Solvable Two-Dimensional Dirac Equation with Matrix Potential: Graphene in External Electromagnetic Field

Abstract

It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of a massless two-dimensional Dirac equation which includes both Hermitian off-diagonal matrix and scalar potentials. Up to now, such two-component wave functions were calculated for different forms of external potentials, though as a rule depending on only one spatial variable. Here, we shall find analytically the solutions for a wide class of combinations of matrix and scalar external potentials which physically correspond to applied mutually orthogonal magnetic and longitudinal electrostatic fields, both depending really on two spatial variables. The main tool for this progress is provided by supersymmetrical (SUSY) intertwining relations, specifically, by their most general—asymmetrical—form proposed recently by the authors. This SUSY-like method is applied in two steps, similar to the second order factorizable (reducible) SUSY transformations in ordinary quantum mechanics.

1. Introduction

The extensive study of two-dimensional massless Dirac equations in the presence of external electromagnetic fields [1,2,3,4,5,6,7] is due to its connection with the properties of electron carriers in graphene and graphene-like materials [8,9,10,11]. The actual task is to find analytically the normalizable solutions of such a Dirac equation where two components of the “spinor” wave function $\Psi \left(\overrightarrow{x}\right)\equiv ({\Psi }_A\left(\overrightarrow{x}\right),{\Psi }_B\left(\overrightarrow{x}\right))$ correspond to two sublattices of graphene. The potentials in the equation have different origins: the off-diagonal matrix term is provided by the electromagnetic vector-potential $\overrightarrow{A}=(A_1(x_1,x_2),A_2(x_1,x_2),0)$ in the “long derivatives” and leads to the magnetic field $\overrightarrow{B}=\overrightarrow{\nabla }\times \overrightarrow{A}$ along the z-axis, while the scalar potential $A_0(x_1,x_2)$ describes the interaction with electrostatic or some other scalar field:

$$[{\sigma }_1(-i{\partial }_1-A_1\left(\overrightarrow{x}\right))+{\sigma }_2(-i{\partial }_2-A_2\left(\overrightarrow{x}\right))+A_0\left(\overrightarrow{x}\right)]\Psi \left(\overrightarrow{x}\right)=0,$$

(1)

where two-dimensional $\overrightarrow{x}\equiv (x_1,x_2),$ the derivatives ${\partial }_i\equiv \frac{\partial }{\partial x_i},$ and the charge is taken as $e=1.$

It is clear that the case of a pure magnetic field in a massless two-dimensional Dirac equation is explicitly solvable. Indeed, when multiplying (1) without the term $A_0$ by ${\sigma }_1$, we obtain a pair of decoupled first order equations which can be easily solved. The presence in (1) of a term proportional to the unity matrix ${\sigma }_0$ prevents this decoupling (an analogous problem appears in the presence of the mass term proportional to ${\sigma }_3$). Different methods have been used [12,13,14,15,16,17,18,19,20,21,22,23] to study such two-dimensional Dirac equations, mainly with strong restrictions on the conditions of the problem. There are problems with only electrostatic or only magnetic fields, as well as the problems with different specific one-dimensional ansatzes for external fields depending on a variable $x_1$ or radial variable r.

The supersymmetrical method inherent in Schrödinger Quantum Mechanics [24,25,26] has become one of the most effective tools in the discussed problems [27,28,29,30,31,32,33]. As a rule, the main ingredient of SUSY Quantum Mechanics, so-called SUSY intertwining relations, has been explored in different forms. In the present paper, a class of external electromagnetic fields is chosen with both magnetic (matrix) and electrostatic (scalar) terms depending effectively on both spatial coordinates. Such progress is possible due to using a particular form of this approach called asymmetric intertwining relations [34,35,36]. To date, this technique has been explored to study the massless two-dimensional Dirac equation with scalar potential as well as the Fokker–Planck equation [37]. More specifically, the procedure includes two steps (see Section 2). First, asymmetric intertwining relations provide SUSY diagonalization of the potential in Equation (1) with both the electromagnetic and electrostatic two-dimensional terms (Section 3). In the second stage, an additional asymmetric intertwining connects the Dirac operator with diagonal potential to its partner, for which the potential is similarly diagonal but with constant elements (Section 4). The solutions of such a Dirac equation can be found analytically; solutions to the initial problem are built by applying intertwining operators of both steps (Section 5). Actually, the whole procedure realizes the asymmetric form of factorizable SUSY intertwining of the second order. Such SUSY intertwinings are known in their standard (symmetric) form [26] in the context of SUSY Quantum Mechanics with Schrödinger operator.

2. Asymmetric Intertwining for Two-Dimensional Dirac Equation in an Electromagnetic Field

We start with the asymmetric intertwining relations

$$D_1{N}_1=N_2{D}_2$$

(2)

for a pair of two-dimensional massless Dirac operators of the form (1) rewritten as general operators with Hermitian matrix potentials

$$D_{1,2}\equiv -i{\sigma }_k{\partial }_k+V_{1,2}\left(\overrightarrow{x}\right);k=1,2;V_i\left(\overrightarrow{x}\right)=\left(\begin{array}{cc}{v}_{11}^{\left(i\right)}\left(\overrightarrow{x}\right)& v_{12}^{\left(i\right)}\left(\overrightarrow{x}\right)\\ v_{21}^{\left(i\right)}\left(\overrightarrow{x}\right)& v_{22}^{\left(i\right)}\left(\overrightarrow{x}\right)\end{array}\right);i=1,2.$$

(3)

Two different intertwining operators have the general matrix form

$$N_1=A_k{\partial }_k+A\left(\overrightarrow{x}\right);N_2=B_k{\partial }_k+B\left(\overrightarrow{x}\right)$$

(4)

with constant matrices $A_k,B_k$ and two $\overrightarrow{x}-$dependent matrices $A\left(\overrightarrow{x}\right),B\left(\overrightarrow{x}\right)$:

$$A\left(\overrightarrow{x}\right)=\left(\begin{array}{cc}{a}_{11}\left(\overrightarrow{x}\right)& a_{12}\left(\overrightarrow{x}\right)\\ a_{21}\left(\overrightarrow{x}\right)& a_{22}\left(\overrightarrow{x}\right)\end{array}\right);B\left(\overrightarrow{x}\right)=\left(\begin{array}{cc}{b}_{11}\left(\overrightarrow{x}\right)& b_{12}\left(\overrightarrow{x}\right)\\ b_{21}\left(\overrightarrow{x}\right)& b_{22}\left(\overrightarrow{x}\right)\end{array}\right).$$

(5)

Expanding Equation (2) in powers of derivatives, we obtain

$$\begin{array}{ccc}& & {\sigma }_k{A}_n-B_k{\sigma }_n+{\sigma }_n{A}_k-B_n{\sigma }_k=0,k,n=1,2,\hfill \end{array}$$

(6)

$$\begin{array}{ccc}& & i{\sigma }_{k}A\left(\overrightarrow{x}\right)-V_1\left(\overrightarrow{x}\right)A_k=-B_k{V}_2\left(\overrightarrow{x}\right)+iB\left(\overrightarrow{x}\right){\sigma }_k,k=1,2,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}& & i{\sigma }_k\left({\partial }_{k}A\left(\overrightarrow{x}\right)\right)-V_1\left(\overrightarrow{x}\right)A\left(\overrightarrow{x}\right)=-B_k\left({\partial }_k{V}_2\left(\overrightarrow{x}\right)\right)-B\left(\overrightarrow{x}\right)V_2\left(\overrightarrow{x}\right),k=1,2.\hfill \end{array}$$

(8)

Equation (6) provide the following form for the constant coefficient matrices $A_k,B_k$:

$$A_1=\left(\begin{array}{cc}a& b\\ d& c\end{array}\right);A_2=\left(\begin{array}{cc}n& -ib\\ id& p\end{array}\right);B_1=\left(\begin{array}{cc}c& d\\ b& a\end{array}\right);B_2=\left(\begin{array}{cc}p& -id\\ ib& n\end{array}\right)$$

(9)

with constant values $a,b,c,d,n,p$. Equation (7) provides the system of eight linear equations for the matrix elements of $V_{1,2}\left(\overrightarrow{x}\right),A\left(\overrightarrow{x}\right),B\left(\overrightarrow{x}\right)$:

$$\begin{array}{ccc}& & (p+ic)(v_{12}^{\left(1\right)}-v_{12}^{\left(2\right)})=0;\hfill \end{array}$$

(10)

$$\begin{array}{ccc}& & (n-ia)(v_{21}^{\left(1\right)}-v_{21}^{\left(2\right)})=0;\hfill \end{array}$$

(11)

$$\begin{array}{ccc}& & (n+ia)v_{11}^{\left(1\right)}+2id{v}_{12}^{\left(1\right)}-(p+ic)v_{11}^{\left(2\right)}=2b_{12};\hfill \end{array}$$

(12)

$$\begin{array}{ccc}& & (n+ia)(v_{21}^{\left(1\right)}-v_{21}^{\left(2\right)})+2id{v}_{22}^{\left(1\right)}-2ib{v}_{11}^{\left(2\right)}=-2a_{11}+2b_{22};\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & (p+ic)v_{22}^{\left(1\right)}-2ib{v}_{12}^{\left(2\right)}-(n+ia)v_{22}^{\left(2\right)}=-2a_{12};\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & (n-ia)v_{11}^{\left(1\right)}+2id{v}_{21}^{\left(2\right)}-(p-ic)v_{11}^{\left(2\right)}=2a_{21};\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & (p-ic)(v_{12}^{\left(1\right)}-v_{12}^{\left(2\right)})+2id{v}_{22}^{\left(2\right)}-2ib{v}_{11}^{\left(1\right)}=-2b_{11}+2a_{22};\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & (p-ic)v_{22}^{\left(1\right)}-(n-ia)v_{22}^{\left(2\right)}-2ib{v}_{21}^{\left(1\right)}=-2b_{21}.\hfill \end{array}$$

(17)

The first two equations of the system indicate the need to highlight four different possibilities for solutions of (10)–(17), bearing the Hermiticity of both potentials in mind:

$$\begin{array}{ccc}\hfill \left(\mathrm{I}\right)& & p+ic=n-ia=0;\hfill \\ \hfill \left(\mathrm{II}\right)& & v_{12}^{\left(1\right)}=v_{12}^{\left(2\right)};v_{21}^{\left(1\right)}=v_{21}^{\left(2\right)};(p+ic)(n-ia)\ne 0\hfill \\ \hfill \left(\mathrm{III}\right)& & v_{12}^{\left(1\right)}=v_{12}^{\left(2\right)};v_{21}^{\left(1\right)}=v_{21}^{\left(2\right)};p+ic=0;\hfill \\ \hfill \left(\mathrm{IV}\right)& & v_{12}^{\left(1\right)}=v_{12}^{\left(2\right)};v_{21}^{\left(1\right)}=v_{21}^{\left(2\right)};n-ia=0.\hfill \end{array}$$

Thus, Equations (6) and (7) are reduced to the system of Equations (12) and (17) in the four possible variants above. We have not yet considered the matrix differential Equation (8). It is convenient to solve it in an indirect form using its combination with the derivative ${\partial }_k$ of Equation (7):

$$V_1\left(\overrightarrow{x}\right)A\left(\overrightarrow{x}\right)-B\left(\overrightarrow{x}\right)V_2\left(\overrightarrow{x}\right)=\left({\partial }_k{V}_1\left(\overrightarrow{x}\right)\right)A_k+i\left({\partial }_{k}B\left(\overrightarrow{x}\right)\right){\sigma }_k.$$

(18)

Below, all four variants are used to obtain the general solution of the initial intertwining relations (2).

3. SUSY Diagonalization by Means of Intertwining

Substitution of $p+ic=n-ia=0,$ i.e., of Variant I above, into other Equations (12)–(17) together with Equation (18) is quite long, but straightforward. These calculations lead to a system of four nonlinear equations, two of them being differential:

$$\begin{array}{ccc}\hfill v_{11}^{\left(1\right)}(i{a}_{11}+b{v}_{11}^{\left(2\right)}+a{v}_{21}^{\left(2\right)})& =& v_{11}^{\left(2\right)}(i{a}_{22}+c{v}_{12}^{\left(2\right)}+d{v}_{22}^{\left(2\right)});\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill v_{22}^{\left(1\right)}(i{a}_{22}+c{v}_{12}^{\left(2\right)}+d{v}_{22}^{\left(2\right)})& =& v_{22}^{\left(2\right)}(i{a}_{11}+b{v}_{11}^{\left(2\right)}+a{v}_{21}^{\left(2\right)});\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill i(v_{12}^{\left(2\right)}-v_{12}^{\left(1\right)})(i{a}_{22}+c{v}_{12}^{\left(2\right)}+d{v}_{22}^{\left(2\right)})& =& 2{\partial }_z(i{a}_{22}+c{v}_{12}^{\left(2\right)}+d{v}_{22}^{\left(2\right)});\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill i(v_{21}^{\left(2\right)}-v_{21}^{\left(1\right)})(i{a}_{11}+b{v}_{11}^{\left(2\right)}+a{v}_{21}^{\left(2\right)})& =& 2{\partial }_{\overline{z}}(i{a}_{11}+b{v}_{11}^{\left(2\right)}+a{v}_{21}^{\left(2\right)}).\hfill \end{array}$$

(22)

Here and below, it is convenient to use the space arguments $\overrightarrow{x}$ of the functions in the form of complex variables $z=x_1+i{x}_2;\overline{z}=x_1-i{x}_2$ and corresponding derivatives $\partial \equiv \frac{1}{2}({\partial }_1-i{\partial }_2);\overline{\partial }\equiv \frac{1}{2}({\partial }_1+i{\partial }_2)$. In particular, the system (19)–(22) takes the compact form:

$$\begin{array}{ccc}\hfill v_{11}^{\left(1\right)}(z,\overline{z})f_1(z,\overline{z})& =& v_{11}^{\left(2\right)}(z,\overline{z})f_2(z,\overline{z});\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill v_{22}^{\left(1\right)}(z,\overline{z})f_2(z,\overline{z})& =& v_{22}^{\left(2\right)}(z,\overline{z})f_1(z,\overline{z});\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill v_{12}^{\left(1\right)}(z,\overline{z})& =& v_{12}^{\left(2\right)}(z,\overline{z})+2i\partial ln\left(f_2(z,\overline{z})\right);\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill v_{21}^{\left(1\right)}(z,\overline{z})& =& v_{21}^{\left(2\right)}(z,\overline{z})+2i\overline{\partial }ln\left(f_1(z,\overline{z})\right),\hfill \end{array}$$

(26)

where two combinations are introduced:

$$\begin{array}{c}\hfill f_1(z,\overline{z})\equiv i{a}_{11}(z,\overline{z})+b{v}_{11}^{\left(2\right)}(z,\overline{z})+a{v}_{21}^{\left(2\right)}(z,\overline{z});\end{array}$$

(27)

$$\begin{array}{c}\hfill f_2(z,\overline{z})\equiv i{a}_{22}(z,\overline{z})+c{v}_{12}^{\left(2\right)}(z,\overline{z})+d{v}_{22}^{\left(2\right)}(z,\overline{z}).\end{array}$$

(28)

One may notice the apparent paradox contained in the system (19)–(22). All these equations are identically fulfilled if

$$i{a}_{11}+b{v}_{11}^{\left(2\right)}+a{v}_{21}^{\left(2\right)}=i{a}_{22}+c{v}_{12}^{\left(2\right)}+d{v}_{22}^{\left(2\right)}=0$$

for arbitrary potentials $V_1,V_2$. The explanation is rather simple; in this case, the Dirac operators $D_1,D_2$ are proportional to the intertwining operators $D_2=C{N}_1,D_1=N_{2}C$ up to some constant matrix C. Therefore, the intertwining relation (2) becomes a trivial identity in such a case.

In the present context, the typical approach [24,25,26,34,36] to using SUSY intertwining relations can be formulated as follows. Let us choose one of the Dirac operators, $D_2,$ such that the corresponding Dirac equation is rather simple and the problem of its analytical solution is more easy. Then, the solutions of the partner Dirac equation with operator $D_1$ are found through the action of the intertwining operator ${\Psi }^{\left(1\right)}=N_1{\Psi }^{\left(2\right)}$ (in turn, the operator $N_2^{†}$ transforms the spinor ${\Psi }^{\left(1\right)}$ into ${\Psi }^{\left(2\right)}$). As the first step in this approach, we choose the potential $V_2\left(\overrightarrow{x}\right)$ as a diagonal matrix:

$$V_2\left(\overrightarrow{x}\right)=diag(v_{11}^{\left(2\right)}(z,\overline{z}),v_{22}^{\left(2\right)}(z,\overline{z}))\equiv diag(v_1(z,\overline{z}),v_2(z,\overline{z}))$$

(29)

with real diagonal matrix elements. Then, due to Equations (23) and (24), the fraction $f_2(z,\overline{z})/f_1(z,\overline{z})$ is a real function. Futhermore, Equations (25) and (26) and the Hermiticity of $V_1\left(\overrightarrow{x}\right)$, i.e., $v_{12}^{\left(1\right)}(z,\overline{z})=v_{21}^{\left(1\right)★}(z,\overline{z})$, lead to the following restriction:

$$f_1(z,\overline{z})f_2^{★}(z,\overline{z})=c$$

(30)

with an arbitrary real constant c. Summarizing these results, the potential $V_1$ is expressed in terms of the components (29), function $f_2(z,\overline{z})$, and real constant c:

$$V_1\left(\overrightarrow{x}\right)=\left(\begin{array}{cc}\frac{|f_2{|}^2}{c}{v}_1& 2i\partial ln\left(f_2\right)\\ -2i\overline{\partial }ln\left(f_2^{★}\right)& \frac{c}{|f_2{|}^2}{v}_2\end{array}\right)=\left(\begin{array}{cc}\pm f^2{v}_1& 2i(\partial ln(f)+i\partial \phi )\\ -2i(\overline{\partial }ln\left(f\right)-i\overline{\partial }\phi )& \pm f^{-2}{v}_2\end{array}\right),$$

(31)

where the function $f_2$ is parameterized as

$$f_2(z,\overline{z})\equiv \sqrt{\pm c}f(z,\overline{z})e^{i\phi (z,\overline{z})}.$$

(32)

In the above, the sign ± corresponds to cases $c>0,c<0$, respectively, and $f(z,\overline{z})$ is a positive function connecting elements of the initial diagonal potential $V_2$ as

$$v_2(z,\overline{z})=f^4(z,\overline{z})v_1(z,\overline{z}).$$

(33)

For the physical system with matrix potential $V_1$ describing a “spin $1/2$ particle” in the external electromagnetic field according to Equation (1), the diagonal elements of $V_1$ define the electrostatic potential

$$A_0\left(\overrightarrow{x}\right)=v_{11}^{\left(1\right)}=v_{22}^{\left(1\right)}=\pm {\left(v_1{v}_2\right)}^{1/2}=\pm f^2\left(\overrightarrow{x}\right)v_1\left(\overrightarrow{x}\right),$$

(34)

while the off-diagonal terms define the magnetic field that is orthogonal to the plane $x_1,x_2$:

$$B_3\left(\overrightarrow{x}\right)=\overrightarrow{\nabla }\times \overrightarrow{A}\left(\overrightarrow{x}\right)=△ln\left(f\left(\overrightarrow{x}\right)\right).$$

(35)

The solutions of Dirac equations with such external fields can be obtained from the two-component solutions ${\Psi }^{\left(2\right)}\left(\overrightarrow{x}\right)$ of Dirac equations with diagonal potential $V_2$ by action of the intertwining operator N; however, the diagonal form of the potential $V_2$ does not yet provide solvability of the corresponding Dirac problem analytically. Thus far, supersymmetric intertwining relations (2) have connected an initial Dirac problem with operator $D_1$ to a Dirac problem with a potential $V_2$ which is diagonal, and as such has a chance of being solvable. By analogy with SUSY separation of variables, this part of the procedure can be called SUSY-diagonalization of the two-dimensional Dirac problem with matrix potential [38].

4. From Diagonal Potential to Constant Potential by Means of Intertwining

Let us now consider variant II with $(p+ic)(n-ia)\ne 0$ for the solution of the system (10)–(17) in the context of the second step of our procedure. Specifically, we consider intertwining relations between two Dirac operators, both with diagonal potential:

$$U_1\left(\overrightarrow{x}\right)=\left(\begin{array}{cc}{v}_1\left(\overrightarrow{x}\right)& 0\\ 0& v_2\left(\overrightarrow{x}\right)\end{array}\right);U_2\left(\overrightarrow{x}\right)=\left(\begin{array}{cc}{m}_1& 0\\ 0& m_2\end{array}\right),$$

(36)

with constant elements $m_1,m_2.$ Here, the potential $U_1\left(\overrightarrow{x}\right)$ is identified with the potential $V_2\left(\overrightarrow{x}\right)$ from the previous step, while the constant partner potential $U_2$ provides solvability of the problem. The intertwining operators $N_1,N_2$ have the same general form (4), and the explicit expressions for matrices $A_k,B_k,A\left(\overrightarrow{x}\right),B\left(\overrightarrow{x}\right)$ are found below, obtained by analytical solutions of the system of equations in Section 2.

We shall consider the system of Equations (10)–(17) sequentially. Equations (10) and (11) are fulfilled automatically. Equations (12), (14), (15) and (17) allow off-diagonal elements of $A\left(\overrightarrow{x}\right),B\left(\overrightarrow{x}\right)$ at (5) to be expressed in terms of $v_1\left(\overrightarrow{x}\right),v_2\left(\overrightarrow{x}\right)$:

$$\begin{array}{ccc}\hfill a_{12}\left(\overrightarrow{x}\right)& =& -\frac{1}{2}(p+ic)v_2\left(\overrightarrow{x}\right)+\frac{1}{2}{m}_2(n+ia);\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill a_{21}\left(\overrightarrow{x}\right)& =& \frac{1}{2}(n-ia)v_2\left(\overrightarrow{x}\right)-\frac{1}{2}{m}_1(p-ic);\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill b_{12}\left(\overrightarrow{x}\right)& =& \frac{1}{2}(n+ia)v_1\left(\overrightarrow{x}\right)-\frac{1}{2}{m}_1(p+ic);\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill a_{12}\left(\overrightarrow{x}\right)& =& -\frac{1}{2}(p-ic)v_2\left(\overrightarrow{x}\right)+\frac{1}{2}{m}_2(n-ia).\hfill \end{array}$$

(40)

Equations (13) and (16) for diagonal elements of $A\left(\overrightarrow{x}\right),B\left(\overrightarrow{x}\right)$ can be written as

$$\begin{array}{c}\hfill b_{11}\left(\overrightarrow{x}\right)=a_{22}\left(\overrightarrow{x}\right)+ib{v}_1\left(\overrightarrow{x}\right)-id{m}_2;\end{array}$$

(41)

$$\begin{array}{c}\hfill b_{22}\left(\overrightarrow{x}\right)=a_{11}\left(\overrightarrow{x}\right)+id{v}_2\left(\overrightarrow{x}\right)-ib{m}_1.\end{array}$$

(42)

Equation (8) in its initial form is convenient to write now in complex coordinates $z\overline{z}:$

$$\begin{array}{ccc}& & 2i\left[\left(\begin{array}{cc}0& \partial \\ \overline{\partial }& 0\end{array}\right)-\left(\begin{array}{cc}{v}_1(z,\overline{z})& 0\\ 0& v_2(z,\overline{z})\end{array}\right)\right]\left(\begin{array}{cc}{a}_{11}(z,\overline{z})& a_{12}(z,\overline{z})\\ a_{21}(z,\overline{z})& a_{22}(z,\overline{z})\end{array}\right)=\hfill \\ & & -\left(\begin{array}{cc}{b}_{11}(z,\overline{z})& b_{12}(z,\overline{z})\\ b_{21}(z,\overline{z})& b_{22}(z,\overline{z})\end{array}\right)\left(\begin{array}{cc}{m}_1& 0\\ 0& m_2\end{array}\right),\hfill \end{array}$$

(43)

where $\partial \equiv {\partial }_z=\frac{1}{2}({\partial }_1-i{\partial }_2),\overline{\partial }\equiv {\partial }_{\overline{z}}=\frac{1}{2}({\partial }_1+i{\partial }_2).$ In components, the matrix Equation (43) is equivalent to the system of linear first-order differential equations:

$$\begin{array}{ccc}\hfill 2i(\partial a_{21}(z,\overline{z}))-v_1(z,\overline{z})a_{11}(z,\overline{z})& =& -m_1{b}_{11}(z,\overline{z});\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill 2i\left(\overline{\partial }{a}_{12}(z,\overline{z})\right)-v_2(z,\overline{z})a_{22}(z,\overline{z})& =& -m_2{b}_{22}(z,\overline{z});\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill 2i(\partial a_{22}(z,\overline{z}))-v_1(z,\overline{z})a_{12}(z,\overline{z})& =& -m_2{b}_{12}(z,\overline{z});\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill 2i\left(\overline{\partial }{a}_{11}(z,\overline{z})\right)-v_2(z,\overline{z})a_{21}(z,\overline{z})& =& -m_1{b}_{21}(z,\overline{z}).\hfill \end{array}$$

(47)

Let us define for convenience

$$a_{11}(z,\overline{z})-i{m}_{1}b\equiv g_1(z,\overline{z});a_{22}(z,\overline{z})-i{m}_{2}d\equiv g_2(z,\overline{z}).$$

(48)

Then, after substitution of Equations (37)–(40) and Equations (41) and (42), the system (44)–(47) takes the form

$$\begin{array}{ccc}& & i(n-ia)(\partial v_1(z,\overline{z}))=v_1(z,\overline{z})g_1(z,\overline{z})-m_1{g}_2(z,\overline{z});\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & i(p+ic)\left(\overline{\partial }{v}_2(z,\overline{z})\right)=-v_2(z,\overline{z})g_2(z,\overline{z})+m_2{g}_1(z,\overline{z});\hfill \end{array}$$

(50)

$$\begin{array}{ccc}& & 4i\left(\overline{\partial }{g}_1(z,\overline{z})\right)=(n-ia)(v_1(z,\overline{z})v_2(z,\overline{z})-m_1{m}_2);\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& & 4i(\partial g_2(z,\overline{z}))=-(p+ic)(v_1(z,\overline{z})v_2(z,\overline{z})-m_1{m}_2),\hfill \end{array}$$

(52)

where the form of last two equations mean that functions $g_1(z,\overline{z}),g_2(z,\overline{z})$ can be expressed in terms of one complex function g:

$$g_1(z,\overline{z})=-\frac{1}{p+ic}(\partial g(z,\overline{z}));g_2(z,\overline{z})=\frac{1}{n-ia}\left(\overline{\partial }g(z,\overline{z})\right),$$

(53)

which satisfies the second order equation:

$$4i(\partial \overline{\partial }g(z,\overline{z}))=-(n-ia)(p+ic)(v_1(z,\overline{z})v_2(z,\overline{z})-m_1{m}_2)$$

(54)

and the first two Equations (49) and (50) become

$$\begin{array}{ccc}& & i(n-ia)(\partial v_1(z,\overline{z}))=-\frac{1}{p+ic}{v}_1(z,\overline{z})(\partial g(z,\overline{z}))-\frac{1}{n-ia}{m}_1\left(\overline{\partial }g(z,\overline{z})\right),\hfill \end{array}$$

(55)

$$\begin{array}{ccc}& & i(p+ic)\left(\overline{\partial }{v}_2(z,\overline{z})\right)=-\frac{1}{n-ia}{v}_2(z,\overline{z})\left(\overline{\partial }g(z,\overline{z})\right)-\frac{1}{p+ic}{m}_2(\partial g(z,\overline{z})).\hfill \end{array}$$

(56)

Thus,

$$\begin{array}{ccc}& & \Omega (\partial v_1(z,\overline{z}))+v_1(z,\overline{z})(\partial g(z,\overline{z}))=-L_1\left(\overline{\partial }g(z,\overline{z})\right);\hfill \end{array}$$

(57)

$$\begin{array}{ccc}& & \Omega \left(\overline{\partial }{v}_2(z,\overline{z})\right)+v_2(z,\overline{z})\left(\overline{\partial }g(z,\overline{z})\right)=-L_2(\partial g(z,\overline{z}));\hfill \\ & & \Omega \equiv i(n-ia)(p+ic);L_1\equiv \frac{m_1(p+ic)}{n-ia};L_2\equiv \frac{m_2(n-ia)}{p+ic}.\hfill \end{array}$$

(58)

Below, we shall solve this system of equations by separately considering different options for the choice of constant parameters.

4.1. Case A: The Parameters Are Real

Let us study the case with real function $g(z,\overline{z})$ and real values of parameters $\Omega ,L_1,L_2$. Taking into account the reality of $v_1(z,\overline{z})$ and $v_2(z,\overline{z})$, it is useful to come back to the Cartesian coordinates $x_1,x_2$ in Equations (57) and (58). Separately, both the real and imaginary parts of (57) are integrated explicitly with two “constants of integration” $s_1\left(x_1\right),s_2\left(x_2\right)$, which are arbitrary real functions of their arguments:

$$\begin{array}{ccc}\hfill exp(g\left(\overrightarrow{x}\right)/\Omega )& =& L_1^{-1}(s_2\left(x_2\right)-s_1\left(x_1\right));\hfill \end{array}$$

(59)

$$\begin{array}{ccc}\hfill v_1\left(\overrightarrow{x}\right)& =& L_1\frac{s_2\left(x_2\right)+s_1\left(x_1\right)}{s_2\left(x_2\right)-s_1\left(x_1\right)}.\hfill \end{array}$$

(60)

Analogously, Equation (58) can be integrated with a similar result:

$$\begin{array}{ccc}\hfill exp(g\left(\overrightarrow{x}\right)/\Omega )& =& L_2^{-1}({\tilde{s}}_2\left(x_2\right)-{\tilde{s}}_1\left(x_1\right));\hfill \end{array}$$

(61)

$$\begin{array}{ccc}\hfill v_2\left(\overrightarrow{x}\right)& =& L_2\frac{{\tilde{s}}_2\left(x_2\right)+{\tilde{s}}_1\left(x_1\right)}{{\tilde{s}}_2\left(x_2\right)-{\tilde{s}}_1\left(x_1\right)}\hfill \end{array}$$

(62)

and arbitrary real ${\tilde{s}}_1\left(x_1\right),{\tilde{s}}_2\left(x_2\right).$ Together, Equations (59) and (61) allow us to connect ${\tilde{s}}_1\left(x_1\right),{\tilde{s}}_2\left(x_2\right)$ with their analogues:

$${\tilde{s}}_1\left(x_1\right)=L_1^{-1}{L}_2(s_1\left(x_1\right)+\delta );{\tilde{s}}_2\left(x_2\right)=L_1^{-1}{L}_2(s_2\left(x_2\right)+\delta ),$$

(63)

with an arbitrary real constant $\delta$. These relations have to be substituted into expression (62).

Using these connections in the second-order differential Equation (54) for the function $g\left(\overrightarrow{x}\right)$ and differentiating it by ${\partial }_1{\partial }_2$, we obtain the following simple third-order equation with separable variables:

$$\frac{s_1^{‴}\left(x_1\right)}{s_1^{\prime }\left(x_1\right)}+\frac{s_2^{‴}\left(x_2\right)}{s_2^{\prime }\left(x_2\right)}=-4L_1{L}_2=-4m_1{m}_2.$$

(64)

After separation ofthe variables in (64) and integration of one-dimensional equations, we have

$$s_1^{″}\left(x_1\right)={\lambda }_1^2{s}_1\left(x_1\right)+{\omega }_1{\lambda }_1^2;s_2^{″}\left(x_2\right)={\lambda }_2^2{s}_2\left(x_2\right)+{\omega }_2{\lambda }_2^2,$$

(65)

where ${\omega }_1,{\omega }_2$ are integration constants and ${\lambda }_1,{\lambda }_2$ are arbitrary constants which satisfy the following relation:

$${\lambda }_1^2+{\lambda }_2^2=-4m_1{m}_2.$$

The solutions of (65) are known:

$$\begin{array}{ccc}\hfill s_1\left(x_1\right)& =& \frac{1}{2}({\sigma }_1{e}^{{\lambda }_1{x}_1}+{\delta }_1{e}^{-{\lambda }_1{x}_1})-{\omega }_1;\hfill \\ \hfill s_2\left(x_2\right)& =& \frac{1}{2}({\sigma }_2{e}^{{\lambda }_2{x}_2}+{\delta }_2{e}^{-{\lambda }_2{x}_2})-{\omega }_2.\hfill \end{array}$$

Because we used derivatives of Equation (54), it is necessary to check the results. Substitution of (65) into (54) gives us two relations between the parameters:

$$\begin{array}{ccc}& & {\omega }_1{\lambda }_1^2+{\omega }_2{\lambda }_2^2+2\delta L_1{L}_2=0;\hfill \\ & & {\lambda }_1^2({\sigma }_1{\delta }_1-{\omega }_1^2)+{\lambda }_2^2({\sigma }_2{\delta }_2-{\omega }_2^2)=0\hfill \end{array}$$

Now, depending on the values of the constants, we can list all possible solutions $s_1\left(x_1\right),s_2\left(x_2\right)$ within Section 4.1. All are expressed in terms of hyperbolic, trigonometric, and exponential functions, and have to be inserted into (63) to find $u_1\left(\overrightarrow{x}\right),u_2\left(\overrightarrow{x}\right)$ according to (60) and (62):

I. 

${\lambda }_1^2>0;{\lambda }_2^2>0;{\sigma }_1{\delta }_1>0;{\sigma }_2{\delta }_2>0$.

By additional translation of $x_{1,2}$ functions, $s_1\left(x_1\right),s_2\left(x_2\right)$ takes the form

$$s_1\left(x_1\right)={\sigma }_{1}cosh\left({\lambda }_1{x}_1\right)-{\omega }_1;s_2\left(x_2\right)={\sigma }_{2}cosh\left({\lambda }_2{x}_2\right)-{\omega }_2$$

with the restriction

$${\lambda }_1^2({\sigma }_1^2-{\omega }_1^2)+{\lambda }_2^2({\sigma }_2^2-{\omega }_2^2)=0.$$

II. 

${\lambda }_1^2>0;{\lambda }_2^2>0;{\sigma }_1{\delta }_1>0;{\sigma }_2{\delta }_2<0$

$$s_1\left(x_1\right)={\sigma }_{1}cosh\left({\lambda }_1{x}_1\right)-{\omega }_1;s_2\left(x_2\right)={\sigma }_{2}sinh\left({\lambda }_2{x}_2\right)-{\omega }_2,$$

with restriction

$${\lambda }_1^2({\sigma }_1^2-{\omega }_1^2)-{\lambda }_2^2({\sigma }_2^2+{\omega }_2^2)=0.$$

III. 

${\lambda }_1^2>0;{\lambda }_2^2>0;{\sigma }_1{\delta }_1>0;{\delta }_2=0$

$$s_1\left(x_1\right)={\sigma }_{1}cosh\left({\lambda }_1{x}_1\right)-{\omega }_1;s_2\left(x_2\right)=\frac{1}{2}{\sigma }_{2}exp\left({\lambda }_2{x}_2\right)-{\omega }_2,$$

with restriction

$${\lambda }_1^2({\sigma }_1^2-{\omega }_1^2)-{\lambda }_2^2{\omega }_2^2=0.$$

IV. 

${\lambda }_1^2<0;{\lambda }_2^2<0;{\sigma }_1={\delta }_1^{★};{\sigma }_2={\delta }_2^{★}$

${\lambda }_1\equiv i{\Lambda }_1;{\lambda }_2\equiv i{\Lambda }_2$

$$s_1\left(x_1\right)={\sigma }_{1}cos\left({\Lambda }_1{x}_1\right)-{\omega }_1;s_2\left(x_2\right)={\sigma }_{2}cos\left({\Lambda }_2{x}_2\right)-{\omega }_2;$$

with restriction

$${\Lambda }_1^2({\sigma }_1^2-{\omega }_1^2)+{\Lambda }_2^2({\sigma }_2^2-{\omega }_2^2)=0.$$

V. 

${\lambda }_1^2>0;{\lambda }_2^2<0;{\sigma }_1{\delta }_1>0;{\sigma }_2={\delta }_2^{★}$

${\lambda }_2\equiv i{\Lambda }_2$

$$s_1\left(x_1\right)={\sigma }_{1}cosh\left({\lambda }_1{x}_1\right)-{\omega }_1;s_2\left(x_2\right)={\sigma }_{2}cos\left({\Lambda }_2{x}_2\right)-{\omega }_2;$$

with restriction

$${\lambda }_1^2({\sigma }_1^2-{\omega }_1^2)-{\Lambda }_2^2({\sigma }_2^2-{\omega }_2^2)=0.$$

VI. 

${\lambda }_1^2>0;{\lambda }_2^2<0;{\sigma }_1{\delta }_1<0;{\sigma }_2={\delta }_2^{★}$

${\lambda }_2\equiv i{\Lambda }_2$

$$s_1\left(x_1\right)={\sigma }_{1}sinh\left({\lambda }_1{x}_1\right)-{\omega }_1;s_2\left(x_2\right)={\sigma }_{2}cos\left({\Lambda }_2{x}_2\right)-{\omega }_2;$$

with restriction:

$${\lambda }_1^2({\sigma }_1^2+{\omega }_1^2)+{\Lambda }_2^2({\sigma }_2^2-{\omega }_2^2)=0.$$

4.2. Case B: $m_2=0.$

The case with $m_2=L_2=0$ and real $\Omega ,L_1,g\left(\overrightarrow{x}\right)$ is considered below. For such a choice, Equation (58) provides us with

$$v_2\left(\overrightarrow{x}\right)=exp(g\left(\overrightarrow{x}\right)/\Omega )=\frac{C}{s_2\left(x_2\right)-s_1\left(x_1\right)},v_1\left(\overrightarrow{x}\right)=L_1\frac{s_1\left(x_1\right)+s_2\left(x_2\right)}{s_2\left(x_2\right)-s_1\left(x_1\right)},$$

(66)

while Equation (54) looks like

$$(s_2^{″}\left(x_2\right)-s_1^{″}\left(x_1\right))(s_2\left(x_2\right)-s_1\left(x_1\right))-({\left(s_1^{\prime }\left({\overrightarrow{x}}_1\right)\right)}^2+{\left(s_2^{\prime }\left({\overrightarrow{x}}_2\right)\right)}^2)=C{L}_1(s_1\left(x_1\right)+s_2\left(x_2\right)).$$

(67)

After differentiation, the latter equation is again amenable to separation of variables, similarly to (64), except with zero on the right-hand side. It has two different solutions depending on the value of separation constant:

$$s_1\left(x_1\right)=a_1{x}_1^2+b_1{x}_1+c_1;s_2\left(x_2\right)=a_2{x}_2^2+b_2{x}_2+c_2$$

(68)

and

$$s_1^{″}\left(x_1\right)={\lambda }^2(s_1\left(x_1\right)+{\omega }_1);s_2^{″}\left(x_2\right)=-{\lambda }^2(s_2\left(x_2\right)+{\omega }_2),$$

(69)

with the latter having three kinds of explicit solutions to insert into (66):

$$\begin{array}{ccc}\hfill s_1\left(x_1\right)& =& {\sigma }_{1}cosh\left(\lambda x_1\right)-{\omega }_1;s_2\left(x_2\right)={\sigma }_{2}cos\left(\lambda x_2\right)-{\omega }_2;\hfill \\ \hfill s_1\left(x_1\right)& =& {\sigma }_{1}sinh\left(\lambda x_1\right)-{\omega }_1;s_2\left(x_2\right)={\sigma }_{2}cos\left(\lambda x_2\right)-{\omega }_2;\hfill \\ \hfill s_1\left(x_1\right)& =& \frac{1}{2}{\sigma }_{1}exp\left(\lambda x_1\right)-{\omega }_1;s_2\left(x_2\right)={\sigma }_{2}cos\left(\lambda x_2\right)-{\omega }_2;\hfill \end{array}$$

and corresponding restrictions for the constants:

$$\begin{array}{ccc}& & {\omega }_2^2-{\omega }_1^2+{\sigma }_1^2-{\sigma }_2^2=0;\hfill \\ & & {\omega }_2^2-{\omega }_1^2-{\sigma }_1^2-{\sigma }_2^2=0;\hfill \\ & & {\omega }_2^2-{\omega }_1^2-{\sigma }_2^2=0;\hfill \\ & & {\lambda }^2({\omega }_2-{\omega }_1)-C{L}_1=0.\hfill \end{array}$$

As for the polynomial solution (68), a few restrictions have to be fulfilled simultaneously:

$$A\equiv 2a_1+2a_2+C{L}_1=0.$$

(70)

Finally, solution (68) leads to two different opportunities for the components of $U_1\left(\overrightarrow{x}\right)$. The first is

$$\begin{array}{ccc}\hfill v_1\left(\overrightarrow{x}\right)& =& -L_1\frac{[C{L}_1(c_2{x}_1^2-c_1{x}_2^2)+2(c_1^2-c_2^2)]}{C{L}_1(c_2{x}_1^2+c_1{x}_2^2)+2{(c_1-c_2)}^2};\hfill \end{array}$$

(71)

$$\begin{array}{ccc}\hfill v_2\left(\overrightarrow{x}\right)& =& -\frac{2(c_1-c_2)}{C{L}_1(c_2{x}_1^2+c_1{x}_2^2)+2{(c_1-c_2)}^2};\hfill \\ \hfill a_1& =& \frac{C{c}_2{L}_1}{2(c_1-c_2)};a_2=-\frac{C{c}_1{L}_1}{2(c_1-c_2)};c_1\ne c_2,\hfill \end{array}$$

(72)

while the second, with $c_1=c_2=0$ and functions $s_1\left(x_1\right)=a_1{x}_1^2,s_2\left(x_2\right)=-(a_1+\frac{1}{2}C{L}_1)x_2^2$, is

$$v_1\left(\overrightarrow{x}\right)=-L_1\frac{a_1{x}_1^2-(a_1+\frac{C{L}_1}{2})x_2^2}{(a_1+\frac{C{L}_1}{2})x_2^2+a_1{x}_1^2};v_2\left(\overrightarrow{x}\right)=-\frac{C}{(a_1+\frac{C{L}_1}{2})x_2^2+a_1{x}_1^2}.$$

(73)

4.3. Case C: $p+ic=0.$

Let us again consider intertwining of two Dirac operators, both with diagonal potential:

$$W_1\left(\overrightarrow{x}\right)=\left(\begin{array}{cc}{w}_{11}^{\left(1\right)}\left(\overrightarrow{x}\right)& 0\\ 0& w_{22}^{\left(1\right)}\left(\overrightarrow{x}\right)\end{array}\right)\equiv \left(\begin{array}{cc}{v}_1\left(\overrightarrow{x}\right)& 0\\ 0& v_2\left(\overrightarrow{x}\right)\end{array}\right);W_2\left(\overrightarrow{x}\right)=\left(\begin{array}{cc}{k}_1& 0\\ 0& k_2\end{array}\right),$$

(74)

and with constant elements $k_1,k_2.$ Here, the matrix potential $W_1\left(\overrightarrow{x}\right)$ is identified with the potential $V_2\left(\overrightarrow{x}\right)$ from Section 3, with the constant partner potential $W_2$ providing solvability of the problem. This means that Case C is an alternative option in relation to Cases B and C from the Section 4.1 and Section 4.2. The difference is that we now take $p+ic=0$, i.e., variant III from Section 2, and it is clear that variant IV can be considered analogously. In this case, the system (12)–(17) is expressed as follows:

$$\begin{array}{ccc}\hfill b_{11}& =& a_{22}+ib{v}_1-id{k}_2;b_{12}=\frac{n+ia}{2}{v}_1;b_{21}=-p{w}_2+\frac{n-ia}{2}{k}_2;\hfill \\ \hfill a_{12}& =& \frac{n+ia}{2}{k}_2=Const;a_{21}=-p{k}_1+\frac{n-ia}{2}{v}_1.\hfill \end{array}$$

The matrix differential Equation (18) takes the form

$$\begin{array}{ccc}\hfill i(n-ia)\partial v_1\left(\overrightarrow{x}\right)& =& v_1\left(\overrightarrow{x}\right)(a_{11}\left(\overrightarrow{x}\right)-ib{k}_1)-k_1(a_{22}\left(\overrightarrow{x}\right)-id{k}_2);\hfill \\ \hfill v_2\left(\overrightarrow{x}\right)(a_{22}\left(\overrightarrow{x}\right)-id{k}_2)& =& k_2(a_{11}\left(\overrightarrow{x}\right)-ib{k}_1);\hfill \\ \hfill \partial a_{22}\left(\overrightarrow{x}\right)& =& 0;\hfill \\ \hfill 2i\overline{\partial }{a}_{11}\left(\overrightarrow{x}\right)& =& \frac{n-ia}{2}(v_1\left(\overrightarrow{x}\right)v_2\left(\overrightarrow{x}\right)-k_1{k}_2),\hfill \end{array}$$

which by convenient definition

$$r_1\left(\overrightarrow{x}\right)\equiv a_{11}\left(\overrightarrow{x}\right)-ib{k}_1;r_2\left(\overrightarrow{x}\right)\equiv a_{22}\left(\overrightarrow{x}\right)-id{k}_2$$

can be reduced to two equations $(r_2\left(\overrightarrow{x}\right)=r_2\left(\overline{z}\right))$

$$\begin{array}{ccc}& & i{k}_2(n-ia)\partial \left(\frac{v_1\left(\overrightarrow{x}\right)}{r_2\left(\overline{z}\right)}\right)=v_1\left(\overrightarrow{x}\right)v_2\left(\overrightarrow{x}\right)-k_1{k}_2,\hfill \end{array}$$

(75)

$$\begin{array}{ccc}& & 4i\left(\overline{\partial }{r}_1\left(\overrightarrow{x}\right)\right)=(n-ia)(v_1\left(\overrightarrow{x}\right)v_2\left(\overrightarrow{x}\right)-k_1{k}_2),\hfill \end{array}$$

(76)

such that both $v_1\left(\overrightarrow{x}\right)$ and $v_2\left(\overrightarrow{x}\right)$ are expressed in terms of one function $\kappa \left(\overrightarrow{x}\right):$

$$v_1\left(\overrightarrow{x}\right)=\frac{4r_2\left(\overline{z}\right)}{k_2(n-ia)}\left(\overline{\partial }\kappa \left(\overrightarrow{x}\right)\right);v_2\left(\overrightarrow{x}\right)=\frac{k_2(n-ia)}{r_2\left(\overline{z}\right)}(\partial \kappa \left(\overrightarrow{x}\right));r_1\left(\overrightarrow{x}\right)=(n-ia)(\partial \kappa \left(\overrightarrow{x}\right)),$$

(77)

where function $\kappa \left(\overrightarrow{x}\right)$ satisfies the following second-order differential equation:

$$i(\partial \overline{\partial }\kappa \left(\overrightarrow{x}\right))=(\partial \kappa \left(\overrightarrow{x}\right))\left(\overline{\partial }\kappa \left(\overrightarrow{x}\right)\right)-k;4k\equiv k_1{k}_2.$$

(78)

Because the right-hand side here is real, the real part of the function $\kappa \left(\overrightarrow{x}\right)$ is a sum of two mutually conjugated functions:

$$\kappa \left(\overrightarrow{x}\right)\equiv \alpha \left(z\right)+\overline{\alpha }\left(\overline{z}\right)+i\xi \left(\overrightarrow{x}\right).$$

Due to Equation (77), the reality of both diagonal elements $v_1\left(\overrightarrow{x}\right),v_2\left(\overrightarrow{x}\right)$ leads to the reality of $(\partial \kappa (z,\overline{z}))\left(\overline{\partial }\kappa (z,\overline{z})\right)$, which, in terms of $\alpha$ and $\xi$, means that

$$\mathit{Im}\left((\partial \kappa \left(\overrightarrow{x}\right))\left(\overline{\partial }\kappa \left(\overrightarrow{x}\right)\right)\right)={\alpha }^{\prime }\left(z\right)\overline{\partial }\xi \left(\overrightarrow{x}\right)+{\overline{\alpha }}^{\prime }\left(\overline{z}\right)\partial \xi \left(\overrightarrow{x}\right)=0,$$

i.e., function $\xi \left(\overrightarrow{x}\right)$ is an arbitrary real function of the specific real argument:

$$\xi \left(\overrightarrow{x}\right)=\Phi \left(i(\alpha \left(z\right)-\overline{\alpha }\left(\overline{z}\right))\right)\equiv \Phi \left(X\right);X(z,\overline{z})\equiv i(\alpha \left(z\right)-\overline{\alpha }\left(\overline{z}\right)).$$

(79)

From Equation (78), we obtain a nonlinear differential equation for the function $\Phi$:

$${\Phi }^{″}\left(X\right)-{\left({\Phi }^{\prime }\left(X\right)\right)}^2+1=\frac{k}{{\alpha }^{\prime }\left(z\right){\overline{\alpha }}^{\prime }\left(\overline{z}\right)}.$$

(80)

The left-hand side in (80) depends only on the variable X, which by definition satisfies the following equation:

$$(\frac{1}{{\alpha }^{\prime }\left(z\right)}\partial +\frac{1}{{\overline{\alpha }}^{\prime }(\overline{z})}\overline{\partial })X(z,\overline{z})=0;$$

therefore, due to (80), we have

$$(\frac{1}{{\alpha }^{\prime }\left(z\right)}\partial +\frac{1}{{\overline{\alpha }}^{\prime }\left(\overline{z}\right)}\overline{\partial })\frac{1}{{\alpha }^{\prime }\left(z\right){\overline{\alpha }}^{\prime }\left(\overline{z}\right)}=0,$$

allowing us to define possible forms of the function $\alpha \left(z\right)$. Indeed, the variables in the latter equation can be separated:

$$\frac{{\alpha }^{″}\left(z\right)}{{\left({\alpha }^{\prime }\left(z\right)\right)}^2}+\frac{{\overline{\alpha }}^{″}\left(\overline{z}\right)}{{\left({\overline{\alpha }}^{\prime }\left(\overline{z}\right)\right)}^2}=0,$$

informing us that exactly two options exist for the function $\alpha \left(z\right)$:

$$a){\alpha }^{\prime }\left(z\right)=\omega ;b){\alpha }^{\prime }\left(z\right)=\frac{i\lambda }{z},$$

(81)

with $\omega$ being an arbitrary constant and $\lambda$ an arbitrary real constant.

For the first option, $r_2\left(\overline{z}\right)$ must be constant:

$$\begin{array}{ccc}\hfill v_1\left(\overrightarrow{x}\right)& =& 4e(1+{\Phi }^{\prime }\left(X(z,\overline{z})\right));v_2\left(\overrightarrow{x}\right)=\frac{{\left|\omega \right|}^2}{e}(1-{\Phi }^{\prime }\left(X(z,\overline{z})\right));\hfill \\ \hfill {\Phi }^{″}\left(X\right)& -& {\left({\Phi }^{\prime }\left(X\right)\right)}^2+1=\frac{k}{{\left|\omega \right|}^2};e\equiv \frac{r_2\left(\overline{z}\right)\overline{\omega }}{k_2(n-ia)},\hfill \end{array}$$

(82)

and the parameters must provide that the constant e is real. Depending on the sign of $(\frac{k}{{\left|\omega \right|}^2}-1)$, one of two solutions is realized:

$$\begin{array}{ccc}& & {\Phi }_1^{\prime }\left(X\right)=-\eta cos(\eta X+\mu );{\Phi }_2^{\prime }\left(X\right)=-\tilde{\eta }ln(cosh(\tilde{\eta }X+\tilde{\mu }));\hfill \\ & & (\frac{k}{{\left|\omega \right|}^2}-1)\equiv {\eta }^2>0;(\frac{k}{{\left|\omega \right|}^2}-1)\equiv -{\tilde{\eta }}^2<0.\hfill \end{array}$$

(83)

For the second option, Equation (77) states that $r_2\left(\overline{z}\right)=\gamma \overline{z}$, constant $\frac{i\gamma }{n-ia}$ must be real, and

$$v_1\left(\overrightarrow{x}\right)=\frac{4{\lambda }^2}{\beta }(1+{\Phi }^{\prime }\left(X\right));v_2\left(\overrightarrow{x}\right)=\frac{\beta }{z\overline{z}}(1-{\Phi }^{\prime }\left(X\right));\beta \equiv \frac{i{k}_2(n-ia)}{\gamma }.$$

(84)

Here, $\alpha \left(z\right)=i\lambda ln\left(z\right),\overline{\alpha }\left(\overline{z}\right)=-i\lambda ln\left(\overline{z}\right)$, and the equation for $\Phi \left(X\right)$ takes the form

$${\Phi }^{″}\left(X\right)-{\left({\Phi }^{\prime }\left(X\right)\right)}^2+1=k{\lambda }^2{e}^{\frac{X}{\lambda }}.$$

(85)

See [39] for more on solutions of this equation in terms of special functions.

5. Wave Functions and Electromagnetic Fields

In the previous sections, we performed two consecutive transformations of the Dirac operator with the matrix potential using first-order intertwining operators, similar to the SUSY intertwining in ordinary quantum mechanics. In this context, such operations are known as second-order reducible (i.e., factorizable) SUSY transformations [26]. Unlike that case, however, for the present problem with the Dirac operator we have used the asymmetrical form of intertwining [34,36] in both steps.

The resulting Dirac equation, with potential which is a diagonal matrix with constant elements at the diagonal (such as $U_2$ in (36)), is amenable to a simple analytic solution. Indeed, one of two components of ${\Psi }^{\left(2\right)}\left(\overrightarrow{x}\right)\equiv {({\Psi }_A^{\left(2\right)}\left(\overrightarrow{x}\right),{\Psi }_B^{\left(2\right)}\left(\overrightarrow{x}\right))}^T$ can be excluded, leading to the second-order equation (Helmholtz equation) for another component:

$$(\Delta +m_1{m}_2){\Psi }_A^{\left(2\right)}\left(\overrightarrow{x}\right)=0;{\Psi }_B^{\left(2\right)}\left(\overrightarrow{x}\right)=\frac{2i}{m_2}\overline{\partial }{\Psi }_A^{\left(2\right)}\left(\overrightarrow{x}\right).$$

(86)

After separation of the variables in (86), its solution can be written as a linear combination with arbitrary complex coefficients ${\sigma }_{k_1{k}_2}$:

$${\Psi }_A^{\left(2\right)}\left(\overrightarrow{x}\right)=\sum _{k_1,k_2}{\sigma }_{k_1{k}_2}exp\left(k_1{x}_1\right)exp\left(k_2{x}_2\right)$$

(87)

where the sum (actually, the integral) is over $k_1,k_2$, that is, arbitrary complex constants such that $k_1^2+k_2^2=-m_1{m}_2$. The coefficients in the sum have to be determined using the boundary conditions for the wave functions.

According to the intertwining relations of form (2), the solutions ${\Psi }^{\left(1\right)}\left(\overrightarrow{x}\right)\equiv ({\Psi }_A^{\left(1\right)}\left(\overrightarrow{x}\right),$${\Psi }_B^{\left(1\right)}\left(\overrightarrow{x}\right){)}^T$ of the initial Dirac equation with potential $V_1\left(\overrightarrow{x}\right)$ can be constructed using the sequential action of two intertwining operators $N_1{\tilde{N}}_1$. The first operator $N_1$ intertwines two Dirac operators: the initial one with potential $V_1$, and one with diagonal potential (29) (see (2)). The second ${\tilde{N}}_1$ analogously intertwines the Dirac operator with potential $V_2\left(\overrightarrow{x}\right)\equiv U_1\left(\overrightarrow{x}\right)\equiv W_1\left(\overrightarrow{x}\right)$ and the operator with either potential $U_2\left(\overrightarrow{x}\right)$ or $W_2\left(\overrightarrow{x}\right)$, depending on whether we follow the exploration described in Section 4.1 and Section 4.2 or Section 4.3. These intertwining operators have the general form (4), and the corresponding explicit expressions for the coefficients $A_k$ and $A\left(\overrightarrow{x}\right)$ are derived in Section 3 and Section 4.

The two-component wave functions ${\Psi }^{\left(1\right)}\left(\overrightarrow{x}\right)$ obtained by the above procedure can describe graphene and similar materials in an external field (i.e., two-dimensional electrostatic plus non-homogeneous orthogonal magnetic). Analytical expressions for the strength of these fields are known from the analytical expression for the initial potential $V_1\left(\overrightarrow{x}\right)$. The strength of electrostatic field is directed along the $(x_1,x_2)$ plane (see (34)):

$$\overrightarrow{E}\left(\overrightarrow{x}\right)=-\overrightarrow{\nabla }{A}_0\left(\overrightarrow{x}\right)=-\overrightarrow{\nabla }\left(f^2\left(\overrightarrow{x}\right)v_1\left(\overrightarrow{x}\right)\right),$$

(88)

and the strength of $B_3$ of the magnetic one (see (35)) is

$$B_3\left(\overrightarrow{x}\right)=▵ln\left(f\left(\overrightarrow{x}\right)\right).$$

(89)

The functions $f\left(\overrightarrow{x}\right)$ are different for the different cases in Section 4.1, Section 4.2 and Section 4.3 and can be calculated from the components $v_1\left(\overrightarrow{x}\right),v_2\left(\overrightarrow{x}\right)$ according to (33). These components are provided by (60) and (62) for Section 4.1, by (66) for Section 4.2, by (82) and (84) for Section 4.3. The explicit expressions for these functions, mainly in terms of trigonometric and hyperbolic functions, lead to corresponding expressions for the electromagnetic strengths in terms of the same elementary functions.

For examples of possible configurations of external fields, we can use the particular case of the polynomial solutions (68) and (70) for $c_1=c_2=0.$ The components $v_1\left(\overrightarrow{x}\right),v_2\left(\overrightarrow{x}\right)$ are provided by (73), while the function $f\left(\overrightarrow{x}\right)$ is defined from

$$f^4\left(\overrightarrow{x}\right)=\frac{v_2\left(\overrightarrow{x}\right)}{v_1\left(\overrightarrow{x}\right)}=\frac{2C}{L_1}{[2a_1{x}_1^2-(2a_1+C{L}_1)x_2^2]}^{-1}.$$

According to (88) and (89), the strengths are

$$\begin{array}{ccc}& & E_1\left(\overrightarrow{x}\right)=4\sqrt{2C{L}_1}{a}_1{x}_1\frac{2a_1{x}_1^2-3(2a_1+C{L}_1)x_2^2}{{[2a_1{x}_1^2+(2a_1+C{L}_1)x_2^2]}^3{[2a_1{x}_1^2-(2a_1+C{L}_1)x_2^2]}^{1/2}},\hfill \\ & & E_2\left(\overrightarrow{x}\right)=2\sqrt{2C{L}_1}(2a_1+C{L}_1)x_2\frac{6a_1{x}_1^2-(2a_1+C{L}_1)x_2^2}{{[2a_1{x}_1^2+(2a_1+C{L}_1)x_2^2]}^3{[2a_1{x}_1^2-(2a_1+C{L}_1)x_2^2]}^{1/2}},\hfill \end{array}$$

and

$$B_3\left(\overrightarrow{x}\right)=2(4a_1+C{L}_1)(2a_1{x}_1^2+(2a_1+C{L}_1)x_2^2){(2a_1{x}_1^2-(2a_1+C{L}_1)x_2^2)}^{-2},$$

with the possibility of choosing arbitrary suitable values for all constant parameters.

6. Conclusions

In this paper, a modification of the well-known method of asymmetrical intertwining relations from SUSY Quantum Mechanics has been used to build a massless two-dimensional Dirac equation with nontrivial matrix potential with solutions that can be found analytically. It was necessary to use factorizable second-order intertwining, which includes two steps: the first allows the Dirac operator to be diagonalized, while the second connects the latter operator with an explicitly solvable Dirac problem containing the diagonal matrix potential with constant elements.