Angular momentum coupling, Dirac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries

We investigate the elementary rearrangements of energy bands in slow-fast one-parameter families of systems whose fast subsystem possesses a half-integer spin. Beginning with a simple case without any time-reversal symmetries, we analyze and compare increasingly sophisticated model Hamiltonians with these symmetries. The models are inspired by the time-reversal modification of the Berry phase setup which uses a family of quadratic spin-quadrupole Hamiltonians of Mead [Phys. Rev. Lett. 59, 161–164 (1987)] and Avron et al [Commun. Math. Phys. 124(4), 595–627 (1989)]. An explicit correspondence between the typical quantum energy level patterns in the energy band rearrangements of the finite particle systems with compact slow phase space and those of the Dirac oscillator is found in the limit of linearization near the conical degeneracy point of the semi-quantum eigenvalues.

1. Introduction. The interest in the geometric and topological properties of parametric families of quantum Hamiltonians was largely stimulated by the seminal paper of Michael Berry [10], who analyzed the evolution of the spin states with spin S = 1 2 ≈ S in the presence of the magnetic field B. The system is governed by the linear HamiltonianĤ = B ·Ŝ.
(1) When B is made to vary adiabatically slowly, following a closed path in the regular domain R 3 B \ {0} of the parameter space, the eigenstates of (1) acquire a geometric phase [10,43]. This phase can be seen as both a consequence and an indicator of the nontrivial topology of the system. To examine the topology more fully, we study the two eigenstate bundles defined by the eigenvalues λ 1,2 (B) with S = 1 2 . Away from the degeneracy at B = 0, and more specifically, over any sphere S 2 1 A particularly instructive example with formal control parameters q of dynamical origin is the phase of the electronic wavefunctions Ψ el in the Jahn-Teller systems. Herzberg and Longuet-Higgins [19] considered such systems within the Born-Oppeheimer approximation, where nuclear coordinates q play the role of control parameters of the separated electronic Hamiltonian H el . Their work anticipated the geometrical phase analysis by Berry [10] and is widely known in molecular physics [32]. The nontrivial topological contribution to the phase of Ψ el is associated with the close loop around the degeneracy point of two electronic potential energy surfaces in the q-space. While being plain control parameters of H el , the nuclear coordinates q are dynamical variables for the complete electronic-nuclear Hamiltonian or, equally, for the separated Hamiltonian H nucl describing the slow vibrations of the nuclei. We like to stress that q are formal and not dynamical control parameters of the fast electronic subsystem because H el has no influence on their evolution. In other words, q get no feedback from the fast system. On the other hand, the reason why dynamical parameters remain influenced explicitly by the fast system in the semi-quantum description of our systems (sec. 2, 3, 4, and 5) is in the nature of their slow-fast separation which applies only on the complement to the open saturated neighbourhood of the degeneracy point (bounded by the base space of ∆, see sec. 1.3) in the formal-dynamical control parameter space. 2 Dynamical variables of slow-fast systems fall in two cathegories with strongly different rates of variation and respective time scales. Under certain conditions, this allows separation into slow and fast subsystems. There is a wast literature on slow-fast dynamical systems which are ubiquitous in applications, see, for example, [8,34,35]. called global. The base space of ∆ is a sphere surrounding the degeneracy point in the total (dynamical-formal) control parameter space. The ∆ bundle is considered in the geometric phase framework. We may call it local. In the example in sec. 2.1, Λ and ∆ are the same bundle, while in sec. 2.2 they are rather different. In sec. 3 and 4, both Λ and ∆ are bundles over a sphere, but one is parameterized by dynamical parameters, while the other-by formal and dynamical control parameters. In the space of all parameters, the base spaces of Λ and ∆ intersect (appendix A.4). They have common closed loops parameterized entirely by dynamical variables for fixed formal control parameters. Furthermore, such loops may be periodic orbits of the slow classical system. Such periodic orbits exhibit dynamical geometric phase given by the Chern index of ∆. Semiclassical quantization [17, Appendix A] provides one possible manifestation of such phase.
We observe further that global bundles Λ form continuous parametric families with regard to regular values of tuning control parameter(s) µ, while there is one single local ∆ bundle for each degeneracy point (tuning parameters are often choosen so that degeneracies occur for critical value µ = 0). When tuning parameters µ have several disconnected open domains of regular values (such as, in the simplest case, two intervals with µ < 0 and µ > 0), the c 1 indices of Λ bundles on each domain may differ and the difference(s) δc 1 or delta-Chern characterize the degeneracy point(s). At the same time, the c 1 indices of ∆ bundles contribute to δc 1 (appendix A.4). While the ∆ bundle construction ignores any differences between the dynamical and formal control parameters, Λ bundles have an important dynamical interpretation. Their index characterizes the number of slow quantum states which can be supported on P and this relates δc 1 to the number of redistributed quantum states, see more in sec. 1.3.
1.3. Quantum, semi-quantum, and classical. Dynamical parameterization enhances considerably the object of our study and places it at crossroads of several powerful mathematical theories. It becomes universally important to many physical applications.
Within the geometric phase setup, we obtain a "fast" quantum system on a finite Hilbert space H fast of low dimension (e.g., two for S = 1 2 ). The Hamiltonian of this system is a combination of operators acting on H fast with coefficients depending on classical dynamical variables (q, p) of the "slow" system and, possibly, additional formal control parameters µ which are also meant to be varied adiabatically slowly (see footnote 2). The eigenstates are obtained as eigenvectors of a low-dimensional Hermitian matrix whose eigenvalues λ(q, p) play the role of classical Hamiltonians governing the dynamics of the slow system corresponding to each fast eigenstate. We call this description semi-quantum.
At the same time, the fast and slow systems can be both treated as quantum and can be described using the Hilbert space H fast × H slow . Since the slow system has a close and well defined classical limit, H slow is typically a much larger space, and, if the underlying slow classical phase space is non-compact, H slow is infinite (Dirac equation, Dirac oscillator in sec. 2.2). Different time-scales (footnote 2) result in a specific structure of the energy spectrum and the localization patterns of the quantum states of such system. The spectrum consists of bands with relatively large density and number of states. Transitions between the states within a band correspond to excitations in the slow degree of freedom, and, in most basic situations, energy alone is enough to separate the bands: the in-band splittings are considerably smaller than the gaps between the bands. In regular cases, the number of energy bands equals the number of semi-quantum eigenvalues. When the latter have a degeneracy, the bands merge and exchange energy levels. The reason for this exchange is in the nature of the dynamical parameterization of the fast system. Near the degeneracy point, due to the interaction between the two subsystems, the separation of the slow and fast dynamics becomes blurred, and certain states localized near the degeneracy may change their character and switch bands. We conclude that the degeneracy of semi-quantum eigenvalues is the cause of both the geometric phase and this redistribution phenomenon.
To realize how considerable the relation and the interplay between the geometric phase and the energy level redistribution is, we like to go a bit further into the structure of the bands. For nondegenerate (typical) singularities, the number of the exchanged states is much smaller than that of the states in the bands. In fact, most of the levels are never exchanged and continue always within the same band. We call them "bulk" states. Their number is given roughly by the symplectic volume of the underlying compact slow classical phase space and may be obtained using an appropriate quantization scheme (sec. 2.1). In the non-compact setting, this can be generalized using fractional formal Chern number (70) in Appendix A. 3. The few levels that can and do get exchanged are the "edge" states. In this context, the correlation diagrams showing how levels continue when parameters cross the degeneracy point become very instrumental and are employed throughout the article (sec. 3, 4, and 5). The Chern index characterizing the bundle Λ of semiquantum eigenvalues over the slow phase space gives, essentially, the number of missing/excessive edge states [11,12,13,21,22,23,25]. More specifically, this index gives the quantity by which the actual number of states in the band differs from that given by the symplectic volume of the underlying classical phase space. The same index, but computed for bundle ∆ gives the geometric phase, and it can be conjectured that computation for all local ∆ bundles gives the number of redistributed states. For further generalization of the analysis of the redistribution phenomenon, an application of the Atiyah-Singer index theorem [2,3,4] and geometric quantization principles [15] seems to be relevant.
The last but not the least, although being the least exploited in our present work, comes the fully classical description. Singularities of the slow-fast classical mechanical system with several (at least two) degrees of freedom are related to the semi-quantum degeneracies and, therefore, to the edge state redistribution. In particular, if the slow-fast system is integrable, such singularities are at the origin of Hamiltonian monodromy [37]. 1.4. Main purpose and outline. Dynamical modifications of the original setup (1) open a large domain of diverse and versatile mathematical theory and applications which go far beyond the original geometric phase analysis. The latter remains, however, a vital organizing tool in the study of different dynamical parametric systems. Our main interest in this work is in the phenomenon of redistribution between the energy bands in the slow-fast systems [12], and so we focus primarily on the full quantum system and its relation to the semi-quantum description. Among different symmetry properties which can be appropriate for concrete physical systems there is one particular property, the time reversal invariance (see note 5), which is considered as rather general due to its relevance across a very wide class of physical systems. In this work, we focus on time-reversal-invariant dynamical modifications. We also prefer uncovering systems which are fundamental and important to atomic and molecular (finite particle) applications. As a consequence, we analyze 2. Three basic examples. We begin with simple modifications of the original setup with Hamiltonian (1) which illustrate sec. 1.1 and 1.3. The details in each example are instructive to follow. They help understanding the key elements in the analysis of the systems in sec. 3, 4, and 5.
2.1. Spin-orbit coupling. One of the simplest and most direct dynamical analogues of (1) is the HamiltonianĤ where N represents the mechanical angular momentum of the system. In atomic physics, this momentum is called orbital and denoted by L, but for us its physical origin can lie elsewhere, e.g., it can be associated with a degenerate molecular vibration, or with the overall rotation of a molecule. Nevertheless, for brevity, we like to call the right hand side of (2a) the spin-orbit coupling term. As discussed in sec. 1.3, dynamically parameterized semi-quantum Hamiltonian (2a), is accompanied by the fully quantum Hamiltonian and the fully classical Hamiltonian Since both N and S Poisson commute with (2c), we can fix respective values 3 of N and S when analyzing systems with Hamiltonians (2). This means that the classical (slow) phase space of the semi-quantum system is the 2-sphere S 2 N , the set of all orientations 4 of N . Furthermore, the phase space of the fully classical system is S 2 ×S 2 , while the (2S + 1)(2N + 1)-dimensional Hilbert space of the corresponding fully quantum system H S,N = H S ⊗ H N is spanned by eigenfunctions |S, σ |N, η ofŜ 1 andN 1 , such that with σ = −S, −S + 1, . . . , S − 1, S and η = −N, −N + 1, . . . , N − 1, N . The above basis implies well separated fast and slow subsystems and is called uncoupled.
2.1.1. The semi-quantum system with spin 1 2 . For a given fast system with spin S, the semi-quantum Hamiltonian (2a) becomes a (2S + 1)-dimensional Hermitian matrix defined on S 2 N . Rewriting (2a) in terms of so(3) ladder operators we can find its matrix from the action ofŜ ± on |S, σ . Specifically, using , we arrive at the S = 1 2 spinor representation of (2a) 3 In this work, unless indicated explicitly, we shall assume N 1 and imply N = N instead of the quantum relation N = N (N + 1), or the semiclassical formula N = (N + 1 2 ). Furthermore, atomic units with = 1 will be used throughout the paper. 4 One pertinent example is the reduced phase space S 2 of the Euler top, the freely rotating rigid body.
in the basis | 1 2 , − 1 2 , | 1 2 , 1 2 . Coordinates on S 2 N are dynamical parameters of (2a), while the third control parameter N (see footnote 3) is formal. In the particular example (2), the spherical symmetry results in the conservation of the norm J of the total angular momentum J = N + S, and in constant eigenvalues of (4) The slow dynamics is trivial. The degeneracy of the two constant eigenvalues (5) themselves is, in turn, achieved for N = 0.
2.1.2. Topologically nontrivial energy bands. Even though the slow dynamics is trivial, the topology of the parametric semi-quantum system with Hamiltonian (4) is not: just like in the original Berry system with Hamiltonian (1), the two semiquantum eigenfunction bundles Λ 1,2 over S 2 N have Chern indices ±1, see Appendix A.1. It is important to uncover how the spectrum of the fully quantum Hamiltonian (2b) reflects this. The SO(3) isotropy of (2b) means that the spectrum is joint with operator Ĵ , and that the eigenstates are labeled by the respective quantum number J along with N and S. Rewriting (2b) aŝ we can see immediately that its spectrum is given by and that for given constant N and S, this spectrum has (2J +1)-degenerated discrete multiplets corresponding to possible values |N − S|, |N − S| + 1, . . . , N + S of J. The spectrum domain [−(N + 1)S, N S] is delimited by the energies of multiplets with J = N − S and N + S, respectively. So, in particular, the upper and lower multiplets of the S = 1 2 system consist of 2N + 2 and 2N levels, respectively. In the limit N 1, as detailed further in sec. 2.3.3, the energies of multiplets are pseudo-symmetric with respect to energy 0.
The two multiplets of the S = 1 2 system are the energy bands corresponding to the two semi-quantum eigenvalues λ 1,2 (N ). Recall that a multiplet of an isolated system with fixed norm N of angular momentum N has 2N +1 levels. This number corresponds to the symplectic volume of the underlying classical phase space S 2 N (plus a quantum correction due to sphere's curvature). The number of levels N 1,2 in the two bands of the S = 1 2 system differs from 2N + 1 by ±1. The difference 2N + 1 − N 1,2 equals the values of Chern indices c 1 of Λ 1,2 , see Appendix A. This is not coincidental. The bands of the coupled system reflect the nontrivial topology of the semi-quantum description [11,21].
2.1.3. Possible deformations. In order to have the nontrivial slow dynamics and split energy bands, the spherical isotropy of (2) should be removed. At the same time, there is an option of retaining its time-reversal 5 invariance under which both angular momenta N and S change sign. In sec. 3 we revisit the simple system [36] with SO(3) broken down to its SO(2) subgroup (axial symmetry) and no T -symmetry, while in sec. 4, we introduce an axially symmetric and Tequivariant deformation of (2). In both cases, we have slow dynamics. The semiquantum eigenvalues are not constant over the phase space S 2 N , and the latter is foliated with typical constant level sets being periodic orbits S 1 . The degeneracy of the bands may still occur locally, at certain points on S 2 N . This brings us to the next section.
2.1.4. Describing and linearizing the slow dynamics. The dynamics of a classical slow system on S 2 N with Hamiltonian λ : S 2 N → R can be described using the Poisson algebra so(3) generated by (N 1 , N 2 , N 3 ) to obtain the Euler-Poisson equations of motionṄ = {N , λ(N )} andṄ = 0. A generic semi-quantum Hermitian 2 × 2 matrix, such as the one we will encounter in sec. 3, has three real control parameters [41,1]. This means that the degeneracy of the eigenvalues λ does typically occur in an isolated point x ∈ S 2 N and for an isolated value of the third formal control parameter. The local study near x uncovers universal features of quantum, semiquantum, and classical slow-fast systems with one slow degree of freedom undergoing degeneracy of their semi-quantum eigenvalues. In other words, while such systems may be very different globally, they are equivalent in their behaviour near the isolated degeneracy point. For the semi-quantum systems, the local study of slow dynamics is based on the linearized equations of slow motion at x.
In the subsequent sections, we deform (2) so that the semi-quantum eigenvalues of the deformed systems have generic degeneracies at one or both poles of the slow phase space S 2 N , and we study the respective linearizations. At the north pole with N 1 = N , the Poisson bracket {N 2 , N 3 } = N 1 = N suggests that in the most basic, lowest order (linear) approximation, the local symplectic coordinates (q, p) of the chart R 2 q,p at this pole should be chosen as while at the origin (q, p) = 0 of the chart, we have The south pole linearization with N 1 = −N differs in the definition of coordinates (q, p) as summarized below 5 The operation T has the same effect on the trajectories of the classical system as reversing time in the equations of motion. A more exact terminology, however, may be momentum reversal.
In classical mechanics, we consider normal and reversing (or reversal) isotropy symmetries of the Hamiltonian function H, depending whether the symplectic form ω remains invariant or covariant [28,29,7]. In quantum mechanics, time reversal action on the Hilbert space [42,6] involves complex conjugation C * times a unitary transformation. So it can be seen that in the concrete example of Pauli matrices, i.e., for spin-1 2 wavefunctions, T in (6) is realized as C * • C y 2 , where C y 2 is rotation by π about axis y (axis 3 in our notation). Although the whole class of reversing symmetries may match Wigner's definition of quantum time-reversal symmetry operation, we like to distinguish our concrete realization of time-reversal proper T from other reversing symmetries.
with standard oscillator creation-annihilation operators Replacing N 1 and N ± in (4) according to (8) gives the spinor forms of (2a) linearized near each pole. Specifically, at the pole with N 1 ≈ N we computê The linearized spinor form (10) is the most basic universal local representation of any dynamical analogue of the geometric phase setup with Hamiltonian (1). Linearization is deeply related to the slow phase space localization (footnote 11) of the quantum eigenstates. In the full quantum description, the slow phase space becomes a set of coherent states localized at points on S 2 N [45], i.e., a set of functions |N, N with all possible orientations of N . Since the slow dynamics for Hamiltonian (2) is trivial, we can associate every state in the 2J + 1 degenerate multiplet of the full quantum system with a specific localized coherent state. Linearization (10) describes what happens to this localized state when formal control parameter µ is varied. In sec. 3 and later, such description will apply to the exceptional edge state(s). We like also to note that linearization (10) can be used to calculate the Chern indices c 1 of the semi-quantum eigenfunction bundle Λ 1,2 over S 2 N >0 . The specific "exceptional point" (see Appendix A.1) where we linearize is coordinatedependent, but the existence of such point (for any coordinates) reflects the nontriviality of Λ.

2.2.
Dirac oscillator. The one-dimensional (1D) Dirac oscillator [33], a variation on the theme of the Dirac equation, is the basic dynamical modification of the geometric phase setup with Hamiltonian (1). Using variables (9) and spin S = 1 2 , the semi-quantum Hamiltonian of this system can be written aŝ The dynamical parameters (q, p) of (11) are symplectic coordinates on the noncompact phase space R 2 (q,p) . We notice immediately that for a particular value of µ, this Hamiltonian corresponds to the linearized spin-orbit Hamiltonian (10). In fact, we will see that linearization of the angular momentum system in sec. 3 provides the correspondence for the entire family (11). The eigenvalues of (11) λ 1,2 = ± n + µ 2 with classical oscillator action 6 n = 1 2 (q 2 + p 2 ) ≥ 0, are distinct as long as we stay away from µ = q = p = 0, where they vanish. The additional third formal control parameter µ is needed to have a handle on the sole degeneracy point of λ 1,2 . The slow dynamics in R 2 (q,p) consists of motion along the circular orbits with constant n and a single equilibrium at n = 0 (and µ = 0). The equations of motions for (q, p) are defined by the Hamiltonian function λ 1,2 , cf. sec. 2.1.4.
The major flaw of the previous example in sec. 2.1, is that the degeneracy of its semi-quantum eigenvalues is not generic dynamically because it coincides with the whole slow phase space contracting to one point (a singularity of the slow dynamical system). In particular, this means that the typical energy level redistribution cannot be observed with such parameterization. The Dirac oscillator with Hamiltonian (11) poses no such problem. Its degeneracy occurs at a regular point n = µ = 0.
The indices c 1 = ±1 of the ∆ 1,2 bundles over the 2-sphere in the parameter space R 3 (µ,q,p) encircling the origin give the number of levels which the two energy bands of the full quantum system gain/loose as µ varies through 0. In other words, c 1 gives the number of redistributed levels δN . The construction and analysis of ∆ 1,2 is analogous to that in the original geometric phase setup [38,43] because the operator form of (11) where we used cyclic components in (3) and (9), reproduces (1) with The relation between B and the concrete parameters of the system defines the sign in the relation between c 1 and δN , see appendices A.2-A.3. On the other hand, comparing the spectra of (10) and (11), these indices can be computed essentially in the same way as for the Λ bundles of the spin-orbit system in sec. 2.1 at constant N > 0 which in turn go back to the original geometric phase setup (with B = N ), see appendix A.1 and A.4. A different calculation, using Λ 1,2 bundles for µ < 0 and µ > 0 with specific boundary conditions [24] yields the same result as "delta-Chern" δc 1 by taking the difference of indices before and after degeneracy for the non-compact classical phase space, see appendix A.3.  Figure 1. Spectrum of the Dirac oscillator with S = 1 2 as function of formal control parameter µ. The energies of the bulk states (blue and green) and of the edge state (red) are given by (13).
The quantum spectrum of (11) can be computed straightforwardly after we realize that the system has a Lie symmetry with generatorn +Ŝ 1 . Using harmonic oscillator wavefunctions |n with n ∈ Z ≥0 as a basis in the infinite-dimensional Hilbert space H slow , we can split H S= 1 2 × H slow into a union of two-dimensional subspaces of eigenfunctions ofn +Ŝ 1 with the same positive half-integer eigenvalue k, , and k > 0, and an exceptional sole function . Replacing (9) for their quantum analogs and recalling the action of operatorŝ we compute the action of (11) on ψ k and find its eigenvalues As illustrated in fig. 1, the bulk states ψ k with k > 0 have a pseudo-symmetric spectrum with upper (positive) and lower (negative) bands separated by at least 2.
The edge state ψ − 1 2 passes between the bands when the formal control parameter µ changes sign. We recognize the redistribution phenomenon described in sec. 1.3 and note the universality of the eigenvalue expression (13) which is encountered, with variations and iterations, across many fields, notably in quantum Hall effect [18] and spin-orbit coupling [36]. Multiplying (11) by −1 alternates the "direction" in which the edge state transfers as µ increases through 0 (top down in fig. 1). Parameterizations in sec. 3 and 4 result in the opposite direction, while both directions occur simultaneously in sec. 5 as two copies of (11) with different signs appear in the linearized semi-quantum 4 × 4 matrix Hamiltonian.
It is also remarkable that for large n, when small quantum corrections to the classical action can be ignored (footnote 6), the quantum bulk energies λ k match the eigenvalues ± µ 2 + I of the semi-quantum Hamiltonian (11), i.e., the semiquantum energies. Examining the semiclassical description of the two classical dynamical systems on the slow phase space R 2 q,p whose Hamiltonians are given by the eigenvalues of (11), we can reveal the reasons why the classical action I is quantized asñ + 1 withñ ∈ R ≥0 , i.e., with a quantum correction of 1. In such systems, phase corrections combine the usual WKB contribution of π and the geometric phase shift [17, Appendix A].
2.3. Spin-quadrupole system and its dynamical modification. The quadratic spin-quadrupole interaction Hamiltonian with five-parameter traceless symmetric matrix Q representing electric quadrupole is of particular interest to our present study. Being invariant under reversal symmetry Hamiltonian (14) was proposed in [31,5,6] as a time-reversal (cf. footnote 5) modification of (1). Our study of T -invariant slow-fast systems is motivated by an attempt to find a dynamical equivalent of the geometric phase analysis in [31,5,6]. Drawing the parallel with (1) requires, naturally, to consider states with halfinteger spin. For such states, the presence of time-reversal invariance of (14) has one important consequence, known as Kramers degeneracy [27,42]: all quantum levels of the system form strictly degenerate doublets whose components are related by the symmetry operation (15). It follows that unveiling the spectrum of (14) requires more states in the fast subsystem. The minimal number of these states is four. They can be realized as four spin components with S = 3 2 which combine into two Kramers degenerate pairs 7 . So just like (1), Hamiltonian (14) for S = 3 2 has typically two distinct eigenvalues λ 1,2 (Q). These eigenvalues will correspond to the semi-quantum eigenvalues, and comparing to the linear system with Hamiltonian (2), we will have again two quantum bands, but each will now be doubly degenerate.
The matrix of Hamiltonian (14) in the spinor basis is of the general quaternionic form with eigenvalues of multiplicity 2, and so it follows that the codimension now is 5. In the parameter space R 5 , the sole degeneracy point 0 is now surrounded by S 4 . The second Chern index c 2 is required to characterize the respective eigenstate bundles ∆ 1,2 . A dynamical extension of (14) can have a slow subsystem with two degrees of freedom and, therefore, four dynamical and one formal control parameter. On the other hand, adding several formal control parameters, we can continue with one slow degree of freedom (sec. 5). Unless the slow phase space is flat, correspondence with [31,6] will require linearization and local analysis.
2.3.1. Time reversal symmetries. In comparison to (2), the spin-quadrupole Hamiltonian (14) has one clear and essential difference: the reversal operation (15) acts exclusively on spin components and does not affect the five formal control parameters of the system, the components of the electric quadrupole Q. As a consequence, (14) is quadratic in S. On the other hand, our dynamical parameters N are engaged by time reversal (6). This makes us to distinguish reversal operations generating an order-4 group Z 2 × Z 2 . Since (2) and its T -equivariant deformations in sec. 4 are not T S -invariant, they cannot be dynamical analogues of (14). We should turn to terms of degree 2 in S, and furthermore, we can introduce "dynamical quadrupole" Q (sec. 2.3.2) as a symmetric rank-2 tensor constructed of slow variables. Depending on the choice of the slow subsystem and on the particular construction, the resulting system may also come out fully T -invariant, but it will be at least T S -symmetric.

2.3.2.
Dynamical "quadrupole" and spin-quadrupole interaction. From the components of the standard rank-1 spherical tensor T 1 (S) T 1 0 (S) = S 1 and T 1 ±1 (S) = ∓S ± , we construct the T S -invariant tensor of rank 2 In the same fashion, we construct T 2 (N ) which models the electric quadrupole Q.
In terms of these tensors 8 , the closest degree-2 analog of (2) can be written aŝ Like (2), it is spherically symmetric. In the classical limit forN and spin-3 2 basis (16), the corresponding semi-quantum spin Hamiltonian can be written as a quadratic form (14) whose real symmetric traceless matrix Q(N ) has elements 9 In the same basis, quantum Hamiltonian (18) is represented as a 4 × 4 matrix operator whose matrix has quaternionic form (17) with So the parameters of the semi-quantum matrix (17) The system has, as expected, a pair of pseudo-symmetric semi-quantum eigenvalues with multiplicity 2. The isotropy of (18) becomes the isotropy of (20) with respect to arbitrary rotations of N .

2.3.3.
Spin-quadrupole and spin-orbit quantum spectra. We find out the structure of dynamical spin-quadrupole quantum bands that correspond to the semi-quantum eigenvalues (20). It is instructive to do this in comparison to the two bands of the spin-1 2 spin-orbit system with Hamiltonian (4) and semi-quantum eigenvalues (5). The linear spin-orbit system with Hamiltonian (2) and the spin-quadrupole system with Hamiltonian (18) are superintegrable. The integrals J = N +S, J 1 , and energy H are associated with the spherical isotropy group SO(3), its axial subgroup SO(2), and time-independence, respectively. For sufficiently large amplitude of the slow (mechanical) angular momentum N > S, with N ∈ Z ≥0 , the quantum spectrum consists of 2S +1 multiplets labeled by half-integer J = N −S, N −S +1, . . . , N +S. The even number 2J + 1 of degenerate levels within each multiplet can be further segregated into J + 1 2 Kramers doublets, each associated additionally with |J 1 | = 1 2 , 3 2 , . . . , J. Once the strict SO (3) isotropy is broken, this additional classification becomes meaningful.
Both (2) and (18) are traceless and their spectra are centered at H = 0. For spin 1 2 and 3 2 , respectively, these spectra split into two bands of positive (H > 0) and negative (H < 0) energy. The difference is in the arrangement of J-multiplets with respect to 0, and in the resulting composition of the bands. Being interested in N S, we can use a small dimensionless parameter x to express Rewriting and renormalizing the linear spin-orbit coupling (2) as we realize that it is a simple function, essentially linear across its domain, with single root ]. Within the same approach, the spin-quadrupole term (18) equals It follows that the energies of J-multiplets in the spectrum of the spin-orbit term increase linearly with J, and so, in particular, the J = N ± 1 2 multiplets of the spin 1 2 system are opposite in energy ±x N 2 . On the other hand, the spectrum of the spin-quadrupole term is quadratic in J, and furthermore, the energies are negative for |x| < / √ 3. So in the particular case of spin 3 2 , the two multiplets with J = N ± 1 2 and x = ± /3 have negative energies, while those with J = N ± 3 2 have positive energies. To acknowledge their additional internal structure, we call the two bands of the spin- 3 2 system superbands, implying that a superband is constituted by several subbands or multiplets.
Knowing the values of J for the multiplets within each superband, we can easily find the number of states with given |J 1 | required to constitute these multiplets. So in particular for S = 3 2 , we can see that typically, for small |J 1 | ≤ N − 3 2 , each band has two such states, one per multiplet. For larger |J 1 | = N − 3 2 + 1, . . . , N + 3 2 , i.e., N − 1 2 , N + 1 2 , and N + 3 2 , we have 3, 2, and one single doublet state, respectively. When the number of doublets is even, i.e., for |J 1 | = N + 1 2 , they split evenly between the bands. Otherwise, the lower band has one extra doublet state with 2 − 2 required to complete the multiplet with J = N − 1 2 , and the upper band takes the sole doublet with maximal |J 1 | = N + 3 2 . We come to the following proposition.
Proposition 1 (spectrum of (18)). When N S, the upper and lower superbands (bands) of the spin- 3 2 system with Hamiltonian (18) are formed by J = N ± 3 2 and J = N ± 1 2 multiplets, respectively. The superbands have an equal number 2N + 1 of Kramers quantum level doublets labeled by the value of |J 1 |. In each superband, the number of doublets with |J 1 | other than N + S and N + S − 2 is the same. The lower band has one extra doublet state with |J 1 | = N + S − 2, while the upper band takes the sole doublet with maximal |J 1 | = N + S as part of its J = N + S multiplet.
The exact quantum spectrum of (18) for concrete N and S can be, of course, obtained if we use quantum expressions for the eigenvalues of all operators in h(x). Thus we should replace J 2 by N 2 (1 + x) 2 + N (1 + x). Alternatively, we can apply the Wigner-Eckart theorem as detailed by eq. (5.71) of [44, chap. 5.4] with reduced matrix elements of T 2 in eq. (35) of [44,Appendix 13]. Analyzing the sign in (21), we confirm the statement of proposition 1 about the number of states in the bands of the spin- 3 2 system. It may also be instructive to see how (21) converges to just two distinct values (20) when S = 3 2 and S/N → 0. The seeming triviality of the superbands of the spin- 3 2 system does also merit a comment. In fact, the individual subbands in these superbands are not trivial. The corresponding eigenstate "superbundles" Λ + and Λ − representing upper and lower superbands consist of two bundles representing the subbands and corresponding to the individual semiquantum eigenstates. The bundles and the subbundles are not trivial, but indices for each superbundle sum up to 0. See more in sec. 5.3.

3.
Original family without time-reversal symmetry. The system with the spin-orbit coupling Hamiltonian (2) has one essential shortcoming: its degeneracy point {N = 0} happens to be the singularity of its slow dynamical system, whose phase space S 2 N contracts to one point. We need a different formal control parameter γ, such that the topology of the phase space is not affected by its variation, while N can simply be fixed.
Additionally, we like to break the isotropy of (2) in order to have regular dynamics on S 2 N and a nondegenerate spectrum within the energy bands. As suggested in [36], this can be most trivially achieved by combining (2) with a one parameter N -independent sub-family of (1). Without any loss of generality, we can addŜ 1 times a γ-dependent factor B(γ). It follows that we are bound to have two limits, one of coupled momenta with dominating Hamiltonian (2a), the other of uncoupled momenta with Hamiltonian (1) and B = (1, 0, 0). These limits and the correlation diagram connecting them are represented in fig. 2.
The coupled limit has already been analyzed in sec. 2.1. The uncoupled system is rather simple. The semi-quantum eigenstates with energies λ 1,2 = ± 1 2 are spin states | − 1 2 and | + 1 2 that remain unchanged over the parameter space S 2 N , thus forming two trivial line bundles over S 2 N with c 1 = 0. The quantum system consists of two multiplets (bands) with energies λ 1,2 . Each multiplet has 2N + 1 degenerate levels, the normal degeneracy of an isolated (uncoupled) quantum rotator with 2N + 1 levels  [36] and was further analyzed in [37,21]. We associate N and S with "slow" mechanical angular momentum and "fast" spin subsystem, respectively. The Hamiltonian can represent two coupled angular momenta in the presence of magnetic field. Both the spin-orbit coupling constant α and the norm B of the magnetic field B = B(1, 0, 0) depend on the formal control parameter γ so that at the boundaries of the parameter domain, we reach the limits of uncoupled (γ = 0) and coupled (γ = 1) momenta occurring in fig. 2. Exploiting conservation of N > 0 and S > 0, we scale the terms in (22) to make the results dimensionless and concise. Using quantum numbers N > 0 and S > 0 as scaling constants (footnote 3) simplifies expressions further as some factors cancel out. The Hamiltonian (22) has nontrivial Lie isotropy group S 1 constituted by simultaneous rotations of N and S about axis 1 (also called axis z elsewhere). The corresponding conserved quantity is the combined projection of N and S on the axis of symmetry. One of the consequences of this symmetry is that the classical analog system is integrable, another consequence is the factorization of the matrix of the quantum HamiltonianĤ into one-and two-dimensional blocks. This all is very similar to the Dirac oscillator symmetry with generator n +Ŝ 1 (sec. 2.2).
3.1.1. Semiquantum energies. The semi-quantum (spinor) matrix representation of Hamiltonian (22) is obtained similarly to (4). It commutes witĥ and its eigenvalues are axially symmetric simplest Morse functions on S 2 N with just one mandatory pair of stationary points, a maximum and a minimum. The action of axial symmetry on S 2 N (rotation about axis N 1 ) has two isolated critical points with extremal values (±N ) of N 1 (poles) at which semi-quantum energies (26) reach their critical values. These energies become degenerate at the south pole (N 1 , N 2 , N 3 ) = (−N, 0, 0) when γ = 1/2. The degeneracy of λ 1,2 has the generic local form of a conical intersection of two 2-surfaces. The eigenfunctions corresponding to λ 1,2 form two nontrivial bundles ∆ 1,2 over a 2-sphere surrounding the isolated degeneracy point in the parameter space (γ, N 2 , N 3 ). These bundles have indices 10 c 1 = ±1, see Appendix A.5. We can also consider bundles Λ 1,2 (γ) with γ = 1 2 of the same eigenvectors over the base S 2 N . In this case, the bundles are trivial (c 1 = 0) when γ ∈ [0, 1 2 ) and nontrivial (c 1 = ±1) when γ ∈ ( 1 2 , 1]. Such bundles represent the energy bands of the quantum system ( fig. 3) and the index change reflects the redistribution of one energy level between the bands ( fig. 2).
On the slow phase space S 2 N , the classical motion goes along the orbits of the axial symmetry which are constant level sets of N 1 . In sec. 3.2 we relate these sets to the orbits of the Dirac oscillator (sec. 2.2). The system has two elliptic equilibria at the poles The slow dynamics under (24) can be described using variables (N 2 , N 3 ) if we distinguish additionally the south (N 1 < 0) and the north (N 1 > 0) hemispheres (charts). The S 1 N1 orbits lie in the base of both the Λ 1,2 and the ∆ 1,2 bundles. In either case, they are associated with a nonzero geometric phase (cf sec. 2.1). The 10 As pointed out in sec. 1.4, all bundle constructions and index computations for the systems in sec. 3 come back to those in the original plain setup with Hamiltonian (1). Specifically, substituting into Hamiltonian (22), and Taylor expanding to the principal order in B result in (1) which we analyze in the standard way [38,43]. All local bundles ∆, including those discussed in sec. 2.2 (with noncompact slow phase space R 2 q,p ), 4, and 5 can be treated similarly. The same applies to any bundle Λ over S 2 N , starting with sec. 2.1 where we identify N with control parameters B of (1). latter can be, in particular, seen as the origin of a specific additional contribution in the semi-classical quantization of slow dynamics (sec. 3.1.2).  3.1.2. Edge and bulk states of the quantum spectrum. The full quantum system is solvable on finite Hilbert subspaces H j spanned by eigenfunctions ψ j of operator J 1 with the same eigenvalue j. Using spin functions | 1 2 , ± 1 2 and spherical functions |N, k = Y N,k with |k| = 0, 1, . . . , N , we construct uncoupled basis spinor functions for all values of k ∈ Z and j = k − 1 2 such that |j| < N + 1 2 , while for the exceptional extremal values j = ±(N + 1 2 ) we have two single functions The eigenvalues of the action of quantum HamiltonianĤ γ (22) on H j with regular j represent bulk eigenstates belonging to different bands. The exceptional functions ψ ±(N +1/2) are eigenfunctions ofĤ γ with eigenvalues So λ + ≡ 1 remains constant, while λ − = 2γ − 1 increases linearly from −1 to 1 as γ sweeps through the interval [0, 1]. Figure 3 shows the spectrum of the system. Considering the localization patterns of the exceptional states, their energies λ ± can be easily understood 11 . The states |N, ±N correspond to coherent states localized maximally around the poles N 1 = ±N of the slow phase space S 2 N , i.e., near the equilibria of the slow system. Their Wigner distribution has a small "round shape" with a maximum at the respective pole and a near Gaussian profile similar to that of the harmonic oscillator ground state. The eigenfunction ψ + is localized near the N 1 = N pole, opposite to the one where the degeneracy occurs. It is the state which is most distant from the conical intersection point and its energy is not affected by the interaction between (coupling of) the bands. On the other hand, the eigenfunction ψ − is localized right where bad things happen. It represents the edge state. This state is critically affected by slow-fast separation breakdown which occurs when γ ≈ 1 2 . 3.2. Linearization at the degeneracy point. The conical intersection of the semi-quantum energy surfaces λ 1,2 (N ; γ) of the original system with Hamiltonian (22) occurs at the N 1 = −N pole on the slow phase space S 2 N . Near this pole, when γ approaches its critical value 1 2 , the dynamics accelerates, the slow-fast separation breaks down locally, and the exceptional highly localized (see footnote 11) quantum state ψ − gets redistributed. We linearize our system near this point following the outline in sec. 2.1.4, and move to the tangent plane R 2 q,p = T N1=−N S 2 N which serves as a symplectic chart of S 2 N with local symplectic coordinates (q, p). In the basic approximation of eq. (8), the Hamiltonian (24) becomeŝ Rescaling the energy, reparameterizing with new formal control parameter and adjusting the phase of one of the basis functions turn it into the standard Hamiltonian (11) of the one-dimensional (1D) Dirac oscillator [33], see sec. 2.2. The two semi-quantum eigenvalues of (11) and (28) are functions on the flat noncompact slow phase space R 2 q,p . It follows that µ → ∞ makes the two bands of the Dirac oscillator completely uncoupled. While the uncoupled limit is unreachable for finite values of formal control parameter µ, near µ = 0 and γ = 1 2 , the two respective systems exhibit the same redistribution phenomenon and their local eigenstate bundles ∆ 1,2 (cf. footnote 10) over a sphere µ 2 + n = const are isomorphic.
Linearization (28) allows making the correspondence of the systems with Hamiltonians (24) and (11) explicit and complete. The 1:1 correspondence of the harmonic oscillator wavefunctions |n of the noncompact slow system and the angular momentum wavefunctions |N, k with k = j + 1 2 can be readily established after noting that the oscillator ground state |0 corresponds to the coherent state |N, −N localized at the south pole, and that the number of nodes of the excited states |n equals the number of nodes in the radial direction (footnote 11)). This suggests n = −N + k. Expanding N 1 near −N to order O((q, p) 4 ) gives the same result, see (8). Furthermore, through the equivalence of the two first integrals, J 1 with 11 Localization in the phase space S 2 N corresponds to the orientation probability of N . The latter is given by the Wigner distribution W of the eigenstates and has no relation to the angular probability distribution used commonly to represent spherical functions, see, for example, [30] and the discussion in [11,16].
value j and n + S 1 with value k , we come to the correspondence of the respective Hilbert subspaces H j and H k , and subsequently-to the equivalence of the edge states and the first N bulk states of the two systems. For k > N we loose nodal correspondence, and of course, beyond k = 2N , the states of the Dirac oscillator find no analogue in the spin-orbit system. The correspondence can be followed as well at the semiquantum level where the circular orbits S 1 N1 ⊂ S 2 N with N 1 < 0 map to harmonic oscillator trajectories in R 2 q,p with sufficiently small n. In other words, the orbits of the Lie symmetries of the two slow classical systems, the axial symmetry and the oscillator symmetry, respectively, are diffeomorphic for n ≤ N . And finally, at the classical level, both systems exhibit the same kind of nontrivial Hamiltonian monodromy [37].
4. Spin-orbit systems with time-reversal symmetry. Unless in the presence of an external magnetic field, molecular and atomic systems are invariant with respect to time reversal (6), see footnote 5. The system with Hamiltonian (22) is not T -invariant for γ < 1. We like to find a T -invariant system which exhibits similar qualitative behaviour with respect to the variation of a single control parameter. Specifically, we want this system to have two bands of approximately 2N states each for all typical values of parameter α ∈ [−1, 1], and we like a redistribution of (few) levels between the bands to occur at the isolated critical value α = 0.  Hamiltonian (22) becomes T -invariant in the limit γ → 1 which is discussed in detail in sec. 2.1. Its upper and lower 2(J + 1)-degenerate bands with J = N + 1 2 and J = N − 1 2 , respectively, include 2N + 2 and 2N − 2 levels (see fig. 2). If we reparameterize (22) so that the new Hamiltonian is identical to (22) for γ > 0 and has the extended parameter domain [−1, 1], we get another T -invariant limit with γ = −1. In comparison to γ = 1, the upper band corresponds now to J = N − 1 2 and has fewer levels. In the spectrum of (29), when γ decreases on the interval γ ∈ [0, 1], i.e., as we follow fig. 2 and 3 right to left, one state is redistributed top down at γ = + 1 2 . For |γ| < 1 2 , both bands have 2N + 1 states. Subsequently, when γ decreases further on the interval γ ∈ [0, −1], one additional state is redistributed top down at γ = − 1 2 . This process is represented by the correlation diagram in fig. 4. The combination of the two onestate redistributions connects one T -limit to another by passing through a family systems with Hamiltonian (29)   It can be conjectured that another passage exists entirely within the class of T -invariants. In such a case, since all quantum states form Kramers degenerate doublets when the norm of the total angular momentum J is half-integer, specifically, when S = 1 2 and N ∈ Z >0 , the two edge states in fig. 4 must become one doublet state. As illustrated in fig. 5, this single doublet state is redistributed, while all other states (bulk) remain within their bands. Mapped between themselves by the T symmetry operation, the two states in the edge doublet are localized at the opposite points on the slow phase space S 2 N . So, if the axial symmetry is preserved, they will be pole-localized coherent states ψ ±J1 with |J 1 | = N + S which we have already encountered in sec. 3.1.2.

The family of T -invariant Hamiltonians.
We construct explicitly the conjectured T -equivariant connection of the α = γ = ±1 limits of Hamiltonian (29) as a one parameter family. The reparameterized spherically symmetric spin-orbital coupling term (2) α N · S N S , with S = 1 2 and α ∈ [−1, 1], defines the two limits with α = ±1 and is the principal term of the family. However, with only this term, the two bands collapse for α = 0 and then change places for α = 0. We must introduce another T -invariant term with a fixed parameter 0 < ε < 1 in order to recover most of the band structure (bulk states) for all values of α. This additional ε-term breaks the spherical symmetry of (2).
Similarly to the systems with Hamiltonians (22) and (29), our T -invariant family of systems can retain the global axial symmetry with first integral J 1 . It can be argued that an approximate S 1 symmetry action on the slow (classical) space can always be introduced and exploited near an isolated degeneracy point of the two semi-quantum eigenvalues. In the presence of this symmetry, the redistributed edge states are strongly localized (footnote 11) near the symmetry axis. In our T -invariant spin-1 2 system with integer N , the edge states correspond to the states ψ ±|J1| with maximal |J 1 | = N + 1 2 (see sec. 3.1. 2) now forming one Kramers doublet. The presence of the global S 1 symmetry, if possible, will greatly simplify the analysis without any loss of generality.
Since any additional terms should not affect the redistribution of the edge state doublet ( fig. 5), we may require these terms to be function of S and N 2 , N 3 (or N ± ) only. Such terms can be called "equatorial" because they vanish, or become maximal when N is aligned with, or is orthogonal to axis 1, respectively, i.e., when N · e 1 = N or N · e 1 = 0.
Under the action of T , the equator becomes the critical set on the slow (classical) space S 2 N . The "equatorial" states with N 1 ≈ 0 are the most distant from the edge states ψ ± , and we can expect the critical value λ crit of the semi-quantum eigenvalue λ(N , α) on the equator to mark the absolute maximum and minimum energy of the "bulk" states in the upper and lower bands (multiplets) with large |α| ≈ 1. As a consequence, in order to remain the absolute maximum and minimum in the transition region, λ crit should be essentially quadratic in α for small |α/ε| 1. In summary, the requirements on the potential ε-term(s) are: (i) to preserve, if possible, the SO(2) symmetry, (ii) to have the equatorial behaviour (on the slow space), and (iii) to provide the essential quadratic dependence of the critical equatorial energy on α. These requirements can be reformulated somewhat differently and more stringently by demanding the quantum spectrum of the ε-term alone (i.e., for α = 0) to be pseudo-symmetric with two edge states remaining at zero energy and with equal numbers of bulk states of positive and negative energy. This means that the two eigenvalues of the corresponding semiquantum system with spin 1 2 are also pseudo-symmetric and have degeneracy points at the poles.
Considering all bilinear (and so necessarily T -invariant) forms in N and S, we come up with two Hermitian axially symmetric terms other than (2) Of these, only (30b) satisfies the above conditions, while the trivial choice (30a) does not conform to requirement (iii). On the same Hilbert space as in (4), the HamiltonianĤ with S = S = 1 2 , α ∈ [−1, 1], and small ε = 0, has spinor representation Its semi-quantum eigenvalues ± ε 2 sin 2 θ + α 2 (33) have critical points at the poles with latitudinal angle θ = 0, π and the equatorial critical set S 1 = {θ = π/2}. The corresponding critical values of semi-quantum energies λ 1,2 (N , α) with θ = 0, π and θ = π/2 are shown by dotted lines in fig. 6. The poles belong to the same critical 2-point orbit of the SO (2) ∧ T -group action on S 2 N and the energy at both poles has the same critical value ±|α|. This value describes the behaviour of the edge states localized (footnote 11) near the poles. The equatorial energy ± √ ε 2 + α 2 has the desired property (iii). This confirms the family (31-32). We analyze the spectrum of (31) using its spinor representation (32). Following the approach in sec. 3.1.2, specifically see (24), we compute the action of (32) on each Hilbert subspace H j with |j| < N + 1 2 and k = j + 1 2 ∈ Z ∩ (−N, N + 1). This gives the energies of the bulk states with Since R j is invariant with respect to sign flipping j → −j, we observe that each λ ±j constitute a pair of Kramers degenerate doublets with energies (34). For nonzero ε, the splitting between the doublets never vanishes. If we consider N 1 ≥ |α|, i.e., being reasonably close to the classical limit for the slow subsystem, the energies of these doublets are nearly opposite, and consequently, the ± signs in (34) correspond to the "bulk" states belonging permanently to the upper and lower bands 12 , see fig. 6.
It remains to find out what happens to the edge states. As before in sec. 3.1.2, they are readily constructed as single states ψ ±(N +S) which are not affected by the nondiagonal terms in (32). Engaging only the diagonal part of the first (principal) term of (31), we obtain  (8).
Replacing N 1 and N ± in (32) according to (8) gives the spinor forms of the Hamiltonian (32) linearized near each polê Their respective operator forms are obtained from (30a), (31), and (8) Similarly to (28), Hamiltonians (36a) and (36b) have first integralsn −Ŝ 1 and n +Ŝ 1 , respectively, with spinor formŝ This follows from direct computation of commutators of (38) and (36). At the same time, it is instructive to see integrals (38) as linearizations ofĴ 1 in (25). To this end recall that in the Holstein-Primakoff approximation [20] near each pole (8) of the slow phase space S 2 N , the angular momentum N 1 equals 13 ∓N ± n, where n is the number of quanta in the local harmonic oscillations about the poles. Consequently, the spinor form (25) of J 1 becomeŝ which is, to a sign and a constant scalar term1N , equivalent to (38). We also recognize the first integral of the Dirac oscillator (sec. 2.2). We further notice that systems with Hamiltonians (36) and respective first integrals (38) are related by time-reversal symmetry T in (6). While this operation maps the polar regions of S 2 N into each other, it interchanges the symplectic coordinates (q, p) in these regions as well as changes their sign. We have T : (S, q, p) → −(S, p, q) and T : a ± → ±ia ∓ .
It can be seen that Hamiltonians (37) are related by this operation and that so are the respective semi-quantum spinor matrices (36), cf. footnote 5. As a result, Hamiltonians (36) are isospectral. Furthermore, to a reparameterization, their spectra are equivalent to that of the Dirac oscillator in sec. 2.2. The eigenstates of either (36a) or (36b) have nondegenerate eigenvalues, with a single "edge" state of energy α and the "bulk" states forming two bands with a pseudo-symmetric spectrum.
We compute the spectrum. Since both (11) and (36b) commute withn +Ŝ 1 , we can work on the Hilbert subspaces H k with k − 1 2 = n ∈ Z >0 of functions ψ k already defined in sec. 2.2. On these subspaces, we find the corresponding bulk state eigenvalues while the single edge state ψ 0 of (36b) has energy α, see fig. 7. The bands are separated at least by ±ε 2/N . Scaling these energies by 2(ε 2 + α 2 )/N > 0 gives the spectrum of the Dirac oscillator in fig. 1 with . 13 Indeed, Taylor expanding and using the semiclassical value N = N + 1 2 , we obtain N 1 /N ≈ ±1 ∓ I = ±1 ∓ n with oscillator action I = 1 2 (q 2 + p 2 ) acting asÎ|n = (n + 1 2 )|n .  fig. 1 and 6. The bulk state energies (red and green lines) are given by (39).
In order to finalize the description of the spectrum, we turn to the spinor Hamiltonian (36a) (linearization at N 1 /N = 1). This operator commutes withn −Ŝ 1 . We should, therefore, work on the Hilbert subspaces of eigenfunctions ofn −Ŝ 1 with eigenvalue k. These subspaces contain the bulk states. The action of (36a) on ψ * n has the bulk eigenvalues identical to (39). The sole edge function ψ * is the eigenfunction ofn −Ŝ 1 with eigenvalue − 1 2 . The energy of this edge state equals α. We conclude that the entire spectrum of the original system with Hamiltonian (29) is reproduced as a sum of two identical spectra (39). Each of these spectra corresponds to a Dirac oscillator, a system without T -invariance. Their sum reproduces Kramers degeneracy of the T -invariant system (footnote 5).

Combining Dirac oscillators and Chern indices.
Our results in sec. 3.2 suggest that the redistribution phenomenon in the T -invariant system can be analyzed entirely by exploiting what we know already for simpler systems in sec. 2.1, 2.2, and 3. The eigenstate bundles Λ 1,2 over S 2 N have Chern indices c 1 = ±1. More specifically, denoting these bundles as Λ ± for positive/negative semi-quantum energies (33) respectively, we find (Appendix A.6 and A.7) c 1 = ∓1 for α > 0 and ±1 for α < 0, cf. fig. 5. The index change |δc 1 | of 2 corresponds to the two edge states (forming one Kramers doublet) exchanged at α = 0. Alternatively, within the standard geometric phase framework, the eigenbundles are constructed locally, near each degeneracy point, after the total degeneracy at α = 0 is removed by the ε-term. The computations are equivalent to those for the original system with Hamiltonian (22) near its sole degeneracy with N 1 = −N and γ = 1 2 and for the Dirac oscillator in sec. 2.2. Two linearizations λ(α, N 2 , N 3 )| N1=±N are required, leading to the analysis of the eigenbundles ∆ 1,2 N1=±N . In this way, choosing an appropriate sign convention, we obtain c 1 = 1 for ∆ 1 defined near either of the poles, giving the total of +2, while for ∆ 2 , the indices have opposite signs totalling up to −2, see Appendix A.7. This sum of indices computed locally near each pole corresponds to the total number of gained/lost levels by the corresponding energy band.

5.
Quadratic spin-orbit coupling. As we explain in sec. 2.3, the T -invariant system in sec. 4 cannot provide a dynamical analogy of the spin-quadrupole systems in [31,5] because the spin-orbit Hamiltonian (31) has no T S symmetry. At the same time, for lack of a better idea and because it seems quite a reasonable thing to begin with when moving into unknown territory, we like to recycle the approach of sec. 4. Recall that in order to construct a two-band system with elementary T -equivariant energy level redistribution, we retained the spherically symmetric spin-orbit term (2) and we used its coefficient α as the principal (sole) parameter of the system. In order to lift the complete collapse of the bands at α = 0, we added an ε-term (30b) with specific properties making the sign and the concrete value of ε unimportant as long as ε = 0. The isotropic Hamiltonian (18) introduced in sec. 2.3.2 seems to be a most natural choice for the quadratic α-term. However, before we attempt constructing the T Sinvariant and, possibly, axially symmetric quadratic ε-term (sec. 5.1), we should pay attention to one important difference between (18) and plain linear spin-orbit Hamiltonian (2). As we show in sec. 2.3.3, the spectrum of the spin- 3 2 system with Hamiltonian (18) and the spectrum of the spin- 1 2 system with plain spin-orbit Hamiltonian (2b) have both two bands. However, while the latter system has bands with different number of states, the bands of the former system have equal number of levels (proposition 1).
More specifically, the bands of the spin-3 2 system with Hamiltonian (18) can be seen as two "super-bands", each having two complete multiplets of 2J + 1 states with a fixed value of J = N − S, . . . , N + S as two "sub-bands". These "subbands" appear naturally for large |α| after adding small perturbation term S · N preserving the time reversal symmetry (6) and the SO(3) isotropy but breaking the T S symmetry. Sub-bands have different J and differ in the number of levels, but each super-band has the same total number of levels. Nevertheless, the super-bands are qualitatively different because the values of J are unique. This makes the limits of α = −1 and α = +1 qualitatively different and when α changes sign, a number of states must be redistributed to rebuild the two super-bands. However, since the total number of states remains unchanged, the redistribution should go both ways. From the detailed classification of states in each band given in proposition 1, we can conjecture that the two Kramers doublets with |J 1 | = N + S and |J 1 | = N + S − 2 have to be exchanged. Assuming that both T S and T N symmetries (sec. 2.3.1) are preserved by the ε-deformation, i.e., that the T symmetry is also present and Kramers doublets remain intact, the entire redistribution is given by the correlation diagram in fig. 8.
In this diagram, the hypothetical values of the super-bands indices c 1 and their decomposition into a sum of two sub-band indices are deduced from the total number of states in each super-band and sub-band. Compared to the previous sections, the actual calculation of these indices is now hampered by the degeneracy of the semi-quantum eigenvalues. The four eigenstates of the T S -symmetric spin-3 2 semiquantum system form two Kramers doublets which correspond to the super-bands. The Λ-bundle for each super-band is, therefore, formed by two semi-quantum eigenstates with degenerate eigenvalue λ : S 2 N → R. Assuming that this Λ-bundle can be decomposed continuously over S 2 N into two respective sub-bundles, these indices can be computed. The linearization in sec. 5.2 may suggest that such decomposition is indeed possible. At the same time, the difference of the indices, or delta-Chern δc 1 = 0, can be confirmed as previously (sec. 2.2) using the linearization in sec. 5.2. The calculation of Chern indices for superbands, i.e., for the rank-2 bundles, is discussed in appendix A.8.

5.1.
The family of quadrupolar spin-orbit Hamiltonians. Using tensors T 2 introduced in sec. 2.3.2 and incorporating spherically symmetric α-term (18), the closest degree-2 analog of (31) can be written aŝ This Hamiltonian is axially symmetric and is invariant under both T S and T N reversal symmetries in sec. 2.3.1. Similarly to its predecessor (30b), the axiallysymmetric ε-term in (40) incorporates the exterior product of S and N (of rank 1). In the basis (16), this term contributes solely to the off-diagonal block of the quaternion-form matrix (17) of (40).

5.1.1.
Semi-quantum energies. The system has, as expected, a pair of pseudosymmetric semi-quantum eigenvalues the same as (33) but with multiplicity 2. Figuratively, multiplicity doubles all features of the semi-quantum system in sec. 4. Thus, critical semiquantum energies in fig. 9 and fig. 6 are the same. In particular, for all α 0 = 0, the eigenvalue λ + (α 0 , N ) has a double minimum at the poles, and a degenerated maximal circle on the equator. Degeneracy occurs only at energy 0 and only when α 0 = 0 and only as conical intersections at the poles {N 1 = ±N }.

5.1.2.
Quantum spectrum. The quantum spectrum of (40) with nonzero ε requires numerical diagonalization (typically of 4 × 4 matrices). As can be clearly seen in fig. 9, for large |α 0 | ε > 0, this spectrum tends to the SO(3) isotropic limit which is analyzed in detail in sec. 2.3.3. Specifically, we can see how the states in this limit regroup into subbands with given total angular momentum J.
It is instructive to consider the spectrum for α 0 = 0, i.e., the eigenstates of the ε-term of (40). These eigenstates, labelled by the absolute value |J 1 | of momentum J 1 , are represented in fig. 10 in the form of an energy-momentum diagram. We notice that, similarly to that of (30b), the spectrum of the ε-term in (40) is pseudo-symmetric (the eigenvalues come either in ± pairs or equal 0), and is nondegenerate on each Hilbert space H J1 spanned by eigenfunctions ofĴ 1 with given fixed eigenvalue J 1 . This follows from the fact that both these ε-terms have imaginary skew-symmetric matrices which are not block-diagonal unless dim H J1 is odd, in which case there is one zero eigenvalue. As fig. 10 illustrates, and in accordance with proposition 1, the two edge state doublets in the spin-3 2 spectrum of the εterm of (40) have unambiguous superband destination in the isotrpic limits with |α 0 | ε. The destination is predefined by the conserved value of |J 1 |. For example, the doublet with the maximal |J 1 | = N + 3 2 joins necessarily the multiplet with the maximal J = N + 3 2 of the J = N ± 3 2 superband.
The elements (see footnote 9) of the linearized matricesQ in (43) are given bỹ andQ In the spinor basis (16), Hamiltonian (43) becomes a matrix operator with quaternionic matrix of the form (17). It has diagonal parameter and the off-diagonal block To a sign and a scalar factor,Ĥ is isospectral to one of Hamiltonians (36). They all are, essentially, Dirac oscillators (11). The sign is important as it defines the transfer direction of the sole edge state in each block (see fig. 7). Linearizations for N 1 = +N or −N are isospectral, T N invariance obliging. Superposition of the spectra of the ±Ĥ blocks for either linearization gives two edge states transferred in opposite directions.

5.3.
Redistribution, Chern indices, and the number of states. Linearization in sec. 5.2 elucidates the point that we have already made when discussing semiquantum energies (42) : our system is a double cover of the one in sec. 4.1 with the two copies having opposite redistribution directions. Due to the multiplicity of (42), the semi-quantum system has four degeneracy points, each point corresponding to one of the four quantum edge states exchanged as suggested by the correlation diagram in fig. 8 and confirmed by the concrete computation represented in fig. 9. The four quantum edge states are grouped into two Kramers doublets, while the respective four degeneracy points form two pairs with the points in each pair related by the T N symmetry operation. Comparing to the redistribution phenomenon in fig. 5 of sec. 4, we have now twice as many degeneracy points and edge state doublets. For each doublet (and respective pair of T N -equivalent points at the poles), the phenomenon is the same as in sec. 4. In general, on the entire interval of the tuning control parameter α 0 , the four individual semiquantum eigenvalues (42) form two bundles Λ ± of rank two, their superscripts ± replicating the sign of (42) and so referring to the upper and lower superbands. There is no further natural decomposition of the ± eigenspaces into a direct sum of one-dimensional subspaces. In fact, as we can see in fig. 9, there is no continuous uniform way to represent the bulk spectrum of (40) as a sum of Dirac oscillator spectra similarly to the way it was possible previously in sec. 3

and 4.
On the other hand, the linearization in sec. 5.2 and the SO(3)-isotropic limit in sec. 2.3.2 and 2.3.3 provide certain grounds for the conjecture that Λ ± at large |α 0 | split into bundles Λ ± 1,2 where the subscripts (1, 2) account for multiplicity 2 of (42). Then in the limit of large |α 0 |, as indicated in fig. 8, we should find c 1 = ±3 for one pair of Λ 1,2 and c 1 ± 1 for the other. These indices correspond to the number of states in the multiplets of the isotropic system, see proposition 1. In both cases, the sum of Chern indices gives index 0 for the rank-2 bundles Λ ± . Continuing from the large |α 0 | limits towards the degeneracy point α 0 = 0, we can assume that c 1 indices of Λ ± remain zero. For the quantum system, such combined zero index means that the total number of states in each superband equals 2(2N + 1), including 2N bulk state doublets plus one edge state doublet, see fig. 8. This number is given by twice the phase space volume of S 2 N without any corrections. On the other hand, within the geometric phase framework, we should consider four local bundles ∆ ± 1,2 over spheres S 2 enclosing one of the two degeneracies. This can be combined with one of the linearizations in sec. 5.2 which have formal control parameter α 0 and dynamical parameters (q, p). For each linearization, we study the bundles ∆ over the base S 2 ⊂ R α0,q,p enclosing the origin. Like in sec. 4.2, Kramers degeneracy ensures that the analysis for linearization at the poles {N 1 = N } and {N 1 = −N } gives the same result. For example, if ∆ N1=N has index c 1 = 1, then so does ∆ N1=−N . Block-diagonal form (45) means that for either linearization, we can construct unambiguously four bundles ∆ ± 1 and ∆ ± 2 , where (1, 2) label Dirac oscillator factor-blocks in (45) and ± refer to the upper and lower superbands. Since the oscillator blocks are of opposite signs, ∆ ± 1 and ∆ ± 2 have indices c 1 = ∓1 and c 1 = ±1, respectively. So, for example, for ∆ + 1 and ∆ + 2 with indices c 1 = −1 and c 1 = +1 this means that one subband of the upper superband looses a state (to the lower superband) while the other subband of the same superband gains a state (from the lower superband). Adding up for two linearizations, we reconstruct lost/gained Kramers doublets. At the same time, the sum of indices for either ∆ + 1,2 ANGULAR MOMENTUM COUPLING 487 or ∆ − 1,2 gives zero, reflecting that the number of states in the superbands remains unchanged.
6. Discussion. The simple Hamiltonian (1) has been widely recognized as relevant in many different physical problems [43]. The geometric phase phenomenon in the parametric family of model systems with this Hamiltonian is directly related to the fundamental mathematical construction of vector bundles, realized as eigenstate bundles over the parameter space, and to the naturally defined connections on these bundles and topological invariants [38]. The parallel discovery of the quantum Hall effect [40,26], topological insulators, and more generally, topological phase transitions and topological phases of matter [39] have arisen considerable interest in topological effects mainly in solid state and high-energy physics. On the other hand, despite being suggested very early, the same year as the paper by Haldane [18] on the Hall effect, the dynamical modification of the geometrical phase setup [36] in finite particle systems with compact phase spaces and its relation to the separation of slow and fast variables and associated rearrangement of energy bands met with little enthusiasm in molecular and atomic physics. One possible reason may be that there are still many important discrete quantum states of these systems that can be studied individually, repeatedly, and with ever increasing accuracy, while large groups of levels, such as polyads, multiplets, shells, and generally-bands require higher excitations, special conditions, and are difficult to reach experimentally, to interpret, and to investigate theoretically. Yet energy bands and their rearrangements are common features of excited molecular systems and their thorough investigation is impending.
Our work reviewed and summarized the universal properties of slow-fast parametric semi-quantum systems with one slow degree of freedom and paws the way for the study of systems with two slow degrees of freedom (four dynamical parameters), specifically the 4-level quaternionic semiquantum systems and their full quantum analogues, and more concretely, the model quadratic spin systems with time-reversal symmetry and spin 3 2 , which are direct dynamical analogues of [31,6]. These systems have a slow phase space P of dimension four supplemented by one formal control parameter. The local semi-quantum eigenstate bundles ∆ and the semi-quantum eigenstate bundles Λ over P are now characterized by the second index c 2 (cf. [13,14]). This brings up the fundamental question of how the value of c 2 and its change are reflected by the numbers of quantum states in the corresponding energy bands and by the redistribution phenomenon, respectively. This question remains yet to be fully addressed. Several compact and non-compact possibilities for P can be envisaged and their analysis promises to be of great interest and importance to mathematical theory and physical applications.
Appendix A. Chern number calculations. This appendix presents explicit calculations of Chern numbers for eigenstate bundles of several semi-quantum systems analyzed in the main body of the article. The relation between Berry setup [10] and topological Chern numbers was immediately recognized by Simon [38] and concrete calculations of Chern numbers for the model Hamiltonian (1) were described about 30 years ago by Avron and co-authors [6] along with more difficult calculations for the quadratic spin Hamiltonian (cf. sec. 2.3). We reproduce these calculations for concrete Hamiltonians following the outline in [21].
A.1. Chern numbers for a spin-orbital coupling system. Let us consider the semi-quantum angular momentum coupling Hamiltonian (2a) for S = 1 2 , which is expressed in the basis cf. (4) and recall that in the semi-quantum setting, we view the "slow" angular momentum operatorN as a classical vector variable N . This brings us within the geometric phase setup [38]. The eigenvalues of the semi-quantum Hamiltonian are obtained straightforwardly. The eigenvector associated with λ + (N ) can be expressed in two ways It should be pointed out that |u + up and |u + down cannot be defined at the north (N) and the south (S) poles of the two-sphere S 2 N of radius N , respectively. In other words, their respective domains are We call the points where the eigenvectors cannot be defined exceptional. On the intersection U + up ∩ U + down , the eigenvectors |u + up and |u + down are related by This relation and (48) determine the eigenvector bundle Λ + over S 2 N associated with eigenvalue λ + (N ). The local connection forms are defined, respectively, as and are related on U up ∩ U down by The local curvature forms are defined, respectively, as Since F + := F + up = F + down on U + up ∩ U + down , the curvature form F + is defined globally on S 2 N . In order to evaluate the first Chern number c 1 of Λ + , we integrate the curvature form F + over S 2 N with spherical coordinates (θ, φ). Let S 2 N + , S 2 N − , and S 1 N denote the northern and southern hemispheres, and of the equator of S 2 N . Then, the integral of F + over S 2 N is calculated using the Stokes theorem, (50), and (52) It follows that the first Chern number for the bundle Λ + is defined and evaluated as In the same manner, we find that the first Chern number c 1 for the eigenspace bundle Λ − assocaited with λ − (N ) equals 1.
A.2. Index for a vector field. The Chern number can be equally calculated locally through the index of the vector field. This is important for further applicaions to linearized problems. Manipulation (54) is valid also when the equator S 1 N is deformed into a small circle Γ(θ 0 ) around the north pole with small constant latitude θ 0 . In the limit θ 0 → 0, the integral of η −1 dη along Γ(θ 0 ) gives the index of a vector field locally defined in the neighbourhood of the north pole [21]. To see this, we take (x, y) = (N 2 , N 3 ) as local coordinates on the northern hemisphere. Then, we can view twice the upper right component N − of H as a vector field W = (X, Y ) = (N 2 , −N 3 ) on the vicinity of the north pole, where W has a singular point (or vanishes) at the north pole. In terms of (X, Y ), we rewrite η as and further obtain Outside of the north pole, we denote the normalized W by w and define v to be the vector field w rotated counterclockwise by π/2, Then, the η −1 dη is rewritten as If we make θ 0 tend to zero, then the right-hand side of the above equation becomes the definition of the index of the vector field W at the singular point; In calculating the index, the linear approximation of W works well. In fact, we see that where A is the Jacobi matrix evaluated at the origin (x, y) = (0, 0). For the vector field W = (N 2 , −N 3 ), one has so that ind(W ) = −1. It then turns out that Eqs. (56) and (57) are put together to show that the Chern number c 1 is determined through the linearization of the Hamiltonian at the exceptional point for the eigenvector. So far we have taken a small circle around the north pole. We may equally take a small circle around the south pole. In the southern hemisphere, we have to take (x, y) = (N 3 , N 2 ) as local coordinates on account of the orientation of the sphere. In this setting, the locally defined vector field determined by twice the upper-right component Then, on the tangent plane to S 2 N at the north pole {N 1 = N }, we can introduce canonical variables by Rewriting the initial Hamiltonian as and rescaling by N 2 , we obtain the linearized Hamiltonian Although µ is a positive number by the initial definition, we will treat µ as a parameter taking values in R. At the same time, if we linearize the Hamiltonian at the south pole {N 1 = −N }, we obtain The eigenvalues of K µ are degenerate if and only if µ = 0 and q = p = 0. The eigenvectors associated with ν + are expressed in two ways as where the domains of |v + up/down are, respectively, On the intersection V + up ∩ V + down , the eigenvectors |v + up/down are related by In what follows, we show that delta-Chern δc 1 , a change in the formal Chern number c 1 for the linearized Hamiltonian with non-compact phase space [24], provides the exact value of the Chern number for the semi-quantum eigenstate bundles of the initial Hamiltonian (46). The main step here is the calculation of δc 1 for noncompact setting where it appears as a mapping degree. To simplify our notation, we introduce variables k = (k 1 , k 2 ) = (q, p)/ √ 2. Then, the model Hamiltonian K µ with non-compact phase space R 2 k is expressed as and eq. (64) is rewritten as This relation determines the complex line bundle associated with ν + , which we call the eigen-line bundle and denote by L + . Local connection forms B + up and B + down for L + are defined to be respectively. From (66), they are shown to be related by where it is to be noted that this relation is independent of µ. The curvature form G + is globally defined on R 2 and evaluated as The Chern number can be formally defined and evaluated, by using (69), as While the formal Chern number (70) is not integer-valued, the difference between the formal Chern number for µ > 0 and that for µ < 0 takes an integer value, which is the same value as in Eq. (57). Figure 11. Circles C ρ and C r of respective radii ρ and r form the boundary of the annulus W ρ,r , and C ρ is also the boundary of disk D ρ .
In what follows, we show that the difference makes sense as a topological quantity. For µ > 0, the origin k = 0 is the exceptional point for |v + up (k) but not so for |v + down (k) . With this in mind, we integrate the curvature form G + on the regions D ρ and W ρ,r shown in Fig. 11 to obtain where use has been made of the relation (68) and the Stokes theorem. For µ < 0, the origin is the exceptional point of |v + down (k) but not so for |v + up (k) . A similar calculation to the above provides Although eqs. (71) and (72) contain locally-defined terms, B + up and B + down , their difference may give a characteristics of the eigen-line bundle L + depending on µ. In fact, using (68) and the equality Cr ζ −1 dζ = Cρ ζ −1 dζ, one can verify that Equation (73) implies that a jump δc 1 in the formal Chern number accompanying the variation of the parameter µ is a topological invariant which is given by the winding number associated with the mapping defined through the transition function ζ : C ρ → U (1). We note also that (73) holds for any µ-independent function ζ.
A.4. Delta-Chern as the index of the ∆-bundle. We turn to the relation between the delta-Chern index δc 1 which is introduced in eq. (73) of sec. A.3 and the index c 1 of the eigenvector bundle ∆ over the sphere S 2 surrounding the origin in the parameter space. The ∆ bundle arises naturally in all geometric phase systems [38,5,43]  Let us consider a generic semi-quantum 2 × 2 Hermitian matrix Hamiltonian H m with one tuning control parameter m ∈ R and dynamical control parameters defining points p on a two-dimensional classical phase space P of the slow subsystem. The total parameter space R × P is of dimension 3. The eigenvalues λ 1,2 of H m are functions R × P → R. In a generic system, they become degenerate [41,1] at isolated points ζ 0 = (m 0 , p 0 ) of R × P . Let us assume that λ 1,2 become degenerate for m 0 = 0. If other isolated non-regular values of m exist, we can always work on a sufficiently small open regular neighbourhood M 0, otherwise M = R. Similarly, we can always work on an open neighbourhood of p 0 ∈ P where p 0 remains a unique degeneracy point, but for simplicity, let us assume for now that p 0 is unique on P . Shifting energy by λ 0 (m, p) = 1 2 (λ 1 + λ 2 ) can always make H m traceless on M × P . Since λ 1,2 remain distinct on M \ 0, the adjusted eigenvalues λ + and λ − are strictly positive and strictly negative on (M × P ) \ ζ 0 and become 0 in the degeneracy point ζ 0 . At the same time, the derivatives ∂λ ± /∂(m, p) ζ0 do not vanish in a typical system [1]. Consequently, the linearization of H m at ζ 0 is of the type (1) with ζ 0 -specific parameters B(ζ 0 ). Furthermore, for a sufficiently general physical interaction between the two dynamical subsystems, this linearization results in a Dirac-oscillator-type system (sec. 2.2) with local symplectic coordinates (q, p) on R 2 q,p = T p i P , {q, p} = 1. Sections 3.2 and 4.2 provide concrete illustrations. For each regular m ∈ M \ 0, the eigenvectors corresponding to eigenvalues λ 1,2 of H m form two rank-1 complex line bundles over P , which we refer to as eigenvector or eigenstate bundles, and which we denote Λ 1,2 in sec. 1.2 and in the rest of the article. Since the Chern number of the combined rank-2 bundle Λ over P , or the eigenspace bundle, remains unchanged for all m ∈ M , it suffices to study one of the Λ 1,2 components. In what follows, we will assume that H m is (made) traceless, and adapting the more informative ± notation, we will work with the Λ + bundle. Λ + constitutes two continuous one-parameter families of bundles Λ +,m<0 and Λ +,m>0 . In other words, bundles Λ +,m over P have a certain fixed topology on each disconnected component of M \ 0. The delta-Chern index δc 1 characterizes the change of this topology at m = 0. For a compact P (with sole point p 0 ), this index can be computed simply as In the non-compact setting, we can either use formal Chern numbers (70) [21,23] or rely directly on (73). Since (73) defines a local number its application does, generally, require linearization at ζ 0 . Figure 12. Base spaces R 2 µ and S 2 r of the Λ ± and ∆ ± eigenvector bundles, respectively, in the parameter space R 3 of the Dirac oscillator (sec. At the same time, we consider local eigenvector bundles ∆ + over sphere S 2 r,ζ0 of sufficiently small radius r surrounding ζ 0 in the combined control parameter space M × P . In this space, as illustrated in fig. 12, typical non-empty intersections of P and S 2 r,p i are circles of radius ρ = ( with fixed m, |m| < r. Indeed, the existence of ∂λ/∂p at p 0 already requires that p 0 is contained in P together with an open saturated disk D = {p ∈ P ; p 0 − p < } of some, possibly small, finite radius > 0. Taking r < , we make sure that S 2 r,ζ0 ⊂ M × P and consequently, that typical non-empty constant-m sets C ρ,m on S 2 r,ζ0 lie entirely within D . We have seen across sec. A.1 and A.3 that Chern and delta-Chern numbers c 1 and δc 1 of line bundles Λ + are computed (for fixed regular values of the tuning parameter m) using integrals (54) and (73) over certain circles in the base space P . This makes C ρ,m central to our analysis here.
Consider the ∆ and Λ bundles of the Dirac oscillator (sec. 2.2), the most basic typical 2 × 2 semi-quantum system. It has traceless Hamiltonian (11) with tuning control parameter µ ∈ M = R and dynamical control parameters (q, p) ∈ P = R 2 .
Its eigenvalues have a sole degeneracy point p 0 = (0, (0, 0)). The following lemma lays the corner stone of our analysis.
Lemma A.1 (δc 1 and c 1 (∆) in the noncompact local setup). The Chern number c 1 of the local eigenvalue bundle ∆ + of the Dirac oscillator (sec. 2.2) equals the delta-Chern index δc 1 of its one-parameter family of bundles Λ +,µ , Proof. As pointed out in sec. 2.2, the local bundle ∆ of the Dirac oscillator can be identified with that of the Berry spin system (1) and of the spin-orbit system (2)-(4)-(46) through a simple GL(3) parameter rescaling Since the determinant of this map (cf. [25, eq. (107)]) is negative, the index (55) and the Chern number c 1 (∆ + ) are of opposite signs, i.e., Further comparing to sec. A.1, we note that now the radius of the base space S 2 N becomes r, and that the transition function in (50) equals As pointed out in sec. A.2, the integration (54) may follow any constant level set of N 1 = 2µ with |2µ| < r, such as shown in fig. 12. Comparing to the delta-Chern computation in sec. A.3, we note that its matrix Hamiltonian (59) is made identical to (11) through (µ, q, p) → − √ 2µ, −q, −p .
In more detail, we can see that in our coordinates, the transition function differs in sign from its original form in (64) while the direction on C ρ is preserved. The sign gets trivially cancelled in (73). On the other hand, the inversion of the parameter space R 3 , which can be seen otherwise as flipping the energy-axis, is important. The positive-energy bundle Λ + of (59) corresponds to the negativeenergy bundle Λ − of the Dirac oscillator (11). Therefore, computing the delta-Chern number for the Λ + bundle of (11), we should use (73) with an additional factor of −1 A similar lemma can be formulated and proven for systems with compact phase space P . We turn to the particular system in sec. 3 with P = S 2 N and tuning control parameter γ ∈ (0, 1) because, like the Dirac oscillator in lemma A.1, it has a sole degeneracy point (γ, p 0 ) = 1 2 , (−N, 0, 0) ∈ [0, 1] × S 2 N . Lemma A.2 (δc 1 and c 1 (∆) in the compact local setup). The Chern number c 1 of the local bundle ∆ + of the basic spin-orbit system in sec. 3 with tuning and dynamical control parameters γ ∈ [0, 1] and p = {N , N = N } ∈ S 2 N , respectively, equals the delta-Chern index δc 1 of its one-parameter family of bundles Λ +,γ over S 2 N . Proof. The Chern numbers c 1 (Λ ± ) equal zero for γ = 0 and are computed in sec. A.1 for γ = 1. Continuing from these limits, we find δc 1 (Λ + ) = −1 − 0 = −1. We can also rely on sec. A.1 and footnote 10 to compute c 1 (∆ + ) like we did in the proof of lemma A.1. The circle C ρ = S 2 N,µ<0 ∩ S 2 (µ,N2,N3) parameterized by (N 2 , N 3 ) is at the centre of the analysis. Omitting the details, c 1 (∆ + ) = −1.
Remark 1 (linearization). By itself, lemma A.2 does not imply any linearization of the semi-quantum Hamiltonian H γ defined on S 2 N . However, the radius r < N of the base space S 2 r of the local bundle ∆ can be chosen sufficiently small r N for linearizing H γ at the degeneracy point N = (−N, 0, 0) T to be viable (see sec. 3.2) and for the Chern index computation to take advantage of such linearization. Delta-Chern δc 1 can as well be defined and computed locally at the (−N, 0, 0) pole using the linearization approach in sec. A.3.
It remains to address the situation with several equivalent isolated degeneracy points (0, p i ) on (0, P ) where λ 1,2 attain the same value. The existence of such points may be related to the presence of symmetries P → P or, more generally, of a nontrivial isotropy group Diff(P ) whose operations leave H m invariant, see sec. 4 and [25] for the concrete examples. In a sufficiently small open neighbourhood of each (0, p i ), the system is generic in the sense of [41,1] and our lemmas apply. Following the ideas in sec. A.2, we can sum the Chern numbers c 1 of the local bundles ∆ +,i in order to match the delta-Chern δc 1 (Λ + ) for the bundle Λ + over the entire P .
A.5. Spin-orbit coupling Hamiltonian in the presence of a magnetic field. We calculate here the Chern numbers for semi-quantum Hamiltonian (22) using the matrix representation (24) in the basis {| 1 2 , 1 2 , | 1 2 , − 1 2 } and with parameter γ replaced by parameter t The eigenvalues (26) of H t are degenerate if and only if N 1 = −N and t = 1 2 . The eigenvector associated with λ + can be expressed in two ways as where the normalization factors N + up/down are given, respectively, by N 1 /N )) and (79a) The exceptional points at which definitions (78) fail are listed below where {N 1 = N } and {N 1 = −N } denote the north and the south poles of the two-sphere S 2 N of radius N . According to (80), the eigenvector |u + down is globally defined on S 2 N for 0 ≤ t < 1 2 , so that the eigenvector bundle Λ + associated with λ + is trivial. On the contrary, for 1 2 < t ≤ 1, both eigenvectors |u + up/down are only locally defined, which means that Λ + is non-trivial. It follows that the topology of Λ + changes when the control parameter t passes the critical value 1 2 . Since the Chern number c 1 is piecewise constant in t, it suffices to evaluate c 1 for t = 0 and t = 1. For t = 0, the Chern number is, of course, c 1 = 0. For t = 1, we have already evaluated the Chern number c 1 = −1 for the eigenvector bundle associated with λ + .
A.6. A family of T -invariant Hamiltonians H α . In this section, we work with the semi-quantum Hamiltonian (31) written in the matrix representation (32) with the basis where 0 < |ε| < 1 is a small non-zero constant. The eigenvalues (33) of H α where (θ, φ) are spherical coordinates on S 2 N , become degenerate if and only if α = 0, θ = 0, π.
The exceptional points at which the eigenvectors fail to be defined are as follows In other words, the domains of |u + up/down are S 2 N without the respective exceptional points. On the intersection of those domains, the eigenvectors are related by The local connection forms A + up/down are defined to be Combining eq. (87) and the above definition yields the relation The local curvature forms are defined to be On account of (89), one has F + up = F + down , so that the curvature form F + is globally defined on S 2 N . We integrate the curvature form F + over S 2 N both for α < 0 and α > 0. In the case of α < 0, after dividing S 2 N into the north hemisphere S 2 N + and the south hemisphere S 2 N − , the integration is performed as follows: where S 1 N denotes the equator of S 2 N and where the Stokes theorem and eq. (89) have been used. It then follows that the first Chern number for the eigenvector bundle Λ + associated with λ + is given by In the case of α > 0, calculation runs in parallel to give It then follows that In a similar manner, the Chern number for the eigenspace bundle Λ − associated with the negative eigenvalue λ − is evaluated to be A.7. Delta-Chern analysis for the linearized Hamiltonians. We calculate now the Chern numbers for the time-reversal invariant semi-quantum Hamiltonians (36) and (36b). which correspond to the linearization of (32) at the north and south poles of the S 2 N sphere, respectively. We rewrite them using the a − = z = q + i p representation as and find the eigenvalues of K (+) The eigenvectors associated with ν + are expressed in two ways as The exceptional points at which the eigenvectors fail to be defined are listed as follows: α α < 0 α > 0 excep. pt. of |v In the same manner as discussed in the subsection A.3, we obtain We need further delta-Chern analysis for K The eigenvectors associated with ν + are expressed in two ways as where N The exceptional points at which the eigenvectors fail to be defined are listed as follows: α α < 0 α > 0 excep. pt. of |v and are related by The local curvature forms are defined, accordingly, to be In order to evaluate the delta-Chern, a similar method with small modification runs in parallel. For the sake of a review of the method, we reproduce the evaluation procedure. For α > 0, the origin is the exceptional point for |v where use has been made of the relation (110) and the Stokes theorem. For α < 0, the origin is the exceptional point of |v + up but not so for |v + down . A similar calculation to the above provides down , the difference between them may have a characteristic of the eigenvector bundle depending on α. In fact, one can verify that where use has been made of the relation (110) and the fact that Cr ξ −1 dξ = Cρ ξ −1 dξ. Eqs. (103) and (114) imply that the respective jumps in the formal Chern number accompanying the variation of the parameter α make sense as topological invariants. For ζ : C ρ → U (1), it is a winding number, but for ξ : C ρ → U (1), it is the negative of the winding number. The Chern number of the eigenvector bundle associate with the positive eigenvalues of the initial Hamiltonian H α is c 1 = −1 for α > 0 and c 1 = +1 for α < 0. Hence, the change in the Chern number in the positive direction of α is −1 − (+1) = −2. The initial semi-quantum Hamiltonian H α has two degeneracy points at the north and the south poles of S 2 N . We have evaluated the change in the formal Chern number for each of the linearized Hamiltonians. The totality of the change is −1 + (−1) = −2, the same result as above.
On the intersection of their respective domains This transition relation determines the eigenspace bundle of rank two over R 2 associatied with the eigenvalue ν + of K µ . The projection map onto the eigenspace associated with ν + P + = 2 k=1 w + k,up w + k,up | = 2 k=1 w + k,down w + k,down | . defines the covariant differential operator P + d with which the basis eigenvectors are operated on in order to define local connection forms. We obtain where A + k,up/down = w + k,up/down | d |w + k,up/down , k = 1, 2.
On the intersection of their respective domains, A + k,up/down are related by The Chern form is defined through where t is a real parameter, 1l denotes the 2 × 2 identity matrix, and where use has been made of F + 1 ∧ F + 2 = 0. Thus, the first Chern form is defined to be and the first Chern number is formally defined to be We are to observe a change in the Chern number against the control parameter µ. On account of (124), under the procedure we have frequently performed, we obtain i 2π where Γ denotes a circle with the center at the origin of R 2 . It then follows that the change in the Chern number for the eigenspace bundle associated with the eigenvalue ν + of the linearized Hamiltonian K µ at the north pole is zero; Similar result can be obtained for the linearization at the south pole.This means that for the Hamiltonian (40) there is no modification of the Chern numbers for superbands. This result is consistent with the conservation of the number of energy levels in superbands. Nevertheless the rearrangement of superbands clearly occurs because of the modification in the decomposition of the superbands into individual irreps demostrated by correlation diagrams (see Fig 8,10). In order to see topological modifications it is sufficient to add small perturbation breaking the Kramers degeneracy of superbands caused by T S symmetry and to discuss the modifications of Chern numbers for individual components of superbands like it is done in the main text.