Point-like Rashba interactions as singular self-adjoint extensions of the Schrödinger operator in one dimension
PPoint-like Rashba interactions as singular self-adjoint extensionsof the Schr¨odinger operator in one dimension
V.L. Kulinskii ∗ and D. Yu. Panchenko
1, 2, † Department for Theoretical Physics, Odessa National University,Dvoryanskaya 2, 65026 Odessa, Ukraine Department of Fundamental Sciences, Odessa Military Academy,10 Fontanska Road, Odessa 65009, Ukraine
Abstract
We consider singular self-adjoint extensions for the Schr¨odinger operator of spin-1 / δ -potential. The other one is the analog of so called δ (1) -interaction. We show that inphysical terms such contact interactions can be identified as the point-like analogues of RashbaHamiltonian (spin-momentum coupling) due to material heterogeneity of different types. Thedependence of the transmissivity of some simple devices on the strength of the Rashba couplingparameter is discussed. Additionally, we show how these boundary conditions can be obtained inthe non-relativistic limit of Dirac Hamiltonian. PACS numbers: 03.65.-w, 03.65.Db ∗ [email protected] † [email protected] a r X i v : . [ m a t h - ph ] D ec . INTRODUCTION Point-like interactions can be described as the singular extensions of the Hamiltonian andare very useful quantum mechanical models because of analytically tractability [1–5]. Theyare equivalent to some boundary conditions at the singular points and represent the limitingcases of field inhomogeneities. Therefore it is important to understand the relation betweenparameters of these BC and the specific physical characteristics of inhomogeneities. In mod-ern nanoengineering the spin control is of great interest [6, 7]. Besides the external magneticfield another interaction which could be used for such controlling is the spin-momentumcoupling [8, 9]. The inclusion of magnetic field and other interactions which influence spindynamics is a natural route for searching spin-dependent singular interactions. The inter-actions which influence spin polarization would give new examples of contact interactionswith applications in condensed matter physics and QFT [10].
II. CONTACT INTERACTIONS FOR SPIN 1/2 CASE
In non relativistic limit spin s = 1 / H = (cid:0) ˆ p − qc A (cid:1) m + q ϕ − q (cid:126) m c ˆ σ · (cid:126) H (1)with σ representing the vector of Pauli matrices and (cid:126) H is the external magnetic field with A is its vector potential and ϕ is the scalar potential. This Hamiltonian acts in space of2-component wave functions: where: Ψ = ψ ↑ ψ ↓ , (2)and ψ ↑ , ψ ↓ are the wave functions of corresponding spin “up-“ and “down-“ states |↑(cid:105) , |↓(cid:105) .The probability current for (1) reads: J w = (cid:126) m Im (cid:0) Ψ † ∇ Ψ (cid:1) − qmc A Ψ † Ψ + (cid:126) m rot (cid:0) Ψ † σ Ψ (cid:1) , (3)with the last term describing the magnetization current.2earing in mind the application to the 1-dimensional layered systems with spatial hetero-geneity we use the conservation of current (3) to derive proper boundary conditions (BCs)for free particle with spin s = 1 / L X = − D x ( 1 + X δ ) + i D x (cid:0) X δ − i X δ (1) (cid:1) + X δ + ( X − i X ) δ (1) . (4)Here symbol D x stands for the derivative in the sense of distributions on the space offunctions continuous except at the point of singularity where they have bounded valuesalong with derivatives [12, 13]: δ ( ϕ ) = ϕ (+0) + ϕ ( − , δ (1) ( ϕ ) = − ϕ (cid:48) (+0) + ϕ (cid:48) ( − X i ∈ R determine the values of the discontinuities of the wave function andits first derivative. The boundary conditions (b.c.) corresponding to each contribution inEq. (5) can be represented in matrix form: ψ (0 + 0) ψ (cid:48) (0 + 0) = M X i ψ (0 − ψ (cid:48) (0 − (6)and conserve the current (we put (cid:126) = 1 , c = 1 and m = 1 / j = 2 Im ( ψ ∗ ψ (cid:48) ) (7)of the Hamiltonianfootnotewe put (cid:126) = 1 and m = 1 / H = − d d x (8)of a spinless particle. Physical classification of all these b.c. on the basis of gauge symmetrybreaking was proposed in [14]. They can be divided into tow subsets. The first one is formedby the matrices: M X = X , M X = − X , (9)which can be associated with potential or electrostatic point-like interactions, e.g. standardzero-range potential is nothing but the limiting case of electrostatic field barrier. Other oneis given by the BC matrices: M X = µ
00 1 /µ , M X = e π i Φ (10)3f point-like interactions of magnetic type. Here X = 2 µ − µ + 1 , e π i Φ = 2 + i X − i X (11)where µ = (cid:112) m + /m − is the mass-jump parameter and Φ is the flux fraction modulo π .Magnetic nature of M X is obvious because it is interpreted as the localized magnetic fluxwhich breaks the homogeneity of the phase of the wave function ψ . Also breaking of thetime reversal manifests itself in scattering matrix [14].The natural question arises as to the consideration a particle with internal magneticmoment, e.g. a particle with spin s = 1 /
2. The very straightforward way for derivation ofcorresponding b.c. is the conservation of current Eq. (3). Therefore we introduce 4-vector(bispinor) of the the boundary values at the singular point:Φ ± = ψ ↑ ψ (cid:48)↑ ψ ↓ ψ (cid:48)↓ ± (12)and boundary condition 4 × M :Φ = M Φ − . (13)Due to the structure of current Eq. (3) for the Hamiltonian (1) we have conservation of allits components: J x = i (cid:16) Ψ † ∂ Ψ ∂x − ∂ Ψ † ∂x Ψ (cid:17) J y = − (cid:16) ∂ Ψ † ∂x σ z Ψ + Ψ † σ z ∂ Ψ ∂x (cid:17) (14) J z = ∂ Ψ † ∂x σ y Ψ + Ψ † σ y ∂ Ψ ∂x Note that here we use expanded form of “curl“ operator in Eq. (3) with explicit derivativesbecause we expect the discontinuity in their values. In fact the very this form follows fromthe Dirac equation in non relativistic limit and the curl-operator appears after collecting thecorresponding terms (see [11]). This difference is important in view of the X interactionswhich breaks the homogeneity in dilatation symmetry [15] because of the mass jump [14, 16].In general J y and J z are non zeroth even if we consider 1-dimensional case, e.g. layered4ystem. The only demand consistent with the hermiticity of the Hamiltonian (1) is theconservation of current components (14).In terms of vector Φ the components of the probability current are represented as follow-ing: J i = Φ † Σ i Φ , i = x, y, z (15)where 4 × i are calculated by comparison of expressions Eq. (14) and Eq. (15):Σ x = 1 i Sp Sp , Σ y = − σ x σ x , (16)Σ z = 1 i σ x − σ x and Sp = − , (17)Thus the conservation of total current gives the conditions for M -matrix: M † Σ i M = Σ i , i = x, y, z (18)Besides trivial solution for M -matrix consisting of two M X , -blocks (no spin-flip), simplealgebra gives the nontrivial 1-parametric solution of Eqs. (18): M r = r r , r ∈ R (19)with M r M r = M r + r . and b.c. of the form ψ ↑ ψ (cid:48)↑ ψ ↓ ψ (cid:48)↓ = M r Φ − = ψ ↑ + r ψ (cid:48)↓ ψ (cid:48)↑ ψ ↓ + r ψ (cid:48)↑ ψ (cid:48)↓ − (20)which defines the spin-flip variant of X -extension. E.g. corresponding scattering matrix for5 r is as following: ˆ S r = 1 k r + 4 k r − i k r i k r k r i k r − i k r − i k r i k r k r i k r − i k r k r (21) k T for Spin - ↑ R for - Spin - ↑ T + R for Spin - ↓ FIG. 1. Scattering of |↑(cid:105) - state on r − X defect Another solution of Eq. (18) is˜ M ˜ r = r
00 0 1 0˜ r , ˜ r ∈ R (22)with the b.c. of the form: ψ ↑ ψ (cid:48)↑ ψ ↓ ψ (cid:48)↓ = ˜ M ˜ r Φ − = ψ ↑ ˜ r ψ ↓ + ψ (cid:48)↑ ψ ↓ ˜ r ψ ↑ + ψ (cid:48)↓ − (23)and can be considered as the δ -potential ( X -extension) augmented with the spin-flip mech-anism. From the explicit form of the boundary conditions, e.g.: ψ ↑ ψ (cid:48)↑ ψ ↓ ψ (cid:48)↓ = M r M X Φ − = µ − ψ ↑ + µ r ψ (cid:48)↓ µ ψ (cid:48)↑ µ − ψ ↓ + µ r ψ (cid:48)↑ µ ψ (cid:48)↓ − (24)6here M X is the block-diagonal matrix of X -extensions. Thus the boundary condition for s = 1 / = ˜ M ˜ r M r M X . (25)In contrast to this the X -extension can not be augmented with the spin-flip mechanismsince it trivially decouples from r -coupling: ψ ↑ ψ (cid:48)↑ ψ ↓ ψ (cid:48)↓ = M r M X Φ − = e i π Φ ψ ↑ + r ψ (cid:48)↓ ψ (cid:48)↑ ψ ↓ + r ψ (cid:48)↑ ψ (cid:48)↓ − (26)In accordance with the spin-momentum nature of the r -couplings the physical reason ofsuch factorization is that X contact interaction does not include spatial inhomogeneity inelectric field potential ϕ . This is quite consistent with the difference between X and X from the point of view of breaking the gauge symmetry [14, 17].On this basis the standard test systems and their transport characteristics can be calcu-lated straightforwardly in order to demonstrate spin-filtering properties. We give here justtwo examples here: the resonator (see Fig. 2,3), and the filter (see Fig. 5,6). The intensity inout FIG. 2. Resonator of spin-flip process, generating the spin- ↓ state from incident spin- ↑ state is shown in Fig 3.These results demonstrate that spin-flip mechanism even at small values of r -coupling can7each high probabilities with increasing the energy of incident particle. Of course this di-rectly follows from the boundary conditions (19) and (22) since the effects depend on both r and the momentum. Comparison of ˜ r − X and r − X cases shows that the last one is Reflection coeff for spin - flop of X - Rashba resonatorr = R ↓ k Reflection coeff for spin - flop of X - Rashba resonatorr = R ↓ k FIG. 3. Intensity of reflected spin- ↓ state for r − X resonator (see Fig. 2) at different values of r . Reflection coeff for spin - flop of X - Rashba resonator r ˜ = R ↓ k Reflection coeff for spin - flop of X - Rashba resonator r ˜ = R ↓ k FIG. 4. Intensity of reflected spin- ↓ state for ˜ r − X resonator (see Fig. 2) at different values of ˜ r . more effective as spin-flipping mechanism.The zone structure for r − X periodic comb can be also calculated in standard way.It strongly depends on r . The lowest states belong to two parabolic zones with differenteffective mass at r < E ± ( k ) = (cid:126) k m ± , m ± = 1 ± r (27)At r = 1 one branch of excitations becomes massless E ( k ) = 2 √ k . Of course this is theremnant of what happens in standard X -structure [3]. More intriguing problem here is theinclusion of the correlation effects due to spin statistics and investigation of phases with8 n out FIG. 5. FilterFIG. 6. Transmission r − X filter intensity for different values of r . magnetic (dis)order in dependence on the intensity of point-like interactions. This way ofresearch may be useful for modeling 1-dimensional magnetic systems [18]. III. PHYSICAL ORIGIN OF THE SPIN-FLIP BOUNDARY CONDITIONS
The spin-flip point interactions introduced above make the spin operator no longer theintegral of motion. There are two obvious physical origins for it a) an external magnetic fieldwith x, y -components and b) spin-momentum coupling (Rashba coupling). The explicit k -dependence of the amplitudes of the spin-flip processes indicates that these interactions aredue to spin-momentum coupling. Thus the physical interpretation of interactions represented9y the b.c. matrices M r , ˜ M ˜ r can be given in terms of the Rashba Hamiltonian [8, 9] (seealso [19] and reference therein). Indeed, the Pauli Hamiltonian Eq. (1) as well as the currentdensity Eq. (3) can be derived as the non relativistic limit for the Dirac Hamiltonianˆ H D = α · (ˆ p − A ) + β m + ϕ (28)where α = α i , i = 1 , , β are the Dirac matrices α = σσ , β = I − I (29)with I being 2 × D = ξη (30)where spinors ξ and η represent particle and hole with respect to the Dirac vacuum statesrespectively [11]. The probability density is: J D = Ψ † D α Ψ D (31)and in non relativistic limit transforms into J = ξ ∗ σ η + η ∗ σ ξ (32)with η = 12 m ˆ v ξ (33)Here ˆ v is the velocity operator. In the absence of external electromagnetic field this isequivalent to the following reduction of the bispinor in 1-dimensional caseΨ D → ξξ (cid:48) (34)so that the boundary element 4-vector (12) appears. Also we refer to the paper [20] wheremass jump matching conditions were derived for the Dirac Hamiltonian in a graphen-likematerial where the speed of light interchanged with the Fermi velocity v F .The expansion of next order generates the spin dependent operator in the Hamiltonian:ˆ H SP = λ σ · ( ∇ ϕ × ˆ p ) (35)10t couples the spin with the momentum due to inhomogeneous background of the electricpotential ϕ . In the limiting case of point-like interaction on the axis when ∇ ϕ → X i , i = 1 , , ϕ can be augmented with the spin-flipmechanism. Thus Eq. (25) defines the one-dimensional analog of the Hamiltonian with thepoint-like Rashba spin-momentum interaction [8]. CONCLUSION
The main result of the paper is that those extensions of the Schr¨odinger operator whichare physically constracted on the basis of the inhomogeneous distribution of the electric fieldpotential ϕ ( x ) can be augmented with the spin-flip mechanism. Note that in Eq. (24) both r -coupling and µ -parameter determine the spin-flip mechanism. This is in coherence withthe results of [17] where X and X extensions were treated on the common basis of thespatial dependent effective mass. In its turn it is caused by the electrostatic field of thecrystalline background. So it is not surprise that these extensions can be combined throughspin-momentum coupling in the Rashba Hamiltonian thus forming the “internal“ magneticfield. In contrast to this pure “magnetic“ X -extension which is due to the external magneticfield does not couple with the Rashba point-like interactions.Thus we can state that one-dimensional analog of the Rashba Hamiltonian is obtained.It is interesting to check this result independently using the Kurasov’s distribution theorytechnique [12] modified correspondingly for spin 1 / EFERENCES [1] Demkov Y N and Ostrovsky V 1988
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