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Edited by: Manuel Gadella, University of Valladolid, Spain

Reviewed by: Silvestro Fassari, Universitá degli Studi Guglielmo Marconi, Italy; Javier Negro, University of Valladolid, Spain

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We consider singular self-adjoint extensions for the Schrödinger operator of spin-1/2 particle in one dimension. The corresponding boundary conditions at a singular point are obtained. There are boundary conditions with the spin-flip mechanism, i.e., for these point-like interactions the spin operator does not commute with the Hamiltonian. One of these extensions is the analog of zero-range δ-potential. The other one is the analog of so called δ^{(1)}-interaction. We show that in physical terms such contact interactions can be identified as the point-like analogs of Rashba Hamiltonian (spin-momentum coupling) due to material heterogeneity of different types. The dependence of the transmission coefficient of some simple devices on the strength of the Rashba coupling parameter is discussed. Additionally, we show how these boundary conditions can be obtained from the Dirac Hamiltonian in the non-relativistic limit.

Point-like interactions can be described as the singular extensions of the Hamiltonian and are very useful quantum mechanical models because of their analytical tractability [

In non-relativistic limit spin

where

Here ψ_{↑}, ψ_{↓} are the wave functions of corresponding spin “up-” and “down-” states |↑〉, |↓〉. The probability current for Eq. (1) is as following:

where the last term describes the magnetization current (see e.g., [

Bearing in mind the application to the 1-dimensional layered systems with spatial heterogeneity we use the conservation of current 3 to derive the boundary conditions (BCs) for the Hamiltonian 1 which model point-like interactions. We use the results of [

Here symbol _{x} stands for the derivative in the sense of distributions on the space of functions continuous except at the point of singularity where they have bounded values along with their first derivatives [

The parameters _{i} ∈ ℝ determine the values of the discontinuities of the wave function and its first derivative. The boundary conditions (b.c.) corresponding to each contribution in Eq. (4) can be represented in matrix form:

and conserve the current^{1}

of the Hamiltonian

of a spinless particle. Physical classification of all these b.c. on the basis of gauge symmetry breaking was proposed in Kulinskii and Panchenko [

and can be associated with point-like interactions of electrostatic nature, e.g., standard zero-range potential is nothing but the limiting case of the potential field barrier. Another one is given by the BC matrices:

and represents the point-like interactions of the “magnetic” type. The parameters of 4 are related with the physical ones:

where _{0} = 2π_{X3} is obvious because of its interpretation as the localized magnetic flux. The last breaks the homogeneity of the phase of the wave function ψ. Also the scattering matrix of this b.c. has no time reversal symmetry [

The natural question arises as to the consideration of a particle with internal magnetic moment, e.g., a particle with spin

and boundary condition 4 × 4-matrix

Due to the structure of current Eq. (3) for the Hamiltonian 1 we have conservation of all its components:

Note that here we use expanded form of “curl” operator in Eq. (3) with explicit derivatives because we expect the discontinuity in their values. In fact, this the very form follows from the Dirac equation in non relativistic limit and the curl-operator appears after collecting the corresponding terms (see [_{2}-interaction which breaks the homogeneity of dilatation symmetry [_{y} and _{z} are different from zero even if we consider 1-dimensional case, e.g., layered system. The only demand consistent with the hermiticity of the Hamiltonian Eq. (1) is the conservation of current components Eq. (14). In terms of vector Φ the components of the probability current are as following:

where 4 × 4 matrices Σ_{i} are calculated by comparison of expressions Eqs. (14) and (15):

The conservation constraint of total current 15 gives the conditions for

Besides trivial solution for _{X2, 3}-blocks (no spin-flip), simple algebra gives the nontrivial 1-parametric solution of Eq. (18):

with

and b.c. of the form

This defines the spin-flip variant of _{4}-extension. E.g., corresponding scattering matrix for _{r} is as following:

The scattering characteristics related to the scattering matrix Eq. (21) are in

Scattering of |↑〉 - state on _{4} defect.

Another solution of Eq. (18) is

with the b.c. of the form:

It can be considered as the δ-potential (_{1}-extension) augmented with the spin-flip mechanism. From the explicit form of the boundary conditions, e.g.,:

where _{X2} is the block-diagonal matrix of _{2}-extensions. Thus the boundary condition for

Note that _{3}-extension can not be augmented with the spin-flip mechanism since it decouples from _{3} contact interaction does not include spatial inhomogeneity in electric field potential φ. This is quite consistent with the difference between _{2} and _{3} from the point of view of breaking the gauge symmetry [

Using the b.c. obtained above the standard test systems and their transport characteristics can be calculated straightforwardly in order to demonstrate spin-filtering properties. We give just two examples. First is the resonator (see _{1}-resonator. Using such device it is possible to create the resonant (quasibound) states in the area between the wall and the defect (see _{4}-filter. Comparison of _{4} cases shows that the last one is more effective as spin-flip mechanism.

Resonator.

Intensity of reflected spin-↓ state for _{4} resonator (see

Intensity of reflected spin-↓ state for

Amplitude of the wave function |↑〉, |↓〉-components in the resonant region.

Filter.

Transmission _{4} filter intensity for different values of

The zone structure for periodic comb (see

where _{4}-comb the lowest states belong to two parabolic zones with different effective masses at _{X4} < 1:

At

Comb structure.

Zone structure of _{1} - comb. Red and green are for “minus” and “plus” branches in Eq. (26) correspondingly.

Zone structure of _{4} - comb. Red and green are for “minus” and “plus” branches in Eq. (27) correspondingly.

Of course this is the remnant of what happens in standard _{4}-structure (see e.g., [

As is known 3D case with the spherical symmetry can be effectively reduced to one dimensional problem on semi-axis

then the limiting value _{2}(ℝ_{+}). The probability current is as following:

so the results for 1D case can be used. Introducing 2-spinor boundary-value vectors:

where

from Eq. (31) we get:

where

the scalar part (first term) corresponds to standard point-like potential b.c. [

and the states |↑〉, |↓〉 evolve independently. There is also spin-dependent repulsive/attractive version of 35:

which might be interpreted as the point-like potential with internal spin so that the sign of the potential depends on the spin-spin orientation of the particle and the center. The vector part (traceless second term) of Eq. (34) describes polarizational contact interactions with the spin-flip b.c.:

These b.c.'s in general describe how spin of an incident particle (e.g., an electron) interacts with the electrostatic potential localized at the singular point. In the absence of the external magnetic field the only mechanism for acting on spin in such situation is the relativistic spin-momentum coupling which we discuss in the following section.

The spin-flip point interactions introduced above make the spin operator no longer the integral of motion. There are two obvious physical origins for it (a) an external magnetic field with

where _{i},

with

where spinors

and in non relativistic limit transforms into

with

Here

so that the boundary element 4-vector 12 appears. Also we refer to the paper [_{F} at the Fermi level serves as the speed of light.

The expansion of next order generates the spin dependent operator in the Hamiltonian:

It couples the spin with the momentum due to inhomogeneous background of the electric potential φ. In the limiting case of point-like interaction on the axis when ∇φ → 0 on both sides of the singular point this term drops out and should be interchanged with the boundary condition for the boundary vector 12 of the Pauli Hamiltonian 1. The conservation of the corresponding probability density current Eq. (3) provides self-adjointess of the boundary conditions for Eq. (1) in the presence of point-like singularity.

As a result, all extensions _{i},

The main result of the paper is that those extensions of the Schrödinger operator which are physically constructed on the basis of the inhomogeneous distribution of the electric field potential φ(_{2} and _{4} extensions were treated on the common basis of the spatial dependent effective mass. In its turn it is caused by the electrostatic field of the crystalline background. So it is not a surprise that these extensions can be combined through spin-momentum coupling in the Rashba Hamiltonian thus forming the “internal” magnetic field. In contrast to this pure “magnetic” _{3}-extension which is due to the external magnetic field does not couple with other Rashba point-like interactions.

Thus we can state that all point-like interactions δ, δ′-local and δ′-nonlocal (in terms of [

VK and DP conceived of the presented idea. VK proposed physical interpretation for the spin-flip matching conditions. VK encouraged DP to investigate the possibility to derive them based on the Dirac Hamiltonian and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Prof. Vadim Adamyan for clarifying comments and discussions.

^{1}here we put ℏ = 1,