Spectrally isomorphic Dirac systems: Graphene in an electromagnetic field

We construct the new one-dimensional Dirac Hamiltonians that are spectrally isomorphic (not isospectral) with the known exactly solvable models. Explicit formulas for their spectra and eigenstates are provided. The operators are utilized for the description of Dirac fermions in graphene in the presence of an inhomogeneous electromagnetic field. We discuss explicit, physically relevant, examples of spectrally isomorphic systems with both nonperiodic and periodic electromagnetic barriers. In the latter case, spectrally isomorphic two- and three-gap systems associated with the Ablowitz-Kaup-Newell-Segur hierarchy are considered.

Figure 1

The graphs of dimensionless functions $g_0^{-}$ (left) and $g_1^{-}$ (right). It is worth noticing that for $V_3=\tilde{m}$, there holds $g_0^{-}=1$ and $g_1^{-}=0$ for $V_2>0$. Hence, $G^{-}$ reduces to the identity operator. $g_0^{-}=1$ and $g_1^{-}=0$ are represented by the black line in each figure. The density plots illustrate the circular domain of radius $\tilde{m}$ where $g_0^{-}$ and $g_1^{-}$ are imaginary.

Figure 2

The potentials ${\tilde{V}}_0^{-}$ (thick line) and ${\tilde{V}}_2^{-}$ (dashed line). The values of parameters $\{A,B,M,\alpha ,m\}$ are in the inset.

Figure 3

(a) ${\tilde{V}}_2^{-}-M$ (dashed line), ${\tilde{V}}_0^{-}$ (solid line) from (37) for $\{B,\alpha ,M,m\}$ specified in the inset. (b) ${\tilde{V}}_0^{+}$ (solid line), $({\tilde{V}}_2^{+}{)}^{\prime }$ (dashed line), $B_z$ from (38) (dotted line) for $B_0=1$ and $\{A,B,\alpha ,m,z_0,d\}$; see the inset.

Figure 4

(a) the vertical axis corresponds to $\sigma (H_M)$, the horizontal axis to $M$, $k=0.6$. Shadowed regions represent allowed energies. (b)–(d) ${\tilde{V}}_2^{-}$ dashed line, ${\tilde{V}}_0^{-}$ solid line. The values of parameters $\{k^2,M,\tilde{m}\}$ are in the inset of each plot.