigenstates (in a basis of practical interest)is not trivial; for three-band systems, it can pose serious difficulties, see for instance Ref.[39]. For higher N – aside from special cases that allow for simple analytical treatment – it isusual to resort to (not always well-controlled) numerical methods or to employ approximateperturbative analytical approaches that decouple the SU( N ) Hamiltonian into effective SU(2)sub-Hamiltonians valid locally in parameter space.In the present work, we therefore reformulate geometrical properties of N -band systemsavoiding energy eigenstates. The transition from point of view (i) to points of view (ii) &(iii) is demonstrated. The latter provide closed-form analytical formulas for the QGT thatcan be used, in principle, for arbitrary N . Over the last few years, some efforts in this spirithave been made for the N = 3 case [40–42]. Very recently, Pozo and de Juan [43] published apaper of much more general scope. They point out that, quite generally, any observable of an N -band system can be computed without energy eigenstates if the eigenenergies are known,and also briefly apply this idea to the QGT. While similar in spirit, our work is less ambitiousin scope. Indeed, we feel that the topic of computing observables and physical quantities morecomplex than the QGT without using eigenstates requires a much more detailed treatment,and therefore defer it to future work [44]. In the present study, we essentially concentrate onthe eigenprojectors and the quantum geometric tensor. In particular, we emphasize the roleof generalized Bloch vectors, an important concept in the quantum information community,and establish links between different strategies that can be used to obtain the QGT.The setup of this paper is as follows. In Section 2, the familiar two-band case involving asimple SU(2) Hamiltonian is reviewed, in order to motivate the eigenprojector approach andoutline the strategy for the N -band generalization.Then, in a first step, we derive a generic formula P α ( x ) = P α ( E α ( x ) , H ( x )) for the eigen-projectors of SU( N ) Hamiltonians, cf. Section 3.1. This formula expresses P α ( x ) as a matrixpolynomial of degree N − E α . Besides its utility for quantum geometry,this formula can be used (a) to expand any function of the Hamiltonian in powers of H ( x )and (b) to analytically construct energy eigenstates of any SU( N ) Hamiltonian. In Section3.2, these results are translated to an equivalent but more convenient vectorial language. Inparticular, expanding the Hamiltonian and the eigenprojectors in SU( N ) Gell-Mann matrices(the generalization of Pauli matrices), one is led to a formalism involving two kinds of SU( N )vectors: the Hamiltonian vector h ( x ) and the (generalized) Bloch vector b α ( x ), the SU( N )analog of the familiar SU(2) Bloch vector on the unit sphere. In this vectorial language, theeigenprojector polynomial formula translates into a function b α ( x ) = b α ( E α ( x ) , h ( x )) thatexpresses the generalized Bloch vectors in terms of the Hamiltonian vector and the energyeigenvalues. 4n a second step, in Section 4.1, we review how to make the transition between eigenstate-and eigenprojector-based expressions of the QGT; this further allows to obtain two differentSU( N ) invariant formulae for the QGT in terms of Bloch vectors, cf. Section 4.2. Section4.3 shows how this may be used for computing the QGT of N -band systems only from theknowledge of h ( x ) and the energy eigenvalues. For the Berry curvature, a closed form formulafor arbitrary N is obtained; applying it to cases of low N , we recover the familiar SU(2) result,observe that the SU(3) formula found in Ref. [40] can be considerably simplified, and writedown new explicit formulas for the SU(4) and SU(5) Berry curvature. It will also become clearthat computing the quantum metric is more cumbersome (though equally straightforward inmethodology) than computing the Berry curvature.In Section 5, we consider in depth the special case of a particle-hole symmetric (PHS)spectrum, meaning an energy spectrum that is symmetric about zero energy. This applies toa plethora of physical systems, in particular those with a chiral and/or charge conjugationsymmetry. In this scenario, the eigenprojectors and Bloch vectors are considerably simplified,and so are the geometrical quantities constructed from them.Section 6 serves to illustrate the formalism by means of explicit examples. Several three-dimensional (3D) models in the class of (linear) multifold fermions are presented, for both N = 3 and N = 4, and compared to the simplest kind of multifold fermion, i.e. a pseudospin S , where S = ( N − /
2. In particular, these models are designed in such a way as toexhibit geometrical and topological properties completely different from a pseudospin S , eventhough they have exactly the same energy spectrum. This emphasizes the fact that, for anygiven pair of Hamiltonians, the indistinguishability of their band structures does not in theleast establish their equivalence. Most strikingly, for some of the models, characterized bya global chiral symmetry, quantum metric and Berry curvature can be continuously tunedwhile maintaining a completely unchanged band structure. This is somewhat reminiscent of2D models with tunable quantum geometry, first discovered in the α − T lattice [45].Finally, we sum up and conclude in Section 7. The properties of two-level/two-band systems are well known, and widely applied in the studyof qubits, in the band theory of graphene, Chern insulators, and countless other systems inatomic and condensed matter physics. The reason why such systems are rather easy todescribe is the very simple structure of the underlying su (2) algebra. In the following, wecompile various key facts that will serve for comparison when we treat the N > i.e. a 2 × H ( x ) with aset of energy eigenvalues { E α ( x ) | α = ±} and (orthonormal) eigenstates {| ψ α ( x ) (cid:105) | α = ±} ,and where the vector x represents a set of parameters. Note that, here and in the following,the x -dependence of quantities of interest is stated explicitly in their definition, but oftenunderstood as implicit afterwards.Without loss of generality, the trivial part of H (proportional to the identity matrix 1 )is set equal to zero such that Tr H = 0. A Hamiltonian of this type can be expanded in thePauli matrices σ = ( σ x , σ y , σ z ): H ( x ) = h ( x ) · σ . (2)The Hamiltonian vector can be written as h = ( h x , h y , h z ) = | h | u h . Here, the unit vec-tor u h ( x ) ≡ (sin θ h cos φ h , sin θ h sin φ h , cos θ h ) is characterized by two Hamiltonian’s angles ( θ h ( x ) , φ h ( x )) whose expressions in terms of the components h x,y,z read cos θ h = h z / | h | and5igure 1: Schematic visualization of the different Bloch sphere’s appearing in the treatmentof SU( N ) Hamiltonians. The familiar SU(2) case is illustrated in the first row, while thesecond row treats the general case of arbitrary N . The Hamiltonian’s Bloch sphere B ( N ) h isthe relevant space for the mapping u h ( x ), and corresponds to a proper unit sphere for allvalues of N . The eigenstate’s Bloch sphere B ( N ) α associated to the mapping | ψ α ( x ) (cid:105) is also aproper unit sphere for all N , but of different dimension than the Hamiltonian’s Bloch sphereif N >
2. The eigenprojector’s Bloch sphere B ( N ) P α associated to the mapping P α ( x ) is only aproper sphere for N = 2. In the N > generalized Bloch sphere ,it is of very complicated shape, see Appendix E.tan φ h = h y /h x . The vector u h ( x ) defines a map from the parameter space to a Hamilto-nian’s Bloch sphere B (2) h , which is simply a unit two-sphere, B (2) h = S , see Fig. 1(a). Theeigenspectrum of the Hamiltonian introduced in Eq. (2) is simply E α ( x ) = α (cid:112) Tr( H ( x )) / α | h ( x ) | , (3)with α = ± . The variation E α ( x ) defines a two-band spectrum in parameter space. It is particle-hole symmetric , i.e. energy levels occur in pairs of opposite sign. Eigenstates vs. eigenprojectors
Since one major focus of this paper is how the N -band eigenstates can be replaced by eigen-projectors, a somewhat detailed discussion of the two-band eigenstates and the complicationsthat arise already in this simple N = 2 case is now in order. This will make clear why it isfavorable to work with eigenprojectors. 6he eigenstate for α = +, i.e. the state | ψ + ( x ) (cid:105) , can be written in either of the forms | ψ + (cid:105) = e i Γ + cos θ + sin θ + e iφ + = 1 (cid:114) (cid:16) h z E + (cid:17) h z E + h x + ih y E + = 1 (cid:112) θ h ) θ h sin θ h e iφ h , (4)and similarly for | ψ − ( x ) (cid:105) . The first equality simply corresponds to the formal parametrizationof a complex-valued two-component unit vector. It expresses the fact that, at each point x inthe parameter space, the eigenstate | ψ + (cid:105) is minimally encoded by a global phase Γ + ( x ) andtwo eigenstate’s angles ( θ + ( x ) , φ + ( x )). These angles can be viewed as defining a map fromparameter space to an eigenstate’s Bloch sphere B (2)+ . Since the angles θ + and φ + can beinterpreted as spherical coordinates, the eigenstate’s Bloch sphere, just like the Hamiltonian’sBloch sphere, is a unit two-sphere: B (2)+ = S , as visualized in Fig. 1(b). Similarly, for | ψ − (cid:105) , one has two eigenstate’s angles ( θ − ( x ) , φ − ( x )) defining a corresponding Bloch sphere B (2) − = S , see Fig. 1(b). The second line in Eq. (4) illustrates the possible singular behaviorof eigenstates. If the eigenstate components are written directly in terms of the correspondingenergy eigenvalue and the components of the Hamiltonian vector h , a singularity is evidently tobe expected when h z ( x ) /E + ( x ) = − x = x in parameter space. Similarly,if the eigenstate components are written in terms of the angles ( θ h , φ h ), the explicit relationtan φ h = h y /h x points towards further possible singular behavior in parameter space when h x ( x ) = 0. As a final subtlety, note that the previous expressions in Eq. (4) reflect a peculiargauge choice.In contrast to the eigenstate | ψ α (cid:105) , the corresponding eigenprojector P α ( x ) ≡ | ψ α (cid:105) (cid:104) ψ α | isan explicitly gauge-independent quantity, and it generically exhibits a less singular behaviorin parameter space. In the N = 2 case, it simply writes as P α ( x ) = 12 1 + 12 E α ( x ) H ( x ) ≡
12 (1 + b α ( x ) · σ ) , (5)where 1 is the 2 × Bloch vector b α ( x ) is given by b α ( x ) = h ( x ) E α ( x ) = α u h ( x ) . (6)It is immediately obvious that the definition of P α (and b α ) is completely unambiguous andunproblematic except at degeneracy points, i.e. when E + ( x ) = E − ( x ) = 0 for some x = x ;this emphasizes the usefulness of the projectors.Of course, the Bloch vector also defines a map b α ( x ) to a space that may be called the eigenprojector’s Bloch sphere B (2) P α ; among the three kinds of Bloch spheres we are dealingwith, it is perhaps the one most frequently referred to as a Bloch sphere in the literature.Importantly, the peculiarity and simplicity of two-band systems, as compared to the moregeneral N -band case, consists in the fact that the eigenprojector’s Bloch sphere is extremelysimple, namely a unit two-sphere: B (2) P α = S , see Fig. 1(c). It is important to keep in mind,however, that the Hamiltonian’s, the eigenstate’s and the eigenprojector’s Bloch spheres are a priori distinct spaces , even though they all happen to correspond to the unit two-sphere S in the SU(2) case. This distinction is necessary to avoid considerable confusion in the SU( N )generalization, as anticipated by Fig. 1(d)–(f).7 last important property of the eigenprojector P α is that it is in some sense a morefundamental object than the eigenstate, namely it permits to construct the eigenstate | ψ α (cid:105) from an arbitrary gauge freedom state | ψ g (cid:105) . More concretely, a state equivalent to Eq. (4) iseasily constructed as | ψ + (cid:105) = 1 (cid:112) (cid:104) ψ g | P + | ψ g (cid:105) P + | ψ g (cid:105) , with | ψ g (cid:105) = cos θ g sin θ g e − iφ g , (7)where the gauge freedom angles ( θ g , φ g ) can be chosen at will at any point in parameterspace x , and independently for α = ± . For example, if cos θ g = 1 and sin θ g = 0, oneexactly recovers the last expression in Eq. (4). Two important remarks are in order aboutthe eigenstate obtained from Eq. (7). First, although this is not immediately obvious, theeigenstate’s angles ( θ + , φ + ) stay unchanged upon varying ( θ g , φ g ) for fixed x . In contrast, theglobal phase Γ + is changing, implying Γ + ≡ Γ + ( x , θ g , φ g ). As a consequence, the possiblesingular behaviors of the wavefunction in parameter space are gauge-dependent. Second, thestate | ψ g (cid:105) must not be orthogonal to the projector P α ( x ), i.e. one requires P α ( x ) | ψ g (cid:105) (cid:54) = 0. Thisconstraint implies that it might be necessary to change the state | ψ g (cid:105) when the parameter x isvarying because it is never guaranteed that a single | ψ g (cid:105) (meaning a fixed gauge) is sufficientto describe a given eigenstate | ψ α (cid:105) over the entire parameter space x . Quantum geometry
Having pointed out sufficient justification for abandoning energy eigenstates in favor of eigen-projectors, this idea may now be applied to quantum geometry. In other words, this sectionserves to explain how the quantum geometric tensor (QGT) , historically first introduced interms of energy eigenstates, can be rewritten in terms of eigenprojectors and Bloch vectors,as is frequently done in the treatment of two-band systems. A similar procedure will then bedeveloped for the
N > (cid:104) ψ α ( x ) | ψ α ( x + d x ) (cid:105) of statesinfinitesimally close in parameter space. Leaving the detailed derivation to Section 4.1, theQGT T α,ij ( x ) associated to an eigenstate | ψ α ( x ) (cid:105) may be written as: T α,ij ( x ) = (cid:104) ∂ i ψ α | ∂ j ψ α (cid:105) − (cid:104) ∂ i ψ α | ψ α (cid:105) (cid:104) ψ α | ∂ j ψ α (cid:105) ,g α,ij ( x ) = Re T α,ij = Re (cid:104) ∂ i ψ α | ∂ j ψ α (cid:105) − (cid:104) ∂ i ψ α | ψ α (cid:105) (cid:104) ψ α | ∂ j ψ α (cid:105) , Ω α,ij ( x ) = − T α,ij = − (cid:104) ∂ i ψ α | ∂ j ψ α (cid:105) , (8)where ∂ i ≡ ∂/∂x i . The real part g α,ij ( x ) of the QGT is the quantum metric tensor andis symmetric under the exchange of indices i, j . The imaginary part Ω α,ij ( x ) is the Berrycurvature tensor and is antisymmetric in the indices i, j . This definition makes it clear thatthe explicit dimension and form of the matrix elements g α,ij and Ω α,ij depend on the chosenset of parameters x .As an illustration, taking x = ( θ h , φ h ) (dim( x ) = 2) and differentiating the state | ψ + (cid:105) [seeEq. (4)] as prescribed by Eq. (8), the associated quantum metric and Berry curvature tensorsare 2 × g + ,ij ( θ h , φ h ) = 14 θ h , Ω + ,ij ( θ h , φ h ) = − θ h − sin θ h , (9)8hich is a well-known result [16]. Taking instead x = ( h x , h y , h z ) (dim( x ) = 3), the quantummetric and Berry curvature tensors are now 3 × g + ,ij ( h ) = 14 | h | (cid:18) δ ij − h i h j | h | (cid:19) , Ω + ,ij ( h ) = − | h | (cid:15) ijk h k , (10)with (cid:15) ijk the Levi-Civita antisymmetric tensor. Note that, in both of these cases, one easilyfinds g − ,ij = g + ,ij and Ω − ,ij = − Ω + ,ij .More generally, the quantum geometric tensor of interest is often related to the explicitdependency of the Hamiltonian vector h ( x ) on some vector of external parameters x , withdim( x ) ≥
2. For example, in condensed matter physics, one will often have x = k , where k represents crystal momentum. In that situation, the corresponding QGT T α,ij ( x ) may easilybe obtained from either T α,kl ( h ) or T α,kl ( θ h , φ h ) by a simple composition rule as T α,ij ( x ) = (cid:88) k,l ( ∂ i y k )( ∂ j y l ) T α,kl ( y ) , (11)with y = h or y = ( θ h , φ h ).Within the eigenstate approach to the QGT [cf. Eq. (8) and point of view (i) of theIntroduction], the derivation of the tensors T + ,ij ( h ) or T + ,ij ( θ h , φ h ) given by Eqs. (9) & (10)makes explicit use of the specific gauge choice that corresponds to the form of | ψ + (cid:105) given byEq. (4). It is not immediately evident that these expressions of the quantum geometric tensorsare gauge-invariant. There is, however, an obviously gauge-invariant formula that containsthe same information as Eq. (8), but encodes it in terms of derivatives of the eigenprojector P α [cf. point of view (ii) expressed in the Introduction]. This formula reads (see Section 4.1for details) T α,ij ( x ) = Tr [( ∂ i P α ) (1 − P α ) ( ∂ j P α )] . (12)For illustration, it is expedient to consider the specific example x = h , insert the expression P α ( h ) as given by Eqs. (5) & (6) into Eq. (12), and to recover the QGT of Eq. (10).The above formula (12) is extremely useful: in combination with Eq. (5), it can imme-diately be used for computing the QGT without eigenstates, assuming the matrix H and itseigenvalues are known. While this is sufficient for computational purposes, it is more instruc-tive and convenient to resort to the vectorial language: equation (5) allows to first obtain aBloch vector-based formula for the QGT, and using further Eq. (6) one gets the correspondingexpression in terms of only the Hamiltonian vector and the energy eigenvalues E α = α | h | [cf.point of view (iii) of the Introduction]. Both forms of writing for the quantum metric andBerry curvature tensors, and with a generic vector parameter x [in the sense of either of Eqs.(9–11)], are summarized in the following expressions: g α,ij ( x ) = 14 b iα · b jα = 14 | h | (cid:34) h i · h j − (cid:0) h · h i (cid:1) (cid:0) h · h j (cid:1) | h | (cid:35) , Ω α,ij ( x ) = − b α · ( b iα × b jα ) = − α | h | h · ( h i × h j ) , (13)with the shorthand notation m i ≡ ∂ i m ≡ ∂ m /∂x i for the partial derivative of a vector m .We will see that the strategy employed to obtain Eq. (13) applies in a completely analogousway to SU( N >
2) systems. It will also become clear that, while the expressions of thequantum metric and Berry curvature tensors in terms of h ( x ) and its parametric derivativesare widely known, the reformulation directly in terms of Bloch vectors b α is actually the onethat naturally pertains to the SU( N ) generalization of Eq. (13).9 Eigenprojectors and Bloch vectors for N -band systems Computing and understanding the quantum geometric properties of N -band systems – i.e. morespecifically N × N Hermitian matrices H ( x ) – is of interest for a wide variety of physical sys-tems, not necessarily restricted to the quantum realm. The issues of eigenstates pointed outin the previous section of course generalize to the case N > | ψ α ( x ) (cid:105) numerically, andperforming direct numerical computation of the geometric tensors. However, this strategyinvolves a necessarily discrete mesh in parameter space and possible difficulties related tospurious gauge dependencies and singularities. One never knows which effective gauge is chosen by the computer; moreover this effective gauge might be different at each point inparameter space and for each eigenstate.In order to avoid such a complete numerical approach, the main goal here is to developsome analytical understanding of the eigenprojectors and geometric tensors, and to providegeneralizations of Eqs. (5), (6) & (13) to N >
2. The strategy to do this is as follows. In afirst step, each eigenprojector P α is written as a polynomial of order N − H , where each monomial coefficient is an explicit function of the associated energyeigenvalue E α . The second step consists in expanding the Hamiltonian and the eigenprojectorsin the basis of Gell-Mann matrices, such that they are described by an SU( N ) Hamiltonianvector h and Bloch vectors b α , respectively. The Bloch vector b α is found to be a polynomialin h , with monomial coefficients that are functions of E α . The last step consists in computingthe corresponding quantum geometric tensors, as is done in Section 4. P α ( H, E α ) Our starting point is the following textbook formula for the eigenprojector P α as a functionof the N × N matrix H and the set { E β , β = 1 , ..., N } of all energy eigenvalues [46]: P α = (cid:89) β (cid:54) = α H − E β N E α − E β , (14)where 1 N is the N × N identity matrix. Note that in the language of matrix theory, P α isknown as the Frobenius covariant of H . The goal is now to eliminate all E β (cid:54) = α from thisformula, such that P α becomes a proper polynomial in H with coefficients that depend onlyon the single eigenvalue E α . Notice that the denominator corresponds to the derivative of theHamiltonian’s characteristic polynomial: p (cid:48) N ( E α ) = (cid:81) β (cid:54) = α ( E α − E β ). For more details on thecharacteristic polynomial p N ( z ), see Appendix A.The products in the numerator and denominator of Eq. (14) can be rewritten as describedin Appendix B. The result is P α = (cid:80) N − n =0 q N − − n ( E α ) H n (cid:80) N − n =0 q N − − n ( E α ) E nα = (cid:80) N − n =0 q N − − n ( E α ) H n (cid:80) N − n =0 q N − − n ( E α ) C n , (15)where the C n are (classical) Casimir invariants [47], which are defined as C n ≡ Tr( H n ) = N (cid:88) α =1 E nα . (16)Obviously, one has C = N . Since one can always assume Tr H = 0 without loss of generality,see also Eq. (22) below, we will have C = 0 throughout. Notice that from the second equalityin Eq. (14), it is immediately clear that Tr P α = 1, as required.10 c n − C − C C − C C C − C − C + C + C C − C Table 1: Coefficients c n determining the polynomial (17).The polynomials q n ( z ) appearing in Eq. (14) are closely related to the Hamiltonian’scharacteristic polynomial and can be formally written as q n ( z ) ≡ n (cid:88) k =0 c k z n − k , (17)with q N ( z ) = p N ( z ). Alternatively, one may construct them by starting from q ( z ) = 1 andusing the simple recurrence relation q n ( z ) = zq n − ( z ) + c n . (18)The c n are derived in Appendix A and listed in Table 1 for n ≤ P α ( E α , H ), as done here for the cases N = 2 to N = 5: P α = 12 E α ( E α + H ) ,P α = 13 E α − C (cid:20)(cid:18) E α − C (cid:19) + E α H + H (cid:21) ,P α = 14 E α − C E α − C (cid:20)(cid:18) E α − C E α − C (cid:19) + (cid:18) E α − C (cid:19) H + E α H + H (cid:21) ,P α = 15 E α − C E α − C E α + C − C (cid:20)(cid:18) E α − C E α − C E α + C − C (cid:19) + (cid:18) E α − C E α − C (cid:19) H + (cid:18) E α − C (cid:19) H + E α H + H (cid:21) . (19)Note that the N = 2 projector in the first line is the same as Eq. (5) discussed above. Forgiven N , each eigenprojector is a polynomial of degree N − N desired.The eigenprojectors can be computed immediately for an arbitrary given Hamiltonian,provided the energy eigenvalues are known. In this context, one needs to distinguish thecases N < N ≥
5. In the former case, it is often simple to find the analytical form of E α ( x ), and in principle it is always possible by using the analytical closed-form expressionsfor the eigenenergies given in Appendix C. In the latter case ( N ≥ H , but Eq. (15) can alwaysbe computed explicitly if the eigenvalues E α (or approximate values thereof) are accessible bysome means. This applies, for example, to the case of particle-hole symmetric energy spectra,discussed in Sections 5 & 6, where closed-form solutions for the eigenvalues exist for N > f ( H ) as a polynomial of order N − Sylvester’s formula : f ( H ) = N (cid:88) α =1 f ( E α ) (cid:80) N − n =0 q N − − n ( E α ) H n (cid:80) N − n =0 q N − − n ( E α ) E nα . (20)Second, Eq. (15) may be employed for constructing energy eigenstates | ψ α (cid:105) that can beused further to compute matrix elements of observables, or the Berry connection and otherquantities of interest. More concretely, similarly to the case N = 2 discussed in Eq. (7), the N -band eigenstate | ψ α (cid:105) in an arbitrary gauge may be obtained as | ψ α (cid:105) = 1 (cid:112) (cid:104) ψ g | P α ( H, E α ) | ψ g (cid:105) P α ( H, E α ) | ψ g (cid:105) , (21)where the gauge freedom state | ψ g (cid:105) can be chosen arbitrarily . The resulting eigenstate | ψ α (cid:105) has now N components that can be minimally encoded by a global phase Γ α and N − eigenstate’s angles ( θ iα ( x ) , φ iα ( x )) ( i = 1 , ..., N − N − eigenstate’s Bloch sphere B ( N ) α , which is depicted,for comparison with the N = 2 case, in Fig. 1(e). α ( h , E α ) The above results on the eigenprojectors can now be translated into a vectorial language; thisproves just as useful as in the N = 2 case. Generalized Bloch vectors and generalized Bloch sphere
Similarly to the N = 2 case, one may define a Hamiltonian’s vector h and a (generalized)Bloch vector b α by expanding the (traceless) Hamiltonian H and the eigenprojector P α as H ( x ) = h ( x ) · λ , (22)as well as [50] P α ( x ) = 1 N N + 12 b α ( x ) · λ . (23)Here, h ≡ Tr { H λ } / b α ≡ Tr { P α λ } , where λ = ( λ , ..., λ N − ) is the vector composedof the N − generalized Gell-Mann (traceless Hermitian) matrices that are the elementarygenerators of the SU( N ) Lie group. Together with the identity matrix 1 N they consitute abasis for the Lie algebra su ( N ) [51]. For the reader’s convenience, the Gell-Mann matrices for N = 3 and N = 4 are listed in Appendix D. Note that the vector b α is also called coherencevector , and that we choose the prefactor in front of b α , although various choices of thisfactor exist in the literature [52–54].In the above, the vectors h and b α have N − u h ≡ h / | h | can be parametrized by N − Hamiltonian’s angles . In other words, u h ( x )defines a map from the parameter space to a Hamiltonian’s Bloch sphere B ( N ) h = S N − , where S N − is the unit ( N − b α ( x ) (for given α ) would similarly seem to define an ( N − The state | ψ g (cid:105) has now N components that are encoded by N − θ ig , φ ig )( i = 1 , ..., N −
1) that may be chosen independently for each eigenstate | ψ α (cid:105) and at each point in parameterspace x . Even though it is far from obvious, it is expected that solely the global phase depends both on thegauge choice and on the parameters x – that is, Γ α ≡ Γ α ( x , θ ig , φ ig ). In contrast, the angles ( θ iα , φ iα ) shouldsolely depend on x . N = 2 case, h and b α areno longer parallel for N >
2, such that u h ( x ) and b α ( x ) are distinct maps.Indeed, the vector b α ( x ) defines a map from parameter space not to an ( N − N − eigenprojector’s Blochsphere B ( N ) P α , or simply the generalized Bloch sphere . For comparing to the N = 2 case, B ( N ) P α is depicted schematically in Fig. 1(f). An understanding of the true geometrical structureof this Bloch sphere is not at all easy to acquire. Many efforts have been undertaken tofigure out its properties for N >
2, which is a surprisingly nontrivial issue, see Refs. [52, 53,55–60] and references therein. For the interested reader, we outline the main results aboutthe generalized Bloch sphere in Appendix E. In summary, the eigenstate’s Bloch sphere, theHamiltonian’s Bloch sphere, and the eigenprojector’s Bloch sphere are all isomorphic spacesfor N = 2, but all different spaces for N > Bloch vectors as a function of the Hamiltonian vector
We now detail the different steps that allow to find the concrete expression b α ( h , E α ) fromthe eigenprojector formula P α ( H, E α ) given by Eq. (15). The only missing ingredient is theexplicit expansion of H n = ( h · λ ) n in terms of generalized Gell-Mann matrices. Using theprevious definition C n = Tr( H n ) and defining vectors η n ≡ Tr( H n λ ) /
2, we can write: H n = ( h · λ ) n = C n N N + η n · λ , (24)where obviously C = N , C = 0, η = 0 and η = h . Inserting into Eq. (15) and using Eq.(23), we obtain the intermediate result: b α = 2 (cid:80) N − n =0 q N − − n ( E α ) η n (cid:80) N − n =0 q N − − n ( E α ) E nα . (25)At this point it remains the task to find the explicit form of the vectors η n ( h ) for n > C n and η n canbe written uniquely in terms of h , have dimensions [energy] n , and therefore necessarily arisefrom products involving the vector h . We therefore require a product identity generalizingthe familiar SU(2) identity ( m · σ )( n · σ ) = m · n + i ( m × n ) · σ to arbitrary N . This can bereadily achieved by focusing on some properties of the Lie algebra su ( N ), which are definedby the (anti)commutation relations of Gell-Mann matrices [61]:[ λ a , λ b ] = 2 if abc λ c , f abc ≡ − i λ a , λ b ] λ c ) , { λ a , λ b } = 4 N δ ab N + 2 d abc λ c , d abc ≡
14 Tr ( { λ a , λ b } λ c ) , (26)where repeated lower indices imply summation (Einstein convention). Here, d abc ( f abc ) arethe totally symmetric (antisymmetric) structure constants of su ( N ), which are a known setof real numbers; for their explicit values in the N = 3 case, see for example Ref. [60]. Notealso that for N = 2, where λ = σ , the d abc vanish identically and f abc = (cid:15) abc , where (cid:15) abc isthe Levi-Civita symbol. From Eq. (26) one easily obtains the product identity λ a λ b = 2 N δ ab N + ( d abc + if abc ) λ c . (27)13ince all generators are traceless, Tr λ c = 0, the convenient trace orthogonality Tr( λ a λ b ) =2 δ ab holds. In order to translate Eq. (27) to its vectorial form, one may use the structureconstants (26) to define dot, star and cross products of vectors: m · n ≡ m c n c , ( m (cid:63) n ) a ≡ d abc m b n c , ( m × n ) a ≡ f abc m b n c , (28)where m and n are ( N − N = 2situations (where d abc = 0) but is crucial for N >
2. Note that a different convention issometimes used for defining the star product, which introduces N -dependent prefactors andis particularly popular in the treatment of qutrits [53, 60]. From Eqs. (27) & (28), one directlyobtains the desired product identity in vector form:( m · λ )( n · λ ) = 2 N m · n N + ( m (cid:63) n + i m × n ) · λ . (29)When computing H n = ( h · λ ) n by applying Eq. (29) repeatedly, it is clear that star productsof the kind h (cid:63) h will appear. We may thus introduce the notation for repeated star productsof a vector with itself: m (0) (cid:63) = m , m (1) (cid:63) = m (cid:63) ≡ m (cid:63) m , m (2) (cid:63) = m (cid:63)(cid:63) ≡ m (cid:63) ( m (cid:63) m ) , m ( k +1) (cid:63) = m (cid:63)(cid:63)... ≡ m (cid:63) m ( k ) (cid:63) . (30)The resulting vectors have the following properties (with n , n ∈ N ): m ( n ) (cid:63) · m ( n ) (cid:63) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m (cid:16) n n (cid:17) (cid:63) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) if n + n even , m ( n ) (cid:63) × m ( n ) (cid:63) = 0 . (31)The former identity follows directly from the total symmetry of the structure constants (26),and the latter is a consequence of the second Jacobi identity, see Appendix F. More generally,from the generic properties of the algebra considered above, it is possible to establish usefulgeneralized vector and scalar Jacobi identities, listed in Appendix G.With all these prerequisites we may now calculate ( h · λ ) n i.e. determine C n ( h ) and η ( h ).In particular, we obtain the following simple recursion relations: C n +1 = 2 h · η n , η n +1 = h (cid:63) η n + C n N h , (32)with initial conditions C = N and η = 0. (A more general but equivalent version ofthis recursion relation is established in Appendix H.) Applying this recursion, for a tracelessHamiltonian matrix, we obtain successively up to n = 4 the important identities C = 0 , η = h ,C = 2 | h | , η = h (cid:63) ,C = 2 h · h (cid:63) , η = C N h + h (cid:63)(cid:63) ,C = 4 | h | N + 2 | h (cid:63) | , η = C N h + C N h (cid:63) + h (cid:63)(cid:63)(cid:63) . (33)14ore generally, for a generic N >
1, the form of the vector η n ( h ) is compactly written as: η n = 1 N n − (cid:88) p =0 C p h ( n − − p ) (cid:63) . (34)The final step required for completing our task of finding the explicit expressions of the Blochvector b α ( h , E α ) consists in substituting Eqs. (33) & (34) into Eq. (25). The Bloch vectorsfor N = 2 to N = 5 are then found to be b α = 22 E α h , b α = 23 E α − C ( E α h + h (cid:63) ) , b α = 24 E α − C E α − C (cid:20)(cid:18) E α − C (cid:19) h + E α h (cid:63) + h (cid:63)(cid:63) (cid:21) , b α = 25 E α − C E α − C E α + C − C (cid:20)(cid:18) E α − C E α − C (cid:19) h + (cid:18) E α − C (cid:19) h (cid:63) + E α h (cid:63)(cid:63) + h (cid:63)(cid:63)(cid:63) (cid:21) , (35)respectively. In the first line, Eq. (6) is recovered. Equation (35) contains the same infor-mation as Eq. (19), and clearly illustrates that the Bloch vector is no longer parallel to theHamiltonian vector h for N >
2. Indeed, for given N , each Bloch vector is a kind of ”vectorpolynomial” of degree N − h , which can be viewed as a con-sequence of the Cayley-Hamilton theorem [48, 49]. Writing down analogous formulas for the N ≥ N , only C n ≤ N and η n 2. For example, for N = 3, allinformation we need is encoded in C , C , h and h (cid:63) . This again follows from the Cayley-Hamilton theorem, which states that q n = N ( H ) = p N ( H ) = 0. From this property it is easyto establish the following useful identities: N = 3 : h (cid:63)(cid:63) = C h , h (cid:63) (cid:63) h (cid:63) = C h − C h (cid:63) ,N = 4 : h (cid:63)(cid:63)(cid:63) = C h + C h (cid:63) , h (cid:63) (cid:63) h (cid:63) = C h , h (cid:63) (cid:63) h (cid:63)(cid:63) = | h (cid:63) | h + C h (cid:63) . (36)In closing this section, some further general remarks on the Bloch vectors are in order. Theusual orthogonality relation P α P β = δ αβ P α and completeness relation (cid:80) α P α = 1 N of eigen-projectors translate to the Bloch vector picture as: b α · b β = 2 (cid:18) δ αβ − N (cid:19) , b α (cid:63) b β = (cid:18) δ αβ − N (cid:19) ( b α + b β ) , b α × b β = 0 , (cid:88) α b α = 0 . (37)15he star product constraint can be viewed as being responsible for the complicated shape ofthe generalized Bloch sphere described in Appendix E. The results of the preceding section can now be used for the discussion of quantum geometry.We first recall the derivation of the quantum geometric tensor (QGT) based on the expansionof the overlap of neighboring eigenstates in parameter space. Second, it is reformulatedin terms of eigenprojectors and then in terms of generalized Bloch vectors. Finally, it isconsidered in detail how to write the QGT in terms of only the Hamiltonian vector h andthe eigenenergy E α . Along this path, we will encounter many expressions for Berry curvatureand quantum metric that are extremely useful for practical computations. The QGT is a tensor quantifying the overlap of two states that are ”infinitesimally close”in parameter space [13]: (cid:104) ψ α ( x ) | ψ α ( x + d x ) (cid:105) = F α ( x ) e iφ α ( x ) , where the modulus F α ( x ) hascome to be known as fidelity [62]. For a given eigenstate | ψ α (cid:105) , the form of the QGT dependson the choice of parameter space, i.e. on the meaning of the vector x (cf. the discussion inSection 2). The fidelity is related to the quantum metric, while the phase φ α ( x ) gives rise tothe Berry curvature. This can be seen by expanding (cid:104) ψ α ( x ) | ψ α ( x + d x ) (cid:105) = 1 + (cid:88) i (cid:104) ψ α ( x ) | ∂ i ψ α ( x ) (cid:105) dx i + 12 (cid:88) ij (cid:104) ψ α ( x ) | ∂ ij ψ α ( x ) (cid:105) dx i dx j + O ( | d x | ) , (38)where ∂ i ≡ ∂/∂x i as before. Noting that Re (cid:104) ψ α ( x ) | ∂ ij ψ α ( x ) (cid:105) = − Re (cid:104) ∂ i ψ α ( x ) | ∂ j ψ α ( x ) (cid:105) andintroducing the Berry connection A α,i ( x ) ≡ i (cid:104) ψ α ( x ) | ∂ i ψ α ( x ) (cid:105) = − Im (cid:104) ψ α ( x ) | ∂ i ψ α ( x ) (cid:105) , oneimmediately finds F α ( x ) = [Re (cid:104) ψ α ( x ) | ψ α ( x + d x ) (cid:105) ] + [Im (cid:104) ψ α ( x ) | ψ α ( x + d x ) (cid:105) ] = 1 − (cid:88) ij [Re (cid:104) ∂ i ψ α ( x ) | ∂ j ψ α ( x ) (cid:105) − A α,i ( x ) A α,j ( x )] dx i dx j + O ( | d x | ) . (39)Now, the quantum metric tensor g α,ij ( x ) is defined [13, 33] by (cid:80) ij g α,ij dx i dx j ≡ − F α , suchthat g α,ij ( x ) = Re (cid:104) ∂ i ψ α | ∂ j ψ α (cid:105) + (cid:104) ψ α | ∂ i ψ α (cid:105) (cid:104) ψ α | ∂ j ψ α (cid:105) . (40)Upon calculating the Berry curvature Ω α,ij ( x ) ≡ ∂ i A α,j − ∂ j A α,i , one can observe that g α,ij and Ω α,ij are parts of a single complex tensor, the quantum geometric tensor (QGT) , firstintroduced by Berry in Ref. [14]: T α,ij ( x ) = (cid:104) ∂ i ψ α | ∂ j ψ α (cid:105) − (cid:104) ∂ i ψ α | ψ α (cid:105) (cid:104) ψ α | ∂ j ψ α (cid:105) ,g α,ij ( x ) = Re T α,ij = Re (cid:104) ∂ i ψ α | ∂ j ψ α (cid:105) − (cid:104) ∂ i ψ α | ψ α (cid:105) (cid:104) ψ α | ∂ j ψ α (cid:105) , Ω α,ij ( x ) = − T α,ij = − (cid:104) ∂ i ψ α | ∂ j ψ α (cid:105) . (41)From a practical point of view, the derivatives acting on energy eigenstates that appear inEq. (41) are not very convenient; in particular, recalling Eq. (21), they might introducespurious effects due to the global phase difference Γ α ( x + d x ) − Γ α ( x ) – the latter might not16e perturbative in d x if the states | ψ α (cid:105) at x and x + d x are not calcuated in the same gauge, i.e. from the same | ψ g (cid:105) .To circumvent this problem, there is an alternative expression of Eq. (41) in terms ofmatrix elements of the parametric velocity operators ∂ i H , which reads as T α,ij = (cid:88) β (cid:54) = α (cid:104) ψ α | ∂ i H | ψ β (cid:105) (cid:104) ψ β | ∂ j H | ψ α (cid:105) ( E α − E β ) . (42)Using the formal identity (cid:104) ψ α | ∂ i H | ψ β (cid:105) = (cid:104) ∂ i ψ α | ψ β (cid:105) ( E α − E β ), valid for α (cid:54) = β , the energiesin the denominator can be eliminated to recover Eq. (41). Equation (42) may further bewritten in an explicitly gauge-invariant form: T α,ij = (cid:88) β (cid:54) = α Tr { P α ( ∂ i H ) P β ( ∂ j H ) } ( E α − E β ) , (43)where we used the identity (cid:104) ψ α | O | ψ α (cid:105) = Tr { P α OP α } = Tr { P α O } . From there, by inserting H = (cid:80) γ E γ P γ , it is straightforward to show that Eq. (43) is equivalent to T α,ij = Tr { ( ∂ i P α ) (1 − P α ) ( ∂ j P α ) } . (44)This equation for the eigenprojectors is very useful. For purely computational purposes, it isalready sufficient to compute the QGT for an N > H whose eigenvaluesare known, Eq. (19) can be immediately combined with Eq. (44). However, similarly to the N = 2 case, it turns out to be much more instructive and convenient to transfer Eq. (44) tothe vectorial language. In order to obtain a Bloch vector picture of the geometric tensors, one can simply substituteEq. (23) for the projector P α in terms of the Bloch vector b α into Eq. (43) or Eq. (44).The most compact expression is obtained in the latter case, using the identities (29) &(37) that imply b α (cid:63) b iα = (1 − /N ) b iα . The QGT can then be expressed through Blochvectors as g α,ij = 14 b iα · b jα , Ω α,ij = − b α · (cid:0) b iα × b jα (cid:1) , (45)for arbitrary values of N (again with the shorthand notation m i ≡ ∂ i m ). For N = 2, Eq. (13)is recovered. For higher N , the only difference is the meaning of the dot and vector products,which have to be interpreted using the structure constants of Eq. (28). For the SU(3) case,an equivalent way of writing Berry curvature and quantum metric was already encounteredin Refs. [40, 41] and [42], respectively; for the SU( N ) case, Pozo and de Juan very recentlyfound an equivalent formula in terms of what they call [43]. While conceptuallyimportant, Eq. (45) is not necessarily in the most practical form for concrete calculations ofthe QGT, as it involves explicit parametric derivatives of the Bloch vector; this is potentiallycomplicated, as cumbersome expressions follow from applying the product rule to Eq. (35).(As we will see below, such complications can be eliminated for the Berry curvature due toorthogonality relations, but not for the quantum metric. )An alternative form of writing the QGT that does not involve such explicit derivatives ofthe Bloch vectors can be obtained by instead inserting Eq. (23) into Eq. (43) and exploiting17he identity (115): g α,ij = (cid:88) β (cid:54) = α S ijαβ ( E α − E β ) , Ω α,ij = − (cid:88) β (cid:54) = α A ijαβ ( E α − E β ) , (46)with S ijαβ = 4 N h i · h j + 1 N (cid:2) ( b α · h i )( b β · h j ) + ( b α · h j )( b β · h i ) (cid:3) + 2 N ( b α + b β ) · ( h i (cid:63) h j ) + 12 (cid:2) ( b α (cid:63) h i ) · ( b β (cid:63) h j ) + ( b α (cid:63) h j ) · ( b β (cid:63) h i ) (cid:3) ,A ijαβ = 2 N ( b α − b β ) · ( h i × h j ) + ( b α (cid:63) h i ) · ( b β × h j ) + ( b α × h i ) · ( b β (cid:63) h j ) . (47)It is straigthforward to check that S ijαβ is symmetric under the exchange of indices i, j (or of α, β ), whereas A ijαβ is antisymmetric under such exchange, and as a consequence (cid:80) α Ω α,ij = 0.While apparently much less compact than Eq. (45), the expressions above appear more con-venient to numerically calculate the QGT. Conceptually, they mainly distinguish themselvesfrom Eq. (45) in that, on the one hand, they involve solely the parametric derivatives of theHamiltonian vector, while on the other hand they also illustrate the interband nature of thetwo geometric tensors, and lastly they also show explicitly the importance of the star product for both the N -band quantum metric and Berry curvature. N -band Berry curvature from the Hamiltonian vector Having established the general expressions (45)–(47) for arbitrary values of N , the final steptowards obtaining a closed-form formula for the QGT in terms of the Hamiltonian vector h and the eigenenergy E α consists in inserting the explicit formula for b α ( h , E α ). In practice,for N > 2, writing down such an explicit analytical formula proves too cumbersome for thequantum metric, but can be achieved using Eq. (45) for the Berry curvature (for details, seeAppendix I). This is illustrated here for N = 3 and N = 4, for comparison with the simple N = 2 expression (13); the N = 5 formula as well as the discussion for arbitrary N is givenin Appendix I.In the N = 3 case, the Berry curvature tensor is found to be given byΩ α,ij = − E α h + h (cid:63) )(3 E α − | h | ) · (cid:2)(cid:0) E α h i + h i(cid:63) (cid:1) × (cid:0) E α h j + h j(cid:63) (cid:1)(cid:3) , (48)with h i(cid:63) ≡ ∂ i h (cid:63) = 2 h (cid:63) h i . This expression recovers the result from Barnett et al. [40] if oneinserts the closed-form energy parametrization (102). Interestingly, Eq. (48) can be simplifiedto the following more compact form (cf. Appendix I):Ω α,ij = − E α − | h | ) (cid:8) E α (cid:2) | h | h · (cid:0) h i × h j (cid:1) + 3 h · (cid:0) h i(cid:63) × h j(cid:63) (cid:1)(cid:3) + (cid:0) E α + | h | (cid:1) h (cid:63) · (cid:0) h i × h j (cid:1)(cid:9) . (49)It may be verified that (cid:80) α E α (3 E α − | h | ) − = 0 and (cid:80) α (3 E α + | h | )(3 E α − | h | ) − = 0 suchthat the Berry curvature sum rule (cid:80) α Ω α,ij = 0 holds.Analogously, for arbitrary N = 4 systems, we haveΩ α,ij = − Q α h + E α h (cid:63) + h (cid:63)(cid:63) )(4 E α Q α − h · h (cid:63) ) · (cid:2)(cid:0) Q α h i + E α h i(cid:63) + h i(cid:63)(cid:63) (cid:1) × (cid:0) Q α h j + E α h j(cid:63) + h j(cid:63)(cid:63) (cid:1)(cid:3) , (50)18ith Q α ≡ E α − | h | / 2. One could proceed to develop further this expression using SU(4)Jacobi identities, to derive the SU(4) analog of Eq. (49). This is outlined in Appendix I,but it leads to a rather lengthy expression that we will not write explicitly. For practicalimplementation, Eq. (50) is already in a convenient form.The procedure for N > N -band systems with particle-hole symmetric energy spec-trum It is worthwhile to consider the above generic results in more detail for special cases of practicalinterest. In the present section, we will therefore restrict to the special class of N × N Hamiltonians with a particle-hole symmetric (PHS) energy spectrum. This is of interestfor many physical applications and corresponds to Hamiltonians characterized by an effectivechiral and/or charge conjugation symmetry. In the case of a PHS spectrum, the Hamiltonian’scharacteristic polynomial is drastically simplified. Consequently, the eigenenergies enteringinto the projector and Bloch vector expansions (19) & (35) are of quite simple form, allowingus to proceed yet a bit further in the analytical treatment than is possible in the case of fullygeneric E α . In particular, the Berry curvature formulas of Section 4.3 simplify considerably,and it becomes possible to write down tractable closed-form expressions for the quantummetric. We define a particle-hole symmetric (PHS) energy spectrum (or band structure) as a (ingeneral x -dependent) set of energy eigenvalues { E α } = {± (cid:15) , ..., ± (cid:15) k , ..., ± (cid:15) N } , if N even , { E α } = { , ± (cid:15) , ..., ± (cid:15) k , ..., ± (cid:15) N − } , if N odd , (51)of the Hamiltonian (22), where (cid:15) k < (cid:15) k +1 . According to the tenfold way symmetry classifica-tion [63], such a PHS spectrum can be caused by a chiral symmetry and/or charge conjugationsymmetry. (Note that some authors use the term ”particle-hole symmetry” instead of ”chargeconjugation symmetry”. We will employ the latter term throughout to avoid confusion; im-portantly, there can be a PHS spectrum (51) in the absence of charge conjugation symmetry.)If the spectrum is PHS, the Casimir invariants (16) take a particularly simple form: C n = 2 (cid:98) N/ (cid:99) (cid:88) k =1 (cid:15) nk ,C n +1 = 0 , (52)where n ∈ N . Conversely, one can use the Casimir invariants to check whether the spectrumof a given SU( N ) Hamiltonian H is PHS. For N even (odd), all odd (even) powers in thecharacteristic polynomial (92) vanish if C n +1 = Tr (cid:0) H n +1 (cid:1) = 0 , ∀ n + 1 ≤ N , in which caseall solutions of the characteristic equation occur in pairs of opposite sign. In other words,if one computes the Casimir invariants C n +1 , 2 n + 1 ≤ N , and if they all vanish, then thespectrum of H is PHS.Let us restrict to N < N = 3 , C =Tr H = 0 by construction) the condition for a PHS spectrum is simply C = 0, i.e. h · h (cid:63) = 019ccording to Eq. (33). For N = 5, one needs C = 0 additionally, which further requires h · h (cid:63)(cid:63)(cid:63) = 0. From Eq. (52) one then immediately obtains N = 2 and N = 3 : (cid:15) = (cid:114) C ,N = 4 and N = 5 : (cid:15) , = (cid:115) C ∓ (cid:114) C − C , (53)where C = 2 | h | and C = 4 | h | /N + 2 | h (cid:63) | , cf. Eq. (33). Any PHS energy spectrum canthus always be written in the form N = 2 : E α = α | h | , α = {± } ,N = 3 : E α = α | h | , α = { , ± } ,N = 4 : E α α = α (cid:115) | h | α | h (cid:63) |√ , α = {± } , α = {± } ,N = 5 : E α α = α (cid:115) | h | α (cid:114) | h (cid:63) | − | h | , α = { , ± } , α = {± } . (54)Note finally that from Eq. (33), the orthogonality relation h (2 n ) (cid:63) · h (2 m +1) (cid:63) = 0 (55)is implied by the vanishing of odd invariants C n +1 . The above properties of PHS spectra can now be used to obtain the eigenprojectors, Blochvectors and quantum geometry for this particular class of Hamiltonians. Here, the cases N = 3 and N = 4 are exposed in detail and, for completeness, the N = 5 case is discussed inAppendix J. Three-band systems Under the constraint of a particle hole-symmetric spectrum, the eigenprojectors and Blochvectors take a particularly simple form: P α = 13 α − (cid:20) ( α − + α (cid:114) C H + 2 C H (cid:21) , b α = 23 α − (cid:18) α u + 1 √ u (cid:19) , (56)with α = 0 , ± , as well as C = √ C = 2 | h | , | h (cid:63) | = | h | / √ u ≡ h / | h | and u ≡ h (cid:63) / | h (cid:63) | ,such that { u , u } form an orthonormal set. It is easily checked that Tr P α = 1 and | b α | = 4 / α , as required by Eq. (37).The Berry curvature is immediately obtained from Eq. (49) as:Ω α,ij = − α − | h | (cid:20) α h · ( h i × h j ) + 3 α | h | h · ( h i(cid:63) × h j(cid:63) ) + 3 α + 1 | h | h (cid:63) · ( h i × h j ) (cid:21) . (57)20t is easily checked that the sum over α of each contribution vanishes. The quantum metricis most compactly expressed as a function of u and u , by inserting Eq. (56) into Eq. (45): g α,ij = 1(3 α − (cid:18) α u i + 1 √ u i (cid:19) · (cid:18) α u j + 1 √ u j (cid:19) . (58)Carrying out the derivatives on u and u , one may express this more explicitly as g α,ij = 1(3 α − (cid:26) α | h | (cid:20) h i · h j − ( h · h i )( h · h j ) | h | (cid:21) + α | h | (cid:2) h i · h j(cid:63) + h i(cid:63) · h j (cid:3) + 1 | h | (cid:20) h i(cid:63) · h j(cid:63) − 43 ( h · h i )( h · h j ) (cid:21)(cid:27) , (59)where it was used that h (cid:63) · h i = − h i(cid:63) · h = − h i (cid:63) h ) · h = 0 due to PHS. This result may becompared with Eq. (13). Evidently, the quantum metric is a rather complicated object, evenunder the constraint of a PHS spectrum. Four-band systems In this case it proves convenient to rewrite the eigenergies as E α α = α E α with E α =( | h | / α | h (cid:63) | / √ / according to Eq. (54). For the projector and the Bloch vector onethen obtains P α α = 14 E α − C (cid:20)(cid:18) E α − C (cid:19) + α (cid:18) E α − C E α (cid:19) H + H + α E α H (cid:21) , b α α = α u α + α √ u ≡ α u + α u | u + α u | + α √ u , (60)where u ≡ h / | h | , u ≡ h (cid:63) / | h (cid:63) | , u ≡ h (cid:63)(cid:63) / | h (cid:63)(cid:63) | and with | h (cid:63)(cid:63) | = | h || h (cid:63) | / √ 2. The vectors { u + , u − , u } form an orthonormal set, however u · u = | h (cid:63) | / ( √ | h | ). It is straightforwardto check that Tr P α α = 1 and | b α α | = 3 / 2, independently of α , α .Using Eq. (50), the Berry curvature in terms of h reads:Ω α α ,ij = − α α √ | h (cid:63) | E α (cid:18) α √ | h (cid:63) | h + α E α h (cid:63) + h (cid:63)(cid:63) (cid:19) · (cid:20)(cid:18) α √ | h (cid:63) | h i + α E α h i(cid:63) + h i(cid:63)(cid:63) (cid:19) × (cid:18) α √ | h (cid:63) | h j + α E α h j(cid:63) + h j(cid:63)(cid:63) (cid:19)(cid:21) . (61)For the quantum metric it is again simple to write it in terms of the unit vectors: g α α ,ij = 14 (cid:20) α u iα + α √ u i (cid:21) · (cid:20) α u jα + α √ u j (cid:21) , (62)however the corresponding expression in terms of h becomes too lengthy to be providedexplicitly.For completeness, the five-band case is discussed in Appendix J. Let us now illustrate the formalism developed above by applying it to concrete N = 3 and N = 4 systems with interesting quantum geometric properties. The vector of parameters x 21s crucial for the form of the quantum geometric tensor (QGT) and can take a large varietyof different meanings depending on the physical system considered. In solid-state physics,where it is common to work in the Fourier space of the real-space periodic lattice, it is knownsince Zak’s seminal paper [64] at the latest that taking x = k to be the crystal momentumis a natural choice for the analysis of Berry phases and related geometrical and topologicalproperties of solids. More generally, in modern condensed matter physics, taking x = q tobe a quasi-momentum of some kind is the starting point for the analysis of many differentkinds of geometrical and topological effects, with various physical meanings attached to q [20,65–69]. It is in this general sense, not restricted to purely crystalline systems, that q shouldbe understood here.While the formalism developed in this paper applies just as well to Hamiltonians set up ona lattice (typically some tight-binding model), the models that are considered in the followingwill, for concreteness, all belong to the class of multifold (Dirac) fermions (or multifold linearband crossings ), terms that we will use as referring to a Hamiltonian of the form H ( q ) = (cid:88) i q i v i · λ , h ( q ) = (cid:88) i q i v i , (63)where the number of terms in the sum represents the dimension of the quasi-momentum q -space; for example q = ( q x , q y , q z ) for a three-dimensional model. The matrices { λ j } are thegeneralized Gell-Mann metrices introduced in the context of Eq. (22). The vectors v i , whichcontain all relevant information about the model, should be understood as q -independentparametric velocities, v i = Tr { ( ∂ q i H ) λ } .For generic velocity vectors v i the energy spectrum of the Hamiltonian (63) is composed of N bands that exhibit a linear band crossing with an N -fold degeneracy at q = 0. Importantly,the existence of such a degenerate point implies a strong coupling between the bands, whichshould be reflected in the quantum geometric properties (Berry curvature and quantum met-ric). In particular, in the familiar case of a three-dimensional (3D) quasi-momentum space,depending on the effective symmetries (charge conjugation, time-reversal and/or chirality)that are implicitly encoded in the velocity vectors v i , the Berry curvature may take the formof an effective topological monopole that gives rise to an integer quantized Berry flux (mea-sured by the first Chern number) through a surface enclosing the degeneracy point. In themore exotic case of a four-dimensional (4D) quasi-momentum space, it was recently shownthat the degeneracy point may act as a tensor monopole, in which case the quantum metricappears as the key ingredient for characterizing another kind of integer topological number,called Dixmier-Douady invariant [70–72]. More generally, the peculiar quantum geometricproperties of linear Dirac Hamiltonians of the form (63) are known to play a key role in manyphysical phenomena, as is evident already from the simplest example of a two-fold degeneracyin graphene and Weyl semimetals. The more complicated case involving multiply degeneratepoints ( N > 2) is currently intensely investigated in many different experimental contexts[73–91].Hereafter, for concreteness, we restrict to a certain class of multifold fermions and discussthe cases N = 3 and N = 4 in detail. Since we wish to focus on instructive models with asimple band structure, we may impose certain constraints on the Hamiltonian, which can beformulated in terms of the v i as discussed in Section 6.1. Section 6.2 will serve to presentexplicit examples for Hamiltonians subject to those constraints; in particular, we will comparethe quantum geometry and topology of pseudospin- S fermions to multifold fermions beyondspin- S . 22igure 2: Schematic illustration of possible energy band structures (in q -space) in the vicinityof the N -fold degeneracy at q = 0, using the examples N = 3 and N = 4. Different bandstructures can be grouped into (a) not PHS, (b) PHS but anisotropic, (c) PHS and isotropic. All multifold fermion models (63) that we will consider are constrained as explained in thefollowing. First, we take the quasi-momentum space to be 3D, i.e. q = ( q x , q y , q z ). Second,we will only consider mutually orthogonal velocity vectors with v i · v j = c δ ij , (64)where c is a positive real constant. In the case N = 2 the energy spectrum is automaticallyparticle-hole symmetric (PHS) due to the traceless character of H . For N = 3 and N = 4,the band structure is not automatically PHS. Instead, different kinds of band structures arepossible, as illustrated schematically in Fig. 2.As a third constraint, we are thus interested in N = 3 and N = 4 models that exhibit anenergy band structure with the two main properties illustrated in Fig. 2(c), namely it shouldbe PHS and isotropic in quasi-momentum space. For N = 2, the orthogonality condition(64) is sufficient to ensure an isotropic spectrum (in q -space), i.e. E ± = ±√ c | q | . Noticehowever that, for N > 2, Eqs. (63) & (64) alone are not sufficient for rendering the bandstructure PHS and/or isotropic; instead, additional constraints on the v i are necessary. Asa last constraint, the energy bands should be non-degenerate for q (cid:54) = 0 and not explicitlydependent on the three velocity vectors v x,y,z .The point we wish to make is that, despite these rather restrictive constraints, it is possibleto find several classes of models that exhibit distinct quantum geometry (sometimes explicitlydependent on v i ), as well as distinct topological properties as reflected by the first Chernnumber.To see how the above constraints on the energy spectrum can be imposed quantitatively,let us first observe that, for N = 3 and N = 4, the identity( v i (cid:63) v j ) · v k = 0 , for any triple ( i, j, k ) , (65)is necessary and sufficient to satisfy the constraint of a PHS spectrum – this simply representsthe identity C = Tr (cid:0) H (cid:1) = 0 in vector form, cf. Eq. (33).23or N = 3, once Eq. (65) is verified, the orthogonality (64) automatically imposes thedesired additional constraint of an isotropic energy spectrum E α ( q ) = α √ c | q | , where α = 0 , ± , (66)implying a Dirac cone along with a zero-energy flat band. For N = 4, however, in additionto the two constraints (64) & (65) on the v i , one further requires the condition that | h (cid:63) | beproportional to | h | , i.e. √ | (cid:80) i,j q i q j ( v i (cid:63) v j ) | = d | q | , for obtaining an isotropic spectrum E α α ( q ) = α √ (cid:112) c + α d | q | , where α = ± , α = ± , (67)as can be verified from Eq. (54). Here, d ≤ c is a non-negative real constant; in thelimiting case d = 0 ( d = c ), the double-cone structure implied by Eq. (67) reduces to twodegenerate Dirac cones (a Dirac cone and two degenerate flat bands). S multifold fermions and beyond We can now distinguish several classes of multifold fermions featuring a band structure ofthe type just described. Models belonging to different classes have distinct geometrical andtopological properties. The standard way of demonstrating that the topological properties ofany two models are distinct consists in showing that they have different effective symmetriesthat place them in different topological classes according to the tenfold way classification[63]. In short, whenever a multifold fermion model exhibits a PHS spectrum, it is due tothe presence of one or both of the following effective symmetries: (a) A chiral symmetry S † H ( q ) S = − H ( q ) of the first-quantized Hamiltonian matrix, represented by a unitary matrix S . (b) A charge conjugation symmetry C † H ∗ ( q ) C = − H ( q ), represented by a unitary matrix C . Since all of our models are PHS, their topological properties can be distinguished accordingto whether only one or both of the matrices S and C can be found, and further according tothe sign of the product CC ∗ = ± N .One obvious class consists of simple pseudospin- S fermions [where S = ( N − / 2] andis considered in Section 6.2.1. This provides both a consistency check for our formalismand a pedagogical example for how the Hamiltonian vectors, Bloch vectors, Berry curvatureformulas et cetera , introduced throughout this paper, can be used in practice. In the languageof the v i , we will see that the class of pseudospin- S fermions encompasses all models for whichthe three orthogonal velocity vectors verify the identity v i × v j = 12 (cid:15) ijk v k , for any triple ( i, j, k ) . (68)All such models possess a global charge conjugation symmetry with CC ∗ = ( − S S +1 andthe quantum geometry is fully independent of the v i ; the Berry curvature takes the form of atopological monopole that gives rise to an integer quantized Berry flux (first Chern number).Two other classes of multifold fermions, going beyond pseudospin- S but exhibiting thesame type of band structure, are considered in Section 6.2.2. The first class, for which wepresent both N = 3 and N = 4 examples, corresponds to models that have a global chiralsymmetry, in which case the velocity vectors verify the identity( v i × v j ) · v k = 0 , for any triple ( i, j, k ) . (69)Importantly, in that situation the quantum geometry is in general anisotropic and tunable bythe velocity vectors, despite the band structure being isotropic and fixed. Correspondingly,24here is no finite Berry flux associated to the Berry curvature. For N = 4, there is a secondclass that corresponds to models such that( v i × v j ) · v k = f (cid:15) ijk , for any triple ( i, j, k ) , (70)with v i × v j (cid:54) = (cid:15) ijk v k for at least one triple ( i, j, k ) and where f is a non-zero positiveconstant (it is a function of c and d ). In this case, there is again a global charge conjugationsymmetry, but now CC ∗ = 1 N in contrast to a pseudospin-3/2. The quantum geometry againappears to be independent of the velocity vectors, and the point q = 0 again corresponds toa topological monopole; its quantized charge, however, is in general different from the one ofa pseudospin-3/2. S multifold fermions The simplest setup for a (pseudo)spin- S multifold fermion involves three spin matrices S x,y,z that satisfy a spin algebra[ S i , S j ] = i(cid:15) ijk S k , for any triple ( i, j, k ) . (71)as well as S = S ( S + 1)1 N with N = 2 S + 1, and in terms of which the Hamiltonian can bewritten as H ( q ) = q x S x + q y S y + q z S z . (72)It may be checked that the matrix C = e iπS y plays the role of an effective charge-conjugationsymmetry. As a consequence, the band spectrum of H ( q ) is particle-hole symmetric. It isgiven by E m ( q ) = m | q | , with m = − S, ..., + S . Eigenstate approach In the conventional eigenstate picture, one may construct the q -dependent spin eigenstates | S, m, q (cid:105) as rotated eigenstates of the form | S, m, q (cid:105) = e iθ q (sin ϕ q S x − cos ϕ q S y ) | S, m (cid:105) , where S z | S, m (cid:105) = m | S, m (cid:105) and cos θ q = q z / | q | , tan ϕ q = q y /q x . One may then use Eq. (42)and the spin eigenstates | S, m, q (cid:105) to obtain the Berry curvature tensor Ω m,ij ( q ) in the 3D pa-rameter space q = ( q x , q y , q z ). Defining the three-component Berry curvature pseudo-vector Ω m ≡ (Ω m,yz , Ω m,zx , Ω m,xy ), it takes the form of a topological monopole [5]: Ω m ( q ) = − m q | q | . (73)The flux of this monopole through any surface enclosing q = 0 results in a topological chargemeasured by the first Chern number C m = − m . As is well known, C m is odd for half-integerspin because CC ∗ = − 1, whereas C m is even for integer spin because CC ∗ = 1. Eigenprojector (Bloch vector) approach As a first simple check, it is now shown for the spin-1 and spin-3/2 cases (correspondingto three-band and four-band Hamiltonians, respectively) that the result (73) for the Berrycurvature can also be obtained without constructing spin eigenstates when our general methodis used. For completing the full QGT we will also compute the quantum metric. Of course, inthe present case of a simple spin- S , using the eigenprojector formalism is not at all necessaryto find the QGT, but the spin Hamiltonian provides a suitable opportunity to get familiarwith the general procedure that can be carried out for an arbitrary Hamiltonian. Indeed,25n more complicated situations (see for example Section 6.2.2), the formalism proves veryadvantageous, avoiding any considerations involving eigenstates.To start with, we rewrite the Hamiltonian (72) in the multifold fermions form of Eq. (63)by defining velocity vectors as v i = Tr( S i λ ), or equivalently S i = v i · λ . Each vector v i has N − S ( S + 1) components, but it appears that only 2 S components are non-zero (usingthe conventional matrix representation for spin matrices in Appendix D). It is clear that thevelocity vectors v i are orthogonal, however their norm depends on S as | v i | = S ( S + 1)(2 S + 1)6 . (74)Likewise, note that the spin matrix commutation relations (71) translate immediately intothe vector identity (68). Recall further that the Hamiltonian vector h and the star productsconstructed from it are key ingredients for the eigenprojector (Bloch vector) formalism. In thepresent case, the Hamiltonian vector is given by Eq. (63) and leads to h (cid:63) = (cid:80) i,j q i q j ( v i (cid:63) v j ), h (cid:63)(cid:63) = (cid:80) i,j,k q i q j q k ( v i (cid:63) v j ) (cid:63) v k , et cetera . Be finally reminded that the star and vector productoperations that we use here very frequently are determined by the character of the associated su ( N ) algebra, cf. Section 3.2, and thus depend on the spin size S .At this point we separate the analysis for the spin-1 and spin-3/2 cases. Consider first thespin-1 case ( N = 3 bands), where each vector v i has N − e j in this 8D space, i.e. writing v i = (cid:80) j v ji e j , one gets from therelation (104) between spin-1 matrices and SU(3) Gell-Mann matrices: v x = √ ( e + e ), v y = √ ( e + e ) and v z = ( e + √ e ). The vectors h and h (cid:63) follow as h ( q ) = 12 ( √ q x , √ q y , q z , , , √ q x , √ q y , √ q z ) , h (cid:63) ( q ) = (cid:32) q x q z √ , q y q z √ , q z − | q | , q x − q y , q x q y , − q x q z √ , − q y q z √ , | q | − q z √ (cid:33) . (75)The spectrum E m = m | q | , where m = 0 , ± 1, is of the form (66) with a normalization factor c = 1, in agreement with Eq. (74). Upon evaluating C = (cid:80) m E m = 2 | q | , the Bloch vectorsfor a pseudospin-1 fermion are readily obtained from Eq. (35): b m ( q ) = 23 m − (cid:18) m h | q | + h (cid:63) | q | (cid:19) . (76)Equivalently, the eigenprojectors follow immediately from Eq. (19) or simply by inserting theBloch vector (76) into Eq. (23): P m ( q ) = 13 m − (cid:20) ( m − + m | q | H + 1 | q | H (cid:21) = 13 1 + 13 m − (cid:18) m h | q | + h (cid:63) | q | (cid:19) · λ . (77)We are now ready to compute the Berry curvature from either of the formulas (45), (49) or(57) – the last one is the most convenient here. Inserting the vectors h and h (cid:63) , one recoversEq. (73) as expected. Note that in obtaining this result, the last term of Eq. (57) vanishes, This can be obtained by calculating on the one hand S = (cid:80) i ( v i · λ ) from Eq. (29), and using on theother hand S = S ( S + 1)1 S +1 ; we also find the velocity vector identities v z (cid:63) v z = 0 and v x (cid:63) v x = − v y (cid:63) v y . (cid:63) · ( h i × h j ) = 0, while h · ( h i(cid:63) × h j(cid:63) ) / | h | = h · ( h i × h j ). We can also quickly compute thequantum metric tensor for each band, from either of Eqs. (45), (58) or (59): g m,ij ( q ) = 2 − m | q | (cid:18) δ ij − q i q j | q | (cid:19) , (78)Drawing on Ref. [92], we can indeed see that this is the spin-1 special case of the quantummetric formula g m,ij ( q ) = S ( S + 1) − m | q | (cid:18) δ ij − q i q j | q | (cid:19) (79)for a generic spin- S fermion.Similarly, considering now the spin-3/2 case ( N = 4 bands), each vector v i has N − e j in this 15-dimensional space one has v x = ( √ e + 2 e + √ e ), v y = ( √ e + 2 e + √ e ), v z = ( e + √ e + √ e ) fromthe relation (105) between spin-3/2 matrices and SU(4) Gell-Mann matrices. The vector h then follows as h ( q ) = 12 ( √ q x , √ q y , q z , , , q x , q y , √ q z , , , , , √ q x , √ q y , √ q z ) , (80)from which one can now obtain h (cid:63) and h (cid:63)(cid:63) . The spectrum E m = m | q | , where now m = ± , ± , is of the form (67) with c = 5 / d = 2. Upon evaluating C = (cid:80) m E m = 5 | q | and using C = 0 we obtain the corresponding Bloch vectors from Eq. (35) as b m ( q ) = 12 m (cid:0) m − (cid:1) (cid:20)(cid:18) m − (cid:19) h | q | + m h (cid:63) | q | + h (cid:63)(cid:63) | q | (cid:21) . (81)Using this expression and either of the formulas provided in Sections 4 & 5, we find that theBerry curvature again takes the expected form (73), and the quantum metric is given by Eq.(79) with S ( S + 1) = 15 / S multifold fermions As announced earlier, the purpose of this last section is to show, using examples with N = 3and N = 4 bands, that there are many possibilities to design Hamiltonians that drastically differ from a simple pseudospin on the level of the eigenprojectors, geometry and topology,even if the band structure is indistinguishable, i.e. of the form (66) & (67). Conveniently, noinformation about eigenstates whatsoever will be required for this discussion. Three-band models In the N = 3 case, we find that a 3D multifold fermion Hamiltonian (63) with a spectrumof the form (66) can only be of two kinds: either it is non-spin-like, which is the case if itexhibits a global chiral symmetry, represented by a matrix S . Or it is spin-like, in which caseit exhibits a global charge conjugation symmetry with CC ∗ = 1 . Chiral models: Two examples for this class are given by the family of Hamiltonians H ( θ ) A , determined by v x = e , v y = e , v z = c θ e + s θ e , and the family H ( θ ) B , determinedby v x = c θ e + s θ e , v y = c θ e − s θ e , v z = e . Here, θ is a free parameter, abbreviations s θ ≡ sin θ and c θ ≡ cos θ are used and we again employ Cartesian basis vectors e j in 8D space.27n matrix form, these models simply read H ( θ ) A = q x − ic θ q z ... q y − is θ q z ... ... , H ( θ ) B = c θ ( q x − iq y ) − iq z ... s θ ( q x + iq y ) ... ... , (82)where the lower left matrix elements are obtained by complex conjugating the upper right ones.For H ( θ ) A , the presence of a chiral symmetry is immediately obvious, with S = diag( − , , − 1) = − − λ + √ λ . For H ( θ ) B , this is less evident, but it is possible to construct a symmetryoperator S = − + c θ λ + 2 s θ c θ λ + √ ( c θ − s θ ) λ . Note that this symmetry is truly globalin the sense that it is q -independent.For the models H ( θ ) A and H ( θ ) B we have c = 1, i.e. their spectrum is θ -independent andexactly the same as for the spin-1 fermion discussed in Section 6.2.1. However, both modelsare far from being a spin-1, since instead of verifying Eq. (68) they share the property (69).Indeed, the same is true for any other chiral model that we can find – assuming each suchmodel to have a spectrum (66). Interestingly, the vector identity (69) appears to provide aquick and direct way to check for a chiral symmetry that is much simpler than the explicitconstruction of a matrix S .Moreover, for any of our chiral models the QGT for all bands α = 0 , ± Ω α ( q ) = (2 − α )( q · µ ) q | q | ,g α,ij ( q ) = 2 − α | q | (cid:20) δ ij − q i q j | q | − − α | q | ( q × µ ) i ( q × µ ) j (cid:21) , (83)where we again introduced a Berry curvature pseudovector Ω α and where µ is a functionof the velocity vectors v i . For the particular example H A , we have µ = ( s θ , c θ , µ = (0 , , 1) for the model H B .There are several notable features of Eq. (83). First, the Berry curvature is completelydifferent from its spin-1 analog (73) and yields zero upon integration over a closed surfacearound the degeneracy, i.e. all Chern numbers C α vanish. Second, the quantum metric containsa part that is exactly like for a spin-1 [cf. Eq. (78)], plus an additional term. Third andimportantly, both Berry curvature and quantum metric are anisotropic for any choice of µ and tunable by the velocity vectors; in particular, the QGT is a function of θ for the Hamiltonian H ( θ ) A . Spin-like models: Two simple examples for this class are the Hamiltonian H C , deter-mined by v x = e , v y = e , v z = e , and the family H ( θ ) D , determined by v x = c θ e + s θ e , v y = c θ e − s θ e , v z = e , or in matrix form H C = − iq x − iq y ... − iq z ... ... , H ( θ ) D = e − iθ q x − iq z ... e iθ q y ... ... . (84)These models have a global charge conjugation symmetry represented by matrices C = 1 and C = diag(1 , − e − iθ , CC ∗ = 1.28or H C and H ( θ ) D we also have a θ -independent spectrum with c = 1, but in contrastto the chiral models above the present models are indeed pseudospin-1 fermions. We maysee this by checking that they verify Eq. (68), which means that they can be written inthe form of Eq. (72), with some spin matrices that are in general different from the standardspin-1 representation (104). Moreover, the Berry curvature vector Ω α ( q ) and quantum metric g α,ij ( q ) of these models (calculated from the same Eqs. as used above) are indeed given byEqs. (73) and (78), respectively (with m → α ), further corroborating their spin-1 nature. Four-band models In the N = 4 case, we find that a 3D multifold fermion with a spectrum of the form (67) canbe of three types. First, it may be of type spin-3/2, in which case it has all the propertiesdiscussed in Section 6.2.1. Here we focus on the two possible types of non-spin-like fermions:either the model exhibits a global chiral symmetry, or it exhibits a global charge conjugationsymmetry with CC ∗ = 1 N . Examples for both classes are provided below. Be aware thatconstructing models that have a PHS and isotropic band structure is now much harder toconstruct than for the N = 3 case. In particular, it is very difficult to verify the isotropycondition stated just before Eq. (67). Chiral models: The Hamiltonian that we will consider as a representative example forthe chiral class is inspired by the 4D topological semimetal model of Ref. [71] and is given by v x = a ( e − e ) + b ( e − e ), v y = a ( e − e ) + b ( e − e ), v z = a [ c θ ( e + e ) + s θ ( e + e )] + b [ c θ ( e + e ) + s θ ( e + e )], where a, b are non-negative real parameters, or in matrixform H ( θ ) E = aq − + be − iθ q z bq − + ae − iθ q z ... − bq − + ae − iθ q z ... ... − aq − + be − iθ q z ... ... ... , (85)where q − ≡ q x − iq y . This model exhibits a global chiral symmetry S = diag(1 , − , − , , c = 2( a + b ) and d = 4 ab , i.e. E α α = α ( a + α b ) | q | ;notice that this is again θ -independent. Fermions with a spectrum of this kind, with twodifferent effective Dirac velocities, have been dubbed birefringent fermions in the 2D case[93]. Two flat bands along with a Dirac cone (two degenerate Dirac cones) appear for a = b ( a = 0 or b = 0), and the band structure of a pseudospin-3/2 fermion is recovered in thespecial case a = 2 b = 1.However, for any possible values of a and b , including the case a = 2 b where the spectrumis the same as for a pseudospin-3/2, the model H ( θ ) E is not at all a pseudospin-3/2; just likethe chiral models that we considered in the N = 3 case, the velocity vectors have the property(69) instead of verifying Eq. (68). Thus, identity (69) indeed appears to be a generic featureof chiral Hamiltonians with a PHS and isotropic band structure.The QGT for the Hamiltonian H ( θ ) E [obtained either from Eq. (45) with the Bloch vector(60), or more directly from Eqs. (61) & (62)] can be written in compact form as Ω α α ( q ) = − α ( q · µ ) q | q | ,g α α ,ij ( q ) = 12 | q | (cid:20) δ ij − q i q j | q | − | q | ( q × µ ) i ( q × µ ) j (cid:21) , (86)29here µ = ( c θ , s θ , N = 3 chiral models, and the same interesting features are present. Moreover, notethe fundamentally different role of the parameters: a and b affect the spectrum but not thequantum geometry, while the role of θ is the very opposite. The model H ( θ ) E thus allows totune band structure and quantum geometry completely independently. Charge conjugation models: For this class, two examples are given by H F = − i ( aq x + bq z ) − i ( bq x + aq y ) i ( bq y − aq z ) ... − i ( bq y + aq z ) − i ( bq x − aq y ) ... ... − i ( aq x − bq z ) ... ... ... ,H G = aq y + bq z − i ( aq x + bq y ) bq x − aq z ... bq x + aq z i ( aq x − bq y ) ... ... aq y − bq z ... ... ... , (87)where the Hamiltonian H F is determined by v x = a ( e + e ) + b ( e + e ), v y = a ( e − e ) + b ( e − e ), v z = a ( e + e )+ b ( e − e ) in vector language, and H G by v x = a ( e − e )+ b ( e + e ), v y = a ( e + e ) + b ( e + e ), v z = a ( e − e ) + b ( e − e ). These models have a globalcharge conjugation symmetry represented by matrices C = 1 and C = diag(1 , − , , − E α α = α ( a + α b ) | q | that is exactlythe same as the spectrum of H ( θ ) E , but H F and H G are in fact neither chiral nor spin-like.We may see this by checking that they violate Eqs. (68) & (69) but satisfy Eq. (70) with f = 2( a + b ).The Berry curvature and quantum metric of these models (calculated in the same way asfor the chiral case above) are given by Ω α α ( q ) = − α (1 + α ) q | q | ,g α α ,ij ( q ) = 12 | q | (cid:18) δ ij − q i q j | q | (cid:19) , (88)which is again completely independent of a and b . Though this result is reminiscent of aspin-like fermion, the models are in fact not of spin-like type; indeed, according to the Chernnumbers obtained from Eq. (88), and also according to the quantum metric, the models differfrom any spin- S fermion. Summary of multifold fermion models For any N , the simplest class of linear band crossings that are PHS and isotropic consistsof pseudospin- S fermions, where S = ( N − / 2. Such models are characterized by a globalcharge conjugation symmetry CC ∗ = ( − N − N , the velocity vectors verify Eq. (68), andthe QGT is of the well-known form (73) & (79), as visualized in Fig. 3.For N = 3 and N = 4, we focused on classes that are not of this spin-like type but exhibitexactly the same spectrum, of the form (66) and (67), respectively. The results are visualizedin Fig. 3 and can be stated as follows: 30a) Charge conjugation class: This class exists for N = 4 but not for N = 3. The Hamilto-nian H has a global charge conjugation symmetry with CC ∗ = 1 N , the velocity vectorsverify Eq. (70), and the QGT is of the form (88).(b) Chiral class: This class exists for both N = 3 and N = 4. The Hamiltonian H has aglobal chiral symmetry, the velocity vectors verify Eq. (69), and the QGT is given byEqs. (83) and (86), respectively. Importantly, the QGT is anisotropic and tunable by avector µ that does not affect the spectrum.Note the important role of the word ”global”. A system with a global chiral symmetry mayexhibit a local ( i.e. q -dependent) charge conjugation symmetry, as is the case for the model H ( θ ) B . Likewise, there can be a local chiral symmetry in the presence of a global chargeconjugation symmetry, as for model H ( θ ) D . The global symmetries appear to be decisive forthe resulting type of monopole ( i.e. for the form of the QGT).Figure 3: Quantum geometry for 3D multifold fermions with a spectrum (66) or (67). In case(a) the Chern number C α is odd for CC ∗ = − N , even for CC ∗ = 1 N , and flips sign within eachparticle-hole conjugate band pair. The quantum metric is determined by ν α = Tr g α ( q ) | q | .In case (b) one has C α = 0 for all bands; the Berry curvature is determined by a number κ α that is identical within each particle-hole conjugate band pair. The QGT is anisotropic andtunable by a vector µ that is a function of the velocity vectors v i .31 Summary and conclusions For a physical system described by a Hermitian N × N Hamiltonian matrix H = H ( x ),where the vector x represents some parameters, there are situations in which it is useful torecall that any eigenstate | ψ α ( x ) (cid:105) ( α = 1 , ..., N ) of H is more fundamentally encoded in the eigenprojector matrix P α ( x ) = | ψ α ( x ) (cid:105) (cid:104) ψ α ( x ) | . We here use the term ”more fundamental” inthe sense that, while | ψ α (cid:105) and P α can always be constructed from one another, P α does notsuffer from gauge arbitrariness and uncontrollable singularities. The inconvenience of theselatter features of eigenstates is apparent already in the simple N = 2 case, as reviewed inSection 2.Consequently, for a given quantity of interest whose standard expression is known in termsof eigenstates, it can prove rewarding to aim for a reformulation in terms of eigenprojectors,which will entirely eliminate the necessity for computing | ψ α (cid:105) explicitly. Such a reformulationcan be done for any physical quantity in principle. The present work demonstrates thisby selecting as a prominent example the special case where the quantities of interest are thequantum metric and Berry curvature tensors (forming together what is known as the quantumgeometric tensor, QGT). Selecting the QGT is natural for two reasons. First, it is relativelysimple (compared to more involved quantities like the orbital magnetic susceptibility), andtherefore serves well for illustrating the main features of the eigenprojector approach, whichare preserved in the treatment of more complicated observables. Second, the inconvenientfeatures of eigenstates become particularly problematic when one is interested in geometricalquantities; therefore, the QGT is one of the observables for which the projector formalism isthe most powerful and beneficial.The most striking qualitative conclusion that can be drawn from the eigenprojector ap-proach is that any quantity of interest can be computed from the knowledge of H and itseigenvalues E α alone , in agreement with recent results of Pozo and de Juan [43] (who useda somewhat different mathematical framework). At the origin of this conclusion is the factthat the eigenprojector is a matrix polynomial of degree N − H , according to the Cayley-Hamilton theorem. This result, though implicitly known for a long time in the form of Eq.(14), is perhaps most useful to physicists when reformulated as an explicit function P α ( E α , H )of a single eigenvalue, as we have done in Section 3.1, see in particular Eqs. (15) & (19). Theseexpressions for P α ( E α , H ) represent our first intermediate result.The realization that the knowledge of H as well as its eigenenergies immediately yields the eigenprojectors implies that any physical quantity of interest can be computed withouteigenstates in the following way:(1) Express the quantity in terms of eigenprojectors.(2) Explicitly insert the function P α ( E α , H ).In the course of carrying out this procedure, it proves extremely enlightening to switchbetween two ”languages” where appropriate. The first language simply employs the relevantmatrices ( H , P α , etc.), while the second one expands those matrices in the (generalized)Gell-Mann matrices. As explained in Section 3.2, in this second (vectorial) language theHamiltonian H is given by a real vector h , and its powers are given by star products of h , which are totally symmetric vector products defined through the underlying Lie algebra.Similarly, the eigenprojector P α is given by a real (generalized) Bloch vector b α . In the vectorlanguage, the analog of the matrix function P α ( E α , H ) is the vector function b α ( E α , h ), seein particular the important result (35). Both languages contain exactly the same information,such that the above protocol can be equivalently formulated as:321) Express the quantity in terms of Bloch vectors.(2) Explicitly insert the function b α ( E α , h ).In the present paper, starting in Section 4, the above protocol was carried out for theQGT, and the systematic application to other, more complicated observables will be publishedelsewhere [44]. The known formula (44) for the QGT in terms of eigenprojectors can be writtenin the vectorial language as described in Section 4.2 [cf. in particular Eq. (45)], which provesquite handy for practical computations and completes step (1). Step (2) was carried out inSection 4.3, where we obtained the Berry curvature tensor Ω α,ij in terms of the vector h and the relevant energy eigenvalue E α only. This generalizes the well-known Berry curvatureexpression (13) to arbitrary N , although it gets rather cumbersome for N (cid:38) i.e. the energy spectrum is particle-holesymmetric (PHS) . In Section 5, it was discussed how the eigenprojectors, Bloch vectors aswell as the QGT simplify in the PHS case. This discussion of PHS spectra can again easilybe extended to more complicated physical observables.As a concrete illustration of all of the above, and especially of the results obtained for theQGT, Section 6 served to present models belonging to the class of multifold Dirac fermions ,characterized by an N -fold degenerate nodal point accompanied by a linear band crossing. Thediscussion of pseudospin- S fermions, conducted in Section 6.2.1, provided us with interestingBloch vector expressions for spin-1 and spin-3 / α − T model[45].To close this paper, we mention a few immediate perspectives of our work. First, asalready mentioned, it is worthwhile to apply the eigenprojector approach to observables thatare not purely of geometric origin. In particular, this will help to unveil and distinguish purelyspectral and geometrical contributions to such observables, similar in spirit to what has beendone for the orbital magnetic susceptibility of two-band models in Ref. [24], superconductingstiffness or non-linear responses and more generalized thermodynamic stiffnesses.Second, it would be interesting to establish a better understanding of the geometry andtopology of multifold fermion models with linear but also quadratic band crossings. Moreconcretely, how to systematically design models that share a PHS and/or isotropic spectrumbut belong to different topological classes according to the tenfold way classification? Howexactly do the quantum geometric properties and their associated topological characterizationdepend on the number N of bands and on the dimensionality of quasi-momentum parameterspace?Finally, note that the validity of the formalism developed in this paper goes beyond thecase of Hermitian Hamiltonian matrices. In particular, the key expressions Eq. (19) for theeigenprojector P α ( E α , H ) and Eq. (35) for the Bloch vector b α ( E α , h ) stay valid for sys-tems where the Hermiticity condition is relaxed [94, 95]. These expressions explicitly encodethe fact that the eigenprojectors retain all the properties (Hermiticity/non-Hermiticiy) andsymmetries of the Hamiltonian. Importantly, for non-Hermitian systems the eigenprojectorsnaturally rewrite as P α ( E α , H ) = (cid:12)(cid:12) ψ Rα (cid:11) (cid:10) ψ Lα (cid:12)(cid:12) in terms of the corresponding left and right33igenstates. When necessary, the left/right eigenstates could be recovered by applying theeigenprojector to some left/right gauge freedom states under an appropriate adaptation ofEq. (21). Acknowledgements We thank Mark-Oliver Goerbig and Andrej Mesaros for fruitful discussions. A Characteristic polynomial The characteristic polynomial of an N × N matrix A is given by [96]˜ p N ( z ) = det( z N − A ) = N (cid:88) k =0 ˜ c k z N − k = N (cid:89) α =1 ( z − a α ) , (89)where a α denotes an eigenvalue of A and ˜ c = 1. According to the Faddeev-Le Verrieralgorithm [96], the coefficients ˜ c k may be computed from the traces s k ≡ Tr A k of powers of A as ˜ c k = − k ( s k + ˜ c s k − + ... + ˜ c k − s )= ( − k k ! Y k ( s , ..., ( − k − ( k − s k ) . (90)The second equality involves (exponential) complete Bell polynomials Y k ( z , ..., z k ) [97, 98],the first few of which read explicitly Y = 1 ,Y ( z ) = z ,Y ( z , z ) = z + z ,Y ( z , z , z ) = z + 3 z z + z ,Y ( z , z , z , z ) = z + 6 z z + 4 z z + 3 z + z . (91)Focusing now on the case where A = H represents an N × N Hamiltonian matrix, we havethe Hamiltonian’s characteristic polynomial p N ( z ) = det( z N − H ) = N (cid:88) k =0 c k z N − k = N (cid:89) α =1 ( z − E α ) . (92)Using the traceless character of the Hamiltonian, cf. Eq. (22), and the Casimir invariantsdefined in Eq. (16), the coefficients are given by c k = ( − k k ! Y k (0 , − C , ..., ( − k − ( k − C k ) , (93)and the first few of them read explicitly c = 1 , c = 0 ,c = − C , c = − C ,c = C − C , c = C C − C ,c = − C 48 + C 18 + C C − C . (94)34 Derivation of Eq. (15) Consider first the numerator of Eq. (14) and note that by explicit multiplication one maywrite (cid:89) β (cid:54) = α ( H − E β N ) = N − (cid:88) n =0 ( − n e n ( E , ..., E α − , E α +1 , ..., E N ) H N − − n , (95)where e n = e n ( E , ..., E α − , E α +1 , ..., E N ) are known as elementary symmetric polynomials [99]. One has e = 1 and all higher e n are determined recursively by Newton’s identities: e n = 1 n n (cid:88) k =1 ( − k − ( C k − E kα ) e n − k , (96)where the C k are the Casimir invariants of Eq. (16) and it was exploited that (cid:80) β (cid:54) = α E kβ = C k − E kα . This may further be rewritten as e n = ( − n n (cid:88) k =0 c k E n − kα , (97)where c k are the coefficients (93) of the characteristic polynomial. If we now define polynomials q n ( z ) ≡ n (cid:88) k =0 c k z n − k , (98)it is clear that q N ( z ) = p N ( z ) is the characteristic polynomial (92), and q n ( E α ) = ( − n e n .Moreover, inserting into Eq. (95), we have (cid:89) β (cid:54) = α ( H − E β N ) = N − (cid:88) n =0 q N − − n ( E α ) H n . (99)Similarly, for the numerator of Eq. (14), exactly the same procedure as above (where H getsreplaced by E α ) leads to (cid:89) β (cid:54) = α ( E α − E β ) = N − (cid:88) n =0 q N − − n ( E α ) E nα . (100)As mentioned in the main text, (cid:81) β (cid:54) = α ( E α − E β ) is equal to the derivative p (cid:48) N ( E α ) of the charac-teristic polynomial. From this one may also show that (cid:81) β (cid:54) = α ( E α − E β ) = (cid:80) N − n =0 q N − − n ( E α ) C n .Combining all of these results, one arrives at Eq. (15). C Closed-form solutions for energy eigenvalues The solutions of the characteristic equation p N ( z ) = 0, with p N ( z ) given by Eq. (92), arethe eigenvalues of H . For N = 2, the solution of p ( E α ) = E α − C / N > 2, the complexity of the function E α ( { C n } ) grows very quickly. For N = 3, thesolutions of p ( E α ) = E α − ( C / E α − C / E α = ( − α | α | ( S + + S − ) + α i √ 32 ( S + − S − ) , α = 0 , ± ,S ± ≡ (cid:32) C ± i (cid:114) C − C (cid:33) . (101)35ote that since C ≥ C , all E α are of course real [100, 102]. This allows to interpret theeigenvalues as lying on a circle, i.e. one may parametrize the solutions using trigonometricfunctions, see for example Ref. [102]: E α = (cid:114) C 13 arccos √ C C + 2( α + 2) π , α = 0 , ± , = 2 | h |√ φ α ,φ α ≡ (cid:104) arccos (cid:16) √ h · h (cid:63) / | h | (cid:17) + 2( α + 2) π (cid:105) . (102)In either of Eqs. (101) & (102) the Casimir invariants can easily be expressed as a functionof h using Eq. (33).For N = 4, there are again several ways to parametrize the solutions of p ( E α ) = E α − ( C / E α − ( C / E α + C / − C / E α = sgn( α )2 √ R + ( − α (cid:115) C − D + sgn( α ) 2 √ C R , α = ± , ± ,R ≡ (cid:112) D + C ,D ≡ B (cid:18) A + √ A − B (cid:19) / + (cid:32) A + √ A − B (cid:33) / ,A ≡ C + C (cid:18) C − C (cid:19) ,B ≡ C − C . (103)Again, the transition to the function E α ( h ) is done by inserting Eq. (33) for the Casimirinvariants.For N ≥ 5, closed-form solutions E α ( { C n } ) of the characteristic equation are unknown[104]. D Gell-Mann and spin matrices Here we list the N = 3 ( N = 4) Gell-Mann matrices and relate them to spin-1 (spin-3/2) ma-trices. Note that the generalization to N ≥ N ( N − / N ( N − / N − N = 3 Gell-Mann matrices [105] are given by λ = , λ = − i i , λ = − , λ = ,λ = − i i , λ = , λ = − i i , λ = 1 √ − . They can easily be related to spin-1 operators, which fulfill the algebra [ S i , S j ] = i(cid:15) ijk S k . Inthe most commonly employed representation of spin-1 operators, we have S x = 1 √ λ + λ ) ,S y = 1 √ λ + λ ) ,S z = 12 ( λ + √ λ ) . (104)The N = 4 Gell-Mann matrices in the defining representation [51, 54] are given by theextended SU(3) Gell-Mann matrices, λ = , λ = − i i , λ = − ,λ = , λ = − i 00 0 0 0 i , λ = ,λ = − i i , λ = 1 √ − , λ = , λ = − i i , λ = ,λ = − i i , λ = , λ = − i i ,λ = 1 √ − . This can easily be used to represent the common definition of spin-3/2 operators: S x = 12 ( √ λ + 2 λ + √ λ ) ,S y = 12 ( √ λ + 2 λ + √ λ ) ,S z = 12 ( λ + √ λ + √ λ ) . (105) E The generalized Bloch sphere We give here a short summary of the concept of a generalized Bloch sphere [ i.e. the SU( N )eigenprojector’s Bloch sphere B ( N ) P α introduced in Section 3.2], drawing largely on Refs. [52,53, 55–60].For an N -dimensional Hilbert space, the three defining properties of a (mixed state) den-sity matrix ρ α representing an N -component (mixed) quantum state | ψ α (cid:105) , where α = 1 , ..., N ,are hermiticity ρ † α = ρ α , probability conservation Tr ρ α = 1 and positive semidefiniteness, ρ α ≥ 0. Pure states have, in addition, ρ α = ρ α , in which case ρ α = P α is an eigenprojector.An expansion in the basis of generalized Gell-Mann matrices analogous to Eq. (23) can bemade for a general (mixed state) density matrix: ρ α = 1 N N + 12 b α · λ , (106)where now | b α | can take various values depending on the pureness of the state. In the purestate case, we have | b α | = (cid:112) N − /N , in agreement with Eq. (37). Considering the vectorspace R N − , and denoting the ( N − b α as Σ ( N ) ρ α , one needs to distinguish between two kinds of boundaries, namely38igure 4: Schematic illustration of the generalized Bloch sphere and other relevant sets, for N = 2 , , 4. The large ( N − W N − , of radius (cid:112) N − /N , corresponds to thehighest permissible length of a Bloch vector. In its interior, the Ball B N − , there is the spaceΣ ( N ) ρ α accessible to a mixed state of type (106), with topological boundary ∂ Σ ( N ) ρ α , and in theinterior of Σ ( N ) ρ α there is the small ( N − V N − , of radius (cid:112) / [ N ( N − B ( N ) P α of Σ ( N ) ρ α , which is the intersection of Σ ( N ) ρ α with the large sphere, isthe generalized (eigenprojector’s) Bloch sphere that interests us in the discussion of Section3.2. For the familiar N = 2 case, these complications are hidden, since all of the relevant setscoincide.its ( N − topological boundary ∂ Σ ( N ) ρ α and its extremal boundary B ( N ) P α . Thelatter is of dimension 2( N − | b α | , i.e. it is thespace of pure state Bloch vectors, or in other words the generalized Bloch sphere , sketched inFig. 4. The dimensionality of B ( N ) P α is in agreement with the fact that one needs 2( N − W N − ≡ (cid:40) r ∈ R N − (cid:12)(cid:12)(cid:12) | r | = (cid:114) N − N (cid:41) (107)that contains all pure state Bloch vectors; in other words, it is the surface of the ball B N − ,the smallest ball that contains Σ ( N ) ρ α . For N = 2, one trivially has ∂ Σ (2) ρ α = B (2) P α = W (and W = S , with the unit two-sphere S ), as illustrated in Fig. 4; but for higher N thethree spaces are different. One has B ( N ) P α ⊂ ∂ Σ ( N ) ρ α and B ( N ) P α ⊂ W N − , i.e. the generalizedBloch sphere is a proper subset of (i) the boundary of the space of mixed states and (ii) thesphere W N − ; in fact, it is their intersection: ∂ Σ ( N ) ρ α ∩ W N − = B ( N ) P α . In other words, if b α corresponds to a pure state, it lies on W N − , but not necessarily the other way around, i.e. not all points on W N − represent physically valid pure states.Analogously, the interior of W N − , i.e. the ball B N − , is not composed of only physicallyvalid mixed states. In fact, calculations [57] of the volume of Σ ( N ) ρ α show that vol(Σ (3) ρ α ) / vol( B ) ≈ . 26 and vol(Σ (4) ρ α ) / vol( B ) ≈ . 12, meaning that most points in B N − do actually not rep-resent physically valid states as soon as N > ρ α ≥ N = 1, this condition is trivially fulfilled. For N = 2, it is equivalent to Tr (cid:0) ρ α (cid:1) ≤ 1, so fora given density matrix there is one additional constraint as compared to N = 1. In the same39ay, whenever N increases by one, one additional constraint involving traces D α,n ≡ Tr( ρ nα )needs to be fulfilled in order to have positive definiteness. For example, for N = 3, thereare the three constraints [52] D α, = 1, D α, ≤ 1, and − D α, + 3 D α, ≤ 1, which can betranslated to constraints on the Bloch vectors via Eq. (106). This restrict the elements of B that represent physically valid states. As a consequence, the relevant sets (including thegeneralized Bloch sphere) acquire a very nontrivial shape already for N = 3 [60]: B (3) P α = (cid:26) b α ∈ R (cid:12)(cid:12)(cid:12)(cid:12) | b α | = 43 , | b α | − b α · ( b α (cid:63) b α ) = 49 (cid:27) ,∂ Σ (3) ρ α = (cid:26) b α ∈ R (cid:12)(cid:12)(cid:12)(cid:12) | b α | ≤ , | b α | − b α · ( b α (cid:63) b α ) = 49 (cid:27) , Σ (3) ρ α = (cid:26) b α ∈ R (cid:12)(cid:12)(cid:12)(cid:12) | b α | ≤ , | b α | − b α · ( b α (cid:63) b α ) ≤ (cid:27) , (108)where the star product is defined in Eq. (28), and the agreement of the first line with Eq.(37) is to be noted. Similarly, more complicated constraints can be obtained for higher N .For more details on this procedure, see Ref. [52].Finally, note that there exists another sphere V N − ≡ (cid:40) r ∈ R N − (cid:12)(cid:12)(cid:12) | r | = (cid:115) N ( N − (cid:41) (109)inscribed inside Σ ( N ) ρ α , i.e. ∂ Σ ( N ) ρ α lies between V N − and W N − .The essence of all of these established facts is visualized schematically in Fig. 4. Note thatthe figure is only a rough sketch supposed to highlight the different sets involved. A visualinsight into the true (very complicated) shape of those sets can be gained by considering two-or three-sections, see for instance Refs. [56, 58–60]. F Proof of orthogonality relation (31) We first prove h × h ( n ) (cid:63) = 0 , ∀ n ∈ N , (110)by induction. The base case h × h (0) (cid:63) = h × h = 0 is obviously verified. We now assume h × h ( n ) (cid:63) = 0 and show that it implies h × h ( n +1) (cid:63) = 0: (cid:16) h × h ( n +1) (cid:63) (cid:17) l = f lkm h k (cid:16) h ( n +1) (cid:63) (cid:17) m = d ijm f lkm h k h i (cid:16) h ( n ) (cid:63) (cid:17) j = 0 . (111)This is easily verified from the second Jacobi identity (114). Equation (110) is thus provenand we use it to prove h ( n ) (cid:63) × h ( n ) (cid:63) = 0 by induction in two variables. The base cases are h ( n ) (cid:63) × h = h × h ( n ) (cid:63) = 0. We now assume h ( n ) (cid:63) × h ( n ) (cid:63) = 0 and show that it implies both h ( n ) (cid:63) × h ( n +1) (cid:63) = 0 and h ( n +1) (cid:63) × h ( n ) (cid:63) = 0: (cid:16) h ( n ) (cid:63) × h ( n +1) (cid:63) (cid:17) l = d ijm f lkm (cid:16) h ( n ) (cid:63) (cid:17) k h i (cid:16) h ( n ) (cid:63) (cid:17) j = 0 , (cid:16) h ( n +1) (cid:63) × h ( n ) (cid:63) (cid:17) l = d ijk f lkm h i (cid:16) h ( n ) (cid:63) (cid:17) j (cid:16) h ( n ) (cid:63) (cid:17) m = 0 . (112)This concludes the proof. 40 SU( N ) Jacobi identities The first Jacobi identity [61, 106] can be written alternatively for the generator matrices, theantisymmetric structure constants and the SU( N ) vectors as[[ λ i , λ j ] , λ k ] + [[ λ j , λ k ] , λ i ] + [[ λ k , λ i ] , λ j ] = 0 ,f ijm f klm + f ikm f ljm + f ilm f jkm = 0 , m × ( n × o ) + n × ( o × m ) + o × ( m × n ) = 0 , ( m × n ) · ( o × p ) + ( m × o ) · ( p × n ) + ( m × p ) · ( n × o ) = 0 , (113)where the third and fourth lines are obtained from the second line depending on whether ornot one keeps a free index. Similarly, the second Jacobi identity is given by[ { λ i , λ j } , λ k ] + [ { λ j , λ k } , λ i ] + [ { λ k , λ i } , λ j ] = 0 ,f ijm d klm + f ikm d ljm + f ilm d jkm = 0 , m × ( n (cid:63) o ) + n × ( o (cid:63) m ) + o × ( m (cid:63) n ) = 0 , ( m × n ) · ( o (cid:63) p ) + ( m × o ) · ( p (cid:63) n ) + ( m × p ) · ( n (cid:63) o ) = 0 . (114)Furthermore, there is the identity[ λ i , [ λ j , λ k ]] = { λ k , { λ i , λ j }} − { λ j , { λ i , λ k }} ,f ijm f klm = 2 N ( δ ik δ jl − δ il δ jk ) + d ikm d jlm − d ilm d jkm , m × ( n × o ) = 2 N [( m · o ) n − ( m · n ) o ] + ( m (cid:63) o ) (cid:63) n − ( m (cid:63) n ) (cid:63) o , ( m × n ) · ( o × p ) = 2 N [( m · o )( n · p ) − ( m · p )( n · o )]+ ( m (cid:63) o ) · ( n (cid:63) p ) − ( m (cid:63) p ) · ( n (cid:63) o ) . (115) H Recursion relations for powers of H Equation (24), together with the product prescription (29), leads to the recursion relations C n = C n − q C q N + 2 η n − q · η q , η n = 1 N (cid:0) C n − q η q + C q η n − q (cid:1) + η n − q (cid:63) η q , (116)which hold for n ≥ q ≥ 0. In obtaining the second line of Eq. (116), we made use of the factthat η n and the structure constants f abc are real by definition, establishing the identity η n × η n = 0 , ∀ n , n ∈ N . (117)Note also that η n can, according to the recursion (116), be expressed as Eq. (34) given in themain text. The orthogonality relation (117) then also agrees with Eq. (31). I Berry curvature formula for N -band systems The Berry curvature formula for arbitrary SU( N ) systems follows from combining Eqs. (25),(34) & (45), together with the total antisymmetry of the triple product m · ( n × o ) = f abc n a o b m c N , the Berry curvature can be written asΩ α,ij = − b α [ q (cid:48) N ( E α )] · N − (cid:88) n,m =1 q N − − n ( E α ) q N − − m ( E α ) τ ( i ) n × τ ( j ) m , (118)with vectors τ ( i ) n ≡ N n − (cid:88) p =0 C p ∂ i h ( n − − p ) (cid:63) . (119)With the notation m i ≡ ∂ i m , the first few τ ( i ) n read τ ( i )1 = h i , τ ( i )2 = h i(cid:63) , τ ( i )3 = C N h i + h i(cid:63)(cid:63) . (120)From Eq. (118) and using the results for b α and q n ( E α ) from Section 3, the N = 2 formula(13) is immediately obtained.For N = 3, analogously, we haveΩ α,ij = − E α h + h (cid:63) )(3 E α − C ) · (cid:2) E α h i × h j + E α (cid:0) h i × h j(cid:63) + h i(cid:63) × h j (cid:1) + h i(cid:63) × h j(cid:63) (cid:3) , (121)or equivalently Eq. (48) in the main text. Importantly, we find that this result can beconsiderably simplified, due to the following identities: h · ( h i × h j(cid:63) ) = h · ( h i(cid:63) × h j ) = h (cid:63) · ( h i × h j ) , h (cid:63) · ( h i × h j(cid:63) ) = h (cid:63) · ( h i(cid:63) × h j ) = h · ( h i(cid:63) × h j(cid:63) ) , h (cid:63) · ( h i(cid:63) × h j(cid:63) ) = (cid:20) − 23 ( h · h (cid:63) ) h + | h | h (cid:63) (cid:21) · ( h i × h j ) . (122)The first two lines are valid for general N and can be proved using the second Jacobi identity(114). The proof of the third line requires the Jacobi indentity as well, but additionally theSU(3)-specific identities (36). Inserting all of these identities into Eq. (121), and exploitingthe characteristic equation E α = C E α + C , one obtains the generic SU(3) Berry curvatureformula (49) provided in the main text.Equivalently, the SU(4) Berry curvature is obtained asΩ α,ij = − (cid:2)(cid:0) E α − C (cid:1) h + E α h (cid:63) + h (cid:63)(cid:63) (cid:3)(cid:2) E α ( E α − C ) − C (cid:3) · (cid:34)(cid:18) E α − C (cid:19) h i × h j + E α (cid:18) E α − C (cid:19) (cid:0) h i × h j(cid:63) + h i(cid:63) × h j (cid:1) + E α h i(cid:63) × h j(cid:63) + (cid:18) E α − C (cid:19) (cid:0) h i × h j(cid:63)(cid:63) + h i(cid:63)(cid:63) × h j (cid:1) + E α (cid:0) h i(cid:63) × h j(cid:63)(cid:63) + h i(cid:63)(cid:63) × h j(cid:63) (cid:1) + h i(cid:63)(cid:63) × h j(cid:63)(cid:63) (cid:21) , (123)which is equivalent to Eq. (50) in the main text. Again, we may exploit several vectoridentities to reduce the number of terms. The first two lines of Eq. (122) still hold, but thethird one is changed [cf. Eq. (36)], and other identities are needed as well, which are againproved by the second Jacobi identity: h (cid:63) · ( h i(cid:63) × h j(cid:63) ) = − 23 ( h · h (cid:63) ) h · ( h i × h j ) + h (cid:63)(cid:63) · ( h i(cid:63) × h j + h i × h j(cid:63) ) , h · ( h i × h j(cid:63)(cid:63) ) = h · ( h i(cid:63)(cid:63) × h j ) = h (cid:63)(cid:63) · ( h i × h j ) , h · ( h i(cid:63) × h j(cid:63)(cid:63) ) = h · ( h i(cid:63)(cid:63) × h j(cid:63) ) = 12 h (cid:63)(cid:63) · ( h i(cid:63) × h j + h i × h j(cid:63) ) , (124)42 t cetera . Putting all of this together, one may derive the SU(4) analog of Eq. (49), whichwill contain nine independent terms instead of the 27 terms in Eq. (50). However, this willgive rise to quite lengthy coefficients, so we deem Eq. (50) to be in the simplest possible formfor practical purposes.Finally, for arbitrary SU(5) systems, the Berry curvature is given byΩ α,ij = − R α h + ˜ R α h (cid:63) + E α h (cid:63)(cid:63) + h (cid:63)(cid:63)(cid:63) )[ q (cid:48) ( E α )] · (cid:104) ( R α h i + ˜ R α h i(cid:63) + E α h i(cid:63)(cid:63) + h i(cid:63)(cid:63)(cid:63) ) × ( R α h j + ˜ R α h j(cid:63) + E α h j(cid:63)(cid:63) + h j(cid:63)(cid:63)(cid:63) ) (cid:105) , (125)where q (cid:48) ( E α ) = 5 E α R α + | h | − | h (cid:63) | , with R α ≡ E α ˜ R α − h · h (cid:63) and ˜ R α ≡ E α − | h | . J Quantum geometry of five-band Hamiltonians with a PHSspectrum In the PHS case (see Section 5), the SU(5) projectors read off Eqs. (19) & (54) are P α =0 = 1 + 2 C C − C H − C − C H ,P α (cid:54) =0 ,α = 1 C + C ( E α − C ) (cid:20) α E α (cid:18) E α − C (cid:19) H + (cid:18) E α − C (cid:19) H + α E α H + H (cid:3) , (126)where the energy was written as E α (cid:54) =0 ,α = α E α with E α ≡ ( C / α (cid:112) C − C / / . Itcan be readily verified that these projectors fulfill the unit trace and completeness relations.In the Bloch vector picture, this translates to b α =0 = − | h | h (cid:63) + h (cid:63)(cid:63)(cid:63) | h | − | h (cid:63) | , b α (cid:54) =0 ,α = α E α R α h + R α h (cid:63) + α E α h (cid:63)(cid:63) + h (cid:63)(cid:63)(cid:63) E α R α + | h | − | h (cid:63) | , (127)where R α ≡ E α − | h | . We only write the Berry curvature explicitly, which is easily obtainedfrom Eq. (125):Ω α =0 ,ij = − − | h | h (cid:63) + h (cid:63)(cid:63)(cid:63) (cid:0) | h | − | h (cid:63) | (cid:1) · (cid:20)(cid:18) − | h | h i(cid:63) + h i(cid:63)(cid:63)(cid:63) (cid:19) × (cid:18) − | h | h j(cid:63) + h j(cid:63)(cid:63)(cid:63) (cid:19)(cid:21) , Ω α (cid:54) =0 ,α ,ij = − α E α R α h + R α h (cid:63) + α E α h (cid:63)(cid:63) + h (cid:63)(cid:63)(cid:63) (cid:0) E α R α + | h | − | h (cid:63) | (cid:1) , · (cid:2)(cid:0) α E α R α h i + R α h i(cid:63) + α E α h i(cid:63)(cid:63) + h i(cid:63)(cid:63)(cid:63) (cid:1) × (cid:0) α E α R α h j + R α h j(cid:63) + α E α h j(cid:63)(cid:63) + h j(cid:63)(cid:63)(cid:63) (cid:1)(cid:3) . (128)The quantum metric can be obtained by combining Eqs. (45) & (127).43 eferences [1] W Ehrenberg and R E Siday. “The Refractive Index in Electron Optics and the Prin-ciples of Dynamics”. In: Proceedings of the Physical Society. Section B doi : .[2] Y. Aharonov and D. 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