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

It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of 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 but, as a rule, depending on one spatial variable only. 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 was provided by supersymmetrical (SUSY) intertwining relations, namely, by their most general - asymmetrical - form proposed recently by the authors. Such SUSY-like method is applied in two steps similarly to the second order factorizable (reducible) SUSY transformations in ordinary Quantum Mechanics.

The extensive study of two-dimensional massless Dirac equation 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 in graphene-like materials [8], [9], [10], [11].The actual task is to find analytically the normalizable solutions of such Dirac equation where two components of "spinor" wave function Ψ( x) ≡ (Ψ A ( x), Ψ B ( x)) correspond to two sublattices of graphene.The potentials in the equation have different origin: the off-diagonal matrix term is provided by the electromagnetic vector-potential A = (A 1 (x 1 , x 2 ), A 2 (x 1 , x 2 ), 0) in the "long derivatives" and leads to the magnetic field B = ∇ × A along the z-axis, while the scalar potential A 0 (x 1 , x 2 ) describes the interaction with electrostatic or some another scalar field: where two-dimensional x ≡ (x 1 , x 2 ), the derivatives ∂ i ≡ ∂ ∂x i , and the charge was taken e = 1.
It is clear that the case of pure magnetic field in massless two-dimensional Dirac equation is explicitly solvable.Indeed, multiplying (1) without the term A 0 by σ 1 , one obtains a pair of decoupled first order equations which can be easily solved.The presence in (1) of term proportional to the unity matrix σ 0 prevents this decoupling (analogous problem would appear also in presence of the mass term proportional to σ 3 ).Different methods were used [15] - [27] to study such two-dimensional Dirac equation, mainly with strong restrictions on the conditions of the problem.There were problems with only electrostatic or only magnetic fields, and also 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 [12], [13], [14] has become also one of the most effective tools in discussed problems [28], [29], [30], [31], [32], [33], [34].As a rule, the main ingredient of SUSY Quantum Mechanics -so called SUSY intertwining relations -was explored in different forms.In the present paper, a class of external electromagnetic fields will be chosen with both magnetic (matrix) and electrostatic (scalar) terms depend effectively on both spatial coordinates.Such a progress is possible due to using a particular form of this approach -so called asymmetric intertwining relations [35], [36], [37].Up to now, this technique was explored to study the massless twodimensional Dirac equation with scalar potential and also the equation of Fokker-Planck [38].
More specifically, the procedure will include two steps (see Section 2).At first, asymmetric intertwining relations will provide SUSY diagonalization of potential in Eq.( 1) with both electromagnetic and electrostatic two-dimensional terms (Section 3).At the second stage, an additional asymmetric intertwining will connect the Dirac operator with diagonal potential with its partner whose potential is also diagonal but with constant elements (Section 4).The solutions of such Dirac equation can be found analytically, and solutions of initial problem are built by applying intertwining operators of both used steps (Section 5).Actually, the whole procedure realizes asymmetric form of factorizable SUSY intertwining of second order.Such SUSY intertwinings, but in their standard (symmetric) form, are known [14] in the context of SUSY Quantum Mechanics with Schrödinger operator.
2 Asymmetric intertwining for two-dimensional Dirac equation in electromagnetic field.
We start with the asymmetric intertwining relations: for a pair of two-dimensional massless Dirac operators of the form (1) rewritten as general operators with Hermitian matrix potentials Here, two different intertwining operators have the general matrix form: with constant matrices A k , B k and two x−dependent matrices A( x), B( x) : Expanding Eq.( 2) in powers of derivatives we obtain: Equations ( 6) provide the following form of constant coefficient matrices A k , B k : with constant values a, b, c, d, n, p, and Eqs.(7) give the system of eight linear equations for the matrix elements of V 1,2 ( x), A( x), B( x) : 12 ) = 0; (2) (1) The first two equations of the system indicates the need to highlight four different possibilities for solution of ( 10) - (17) with Hermiticity of both potentials in mind: Thus, Eqs.( 6), (7) are reduced to the system of Eqs.( 12) -( 17) in four possible variants above.We have not considered yet the matrix differential equation (8).It is convenient to solve it in an indirect form by using its combination with derivative ∂ k of equation ( 7): Below, all four variants will be used to obtain the general solution of initial intertwining relations (2).
3 SUSY-diagonalization by means of intertwining. 22 22 (ia 11 + bv 12 Here and below, it is convenient to use the space arguments x of the functions in the form of complex variables z = x 1 + ix 2 ; z = x 1 − ix 2 and corresponding derivatives In particular, the system ( 19) -( 22) takes the compact form: where two combinations are introduced: One may notice the apparent paradox contained in the system ( 19) - (22).All these equations are identically fulfilled if 12 + dv In the present context, the typical way [12], [13], [14], [35], [37] of 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 so that the problem of its analytical solution is more easy.Then, the solutions of partner Dirac equation with operator D 1 will be found by the action of intertwining operator Ψ (1) = N 1 Ψ (2) (in its turn, the operator N † 2 transforms the spinor Ψ (1) into Ψ (2) ).As the first step on this way, we choose the potential V 2 ( x) as a diagonal matrix: with real diagonal matrix elements.Then, due to Eqs.( 23), (24), fraction is a real function.Futhermore, Eqs.( 25), (26) and the Hermiticity of 12 (z, z) = v (1)⋆ 21 (z, z), lead to the following restriction: with an arbitrary real constant c.Summarizing these results, potential V 1 is expressed in terms of components (29), function f 2 (z, z) and real constant c : where the function f 2 was parameterizing as: Above, the sign ± corresponds to cases c > 0, c < 0, respectively, and f (z, z) is a positive function.This function connects elements of initial diagonal potential V 2 as: For the physical system with matrix potential V 1 describing "spin 1/2 particle" in external electromagnetic field according to Eq.( 1), the diagonal elements of V 1 define the electrostatic potential while the off-diagonal terms define the magnetic field which is orthogonal to the plane x 1 , x 2 : The solutions of Dirac equation with such external fields could be obtained from the two-  [39]).
4 From diagonal potential to the constant one by means of intertwining.
Let us consider now the variant II with (p + ic)(n − ia) = 0 for solution of the system (10) - (17) in the context of the second step of our procedure.Namely, let us consider intertwining relations between two Dirac operators, both with diagonal potential: with constant elements m 1 , m 2 .Here, potential U 1 ( x) is identified with potential V 2 ( x) of the previous step, and the constant partner potential U 2 will provide 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( x), B( x) will be found below by analytical solution of the system of equations which were obtained in Section 2.
We shall consider the system of equations ( 10) -( 17) sequentially.Eqs.( 10) and ( 11) are fulfilled automatically.Eqs.( 12), ( 15) and ( 14), ( 17) allow to express off-diagonal elements of A( x), B( x) at (5) in terms of v 1 ( x), v 2 ( x) : Eqs.( 16) and ( 13) for diagonal elements of A( x), B( x) can be written as: Eq.( 8) in its initial form is convenient to write now in complex coordinates z z : 2i where . In components, matrix equation ( 43) is equivalent to the system of linear first order differential equations: Let us define for convenience: Then, after substitution of Eqs.( 37) - (40) and Eqs.( 41), (42), the system (44) -(47) takes the form: where the form of last two equations means that functions g 1 (z, z), g 2 (z, z) can be expressed in terms of one complex function g: which satisfies the second order equation: and the first two equations (49), (50) become now: Thus, Below, we shall solve this system of equation by considering separately different options for the choice of constant parameters.

4.1
The case A: Parameters are real.
Let us study the case with real function g(z, z) and real values of parameters Ω, L 1 , L 2 .
Taking into account the reality of v 1 (z, z) and v 2 (z, z), it is useful to come back to Cartesian coordinates x 1 , x 2 in Eqs.( 57) and (58).Separately, both real and imaginary parts of (57) are integrated explicitly with two "constants of integration" s 1 (x 1 ), s 2 (x 2 ), which are arbitrary real functions of their arguments: Analogously, Eq.( 58) can be integrated as well with similar result: and arbitrary real s1 (x 1 ), s2 (x 2 ).Eqs.( 59), (61) together allow to connect s1 (x 1 ), s2 (x 2 ) with their analogues: with an arbitrary real constant δ.These relations have to be substituted into expression (62).
Using these connections in second order differential equation (54) for function g( x) and differentiating it by ∂ 1 ∂ 2 , we obtain the simple third order equation with separable variables: After separation of variables in (64) and integration of one-dimensional equations, we have: where ω 1 , ω 2 are integration constants, and λ 1 , λ 2 -arbitrary constants which satisfy relation: Solutions of (65) are known: Because we used derivatives of Eq.( 54), it is necessary to check results.Substitution of ( 65) into (54) gives two relations between parameters: Now we can list, depending on the values of the constants, all possible solutions s 1 (x 1 ), s 2 (x 2 ) within this Subsection.All of them are expressed in terms of hyperbolic, trigonometric and exponential functions, and they have to be inserted into (63) to find u 1 ( x), u 2 ( x) according to (60), (62): 0 By additional translation of x 1,2 functions s 1 (x 1 ), s 2 (x 2 ) takes the form: with restriction: with restriction: with restriction: with restriction: with restriction: with restriction:
The case with m 2 = L 2 = 0 and again real Ω, L 1 , g( x) will be considered below.For such a choice, Eq.(58) gives: and Eq.(54) looks like: The latter equation after differentiation is again, similarly to (64), amenable to separation of variables but with zero in the r.h.s.It has two different solutions depending on the value of separation constant: and The latter one has three kinds of explicit solutions to insert into (66): with restrictions for constants, correspondingly: As for the polynomial solution (68), a few restrictions have to be fulfilled simultaneously: Finally, solution (68) leads to two different opportunities for the components of U 1 ( x).The first one: (71) (72) and the second, with c 1 = c 2 = 0 and functions , is: 4.3 The case C: p + ic = 0.
Let us consider again intertwining of two Dirac operators both with diagonal potential: with constant elements k 1 , k 2 .Here, matrix potential W 1 ( x) will be also identified with potential V 2 ( x) of Section 3, and the constant partner potential W 2 will provide solvability of the problem.This means that the case C is an alternative option in relation to cases B and C of the previous two Subsections.The difference with them is that we will take now p + ic = 0, i.e. variant III of Section 2 (it is clear that the variant IV can be considered analogously).In this case, the system ( 12) -( 17) is expressed as: Matrix differential equation ( 18) takes a form: which by convenient definition, can be reduced to two equations (r 2 ( x) = r 2 (z)): so that both v 1 ( x) and v 2 ( x) are expressed in terms of one function κ( x) : where function κ( x) satisfies the following second order differential equation: Since the r.h.s.here is real, the real part of the function κ( x) is a sum of two mutually conjugated functions: Due to Eq.( 77), the reality of both diagonal elements v 1 ( x), v 2 ( x) leads to reality of (∂κ(z, z))( ∂κ(z, z)) that means in terms of α and ξ : i.e. function ξ( x) is an arbitrary real function of the specific real argument: From Eq.(78), we obtain nonlinear differential equation for the function Φ : The l.h.s. in (80) depends only on the variable X which satisfies the following equation by definition: and therefore, due to (80): allowing to define possible forms of the function α(z).Indeed, variables in the latter equation can be separated: ᾱ′′ (z) ( ᾱ′ (z)) 2 = 0 informing us that exactly two options exist for the function α(z) : with ω -an arbitrary constant and λ -an arbitrary real constant.
For the first option, r 2 (z) must be constant, and parameters must provide that the constant e is real.Depending on the sign of ( k |ω| 2 − 1), one of two solutions is realized: For the second option, Eq.(77) gives that r 2 (z) = γ z, constant iγ n−ia must be real, and: Here, α(z) = iλ ln(z), ᾱ(z) = −iλ ln(z), and equation for Φ(X) takes the form: See [40] about 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 Dirac operator with matrix potential using the first order intertwining operators similar to the SUSY intertwining in ordinary Quantum Mechanics.In that context, such operation is known as a second order reducible (i.e.factorizable) SUSY transformation [14].Unlike that case, for the present problem with Dirac operator, the asymmetrical form of intertwining [35], [37] was used in both steps.
The resulting Dirac equation with potential which is a diagonal matrix with constant elements at the diagonal (like U 2 in (36)) is amenable to a simple analytic solution.Indeed, one of two components of Ψ (2) ( x) ≡ (Ψ A ( x), Ψ B ( x)) T can be excluded leading to the second order equation (Helmholtz equation) for another component: After separation of variables in (86), its solution can be written as a linear combination with arbitrary complex coefficients σ k 1 k 2 : Ψ (2) where sum (actually, integral) is over k 1 , k 2 -arbitrary complex constants such that k 2 1 +k 2 2 = −m 1 m 2 .Coefficients in the sum have to be determined by boundary conditions for the wave functions.
The modification of well known method of SUSY Quantum Mechanics -asymmetrical in- Funding.
This research received no external funding.

V 1 , V 2 .
The explanation is rather simple: just in this case, Dirac operators D 1 , D 2 are proportional to the intertwining operators D 2 = CN 1 , D 1 = N 2 C up to some constant matrix C. Therefore, intertwining relation (2) becomes a trivial identity in such a case.
tertwining relations -was used to build the massless two-dimensional Dirac equation with nontrivial matrix potential whose solutions can be found analytically.It was necessary to use factorizable second order intertwining which includes two steps: the first step allows to diagonalize the Dirac operator, and the second one -to connect the latter operator with explicitly solvable Dirac problem containing the diagonal matrix potential with constant elements.Author Contributions.Conceptualization, writing and editing -M.V.I. and D.N.N.Both authors have read and agreed to the published version of the manuscript.