Armchair graphene nanoribbons: $\mathcal{PT}$-symmetry breaking and exceptional points without dissipation

We consider a single-layer graphene nanoribbon with armchair edges and with a longitudinally constant external potential, pointing out that it can be described by means of an effective non-Hermitian Hamiltonian. We show that this system has some features typical of dissipative systems, namely, the presence of exceptional points and of $\mathcal{PT}$-symmetry breaking, although it is not dissipative.

Figure 1

Coalescing of two eigenvalues along the real line for $\varepsilon =1/5$; the arrows denote the direction of increasing $\alpha$ values ($\Delta \alpha \equiv \alpha -0.060\hspace{0.5em}953$). In the inset (a), the imaginary parts of the coalescing eigenvalues are shown together with a fit of the form $\alpha =c_1+c_2[\mathrm{Im}\hspace{0.5em}(k_{x}a)]{}^2$. The inset (b) shows the behavior of the square of the transverse momentum together with a linear fit.

Figure 2

The momenta (top panel) $\Delta k_x^{(1,2)}\approx k_x^{(1,2)}-0.199\hspace{0.5em}894\hspace{0.5em}795/a$ and the energy (bottom panel) $\Delta E\approx E+0.200\hspace{0.5em}651\hspace{0.5em}930\hspace{0.5em}{v}_F/a$ of two coalescing modes as functions of $\cos \theta$, estimated from $(detR-2)/\sqrt{1-detR}$ [see Eqs. (14) and (15)] for the potential (10) with $\alpha =\pi /2$ and $\varepsilon =-a\hspace{0.5em}E/v_F$. The lines in the upper graph follow the prediction (16): the asymptotic velocities $v_{\pm }$ are obtained by considering $\Delta E\gtrsim 2\times {10}^{-5}\hspace{0.5em}{v}_F/a$ and $\mu$ obtained from the linear fit of the energy versus $\cos \theta$ shown in the bottom graph.