Theory of field-modulated spin-valley-orbital pseudospin physics

Pioneering studies in transition metal dichalcogenides have demonstrated convincingly the co-existence of multiple angular momentum degrees of freedom -- of spin (1/2 $s_z = \pm 1/2$), valley ($\tau = K, K'$ or $\pm 1$), and atomic orbital ($l_z = \pm 2$) origins -- in the valence band with strong interlocking among them, which results in noise-resilient pseudospin states ideal for spintronic type applications. With field modulation a powerful, universal means in physics studies and applications, this work develops, from bare models in the context of complicated band structure, a general effective theory of field-modulated spin-valley-orbital pseudospin physics that is able to describe both intra- and inter- valley dynamics. Based on the theory, it predicts and discusses the linear response of a pseudospin to external fields of arbitrary orientations. Paradigm field configurations are identified for pseudospin control including pseudospin flipping. For a nontrivial example, it presents a spin-valley-orbital quantum computing proposal, where the theory is applied to address all-electrical, simultaneous control of $s_z$, $\tau$, and $l_z$ for qubit manipulation. It demonstrates the viability of such control with static field effects and an additional dynamic electric field. An optimized qubit manipulation time ~ O(ns) is given.


I. INTRODUCTION
The discovery of spin degree of freedom (DoF) in the Stern-Gerlach experiment has opened up a new era in quantum physics. Striking spin phenomena include spin Hall effect [1][2][3][4][5] and spin-dependent transport such as giant [6,7] or colossal [8,9] magnetoresistance, to name a few. Effective field modulation with Rashba [10] or Zeeman [11,12] effects plays a crucial role in pioneering studies and device proposals, including, for example, spin FETs, [13] spin quantum computing [14,15], and so on, in the category of spintronics. [16,17] With the rise of 2D materials [18][19][20] recent years have seen a rapid expansion of research from spin to angular momenta on various length scales. Notably, in 2D crystals of hexagonal symmetry, "valley pseudospin" -a binary electron DoF has emerged, which derives from the existence of doubly degenerate, time-reversal-conjugated energy band valleys at Dirac corners (K and K′) of Brillouin zone. [21][22][23] Exotic topological transport phenomena arise due to the valley DoF, such as valley Hall effect [21,23,24] in graphene [25][26][27] and transition metal dichalcogenides (TMDCs) [28]. In these materials, electron "valley" magnetic moments or angular momenta [21,29] are manifested on the unit-cell scaled orbital motion, and can interact with an in-plane electric field in the form known as valley-orbit interaction (VOI) ( k  = in-plane electron wave vector; //   = in-plane electric field; and    = valley magnetic moment). [30,31] Such interaction is similar to the spin-orbit interaction (SOI) and constitutes a useful mechanism for applications in the category of valleytronics.
Among 2D materials, TMDCs stand out as a unique family characterized by the presence of strong SOI and plural angular momentum DoFs -of spin, valley, and atomic orbital origins. Pioneering studies [32][33][34][35] have convincingly demonstrated the existence of rich quantum physics in TMDCs from intriguing interplay among co-existing DoFs and SOI. With TMDCs, the spectrum of spintronic type physics is broadened for varied applications. Figure 1 summarizes important elements in single-particle, spintronic type physics in solids, in the four categories: spin, valley, spin-valley, and spin-valley-orbital (SVO), with the variety summarized here hosting a vast range of possibilities, including all-spintronic and valleytronic circuits. The figure places an emphasis on field control or modulation of the physics. In general, electrical fields, as well as magnetic fields in vertical [30,[40][41][42] or in-plane directions [30,43] can be introduced and coupled to the various magnetic moments (or angular momenta), in order to tune the physics. and a corresponding interaction with the magnetic moment. On the other hand, the similarity exhibited is superficial, since the two mechanisms differ fundamentally in physics: SOI has a relativistic origin, whereas VOI is a pseudo-relativistic effect determined by the band-structure physics. In addition, while in the SOI case both the spin magnetic moment ( s   ) and electric field (  ) can be arbitrarily oriented, in the VOI case the valley magnetic moment (    ) derives from the circulating current inside each hexagon of the honeycomb lattice and, thus, always points out of plane (//ẑ ), which constrains the corresponding  ( eff B  ) to be in-plane (out-of-plane), e.g., VOI a valley index-conserved interaction. Overall, the availability of and the flexibility in modulation via SOI or VOI have profound implications for industrial applications, e.g., electrical gate-controlled ICs.
Apart from the control, another critical issue -state coherence faces spintronic type applications. Generally speaking, robust state coherence is required for applications in a noisy setting, in particular those at the room temperature. In connection with this respect, as well as for applications in general, TMDCs exhibit the following band structure features with important implications. [29,[39][40][41][42][44][45][46][47] In the monolayer case, they have a unit cell consisting of one transition metal atom (M) and two chalcogen ones (X2), a semiconductor band structure with direct band gap (1-2 eV) at show a unified methodology for electrical manipulation of electrons, based on the interaction between a pseudo-magnetic field eff B k      (  = static electric field and k  = electron wave vector) and magnetic moments, e.g., s   in (a) and    in (b), with the interaction mechanism being SOI in (a) for semiconductors such as InAs [36], InSb [37], InGaAs [38], and etc., and VOI in (b) for 2pz electrons in graphene. In (c), for conduction band electrons in TMDCs, spin and valley DoFs co-exist, but the atomic orbital DoF is basically frozen at d0 with feeble components of d±1 and p±1, [39] which induce a weak L   as well as SOI coupling between spin and atomic orbital DoFs ( Dirac points, and valence (conduction) band edge states primarily derived from the d±2 (d0) orbital of M. Due to the SOI in M, spin-orbit splitting occurs at band edges, with the splitting much more pronounced in the valence band (0.1-0.5 eV) than in the conduction band (3-50 meV). The existence of band gap makes it possible to create electric gate-defined confining structures, e.g., quantum dots [48][49][50][51] or wires [52] useful for general applications.

Figures 1(c) and 1(d)
summarize the implications of foregoing band structure features for pseudospin physics in TMDCs. They show the coupling among spin, valley, and atomic orbital DoFs, in the conduction and valence bands, respectively. Due to such coupling, novel pseudospin states emerge near the gap, as experimentally confirmed by the generation of valley polarization with optical excitations [32][33][34]. Notably, as shown in Figure 1(c), since spin and valley in the conduction band are only weakly SOI-coupled, they can be used nearly independently and simultaneously. [41] Such advantage has recently been exploited, resulting in unique spinvalley quantum computing proposals [53][54][55] and versatile electron qubit schemes. [43,56] On the other hand, as indicated in Figure 1(d), a distinct type of pseudospin physics exists in the valence band. At the valence band maximum (VBM), a Kramers pair of states, denoted as |K> (or |VBM, K>) and |K'> (or |VBM, K'>) throughout the work, are formed at K and K' and characterized by opposite values of quantum indices, (1/2 sz = 1/2, τ = 1 or K, lz = 2) and (1/2 sz = -1/2, τ = -1 or K', lz = -2), respectively, where sz,, τ, and lz refer to spin, valley, and atomic orbital indices of the electron, respectively. Such pair of states define a unique "spin-valley-orbital pseudospin", extremely noise-resilient due to strong SOI-induced interlocking among sz, τ, and lz against individual index fluctuations. [29] Experimentally [57][58][59] and theoretically, [60] the valley lifetime of holes is reported to be enhanced over that of electrons by 10-100 times reaching O(10 µs) at 5-10 K. Such advantage fosters quite an exciting promise for pseudospin-based studies, applications at low temperatures such as quantum computing, and also roomtemperature devices such as pseudospin filters and FETs, and has motivated researchers from a wide range of disciplines.
This work searches a theory for spin-valley-orbital pseudospin physics studies and applications. Concerning the latter, the following nontrivial issue is to be addressed, namely, while the pseudospin coherence is a key advantage, the underlying mechanism for coherence -sturdy interlocking among existing DoFs -also poses a tremendous challenge to the control of pseudospins, especially in the case of pseudospin flip manipulation. In view of such issue, this work proceeds as summarized in the following. Overall, it formulates a general theoretical framework for the pseudospin physics in external fields, in the context of complicated TMDC band structure. It starts by setting up multi-band "bare models", which account for effects of elastic valley-flip scattering due to impurities in the bulk or boundaries of quantum structures. Inclusion of such scattering, when combined with that of spin-and atomic orbital-mixing mechanisms as well as field effects, enables the description of general pseudospin control including pseudospin flipping. Bare models are then reduced to an effective valence band theory encapsulating the low-energy SVO physics including linear response of pseudospins to external fields. Based on the theory, it discusses Rashba and Zeeman type effects in electric and magnetic fields, respectively, of arbitrary orientations. Two paradigm configurations of static external fields are identified for pseudospin control, with one involving only vertical fields and the other in-plane fields. For an example of applications, spin-valley-orbital based quantum computing is proposed, with qubits formed of quantum dot-confined holes. The theory is applied to address the challenge in all-electrical, simultaneous quantum control of spin, valley, and atomic orbital indices for qubit manipulation, and demonstrate the viability of such control with an additional dynamic, in-plane electric field in both configurations. An optimized qubit manipulation time ~ O(ns) is given. This paper is organized as follows. To prepare for the whole discussion of the work, Sec. II introduces elastic valleyflip scattering. Sec. III presents the symmetry perspective of SVO physics in external fields, and demonstrates the two configurations of interest for pseudospin control. Sec. IV presents bare models and the main result -effective theory of field-modulated SVO physics, with a discussion of Rashba and Zeeman type field effects. Sec. V presents the SVO-based quantum computing -qubit states, and qubit manipulation via external field modulation. Sec. VI concludes the study. Appendix A summarizes a few important matrix elements used in this work. Appendix B provides a supplement of certain mathematical details for bare models. Appendix C summarizes the main theoretical tool of this work ̶ Schrieffer-Wolff reduction, and applies it to the derivation of effective theory, as well as systems of dynamic electric field-driven qubits. Expressions of coupling parameters in the theory in terms of bare ones are derived. Appendix D presents a discussion of elastic scattering, including both valley-conserving and valleyflipping ones that enter bare models.

II. ELASTIC VALLEY-FLIP SCATTERING
For complete pseudospin control, one must be able to "rotate" a pseudospin arbitrarily, in the two-state space expanded by {|K>, |K'>}. This includes the pseudospin flip |K> ↔ |K'> as an important type of manipulation. As such flip consists partially of reversing the valley index, the existence of a mechanism to couple opposite valleys, or flip valley, is a necessary condition for complete pseudospin control.
Elastic carrier scattering can change the wave vector and compensate for the difference between K and K', providing valley-flip coupling. Such scattering occurs spontaneously at impurities or, in a more controlled fashion, at boundaries in quantum structures. We denote Uelastic as the corresponding scattering potential energy.
For quantum structures, we focus specifically on the armchair nanoribbon-based quantum wires (QWs) and quantum dots (QDs) with confinement potentials shown in Figure 2. In these structures, since the wave vectors at K and K' are normal to the armchair edge, the edge scattering can effectively provide the wave vector difference needed for valley flip. In the case of QDs, the scattering can be optimized using a triangular QD with all armchair edges [61]. For a similar purpose, a sharp confining boundary is preferred over a graded one. In this work, however, we do not attempt to maximize valley-flip scattering. Instead, we focus on structures with an intermediate coupling, for example, a rectangular QD defined by sharp armchair edges and graded zigzag edges, with a quadratic confining potential profile in association with graded edges as shown in Figure 2. Such structures allow for Brillouin zone of underlying bulk lattice. The confinement potential UQW is taken to be piecewise constant. A K-electron with wave vector "K+k" is scattered into a K'-electron with wave vector "K'+k'". (b) Illustration of the QD potential profile, which consists of a harmonic potential (Uquad) in the armchair direction and UQW in the zigzag direction.
an analytical treatment as well as possible experimental realization, and the corresponding study should be sufficiently informative for assessing general quantum structures.
In general, all-gate patterning technique may be applied to electrostatically define nanoribbons or QDs, depending on the availability of advanced lithography facilities with sharp, lateral pattern defining capacity. [48][49][50][51][52] In a somewhat varied approach, the QD may be gate-patterned in an already grown armchair nanoribbon, with ribbon edges serving as boundaries of the QD on two sides. This approach would require a passivation of the surface states [62] on armchair edges. In yet another alternative, the QD may be fabricated in a lateral TMDC-based heterostructure, where the valence band offset between materials serves to confine a hole. [63,64]. Overall, in general quantum structures, Uelastic can include both electric gate-induced confinement potential and valence band offset.
In this presentation, Uelastic is taken to be nonmagnetic and an even function under the reflection z → -z. Generalization of the theory to arbitrary Uelastic is possible at the cost of increased presentation complexity. Specifically, we consider where Ri denotes impurity position, in the case of a bulk with a random, dilute distribution of identical impurities; in the case of a nanoribbon, where 0 V is the barrier height and Wy is the y-dimension; or (iii) in the case of a QD. The harmonic potential energy Uquad(x) provides the x-confinement and gives a corresponding xdimension Wx ~* 1/2 ( / ) x m   ( * m = hole effective mass = O(me); me = electron mass in vacuum; ωx = frequency parameter for the harmonic potential). In practical quantum structures, Uelastic in (ii) and (iii) is defined basically with a unitcell scale resolution, meaning that Uelastic actually varies insignificantly in a unit cell.

III. SYMMETRY PERSPECTIVE
For a SVO pseudospin, with more DoFs than just valley involved, the elastic scattering mechanism alone is insufficient to flip such pseudospin. In the case of nanoribbons, due to the insufficiency, energy subbands are always valley-polarized in spite of the ribbon edge scattering. [62] We provide an analysis below for such valley rigidity, show that it has a symmetry origin, and demonstrate configurations of external fields that can successfully break the symmetry and lift the rigidity, effecting a pseudospin-flip coupling for pseudospin control.

Vertical configuration
This configuration consists primarily of a static, vertical electric field z  . The following explains the role of z  in symmetry breaking.
We use an armchair nanoribbon for the discussion. When free of external fields, it has the symmetry of time-reversal (T), and mirror reflection with respect to the layer plane (Mz) as well as the center axis (My). When z  ≠ 0, My and T are preserved but Mz is broken.
For z  = 0, energy eigenstates are valley-degenerate and denoted as |K, kx, n> and |K', kx, n> (n = subband index, and kx = wave vector in the x-direction). In the general case where z  may be finite, it can be shown that the common eigenstates of both energy and My can be written in the following forms , ( ) , , ' , , , with My |+, kx>y = |+, kx>y, My |-, kx>y = -|-, kx>y. (6) Eqn. (5) expresses a possible occurrence of mixing between subbands when z  ≠ 0. For z  = 0, the mixing vanishes, and it reduces to the simple result where Cn = 1 and Cm≠n = 0 for a certain subband of index "n", for example. Above, a subscript "y" is attached to the state to indicate that the pseudospin is "polarized in the y-direction", as implied by Eqn. (6).
Eqn. (7) implies the following. For z  = 0, with Mz a symmetry of the system, the equation constrains |+, kx>y and |-, kx>y or, the corresponding basis states -|K, kx, n> and |K', kx, n> for example -to be degenerate. For z in the leading order when kx ~ 0.
The requirement of fields varies in the case of a QD, as shown in the top graph of Figure 3 with frequency ωac, is introduced into the configuration. Secondly, for the ac field to work effectively, the carrier must be in resonance with ac  . Therefore, a vertical magnetic field Bz is further included to Zeeman-split |K> and |K'>, with the corresponding Larmor satisfying the resonance condition. Sec. V provides more details, when demonstrating a complete pseudospin control in the QD case.

In-plane configuration
This configuration consists primarily of a static, in-plane magnetic field. We again use the nanoribbon as an example, and take the magnetic field in the x-direction (Bx). With the field, the spin Zeeman interaction is introduced into the system ( s  = Pauli spin operator), which can flip a spin and hence assist the |K>-|K'> coupling. Other magnetic effects, e.g., the Landau orbital quantization, cannot directly induce pseudospin flip and, hence, would only produce higher-order corrections.
In the presence of Zeeman interaction, the composite MzMy and TMy are both symmetry elements of the system. For example, when applying to a spin, MzMy ~ szsy and thus commutes with sxBx.
In the in-plane configuration, electric pseudospin control can be achieved by creating an electric coupling between "+" and "-" bands with, for example, the VOI derived from a static electric field y  . Based on Eqn. ( giving a coupling between the "+" and "-" states.
In the case of a QD, the coupling vanishes because <kx>QD = 0. One can again solve the issue by introducing into the configuration an ac electric field ac  in the x-direction, with the frequency ωac satisfying the resonance condition as discussed in Sec. V. The overall field configuration is shown in the top graph of Figure 3(b).
The above symmetry-based analysis not only yields useful configurations for pseudospin control, it also sets up a constraint on the construction of effective theory -the theory should incorporate correct symmetry and reproduce the same symmetry-breaking phenomena demonstrated above, as we proceed to the next section and present the theory.

Figure 3 (a)
Subband structure with Rashba type energy splitting, in a WSe2 armchair nanoribbon in the vertical-field configuration, with εz = 10 mV/a (solid black arrow) and Wy = 9 a. (b) Subband structure with Zeeman type energy splitting, in a WSe2 armchair nanoribbon in the in-plane-field configuration, with Bx = 1 T (solid black arrow) and Wy = 9 a. The tight-binding model parameters here are adopted from References 39 and 42. Red thin dashed lines portray additional confinement, besides that provided by ribbon edges, for a QD. Red dashed arrows denote additional fields required for pseudospin control in the QD structure, in the two configurations.

IV. THE EFFECTIVE THEORY
6 Valence (conduction) band edge states in a TMDC crystal are primarily composed of d±2 (d0) orbitals of the metal atom with the symmetry of even parity under Mz. However, as the pseudospin flip |K> ↔ |K'> requires multiple quantum index mixing, it generally involves plural intermediate states both near and distant from the band gap which derive from, besides d0 and d±2, also d±1 with odd parity under Mz. Figure 4 summarizes the TMDC band structure, with a tabulation of band edge states at Dirac points both near and away from the gap. It describes the symmetry of their wave functions in terms of the quasi-atomic orbital notations, e.g., D0, P0, D±2 and D±1, with corresponding wave functions (Lower-cased letters "p" and "d" are reserved for true atomic orbitals.) These notations of ours correspond to the standard group irreducible representations (IR) A1, A2, E1± and E2∓, respectively, of C3h -the 2D hexagonal symmetry group, and are introduced here to describe states and specify in particular their transformation properties under C3h symmetry operations. For example, |VBM, K> and |VBM, K'> have IR indices D2 (or E1+) and D-2 (or E1-), respectively, in our (standard) notations. The figure also presents the primary constituent atomic orbital of each state, e.g., d0, d±2 and etc. As it shows, they are closely correlated with the corresponding IR indices D0, D±2 and etc., justifying the quasi-atomic orbital notations introduced by us. However, the correlation breaks down when atomic p-orbitals of chalcogen (X) are primary, due to the following reason. In our convention, the metal ion (M) is taken to be the center about which one performs a symmetry operation. Therefore, in the case of p-orbitals the correlation would hold if they belong to M but would not if they belong to X. Overall, the IR index specifies the wave function symmetry of a state with respect to the metal ion.

Figure 4
Irreducible representations and atomic orbital characters of valence and conduction band states at K and K', based on TMDC band structure calculations, e.g., Reference 47. Lower-cased letters "p" and "d" denote primary constituent atomic orbitals of states while upper-cased letters "P" and "D" denote corresponding irreducible representation indices of states. "c" denotes the bottom conduction band, "c+1" the next conduction band, and etc. Here, d-orbitals come from metal M while p-orbitals from chalcogen X.
Nontrivial elements of C3h consist of C3 and Mz, with C3 the three-fold rotation and Mz the mirror reflection with respect to x-y plane. Table 1 tabulates transformation properties of various states under C3 and Mz as well as the correspondence between our and standard group-theoretical notations.
C3h irreducible representation (standard notation) Under C3 and Mz, the states are transformed as follows: (1) Here, the notations | , The theory is intended to cover both intra-and intervalley electron dynamics, with the inter-valley part describing the |K>-|K'> coupling. IV-1 presents the theory and then discusses important field effects based on the theory. IV-2 describes bare models. IV-3 provides expressions of effective coupling parameters.

IV-1. Theory and field effects
The theory describes the quantum mechanics of nearband-edge valence band states, in the pseudospin state space expanded by {|K>, |K'>}. A general state in the space is expressed as are envelope functions. They are governed by the following wave equation where Heff is the Hamiltonian. We divide Heff into diagonal (pseudospin-conserving) and off-diagonal (pseudospinflipping) parts, i.e., Each part is presented and discussed below. A number of coupling parameters are present in Heff and reflect the existence of rich physics in the pseudospin space. Overall, five primary ones, , and two secondary ones, characterize Heff as well as the linear response of a pseudospin to external fields, with secondary parameter-dependent Hamiltonian terms taken to be "corrections", as they are dominated by corresponding primary ones (see Appendix D). Exact role of each parameter will become clear below. Expressions of these parameters are presented below in IV-3 and Appendix C.
The diagonal part governs the intra-valley dynamics and is given by Pauli "pseudospin" operator in the pseudospin state space.
Specifically, 1 0 in the basis {|K>, |K'>}, and etc.) The first term describes the orbital part of dynamics, in external fields / /   and Bz, and the potential Uelastic. The remaining terms predict two important field effects on the pseudospin part.
Here, / /  is the corresponding Rashba effect constant. With a similar argument, this effect exists in the nanoribbon case between subband states of opposite valleys, e.g., |K, kx, n> and |K', kx, n> (n = subband index) in both armchair and zigzag nanoribbons.
As the effect arises out of the VOI, , VOI eff R is the only relevant coupling parameter in the effect.

Vertical Zeeman effect due to Bz
. For ε = 0 and B = Bzẑ , it results in a Zeeman type splitting " , between |K, kx, ky> and | K', kx, ky> in the bulk case as well as between corresponding subband states in both armchair and zigzag nanoribbons, with eff g  the corresponding g-factor in this effect. eff g  consists of two parts, namely, "ge" and "gvalley- Note that eff g  is the only relevant coupling parameter in the present effect.
The off-diagonal part of eff H is given by ( , |K>-|K'> coupling and involves quite a few coupling parameters, namely, (i) In the case of a bulk with random, dilute distribution of identical, short-range impurities on M sites, Basically, we do not distinguish between IR-flipped and conserved potentials in this case. (ii) In the case of quantum structures, ( ) , ("a" = lattice constant).
The lengthy expression of ( (ii) Presence of the common factor " 2iKy e  " throughout 9 the expression: This comes from the need to compensate for wave vector difference between |K> and |K'> in the flipping.

(iii)
Presence of valley-flipping potential energy functions throughout the expression: In particular, when implying the absence of any pseudospin flipping, as we would expect, for example, in the trivial case of a defect-free bulk. (iv) Presence of the momentum operator    up to the first order: Being a low-energy theory, Heff is primarily valid in the vicinity of Dirac points. As will become clear later, its derivation based on the Schrieffer-Wolff reduction involves a perturbation -the " k p (v) Separation of Uelastic into "IR-diag" and "IR-flip" components: Quantum path Classes A and B have different structures since they involve distinct scattering, namely, IR -conserved andflipped ones, respectively. This results in a corresponding difference in the functional forms of derived Hamiltonian terms, as is manifested in, for example, Below we consider the case of an armchair nanoribbon in the x-direction, for which Eqn. (21)  For ε = z ẑ and B = 0, a coupling between subband states |K, kx, n> and | K', kx, n> exists and is given by Above, is the leading-order Rashba effect constant in the hard-wall limit where where y v is the Pauli operator in the subspace. Due to the coupling, energy eigenstates in the subspace are given by |+, kx, n>y and |-, kx, n>y with a Rashba-split subband structure, in agreement with the result shown in Figure 3(a). Our theory yields a linear kx energy splitting " 2 x k   " for states near kx = 0. In addition, Eqn. (23) predicts an oscillatory and decaying behavior in the energy splitting when increasing y W . Such prediction is numerically confirmed by the same tight-binding calculation used to obtain Figure 3 (a).
In Eqn. (22), subband state wave functions in the hard-wall limit are given by in the hard-wall limit. Due to the coupling, energy eigenstates in the subspace expanded by {|τ, kx, n>'s, τ = K, K'} are given by |+, kx, n>x and |-, kx, n>x with a Zeeman-split subband structure, in agreement with the result shown in Figure 3   and Mz, if we ignore external fields. Moreover, if Uelastic is taken to be an even function of y, it also respects My, in consistency with our choice of x-axis in the armchair direction.

IV-2. Bare models
We introduce below only "minimal" bare models essential for deriving primary parameter-dependent Hamiltonian terms in eff H . Appendix B presents certain mathematical details of the models and also an extension that can generate secondary terms.
As illustrated earlier in Figure 5, quantum paths are divided into two classes -Class A of "IR-conserved" nature and Class B of "IR-flipped" nature. They will be identified and presented below for each configuration, according to the two following rules. Firstly, they contribute terms to

Twelve-state model for the vertical configuration
In the vertical configuration, the coupling between |K> and |K'> comes primarily from the four-step quantum paths consisting of 1) valley-flip scattering, 2) SOI-induced spin flipping, 3) k p    coupling, and 4) z  -induced parity mixing.
With an analysis based on permutation of the four steps, such paths involve ten intermediate states , ' VBM K  in place of |K> and |K'>, respectively, to explicitly indicate their locations at the valence band maximum. φ7-φ12 are time reversal conjugates of φ1-φ6. This is essential to ensure that the model so constructed satisfies the T-symmetry, in the absence of any magnetic field. In the model, quantum paths for the |K>-|K'> coupling are classified into four types, according to the intermediate states involved, as depicted in Figure 6. Corresponding contributions from them to the coupling are all given by fourth-order perturbation-theoretical expressions. In contrast, other contributions that involve intermediate states outside the twelve-state space, such as those of (D2, sz = ±1, K) or (D-2, sz = ±1, K'), are generally of higher order. An example is given below which is fifth-order and O((ħ/me) k p    /  ) smaller than leading, fourth-order ones (  = O(eV)).

Eight-state model for the in-plane configuration
In the presence of an in-plane magnetic field / / B  , the coupling between |K> and |K'> comes from two-step quantum paths consisting of 1) elastic scattering and 2) magnetic fieldinduced spin flipping, or three-step ones consisting of 1) elastic scattering, 2) magnetic field-induced spin flipping, and 3) k p    coupling. Corresponding quantum paths are classified into four types as depicted in

IV-3. Effective coupling parameters
A Schrieffer-Wolff transformation is performed on both models, reducing them to corresponding effective theories in the small space expanded by {|K>, |K'>}. See Appendix C. The reduction obtains coupling parameters in the effective theory in terms of both band structure and "bare coupling" parameters, providing an important revelation to the connection between the SVO physics and underlying band structure.
From the reduction of twelve-state model, (4

V. SPIN-VALLEY-ORBITAL QUANTUM COMPUTING
Spin-valley-orbital quantum computing is proposed here with QD-confined holes as qubits. In such scheme the qubit state space is expanded by the Kramers pair of QD ground states, one labeled as |K>QD or |K, m = 0, n = 0> with lz = 2, sz = 1 and the other |K' >QD or |K', m = 0, n = 0> with lz = -2, sz = -1. "m" and "n" refer to the quantum labels for hole confinement in x-and y-directions, respectively.
Qubit states and all-electrical manipulation are discussed in V-1 for the vertical configuration where ε =ẑ z  and B = Bẑ z , and in V-2 for the in-plane configuration where ε = εyŷ and B = Bxx . In V-3, we compare manipulation rates in the two configurations. In V-4, we briefly remark on issues of qubit initialization, readout, and qugates in the scheme.

V-1. The vertical configuration
The physics of qubits in this configuration is controlled by three Hamiltonian terms, as summarized below: 1) the potential energy "Uelastic" (= UQD) in confines the carrier and determines qubit states; 2) the Bz-induced vertical Zeeman term H generates a Larmor precession in the Bloch sphere representation around the "z-axis going through |K>QD and |K'>QD", providing one type of qubit manipulation; and 3) the εz-induced vertical Rashba term provides another type of manipulationa rotation around the "y-axis" of Bloch sphere. 2) and 3) combined together accomplish an arbitrary qubit manipulation.

Qubit states
Let QW  = QW quantization energy (in the y-direction), H due to two considerations. Firstly, we work within the regime where the QD confinement dominates over the Landau orbital confinement. Secondly, the magnetic field is primarily introduced to provide the Larmor precession for qubit manipulation. As will be shown below, the manipulation rate obtained in the present approximation scales with the Zeeman energy , Z eff E  in the leading order. Inclusion of the Landau orbital effect here would only produce the next-order correction in the discussion of manipulation.
Next, we discuss the effect of ( ) off diag eff H  for qubit manipulation. In the hard-wall approximation, we obtain where Eqn. (44) shows that the mixing between |K>QD and |K'>QD scales, in the limit of weak Bz, with x QD k . This result has two implications. Firstly, it vanishes since  for an energy eigenstate due to the Ehrenfest theorem, indicating a protection for the state from pseudospin flipping. Secondly, when a pseudospin flipping manipulation is intended, it suggests the application of an ac auxiliary electric field in the x-direction, which can generate a finite d x dt as discussed next.

Larmor precession, Rabi oscillation, and qubit manipulation
In the qubit state space, the Hamiltonian in the leading order is given by (with E0,0 omitted from the diagonal terms) where L   is the Larmor frequency given by , / .
Next, consider the application of an ac in-plane electric field . We provide a relatively intuitive discussion within the adiabatic approximation [67] for this regime. In the ac field, the total QD confinement potential in x-direction becomes timedependent, with the center 0 ( ) x t being oscillatory: . Within the adiabatic approximation, it results in the following dynamical qubit state, namely, a harmonic oscillator ground state with wave function centering around 0 ( ) x t . This leads to Appendix C provides an alternative derivation with the Schrieffer-Wolff reduction.  ' , cos , sin , (51) in the rotating reference frame. Let

V-2. The in-plane configuration
A close analogy exists between the qubit physics here and that in the vertical configuration. In particular, 1) Bx induces an in-plane Zeeman effect, by which the pseudospin is quantized into states denoted below as , which enables, in the presence of an ac electric field in the x-direction, a rotation around the "z-axis".

Qubit states
We perform a perturbation-theoretical analysis in the regime where and the corresponding wave solution is, in the case where the initial state (59) in the rotating reference frame. Results in V-1 and V-2 are summarized in Figure 8, which shows the time evolution of qubit states in the Bloch sphere, in both the lab and rotating reference frames.

V-3. Comparison between configurations
We compare manipulation rates in the two configurations. In the resonance condition where ac L     in Eqn. (50) and in Eqn. (57), it shows that the ac electric field strength, and the | Therefore, under the same ac electric field, we obtain the following ratio Upper graphs depict corresponding transitions between qubit states, which are effected by ac electric field-induced Rabi oscillations, based on the SOI and VOI mechanisms in the vertical and in-plane configurations, respectively. , Above, We also note a few points in the numerical estimation of Rabi frequencies given earlier. Firstly, the dc electric field strengths there were chosen to be as large as possible in order to obtain favorable Rabi frequencies while at the same time it does not invalidate in a qualitative way the theoretical analysis presented. For example, while trying to optimize (4) , z SOI eff R   in the vertical configuration, a conservative z  = 10 mV/a was used which makes " z e a  " two orders of magnitude below atomic energy level spacing (~O(eV)), in order to avoid a strong z  -induced atomic orbital mixing. On the other hand, y  = 5 mV/a was taken in order to maximize y  RVOI,eff in the inplane configuration. In fact, at y  = 5 mV/a, the corresponding potential energy across the QD, "e y  Wy", is comparable to the quantization energy in the y-direction, and the quantum state wave function Yn(y) may be modified quantitatively if not qualitatively. However, we do not expect such modification to affect the order of magnitude of Rabi frequencies estimated. In passing, we note that both the vertical and in-plane electric field strengths envisioned here are experimentally accessible. In particular, stronger vertical and in-plane field strengths at 200 mV/aBLG (aBLG = interlayer spacing in AB-stacked bilayer graphene) [69] and 10 mV/Å [70], respectively, have been experimentally demonstrated. Secondly, at the above field strengths, we have (4) , , which is comparable to that in the spin qubit case [71] and much shorter than the decoherence time ~ O(10 μs) mentioned earlier in TMDCs at 5 -10 K [57][58][59] by a factor of 10 -4 -10 -5 , allowing for successful error correction [72].

V-4. Initialization, readout, and qugates
The SVO pseudospin qubit naturally shares properties of spin or valley qubits. As such, for initialization, readout, and qugate implementation one may adapt the methods previously developed for spin or valley qubits. For example, one may initialize the qubit by placing a "pseudospin valve" -the analogy to a spin valve in close proximity [14]. For readout, the spin-to-charge conversion scheme [73,74] could be adapted here as well. Last, in order to implement a two-qubit gate (qugate), one could place two qubits side by side, and make use of the electrically-tunable exchange coupling J between localized pseudospins to perform a SWAP operation [14,15]. Overall, all-electric, universal SVO-based quantum computing is therefore feasible according to Divincenzo's criteria [72].
Last, we note that SVO qubits and qugates envisioned here can be realized with gated structures. This makes the corresponding quantum computing scalable. Combined with the optimized electrical manipulation time ~ O(ns) in the case of in-plane configuration and the experimentally observed, much longer SVO decoherence time, favorable characteristics are implied for SVO-based quantum computing.

VI. SUMMARY
In summary, for an insightful understanding and applications in spin-valley-orbital pseudospin physics, this work has formulated an effective theory, with important field effects included. Based on the theory, the linear response of a SVO pseudospin such as Zeeman and Rashba type effects has been discussed, with a clear connection established among the underlying band structure, external fields, and pseudospin physics.
Specifically, the work has investigated the pseudospinflip coupling for pseudospin control, based on bare models that elucidate quantum paths leading to the coupling. Reduction of bare models yields the effective theory as well as expressions of effective coupling parameters in terms of band structure and bare coupling parameters. Two configurations, one with vertical and the other in-plane fields, are identified as of particular interest for pseudospin manipulation. The manipulation is shown, in the context of SVO-based quantum computing, to be achievable via electrical interaction mechanisms -SOI or VOI, and magnetic Zeeman effects. Overall, an optimized electrical manipulation time ~ O(ns) is given.
In conclusion, field-modulatable spin-valley-orbital physics carries numerous promises. Together with the distinct electron-based spin-valley physics in the same material, it brings the rather appealing prospect -versatile spintronic type applications in a single material with flexible principles as well as carrier species.

ACKNOWLEDGMENT
We would like to acknowledge the financial support of MoST, ROC through the Contract No. 107-2112-M-007 -023.

MATRIX ELEMENTS
The information of state symmetry as given in Figure 4 and Table 1 helps the evaluation of matrix elements. Below we provide examples of matrix elements used in our work and evaluated with this information: (1) Matrix elements involving the spin operator, such as where the momentum matrix elements nm P and ij P are both imaginary numbers. In addition, we have Here, Apart from the Class A paths in Figures 6 and 7, it can be verified that Class B paths exist at the third and fourth orders.
These additional paths lead to (3, In some cases, a conjugated relation exists based on which Class B paths can be built from Class A ones in a systematic way. Consider the following Class A path: (VBM, K'), which is shown in Figure 7 as a Type-II path for the (3)// eff g -dependent term. In this path, the k p    coupling beween the states of (D2, ↓, K) and (D-2, ↓, K) comes from the "k-p+" term, as can be verified using Appendix A. However, through the alternative "k+p-" term, it can instead connect the (D2, ↓, K) state to an (D0, ↓, K) state, and then arrive at (VBM, K') via the Uelasticinduced valley-flip scattering. With (VBM, K') a state of D-2, this generates an alternative path -a Class B one where the irreducible representation index is varied from D0 to D-2 during the valley flip, and thus contributes to the ( in the vertical case; and   in the in-plane case.
Last, we note that the bare Hamiltonian operators in the extension remain the same forms as those in minimal models and so will not be redundantly presented.

APPENDIX C THE SCHRIEFFER-WOLFF REDUCTION
The Schrieffer-Wolff (SW) reduction provides a way to obtain from a bare model the effective Hamiltonian in a reduced subspace. [76] In C-1, we summarize the SW reduction in general. In C-2, we apply the method to the case of ac field-driven qubits, which was discussed in V-1 and V-2 of the main text in the adiabatic approach, for a verification of the approach. In C-3, the method is applied to the derivation of effective coupling parameters.

C-1. General result
We consider a general Hamiltonian in the perturbation theory,

C-2. AC-field driven qubits
Consider now the QD envisioned in our work, which is subject to the potential energy ( ) ( ) For simplicity, we switch the notation (x',y',t') back to (x,y,t). Then, overall, correct up to O( ac  ), the transformation replaces the ac potential energy "e ac  xcos(ωact)" by "-px∂tx0" in the Hamiltonian. Below, we apply the result of Eqn. (C5).
In the vertical-field case, the QD ground states     (48) and (56) obtained in the adiabatic approximation, respectively.

Effective mass m *
In the k p    theory, an effective (or "re-normalized") mass consists of the "bare" mass and second-order corrections due to the perturbation " We apply the twelve-state model first. We take εz = 0, Bz = 0, H0 = Hband, and the perturbation H1 =  In order to obtain secondary parameters, one performs the SW reduction on extended bare models. We provide results below without giving details. (4,corr) ,  SOI eff R in the vertical Rashba effect Complete expressions are quite lengthy and so we only provide typical leading-order terms: (