Unified Description of the Intrinsic Spin-Hall Effects

The intrinsic spin-Hall effects (SHE) in $p$-doped semiconductors [S. Murakami et al., Science 301, 1348 (2003)] and two-dimensional electron gases with Rashba spin-orbit coupling [J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004)] have been the subject of many theoretical studies, but their driving mechanisms have yet to be described in a unified manner. The former effect arises from the adiabatic topological curvature of momentum space, from which holes acquire a spin-dependent anomalous velocity. The SHE in the Rashba system, on the other hand, results from the momentum-dependent spin dynamics in the presence of an external electric field. The two effects clearly appear to originate from distinct mechanisms. Our motivation for this article is to address this apparent disparity and, in particular, to seek a unifying description of the effects. In this endeavor, we consider the explicit time-dependence of SHE systems starting with a general spin-orbit model. We find that by performing a gauge transformation of the general model with respect to time, a well-defined gauge field appears in time space which has the physical significance of an effective magnetic field. This magnetic field is shown to precisely account for the SHE in the Rashba system in the adiabatic limit. Remarkably, by carefully analyzing the equation of motion of the general model, this field component is also found to be the origin of the anomalous velocity due to the momentum space curvature. Our study therefore unifies the two seemingly disparate intrinsic SHEs under a common adiabatic framework.


I. INTRODUCTION
The spin-Hall effects (SHE) are a family of phenomena in condensed matter systems in which an applied longitudinal electric field gives rise to a transverse spin current. The spin current arises from the transverse separation of spin species in the system, and the physics driving the separation mechanism can be rather distinct across different systems. with Rashba spin-orbit coupling. 7 Both effects are finite in the absence of disorder, and are characterized by a pure transverse spin current generated from the spin-orbit coupling (SOC) in the band structure of the system. Previous studies 8,9 have shown that the SHE described in Ref. [7] for infinite Rashba systems vanishes when one includes vertex corrections to model the effects of impurity scattering. However, the effect may still be manifested in finite-sized systems 9,10,11,12 such as in mesoscopically confined 2DEGs [10] (e.g. in quasi one-dimensional quantum wires), or in the presence of magnetic impurities. 13 In this work, we will analyze the SHE in Rashba systems without considering vertex corrections (i.e. in line with the original treatment in Ref. [7]). Instead, we will provide a phenomenological explanation of the vanishing SHE based on our own analysis later in the paper. In constrast, the SHE in the Luttinger hole system is robust to vertex corrections. 14 It is intriguing that the two SHEs in Refs. [6] and [7], although both intrinsic in nature, appear to originate from distinct mechanisms. The former effect in Ref. [6] is an adiabatic effect described via a gauge potential which arises from the relaxation of hole spins to an effective magnetic field in momentum ( k) space. The topological Berry curvature 15 of the gauge potential has the physical significance of a magnetic field in momentum space, and affects the trajectory of carriers in much the same way as a classical magnetic field does in real space. Here, the resulting magnetic Lorentz force in momentum space manifests itself as an additional (anomalous) velocity in real space. The semiclassical equations of motion of carriers in the presence of the Berry curvature have been derived previously, 16 and will be revisited for the Luttinger system in this article. It is found that the real space trajectory of holes along the transverse direction is spin-dependent, thus resulting in a finite SHE. The SHE in Rashba systems, 7 on the other hand, was derived originally from a semiclassical analysis of electron spin dynamics in a Rashba 2DEG system, with no apparent relation to the k-space topology. In the analysis, 7 electrons were found to gain a momentum-dependent out-of-plane spin polarization in the presence of Rashba SOC and an external electric field, leading to a transverse separation of spins. It is often stated in the literature that the effect arises from the k-anisotropic precession of spins. 8,14,17,18 As part of the motivation for this paper, we will clarify the mechanism and show that the effect is in fact an adiabatic effect in which spins become aligned to momentum-dependent effective magnetic fields.
From a heuristic viewpoint, the physical mechanism of the two SHEs [6,7] are clearly distinct: in the former, carriers acquire a spin-dependent anomalous velocity, whilst in the latter they acquire a momentum-dependent spin-polarization. Our motivation for this paper is two-fold. Firstly, we ask whether the SHE in Rashba systems can also be formulated within an adiabatic framework, and, secondly, whether the physical mechanisms of the two SHEs can be unified. In order to describe the Rashba SHE under an adiabatic formulation, it is instructive to make note of several points: (1) the adiabatic Berry curvature of momentum space of the Rashba system vanishes except as a δ-function singularity at k = 0. 19 Therefore, the spin-dependent anomalous velocity in the Rashba system vanishes for electrons with k = 0 and thus does not contribute any transverse spin current. In contrast, the spin-Hall current in the Luttinger system results from the Dirac monopole curvature (∼ k/| k| 3 ) of momentum space, (2) in the Rashba SHE, spins become tilted out-of-plane which appears contradictory to the adiabatic regime whereby they are assumed to follow perfectly the inplane Rashba field, and (3) although the Berry curvature in the Rashba system exists only at a singular point in k-space, the resulting Berry phase is finite, and previous studies have shown that the spin-Hall conductivity in the Rashba system is related to the Berry phase through the Kubo formula. 4,17 In this article, we find that the above remarks (1)-(3) can be consolidated into a consistent adiabatic theory that emerges from a consideration of the explicit time-dependence of SHE systems. In particular, a gauge field A 0 (t) naturally appears in time space upon applying a unitary transformation to the system, which has the physical significance of a magnetic field in the transformed system. This magnetic field couples to the electron spin, and is shown to precisely account for the SHE in the Rashba system in the adiabatic limit; here, this limit amounts to the spins following the direction of the sum of the Rashba field and the new effective field. Furthermore, the Berry phase can be equivalently expressed in terms of the adiabatic components of the gauge field A 0 (t). Thus, a gauge field description can be attributed to both intrinsic SHEs, although their respective gauge fields are defined in different spaces (momentum and time).
Having identified that both intrinsic SHEs arise from gauge fields in the adiabatic limit, we finally embark on the problem of unifying the physical origin of the two effects. In the presence of an external electric field, the momentum and time spaces become coupled through the usual drift equation of charged carriers. Remarkably, by analyzing carefully the equations of motion of a general SOC model, it is found that the anomalous velocity due to the Berry curvature in momentum space is in fact a direct result of the effective magnetic field component arising from A 0 (t). In this sense, the common origin of the two seemingly disparate SHEs is clarified.

II. THEORY
A. Carrier dynamics in the presence of Berry's curvature in momentum space in spin-orbit coupling systems

Holes in the Luttinger system
Let us briefly review the mechanism for the SHE of holes in p-doped semiconductors reported in Ref. [6]. The effective Luttinger Hamiltonian for holes in the valence band of conventional semiconductors is given by 20 where γ 1 , γ are valence-band parameters defining the effective hole masses,k is the momentum operator, S is the vector of spin-3/2 matrices, and V = V (r) is the potential energy (in our notation, a hat (ô) signifies an operator while an over-arrow ( v) signifies a vector).
The holes described by (1) have a well-defined chirality,λ = −1k · S/| k|. Because of the chirality-squared term in the Hamiltonian, states with opposite signs of chirality (λ = ±1/2 and λ = ±3/2) are degenerate (they correspond to the light-hole and heavy-hole bands, respectively). In the presence of an external electric field E, the potential energy term is where −e is the electron charge. Parameterizing the momentum vector k = | k|(sin θ cos φ, sin θ sin φ, cos θ), we proceed to define a 4 × 4 unitary matrix U( k), which aligns the reference spin axis to be along the direction of k, i.e. it satisfies the In the last term of (3), the position operatorr = i∂ k acts as a partial derivative in momentum space, and we obtain from the k-dependence of U: Thus, under the local transformation, the position operator transforms into covariant form, is a gauge field in reciprocal space.
Thus far, the transformationĤ Lutt. →Ĥ ′ Lutt. is exact. Being a pure gauge field, A( k) induced by the transformation has no associated curvature. Assuming adiabatic transport, in which we neglect mixing between the light-hole and heavy-hole bands, and applying an Abelian approximation within each hole band, we are left with only the diagonal gauge field components of the respective 2 × 2 hole band subspaces. Explicitly, the Abelian gauge fields are given by where the superscript ad. denotes adiabatic transport, and λ is the hole chirality. The corresponding gauge invariant quantity (which is thus related to a real physical effect) is the curvature tensor Ω( k), defined by The curvature Ω( k) above is frequently called the Berry curvature in momentum space. In the present case, a simple calculation reveals that the Berry curvature is i.e. it is a Dirac monopole with strength eg = λ. It turns out that the k-space curvature (7) has important implications on carrier dynamics. In particular, Ω( k) can be regarded as a magnetic field in k-space, which gives rise to a k-space Lorentz-type force. The modified semiclassical equations of motion for carriers in the presence of a non-trivial curvature in k-space have been derived elsewhere to be: 16 where the over-dot signifies time differentiation, and ǫ is the energy eigenvalue of the system.
The final term in (9) is the Lorentz-type force in k-space, and is equivalent to an additional velocity of electrons corresponding to the so-called anomalous Karplus-Luttinger term. 22 Substituting the expression for the curvature (7) into the equation of motion, the anomalous velocity component is given by which is perpendicular to both the applied electric field E and k. Since the chirality of the holes has sign λ > 0 (< 0) for hole spins (anti-)parallel to the electron momentum, the anomalous velocity is also perpendicular to the spin S, and points along opposite directions depending on the sign of the chirality. This transverse separation of the spins gives rise to the SHE of holes in the Luttinger system.

Conduction electrons in the Rashba system
We now analyze the Berry curvature in momentum space for the case of the linear Rashba SOC, 23,24 which is present in two-dimensional electron gases formed in semiconductor heterostructures. We begin with the generalized spin-orbit Hamiltonian, where m is the effective electron mass, γ is the SOC strength, σ = {σ i } is the vector of  (1) can also be transformed to be of this general form when re-cast in terms of the SO(5) Clifford algebra as was done in Ref. [28], although this representation is in a 5-dimensional space rather than the usual spin-1 2 space. The single particle eigenstates of the Hamiltonian are of the form |ψ ± = exp i k · r χ ± ( k), i.e. a product of the spatial plane wave state and the spinor part which encodes the electron spin state. For any k, the spin degeneracy is lifted between the two eigenstates |ψ ± , which have corresponding spin-orbit energy eigenvalues of ǫ ± = ±γ| B( k)|. Let us rotate the reference spin axis such that it points along the direction of the spin-orbit field B( k), i.e. we diagonalize the Hamiltonian with respect to B( k). By parameterizing the spin-orbit field in terms of spherical angles, B = | B|(sin θ cos φ, sin θ sin φ, cos θ) ≡ | B| n, where θ and φ are explicit functions of k, the diagonalization may be achieved through the SU(2) rotation matrix U = U( k) given by Eq. (2) but with the replacements S y → σ y /2, S z → σ z /2.
However, the choice of U for the diagonalization is not unique: for convenience we shall adopt another rotation matrix given by 21 where m = sin θ 2 cos φ, sin θ 2 sin φ, cos θ 2 . The effective, diagonalized Hamiltonian is given byĤ The σ z Pauli matrix in the diagonalized spin-orbit term represents the two spin states either parallel (the ground state) or anti-parallel (the ǫ + state) to the spin-orbit field B( k). In the last term of (13), the position operatorr = i∂ k is transformed into the covariant form of Eqs. (4) and (12), the gauge field components can be represented in terms of the m-vector and the Pauli spin matrices, i.e.
and in terms of the n-vector by replacing A k i in the above equation with where i = x, y, z are real space coordinates. Up to this point, the transformation of the Hamiltonian is general. We now impose the adiabatic approximation, in which mixing between the two eigenstates of the diagonalized Hamiltonian is neglected. Mathematically, this corresponds to retaining only the diagonal terms of A( k), i.e. the σ z coefficients in Eq. (14), from which we obtain an Abelian gauge field known as the Berry connection, A ad. ( k, s).
A ad. ( k, s) has two values, representing the two spin states, s = ±1, of the diagonalized Hamiltonian (we denote the ground state as s = +1), and which correspond to the diagonal terms of A k i ( k). Explicitly, the Abelian gauge field is given by The curvature tensor Ω( k) of this connection, defined by Eq. (6) The above relation is general and applies to any SOC system. One can transform the curvature from B-space to any other space (e.g. k-space) by using the relation: 29,30 where ǫ ijk is the Levi-Civita symbol. Generally, the curvature in momentum space Ω( k) is not the Dirac monopole field (although it still is for the case of the Luttinger Hamiltonian).
The actual form of Ω( k) depends on the k-dependence of the effective magnetic field. We saw in Eq. (9) how this curvature Ω( k) gave rise to an anomalous velocity which resulted in the SHE in p-doped semiconductors. However, the same reasoning cannot be applied to the SHE in the Rashba system, as Ω( k) in this system is vanishing (for k = 0) as we outline below.
The Hamiltonian in the presence of the Rashba SOC is given by 23,24 where α is the Rashba spin-orbit coupling parameter expressed in units of eVm. The effective magnetic field is given by B R (k) = (k y , −k x ), and the eigenvectors are | k, ± = 1/ √ 2 exp i k · r (∓ik −1 (k x − ik y ), 1) T , with the corresponding energy eigenvalues of ǫ ± = ±αk where k = | k| = k 2 x + k 2 y is the in-plane wave-vector magnitude. In momentum space, the effective magnetic field B R is directed along θ = π/2 and φ = tan −1 (−k x /k y ), and from Eq. (16) the Berry connection is given by A ad. ( k, ±) = ± 1 2k 2 (−k y , k x , 0). Evidently, the curvature (6) of this connection is trivial, i.e. Ω( k) = 0 over the entire k-space, except at the singularity point at k = 0 where the k z -component of the curvature has non-vanishing value ±π, i.e. the curvature is of the form Ω( k, ±) = (0, 0, ±πδ( k)). 19 Thus, conduction electrons having a finite momentum in the 2DEG plane do not experience any Lorentz-type force in k-space, as is the case for holes in the Luttinger system. Furthermore, even if this force existed, it would only separate the Rashba SOC eigenstates (whose spins lie entirely in-plane) in the transverse direction, and thus cannot explain the out-of-plane spin polarization acquired by the electrons in the SHE. Now the question arises as to whether the SHE in Rashba systems can be described within a gauge field framework. In particular, the out-of-plane spin polarization seems to suggest the presence of an additional magnetic field in the system. It turns out that such a gauge formulation does exist, but one must turn to another parameter space, namely the time space. When considering the temporal evolution of a quantum system, the unitary transformation is explicitly time-dependent, i.e. U = U(t). In SHE systems, the t-dependence of the unitary transformations naturally arises due to the acceleration of carriers in the presence of an electric field: the electron wave-vector k changes linearly in t, and consequently B(k) acquires a time-dependence. To incorporate the explicit time-dependence of the system quantum mechanically, we switch to the interaction picture. 31 In this picture, the original Hamiltonian (11) is split into two parts,Ĥ =Ĥ 0 +Ĥ 1 , wherê governs the time evolution of the operators, and whereĤ I (t) = e iĤ 0 t/ Ĥ 1 e −iĤ 0 t/ . For the case of linear (e.g. Rashba) spin-orbit coupling, Higher order spin-orbit terms (∼ k n , n ≥ 2) generally lead to correspondingly higher order partial derivatives of the spin-orbit field inĤ I (t). The Hamiltonian (23) governing the state vector evolution in the interaction picture is that of an electron subject to an explicitly timedependent magnetic field, which we denote as B(t). Analogous to our previous treatment, we proceed to diagonalize the Schrödinger equation (22)  This transformation aligns thez-axis to be parallel to the instantaneous magnetic field B(t), i.e.
whereǫ = i ∂ t is the energy operator. On the left-hand-side, the local transformation diagonalizes the time-dependent Zeeman term as required. On the right-hand-side, we obtain from the time-dependence of U a gauge field A 0 (t) ≡ −iU(t)∂ t U † (t) related to the temporal evolution of the system. From the relations in Eqs. (14) and (15), we can express the gauge field as A 0 (t) = A t · σ, where A t = m ×˙ m = 1 2 n ×˙ n + ( A t · n) n. Thus, the term A 0 (t) represents an additional Zeeman-like term, indicating the presence of an effective magnetic field in the rotating frame. We elucidate the origin of this field in more detail below. The unitary transformation U(t) we invoked defines an instantaneous angular velocity ω l = ω l (t) of the coordinates (in the laboratory frame, l) as it follows the time-dependent magnetic field. In the rotating frame r, this vector is given by ω r , where σ · ω r = U σ · ω l U † . Sincė , the Zeeman-like term A 0 (t) in (24) thus yields − /2( σ · ω r ), which corresponds to an effective magnetic field − ω r (omitting a scaling factor) in the rotating frame. This translates into an effective magnetic field B t = − ω l in the laboratory frame. If we now denote by n = n(t) the unit vector pointing along the direction of the magnetic field at time t, we have the equation of motion˙ n = ω l × n. Performing a post cross product on both sides by n, one arrives at the expression for the angular velocity ω l = n ×˙ n + ω l · n n, or, in terms of the effective magnetic field, Thus, the effective magnetic field arising from the gauge field A 0 (t) of the unitary transformation has a component along˙ n× n and along n. Note that it does not have any component along˙ n. As we noted previously, the unitary rotation matrix used by us (12) is not unique.
Specifically, different rotation matrices U i , each specifying distinct angular velocities ω l i , can be used to align the referencez-axis with the instantaneous magnetic field B(t); the freedom of choice here lies in determining the trajectory of the remainingx,ỹ-axes. The second term on the right-hand-side of Eq. (25) reflects the particular choice of the gauge transformation U i . It is not an invariant of the gauge transformation (its magnitude being dependent on the particular gauge choice), and does not represent a physical field. However, the first component˙ n × n of the effective magnetic field is invariant with respect to the gauge transformation, depending only on the time-dependence of the magnetic field B(t).
This term can be understood to be a direct consequence of the time-dependent rotation of the axes. 33,34 The same expression can be derived classically by directly comparing the spin vector in adjacent time frames 33 -as a complement to the quantum derivation, the classical treatment is shown in detail in the Appendix. The˙ n × n component represents a physical magnetic field which couples to the electron spins, 33,34 and, as we show below, is precisely the component which leads to the SHE in Rashba 2DEG systems.

III. ANALYSIS
A. The intrinsic SHE due to Rashba SOC The Hamiltonian of conduction electrons in the Rashba system is given by Eq. (19).
Following our analysis above, the time-dependence of the effective Rashba field B R due to the electrons' motion in momentum space necessarily gives rise to a secondary component −p x , 0) is the unit vector in the direction of B R . We assume a longitudinally applied electric field along thex-direction E = E xx , so that˙ n = p −1 (0, eE x , 0). Because B R is strictly in-plane (i.e. it lies in thex,ỹ-plane of the 2DEG), the term˙ n × n represents an out-of-plane magnetic field component which is along thezdirection by convention. Next, we apply the adiabatic condition for the electron spins. In the ideal adiabatic limit, the magnetic field | B R | is infinitely strong, so the spins always remain aligned to it as it varies with time. In reality, | B R | is finite and there is a non-zero secondary component B ⊥ , and the relevant condition is | B R | ≫ | B ⊥ |, i.e. the electron spin is primarily aligned to B R , but with a small deviation along B ⊥ . In terms of the parameters of the Rashba system, the adiabatic condition reads as Inserting typical values for the Rashba parameter α = 10 −11 eVm and the Fermi wavevector k = 10 8 m −1 , we arrive at the condition E x ≪∼ 10 5 Vm −1 , which usually holds true in experiments. Assuming that the spin of electrons follow the direction of the net effective magnetic field, B Σ , which is the sum of the spin-orbit field B R and the secondary component, the classical spin vector is given by where ± represents spin aligned parallel (+) or anti-parallel (−) to the net field. To first order, the component of the spin along thez-direction is where, to be consistent in units, the magnetic field in the denominator is defined in terms of its equivalent angular velocity. Note that in the convention above the + corresponds to the ground state ǫ − , whilst − corresponds to the eigenstate ǫ + . In the adiabatic limit, the magnitude of B Σ approaches that of B R , and applying this limit to Eq. (28), we obtain for the out-of-plane spin polarization Eq. (29)  . Summing over the Fermi surfaces of the two eigenstates yields an intrinsic spin-Hall (sH) conductivity of σ sH ≡ j z y /E x = −e/8π, where j z y = /4{s z , v y } is the transverse spin-current. From our analysis above, we have clarified that the SHE in Rashba systems occurs as a result of an adiabatic process, in which electrons' spins become aligned to momentum-dependent magnetic fields that arise from the time-dependence of the system. The effect is therefore not due to the precessional behavior of spins, as is often stated in the literature.

Berry's phase
We alluded earlier to previous work which related the intrinsic spin-Hall conductivity in Rashba systems to the k-space Berry phase of electrons through the Kubo formula. 4 It was found there that σ sH = eϕ ± /8π 2 , where ϕ ± is the Berry phase of electrons, The natural parameterization for the vector k is the time variable t, and rewriting the line integral above in terms of t we obtain Thus the Berry phase and hence the intrinsic spin-Hall conductivity of the Rashba system can be written equivalently in terms of the time component of the adiabatic gauge field, A ad. 0 (t).

Effects of disorder
Previous studies have shown that the intrinsic SHE in infinite Rashba systems vanishes in the presence of disorder. 8,9 Specifically, the vertex correction was shown to exactly cancel the intrinsic conductivity of e/8π even in the weak scattering limit. We provide a heuristic argument based on our analysis for the vanishing SHE. In the presence of disorder, the scattering provides a braking effect which cancels the acceleration of carriers on average in the steady state. 12 This implies that in the steady state we have ˙ k = 0, i.e. there is no net change in the momentum and thus the magnetic field component B ⊥ =˙ n × n averages out to zero. Note, however, that this picture is an oversimplification, 36 and that the SHE in Rashba systems does not vanish in general. For example, the SHE persists in finite-sized systems 9,10,11,12 and in the presence of spin-dependent impurities. 13

B. The intrinsic SHE due to linear Dresselhaus SOC
The case for the linear Dresselhaus spin-orbit coupling is also easily verified by our analysis. The Dresselhaus spin-orbit Hamiltonian is given by where β is the Dresselhaus SOC strength and B D is the effective Dresselhaus SOC field.
Here we have n = p −1 (p x , −p y , 0), and we find that (˙ n × n) z = +eE x p y /p 2 . Consequently, the out-of-plane spin polarization s z has the same magnitude but opposite sign compared to the Rashba SOC case. This is in agreement with previous theoretical studies 4 which predicts the spin-Hall conductivity in this system to be σ sH = e/8π.

IV. DISCUSSIONS
Having established the Rashba SHE as an adiabatic effect, we now identify two common traits of the two intrinsic SHEs; adiabaticity and time-dependence. For the Rashba  Thus, at first glance it appears that the two effects are rather independent phenomena.
However, an interesting duality exists between the two effects. In the former Luttinger case, a spin-dependent anomalous velocity pushes opposite spin species to opposite lateral sides of the sample. The magnetic field responsible for this effect is the Berry curvature defined by the k-space gauge field. On the other hand, in the Rashba system, a momentum-dependent magnetic field polarizes electrons along opposite directions out-of-the-plane depending on their transverse propagation direction. The magnetic field responsible for this effect is defined by the t-space gauge field. Given this duality, it would be tempting to ask whether there is any underlying relation between the two pictures.
We proceed to consolidate the link between the two effects by investigating the connection between the anomalous velocity due to the Berry curvature, and the presence of the B ⊥ term.
In this endeavour, we employ the reciprocal space analogue of the analysis by Aharanov and Stern 33 of the origin of the Berry's curvature in real space. We consider again the general spin-orbit Hamiltonian in Eq. (11). The velocity along the i-th coordinate is given by When the magnetic field is time-dependent, the spins see an additional magnetic field B ⊥ .
Assuming that spins align to B Σ = B + B ⊥ , the spin vector, to first order, is given by Writing the spin-orbit field B = | B| n, the partial derivative in Eq. (33) can be expanded into its magnitude and directional parts as (∂| B|/∂p i ) n + | B|∂ n/∂p i . Taking the adiabatic limit | B Σ |/| B| → 1, the second term in the velocity expression becomes Writing˙ n =k j ∂ n/∂k j , where the summation over j is implicit, and rearranging the terms we then get The first term in the above equation represents a velocity term that is due to the inhomogeneity of the spin-orbit field B in momentum space, i.e. it is the reciprocal space analogue of the Stern-Gerlach force. Remarkably, the second term in Eq. (35) is the anomalous velocity of electrons due to Berry's curvature in k-space. This becomes clearer when written in terms of the magnetic field vector B = | B| n, v anom.
We find that this is exactly the anomalous velocity component in Eq.

V. SUMMARY
The primary motivation for this paper is to establish the link between the two intrinsic SHEs reported in Refs. [6,7], which has not been clarified hitherto. We first considered the intrinsic SHE in the Luttinger system, which is driven by the spin-dependent anomalous velocity due to the non-trivial curvature of momentum space. However, this theoretical picture is not applicable in the planar Rashba system. Instead, the SHE in the Rashba system was shown to arise from spins acquiring a component (in the adiabatic sense) along an additional effective magnetic field B ⊥ , arising from the time-dependence of the system.
This field component was shown to be described by a gauge field in time space. Finally, we showed that in the adiabatic limit, B ⊥ is also the origin of the anomalous velocity due to the momentum space Berry curvature. Thus, we conclude that the intrinsic SHEs in the two systems are simply different manifestations of B ⊥ , and that this term provides a unifying link between the two effects. where g is the coupling factor. To solve the above equation, we freeze the time-dependence by transforming to a rotated coordinate frame at each point in time, such that thez-axis is aligned with the magnetic field. A spin vector s defined relative to the coordinate frame at time t, is expressed as the vector s ′ = s + s × ω(t)dt in the coordinate frame at time t + dt, where ω(t) is the generator of infinitesimal rotations (see Fig. 1). The choice of ω(t) is not unique; however, specifically choosing ω(t) =˙ z × z where z is the unit vector n = B/| B| as seen in the rotated frame, coincides with the parallel transport of the coordinate frames. 33,35 Suppose we have a vector representing the spin, s(t), in the rotated frame at time t. At time t + dt, this vector becomes [relative to frame t + dt] s(t + dt) + s(t + dt) × ω(t)dt where s(t + dt) ≈ s(t) + g( s(t) × | B| z)dt. For infinitesimally small dt, we may write s(t + dt) [in frame t] ≈ s(t + dt) + s(t + dt) × ω(t)dt [in frame t + dt]. The right-hand-side of the resulting Therefore, as seen in the laboratory frame, there is an additional, effective magnetic field B eff. = B(t) + g −1˙ n × n.