Form-preserving Darboux transformations for $4\times 4$ Dirac equations

Abstract

Darboux transformation is a powerful tool for the construction of new solvable models in quantum mechanics. In this article, we discuss its use in the context of physical systems described by $4\times 4$ Dirac Hamiltonians. The general framework provides limited control over the resulting energy operator, so that it can fail to have the required physical interpretation. We show that this problem can be circumvented with the reducible Darboux transformation that can preserve the required form of physical interactions by construction. To demonstrate it explicitly, we focus on distortion scattering and spin-orbit interaction of Dirac fermions in graphene. We use the reducible Darboux transformation to construct exactly solvable models of these systems where backscattering is absent, i.e., the models are reflectionless.

Appendix

We shall find asymptotic form of the matrix $U_xU^{-1}$, where U is given in (54). We have

$$\begin{aligned} (\partial _xU)U^{-1}=\frac{1}{{\mathcal {D}}(x)}\left( \begin{array}{ll} f_{11} &{} f_{12} \\ f_{21} &{} f_{22} \end{array}\right) , \end{aligned}$$

(A-1)

where the matrix elements and the denominator are as follows:

$$\begin{aligned} f_{11}= & {} ic_1+c_2\tanh (z_1)-i\,a^*\,\kappa _{\epsilon _2}(\text {Im}\,a+\kappa _{\epsilon _1}\tanh (z_1))\tanh ({z}_2), \end{aligned}$$

(A-2)

$$\begin{aligned} f_{22}= & {} i{\widetilde{c}}_1+{\widetilde{c}}_2\tanh ({z}_2)+i\,a\,\kappa _{\epsilon _1}(\text {Im}\,a-\kappa _{\epsilon _2}\tanh ({z}_2))\tanh (z_1), \end{aligned}$$

(A-3)

$$\begin{aligned} f_{12}= & {} i (w-\epsilon _1) (\kappa _{\epsilon _2}^2+\text {Im}\,a\, \kappa _{\epsilon _1} \tanh (z_1)-\kappa _{\epsilon _2} (\text {Im}\,a+\kappa _{\epsilon _1} \tanh (z_1))\tanh ({z}_2)), \end{aligned}$$

(A-4)

$$\begin{aligned} f_{21}= & {} i (v-\epsilon _2) (\kappa _{\epsilon _1}^2+ \text {Im}\,a\,\kappa _{\epsilon _1}\tanh (z_1)-\kappa _{\epsilon _2}(\text {Im}\,a+\kappa _{\epsilon _1} \tanh (z_1)) \tanh ({z}_2)). \end{aligned}$$

(A-5)

and ${\mathcal {D}}(x)$ is given in (57). For the sake of completeness, we put the formula here again

$$\begin{aligned} {\mathcal {D}}(x)=(v-\epsilon _{2})(w-\epsilon _{1})+\left( \kappa _{\epsilon _{1}}\tanh (z_{1})+{\mathrm{Im}}\,a\right) \left( \kappa _{\epsilon _{2}}\tanh ({z}_{2})-{\mathrm{Im}}\,a\right) . \end{aligned}$$

(A-6)

The $c_1$ and $c_2$ constants are given as

$$\begin{aligned} c_1= & {} \text {Im}\,a^2\text {Re}\,a-i\text {Im}\,a\kappa _{\epsilon _2}^2-\text {Re}\,a(w-\epsilon _1)(v-\epsilon _2), \end{aligned}$$

(A-7)

$$\begin{aligned} c_2= & {} i\text {Im}\,a\text {Re}\,a+\kappa _{\epsilon _2}^2+(w-\epsilon _1)(v-\epsilon _2),\nonumber \\ {\widetilde{c}}_1= & {} c_1^*|_{\kappa _{\epsilon _2}\rightarrow \kappa _{\epsilon _1}},\quad {\widetilde{c}}_2=c_2^*|_{\kappa _{\epsilon _2}\rightarrow \kappa _{\epsilon _1}}. \end{aligned}$$

(A-8)

Now, we calculate the limit $x\rightarrow {{{\pm }}}\infty$ of the matrix $(\partial _xU)U^{-1}$,

$$\begin{aligned} \begin{aligned}&w_{{{\pm }}}=\lim _{x\rightarrow {{\pm }}\infty } (\partial _xU)U^{-1}\\&\quad =\frac{1}{{\mathcal {D}}^{{\pm }}}\left( \begin{array}{cc} ic_1{{\pm }} c_2\mp i\,a^*\kappa _{\epsilon _2}(\text {Im}\,a{{\pm }}\kappa _{\epsilon _1}) &{} i (w-\epsilon _1) (\kappa _{\epsilon _2}^2{{\pm }}\text {Im}\,a \kappa _{\epsilon _1} \mp \kappa _{\epsilon _2} (\text {Im}\,a{{\pm }}\kappa _{\epsilon _1})) \\ i (v-\epsilon _2) (\kappa _{\epsilon _1}^2{{\pm }} \text {Im}\,a\kappa _{\epsilon _1}\mp \kappa _{\epsilon _2}(\text {Im}\,a{{\pm }}\kappa _{\epsilon _1})) &{} i{\widetilde{c}}_1{{\pm }}{\widetilde{c}}_2{{\pm }} i\,a\,\kappa _{\epsilon _1}(\text {Im}\,a\mp \kappa _{\epsilon _2}) \end{array}\right) , \end{aligned}\nonumber \\ \end{aligned}$$

(A-9)

where

$$\begin{aligned} {\mathcal {D}}^{{\pm }}=-\text {Im}{a}^2{{\pm }}\kappa _{\epsilon _2}(\text {Im}\,a{{\pm }}\kappa _{\epsilon _1} )\mp \text {Im}\,a \kappa _{\epsilon _1}+(v-\epsilon _2) (w-\epsilon _1). \end{aligned}$$