aa r X i v : . [ h e p - t h ] F e b Pure Gauge Spin-Orbit Couplings
M. S. Shikakhwa
Physics Group, Middle East Technical University Northern Cyprus Campus,Kalkanlı, G¨uzelyurt, via Mersin 10, Turkey
Planar systems with a general linear spin-orbit interaction (SOI) that can be cast in the formof a non-Abelian pure gauge field are investigated using the language of non-Abelian gauge fieldtheory. A special class of these fields that, though a 2 × θ -dependentpotential with the unitary transformation relating the two being again the space-dependent rotationoperator characteristic of R+D SOI. I. INTRODUCTION
Planar spin one-half systems with spin-orbit interaction (SOI) have been receiving much attention recently dueto their relevance to the important applied field of spintronics [1]. In these systems, the SOI acts as a momentum-dependent effective magnetic field that gives rise to spin precession, therefore, manipulation of the spin can be achievedeven in the absence of a magnetic field by tuning the electric field that generates the SOI. So far, two major types ofSOC mechanisms have been receiving the most interest; the one resulting from the structure inversion asymmetry, orthe Rashba SOI (R) [2] , and the one due to the bulk inversion asymmetry known as Dresselhaus SOI (D)[3]. Systemswhere both these SOI are present, especially when they are of equal strength, have special importance (see below).A useful approach to systems with SOI is the non-Abelian gauge field formalism of SOI [4–7] (see also the review [8]). Here, one expresses the SOI interaction as a coupling of the momentum of the particle to a synthetic non-Abeliangauge field. The advantage of this approach is that it provides an elegant and general approach to the problem whereone can carry over the well-established machinery of classical gauge field theory to the treatment and apply ideas likegauge transformations, gauge-covariance ...etc. One has to keep in mind here, however, that unlike the case in particlephysics, the gauge fields in these models, being proportional to the electric fields, are physical. Therefore, a gaugetransformation corresponds to a transformation of one physical configuration of these fields to another. With thistransformation representing a unitary transformation on the Hamiltonian as we will show below, the correspondingHamiltonians are unitarily-equivalent and thus iso-spectral.An important symmetry was recently discovered in planar SOI systems where equal-strength R and D SOI’s (R+DSOI) are present [9]. In such systems, the spin is rotated about a fixed direction but with position-dependent anglesuch that the projection of the spin of the particle along this direction is constant. This results in a long-livingspin polarization. This phenomenon that became known as Persistent Spin helix (PSH) is important in spintronicsapplications and was recently observed experimentally [10, 11]. A gauge field treatment specific only to R+D SOI’swas provided in reference [12], where it was seen that the gauge field in this case is merely a pure gauge, and the PSHwas seen to emerge upon gauging it away . No general treatment valid for any linear SOI is present in the literature,however. Moreover, spin one-half systems confined to a ring with R+D SOI were considered in[13], but without theuse of the gauge-field approach. A gauge field treatment with only one or the other of R and D SOI’s are present at atime on a ring was carried out in [14]. The present article provides a general gauge field-based treatment valid for anylinear planar SOI that can be expressed as a pure gauge, both for a planar system as well as for a particle confinedto a ring. The importance of this general treatment for a planar system is that it makes manifest the fact that theHamiltonian in the presence of a linear SOI is not, in general, gauge-covariant as far as the non-Abelian gauge fieldrepresenting the SOI is concerned. Therefore, the use of gauge transformation as done in the literature should bemade with caution. We demonstrate the intricacy of this point on general grounds and show how to handle it. Thisfact becomes important when one considers the case of a ring with pure gauge SOI. Moreover, we address the issueof the conservation of the Coulomb gauge condition satisfied by the non-Abelian gauge field upon gauging away thefield, especially in the ring case.In section II, we introduce the Hamiltonian of a spin one-half particle in a general linear SOI that can be expressedas a pure gauge. We, noting the absence of gauge-covariance of this Hamiltonian, discuss how we can still utilizethe notion of gauge transformations to gauge away the gauge field from the Hamiltonian. In section III, we considerthe special class of pure gauge fields that, though a 2 × II. PLANAR SYSTEM WITH PURE GAUGE SOI
The spin-orbit interaction (SOI) emerges upon considering the O (cid:0) c (cid:1) expansion of the Dirac Hamiltonian for aspin one-half particle in a scalar potential V ( x ): H = p m + V ( x ) + e ~ m c σ · ( E ∧ p ) = p m + V ( x ) + e ~ m c p · ( σ ∧ E ) (1)where ∇ ∧ E = 0 was assumed. The SOI can be expressed in the form of a minimal coupling of an SU (2) field to themomentum operator by defining the SU (2) gauge field W ai by [15] − gW ai ≡ e ~ mc ǫ iaj E j . (2)where i, a = 1 ... σ ’s are the the Pauli matrices. Using W , the Hamiltonian can be expressed as ( the scalarpotential V ( x ) is dropped now on) : H = p m − gm p · W = p m − gm W · p . (3)where we have noted that ∇ ∧ E = 0 implies ∇ · W = 0. Completing the square, we can put this into the form H = H ( W ) = ( p − g W ) m − g m W · W (4)where W i = W ai σ a is the i th component of the field W . The above is the Hamiltonian of a spin one-half particlecoupled to the SU (2) gauge field W . The spin-independent term quadratic in the gauge field, i.e., − g m W · W = − g m W ai W ai = − e ~ m c E , evidently breaks the SU (2) gauge symmetry. Some works [5, 16] in the literature eitherdrop or absorb this term into the scalar potential thus ending up with a gauge-covariant Hamiltonian: H GC = H GC ( W ) = ( p − g W ) m (5)This approach is controversial [15, 17], and so we will in this work focus on the non gauge-covariant case. For reasonsthat will become clear shortly, we now restrict our consideration to a special class of gauge fields W ai ; those that areplanar i = 1 , G i,j ≡ ∂W i ∂x j − ∂W j ∂x i − i g ~ [ W i , W j ] = 0 (6)Recall that under gauge transformations -a clear discussion of non-Abelian gauge transformations within the frame-work of the Hamiltonian formalism is given in [15] - , the gauge field W transforms as: W ′ = U W U − + i ~ g U ∇ U − (7)with the unitary operator U having the general form U = exp( i Λ ( x ) · σ ) (8)where Λ ( x ) is an arbitrary vector field. It is an established result [18] that when W is a pure gauge, then it is alwayspossible to find a gauge transformation that sets W to zero: W → W ′ = U W U − + i ~ g U ∇ U − = 0 (9)Thus making it possible to write down a general expression of W in terms of U : W = i ~ g U − ∇ U (10)For such a gauge field, implementing the gauge transformation, Eq.(9), removes W and takes the gauge-covariantHamiltonian, Eq.(5), to that of a free particle. This transformation is at the same time equivalent to implementing aunitary transformation on the Hamiltonian; U H GC ( W ) U − = H GC ( W ′ = 0) = H ′ free (11) H GC and H ′ free are unitarily-equivalent and isospectral, and the corresponding eigenfunctions are related as: ψ ′ n = U ψ n (12)As for our relevant non gauge-covariant Hamiltonian, Eq.(4), a gauge transformation in the sense of Eq.(7), indeedtakes the Hamiltonian to that of the free particle Hamiltonian as well. The issue, however, is that here, in contrastto the gauge-covariant case, this gauge transformation can not be implemented through a unitary transformation on H ,i.e. U H ( W ) U − = p m − g m W · W ≡ H ′ = H ( W ′ = 0) (13)In other words, the unitary transformation in the non gauge-covariant case is not a gauge transformation !. Note,however, that the following relation still holds: H ( W ) = U − (cid:18) p m − g m W · W (cid:19) U ≡ U − H ′ U (14)which implies that the eigenfunctions of H ( W ) and H ′ are related as: ψ n = U − ψ ′ n (15)Note also that although H ′ ≡ U H ( W ) U − is not the free particle Hamiltonian, it is spin-independent since the termquadratic in the gauge field W is proportional to the identity as we have shown before. III. PSH IN A PLANE
We introduce a special class of the pure gauge fields, the Abelian but at the same time 2 × Λ ( x ) appearing in U , Eq.(8), is of the general form Λ ( x ) = λ ( x ) ˆ n (16)with ˆ n a constant unit vector, then Eq.(10) gives us the following special form of W : W = − ~ g ( ˆ n · σ ) ∇ ( λ ( x )2 ) (17)which is evidently Abelian: [ W i , W j ] = 0 (18)The unitary operator U , Eq.(8), that defines - and at the same time gauges away- this Abelian gauge field has aspecial form now: U = exp i { ( λ ( x )2 )( ˆ n · σ ) } (19)The above is just a rotation operator about the fixed direction defined by ˆ n , but with the space-dependent angle λ ( x ).Eqs.(17) and (19)are, respectively, the most general form and the corresponding unitary transformation for aplanar, Abelian pure gauge field. These,with the simple relation between the rotation angle λ ( x ) and the generalform of the gauge field are reported for the first time to the best of our knowledge. Note that it is this specific formof the gauge field that is behind the phenomenon of PSH as we will show below.Here, we first introduce the general algorithm for gauging away any Abelian gauge field of the form given above, andthen apply this algorithm to the case of an electron gas subject to an equal-strength R + D SOI in which the PSH wasdiscovered. To this end, if we read the gauge and unitary transformations, Eq.(7), and Eqs.(13)-(15), for the Abelianpure gauge, Eq.(17), we see that the eigenfunctions ψ of the SOI Hamiltonian H ( W ) are constructed from those ofthe spin-independent Hamiltonian H ′ by just multiplying them by the space-dependent rotation about ˆ n , Eq.(19),(recall that U − = U † = exp − i { ( λ ( x )2 )( ˆ n · σ ) } ). The space-dependence of this rotation is such that it guarantees thatthe spin projection about ˆ n is conserved. If H ′ has propagating solutions, then the spin direction of the particle willprecess about ˆ n as it propagates, with the angle of precession being λ ( x ). It is the appearance of this phenomenon ina free electron gas that gave rise to the name PSH. Note that both the unit vector ˆ n and the precession angle λ ( x )are fixed by the specific form of the gauge field , i.e. by the SOI. In the same manner, for non-propagating solutionsof H ′ , in the corresponding solutions of H ( W ) we will have the spin of the particle rotated about ˆ n with differentangles according to the spacial distribution of the eigenfunctions.We now apply the above formalism to the case when R + D SOI’s with equal strengths are present. Let us recallthat the Hamiltonian for the RSOI is [2], H R = p m + α ~ ( p y σ x − p x σ y ) , (20)and that for the DSOI coupling[3] is H D = p m + β ~ ( p x σ x − p y σ y ) (21)here α and β are the strengths of the SOI couplings and are taken as mere constants. The special case when both theRSOI and DSOI are present and have equal strengths, i.e. α = ± β will be our focus as it is the system where PSHphenomenon appears. The Hamiltonian for this case can be cast in the form: H R + Dα = ± β = p m + α ~ ( σ x ∓ σ y ) ( p y ± p x ) (22)To apply the formalism developed above to this case, we note that this Hamiltonian is of our standard form given byEq.(3) upon identifying the components of W as : − gm W x | α = ± β = ± α ~ ( σ x ∓ σ y ) = ± α ~ ( ˆ n ± · σ ) (23) − gm W y | α = ± β = α ~ ( σ x ∓ σ y ) = α ~ ( ˆ n ± · σ )with the unit vector ˆ n ± characteristic of the R+D SOI (at α = ± β ) given by:ˆ n ± = ˆ i ∓ ˆ j W x , W y ] = 0 and since α is a constant, it is a pure gauge. Putting it in the form givenby Eq.(17)we have W α = ± β = − ~ g ( ˆ n · σ ) (cid:16) mα ~ ˆ j ± mα ~ ˆ i (cid:17) = − ~ g ( ˆ n · σ ) ∇ ( λ ( x )2 ) (25)thus identifying ∇ ( λ ( x )2 ) | α = ± β = (cid:16) mα ~ (cid:17) ( ± ˆ i + ˆ j ) (26)Here, we note that in this specific case, the gauge-symmetry breaking term in the Hamiltonian is just a constant since g m W · W = mα ~ . Therefore, the eigenfunctions ψ n = U − ψ ′ n of the Hamiltonian, Eq.(22), are just those of a spinone-half free particle multiplied by the position-dependent rotation U about the fixed axis ˆ n ± = ˆ i ∓ ˆ j . Now, whatremains is to integrate to find λ and then construct U given in Eq,(19)that removes W . With λ ( x )2 | α = ± β = mα ~ ( y ± x ) (27)we have U R + D = exp i { mα ~ ( y ± x )( ˆ n ± · σ ) } (28)One can readily check that this U indeed takes W given by Eq.(25) to zero. The above equation has been derived inreference [12]. The value added in this derivation is that it is a special case of a general formalism of any linear SOIthat is expressible as a pure gauge field. IV. PSH ON A RING
The Planar Hamiltonian with SOI expressed as a gauge field in polar coordinates reads: H = − ~ m (cid:18) r ∂∂r ( r ∂∂r ) + 1 r ∂ ∂θ (cid:19) + i ~ gm (cid:18) W r ∂∂r + 1 r W θ ∂∂θ (cid:19) (29)The construction of the Hermitian ring Hamiltonian is not a trivial issue. We will follow the approach introduced in[19, 20] (see also [21])to do this. We start from the above Hamiltonian and assume that a strong radial potential V ( r )-not shown in the Hamiltonian- confines it within a narrow layer about a ring of radius a , say. In the limit of a verystrong potential the particle is then pinned to the ring. As was shown in [19, 20] , the Hermitian radial momentumoperator is this case is not simply − i ~ ∂∂r , rather it is the operator p r ≡ − i ~ ( ∂∂r + 12 r ) (30)So, we can express the planar Hamiltonian, Eq.(29), in terms of p r as: H = ( p r ) m − ~ m (cid:18) r ∂ ∂θ (cid:19) − ~ mr − gm W r p r − i ~ gm W r r + i ~ gm (cid:18) r W θ ∂∂θ (cid:19) (31)When the particle is pinned to the surface, the radial degree of freedom is frozen, and so we can drop the momentumoperator p r from the above Hamiltonian. Setting also r = a and substituting (for a constant W and as a result ofthe condition ∇ · W = 0) W r r = − r ∂∂θ we have the Hermitian ring Hamiltonian as: H ring = − ~ ma ∂ ∂θ − ~ ma + i ~ gma W θ ∂∂θ + i ~ g ma ∂W θ ∂θ (32)Note that while any reference to W r dropped from the Hamiltonian along with p r , this component is still there ! it isjust not ”seen” on the ring. In fact, it should always be there to preserve the condition ∇ · W = ∂W r ∂r + W r a + a ∂W θ ∂θ = 0,which should be met everywhere in the plane including on the ring. The above is now the most general Hamiltonianwith a linear and constant SOI for a spin one-half particle on a ring. It can be also put in the form: H ring = − ~ ma (cid:18) a ∂∂θ − ig ~ W θ (cid:19) − ~ ma − g m W θ (33)The above, of course, is the non gauge-covariant Hamiltonian on the ring constructed starting from the non gauge-covariant planar Hamiltonian. − g m W θ is the gauge-symmtry breaking term. For an Abelian W of the form given inEq.(17), we have W r and W θ as: W r = − ~ g ( ˆ n · σ )ˆ r · ∇ ( λ ( x )2 ) = − ~ g ( ˆ n · σ ) ∂∂r λ ( r )2 (34) W θ = − ~ g ( ˆ n · σ )ˆ θ · ∇ ( λ ( x )2 ) = − ~ g ( ˆ n · σ ) 1 a ∂∂θ λ ( r )2 (35)Similarly, the gauge transformation, Eq.(7) reads for the polar components of the above W : W ′ r = W r + i ~ g U ∂∂r U − (36) W ′ θ = W ′ θ + i ~ g U a ∂∂θ U − For U in Eq.(19), which is U in the planar case, the above gauge transformation sets both polar components of W to zero. Applying this transformation to the ring Hamiltonian, Eq.(33), will take it , therefore, to H ring ( W = 0) : H ring ( W = 0) = − ~ ma ∂ ∂θ − ~ ma (37)which is - up to a constant- that of a ”free” particle on a ring. In this case, too, as in the planar case ( see Eq.(13) andthe discussion below it) this gauge transformation is NOT equivalent to a unitary transformation on the Hamiltonian.This is because: U H ring U − = − ~ ma ∂ ∂θ − ~ ma − g m W θ = H ring ( W = 0) (38)Note that even in the case when W is a constant Cartesian vector, W θ need not and generally is not a constant; it isa function of θ . The above, therefore, is the Hamiltonian of a particle on the ring subject to a θ -dependent potential.Regardless of the absence of gauge-covariance, we still have the following transformation holding: H ring = U − (cid:18) H ring ( W = 0) − g m W θ (cid:19) U ≡ U − H ′ ring U (39)So, the eigen functions ψ n of the SOI Hamiltonian on the ring H ring are related to those of H ′ ring defined in the aboveequation as ψ n = U − ψ ′ n with U again the same operator defined in Eq.(19). This means that they are related by aposition-dependent rotation about the fixed axis ˆ n , thus the spin projection along this direction is still conserved. Itis important to note her that we should gauge away W r too along with W θ , even though it is not ”seen” in the ringHamiltonian. This is because the condition ∇ · W = ∇ ∧ E = 0 will be violated otherwise.It is now straightforward to apply the above formalism to the R+D SOI when α = ± β . The gauge field componentsin polar coordinates are now given as: W r = − ~ g ( ˆ n · σ ) ∂∂r λ ( r )2 = − ~ g ( ˆ n ± · σ ) (cid:16) mα ~ (cid:17) (sin θ ± cos θ ) (40) W θ = − ~ g ( ˆ n · σ ) 1 a ∂∂θ λ ( r )2 = − ~ g ( ˆ n ± · σ ) (cid:16) mα ~ (cid:17) (cos θ ∓ sin θ )The ring Hamiltonian, upon substituting the above expressions in Eq.(33) will thus read: H ringR + D = − ~ m (cid:18) a ∂∂θ + imα ~ ( ˆ n ± · σ )(cos θ ∓ sin θ ) (cid:19) − ~ ma − mα ~ (1 ∓ sin 2 θ ) (41)The explicit form of U, Eq.(19), is straightforward to calculate and is given by: U ringR + D = exp i { ( mα ~ )( ˆ n ± · σ ) a (sin θ ± cos θ ) } (42)which is just -as expected- the polar coordinate form of U in the plane found earlier, Eq.(28), evaluated at r = a .This transformation, when applied to the Hamiltonian, Eq.(41), gives : H ′ ringR + D ≡ U − H ringR + D U = − ~ ma ∂ ∂θ − ~ ma − mα ~ (1 ∓ sin 2 θ ) (43)This Hamiltonian was obtained in [13]without using the gauge field formalism, however. The solutions are generallythe Mathieu functions [22]. The eigenfunctions ψ ring of the original SOI ring Hamiltonian,Eq.(41), can be constructedfrom those of the above H ′ ringR + D as : ψ ringR + D = ( U ringR + D ) − ψ ′ ringR + D (44) V. CONCLUSIONS
In this work, we have considered the question of systematically formulating the theory of SOI of a spin one-halfparticle in termsof a pure ( thus can be gauged out) non-Abelian gauge field W . The corresponding Hamiltonian isnot gauge-covariant due to the presence of a term ∼ W · W that breaks the gauge symmetry. As a consequence, wehave shown that the gauge transformation that gauges it out is not equivalent to a unitary transformation of theHamiltonian as is the case with a gauge-covariant Hamiltonian. This latter unitary transformation removed W onlypartially from the Hamiltonian. Nevertheless, the eigenfunctions of the two unitarily-equivalent Hamiltonians arerelated by this unitary transformation that partially removes W ( see Eqs.(11)-(15)).A general form (Eq.(17)) was constructed for a sub-class of these gauge fields that , though 2 × ∇ · W = 0,it was argued. Again, in this case also, the unitary transformation that partially removed the θ -component of thegauge field was shown to be the same as that in the planar case, and it also simultaneously removed the ’unseen”radial component of the gauge field; again to preserve the gauge condition. The general formalism was again appliedto the the case of equal strength R+D SOI, and the Hamiltonian of a spin one-half particle subject to an angularpotential reported in the literature - not using the gauge field approach- was obtained . The Unitary transformationthat related the eigenfunctions of the two Hamiltonians was seen again to be the same as the one in the correspondingplanar case. [1] I. ˇZuti´c, J. Fabian and S. Das Sarma, Rev. Mod. Phys. , 323 (2004).[2] E. I. Rashba, Sov. Phys. Solid State , 1109 (1960).[3] G. Dresselhaus, Phys. Rev. , 580 (1955).[4] J. Fr¨ohlich and U. M. Studer, Rev. Mod. Phys. , 733 (1993).[5] P-Q. Jin, Y-Q. Li and F-C. Zhang, J. Phys. A: Math. Gen. , 7115 (2006).[6] C. A. Dartora and G. G. Cabrera, Phys. Rev. B , 012403 (2008).[7] V.P. Mineev and G. E. Volovik, J.Low Temp. Phys. , 829 (1992).[8] B. Berche and E. Medina, Eur. J. Phys. (2013) 161.[9] B. A. Bernevig, J. Orenstein and S-C. Zhang, Phys. Rev. Lett. , 236601 (2006).[10] J. D. Koralek et. al. , Nature , 610-613 (2009).[11] M. P. Walser, C. Reichl, W. Wegscheider and G. Salis, Nature Physics , 757–762 (2012).[12] S-H. Chen and C-R. Chang, Phys. Rev. B , 045324 (2008).[13] J.S. Sheng and K. Chang, Phys. Rev. B , 235315 (2006).[14] B.Berche et. al., Eur. J. Phys. , 1267 (2010)[15] M.S.Shikakhwa,S. Turgut, N.K.Pak, J. Phys. A: Math. Gen. , 105305 (2012).[16] I. V. Tokatly, Phys. Rev. Lett. , 106601 (2008).[17] E. Medina, A. Lopez and B. Berche, Eur. Phys. Lett. , 47005 (2008).[18] S. Weinberg, The Quantum Theory of Fields, Vol.1, Foundations , Cambridge University Press, Cambridge, 1995; and
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