Beyond the constant-mass Dirac physics: Solitons, charge fractionization, and the emergence of topological insulators in graphene rings

The doubly connected polygonal geometry of planar graphene rings is found to bring forth topological configurations for accessing nontrivial relativistic quantum field (RQF) theory models that carry beyond the constant-mass Dirac-fermion theory. These include the generation of sign-alternating masses, solitonic excitations, and charge fractionization. The work integrates a RQF Lagrangian formulation with numerical tight-binding Aharonov-Bohm electronic spectra and the generalized position-dependent-mass Dirac equation. In contrast to armchair graphene rings (aGRGs) with pure metallic arms, certain classes of aGRGs with semiconducting arms, as well as with mixed metallic-semiconducting ones, are shown to exhibit properties of one-dimensional nontrivial topological insulators. This further reveals an alternative direction for realizing a graphene-based nontrivial topological insulator through the manipulation of the honeycomb lattice geometry, without a spin-orbit contribution.

Figure 1

Aharonov-Bohm spectra for hexagonal armchair graphene rings. (a) Tight-binding spectrum for a class-I nanoring with ${\mathcal{N}}_W=15$. (b) TB spectrum for a class-II nanoring with ${\mathcal{N}}_W=16$. The armchair graphene rings are semiconducting (${\mathcal{N}}_W=15$) and metallic at $\Phi =(\pm j+1/2){\Phi }_0$, $j=1,2,3,...$ (${\mathcal{N}}_W=16$). The two lowest-in-energy six-membered bands are shown. The hole states (with $\varepsilon <0$, not shown) are symmetric to the particle states (with $\varepsilon >0$). Insets: schematics of the Higgs fields $\varphi \left(x\right)$ employed in the DKP modeling. $\varphi \left(x\right)$ is approximated by steplike functions $m_i^{\left(n\right)}$; $i$ counts the three regions of each arm ($L_1^{\left(n\right)}=L_3^{\left(n\right)}=a$ and $L_2^{\left(n\right)}=b$), and $n$ ($n=1,...,6$) counts the hexagon's arms. The nonzero (constant) variable-mass values of $\varphi \left(x\right)$ are indicated by yellow (red) color when positive (negative). These resulting DKP spectra [(c) and (d)] reproduce the TB ones in (a) and (b), respectively. The parameters used in the DKP modeling are (c) $a=2a_0$, $b=28a_0$, $m_1^{\left(n\right)}=m_3^{\left(n\right)}=0.06t/v_F^2$, $m_2^{\left(n\right)}=0.13t/v_F^2$ [see corresponding schematic inset in (a)] and (d) $a=7a_0$, $b=15a_0$, $m_1^{\left(n\right)}=m_3^{\left(n\right)}=0$, $m_2^{\left(n\right)}={(-1)}^n{m}_0$ with $m_0=0.18t/v_F^2$ (see schematic inset). The inset in (c) shows the spectrum for a free massive Dirac fermion with a constant mass $\mathcal{M}=0.13t/v_F^2$. Note the six-membered braided bands and the “forbidden” band [within the gap, in (b) and (d)]. $a_0=0.246$ nm is the graphene lattice constant and $t=2.7$ eV is the hopping parameter.

Figure 2

Wave functions for an excitation belonging to the “forbidden” solitonic band. (a) $A$-sublattice (red) and $B$-sublattice (blue) components of the TB state with energy $\varepsilon =0.12507\times {10}^{-2}t$ at $\Phi ={\Phi }_0/3$, belonging to the forbidden solitonic band of the class-II nanoring with ${\mathcal{N}}_W=16$ [see Fig. 1]. (b) Upper (red) and lower (blue) spinor components for the corresponding state (forbidden band) according to the DKP spectrum [see Fig. 1], reproducing the TB behavior of the class-II nanoring with ${\mathcal{N}}_W=16$ ($m_0=0.18t/v_F^2$). The TB and DKP wave functions for all states of the solitonic band are similar to those displayed here. The wave functions here represent trains of solitons. For contrast, see Fig. 10 in Ref. [20] which schematically portrays the spinor ${\Psi }_S$ for a single fermionic soliton attached to a Higgs field with a smooth kink-soliton analytic shape ${\varphi }_k\left(x\right)={\varphi }_{0}tanh\left(\sqrt{\xi /2}{\varphi }_{0}x\right)$. ${\varphi }_k\left(x\right)$ is a solution [14, 20] of the Lagrangian in Eq. (2). (c) $A$-sublattice (red) and $B$-sublattice (blue) components of the TB state with energy $\varepsilon =0.55636\times {10}^{-2}t$ at $\Phi ={\Phi }_0/3$, associated with the metallic (class-III) nanoring with ${\mathcal{N}}_W=14$ (see corresponding spectrum in Fig. 1(b) of Ref. [20]). In contrast to the localized-at-the-corners topological-insulator wave functions of the semiconducting (class-II) ring in (a), the metallic-aGRG (class-III) wave functions in (c) do not exhibit any localization features and are thus devoid of any TI characteristics. DKP densities in units of ${10}^{-3}/d$, where $d=0.35a_0$.

Figure 3

Spectra and wave functions of a mixed ring with arms belonging to two different classes. Two contiguous arms [5 and 6 in (b)] belong to class I (semiconducting ribbons, ${\mathcal{N}}_W=15$) and the remaining arms belong to class III (metallic ribbons, ${\mathcal{N}}_W=17$). (a) The TB Aharonov-Bohm spectrum. (b) Schematic of the $\varphi \left(x\right)$ field in the DKP model, yielding the best reproduction [not shown, but see the wave function in (d)] of the TB spectra in (a). Note the unequal size of the colored boxes specifying $\varphi \left(x\right)$. The $m_{0,2}$ mass (class-I arms) is much larger than the $m_{0,1}$ one (class-III arms). Positive (negative) mass values are indicated in yellow (red). (c) TB wave function for a state of the braid band at $\Phi ={\Phi }_0/3$ ($\varepsilon =0.1995\times {10}^{-2}t$), for the $A$ (red) and $B$ (blue) sublattice components. (d) The upper (red) and lower (blue) spinor components of the DKP state, corresponding and showing agreement with the TB state in (c). The arrows [in (b)–(d)] indicate the single hexagon corner where the soliton is localized. Parameters for the DKP model are $m_1^{\left(n\right)}=m_3^{\left(n\right)}=0$ and $m_2^{\left(n\right)}={(-1)}^n{m}_{0,1}$ for $n=1,2,3,4$, and $m_2^{\left(n\right)}={(-1)}^n{m}_{0,2}$ for $n=5,6$, with $m_{0,1}=0.01t/v_F^2$ and $m_{0,2}=0.20t/v_F^2$. $a=10a_0$ and $b=10a_0$. In (c), the indices denote the arms (left) and corners (right). DKP densities in units of $1/d$, where $d=0.36a_0$.