Generalized chiral symmetry and stability of zero modes for tilted Dirac cones

While it is well known that chirality is an important symmetry for Dirac-fermion systems that gives rise to the zero-mode Landau level in graphene, here we explore whether this notion can be extended to tilted Dirac cones as encountered in organic metals. We find that there exists a “generalized chiral symmetry” that encompasses tilted Dirac cones, where a generalized chiral operator $\gamma$, satisfying $\gamma {}^{†}H+H\gamma =0$ for Hamiltonian $H$, protects the zero mode. We use this to show that the $n=0$ Landau level is $\delta$-function-like (with no broadening) by extending the Aharonov-Casher argument. We confirm numerically that a lattice model that possesses generalized chirality has an anomalously sharp Landau level for spatially correlated randomness.

Figure 1

(a) A lattice model possessing tilted Dirac cones. Hopping energies: $t$, solid thick lines; $-t$, solid thin lines; $t^{\prime }$, dotted lines. A unit cell is indicated by the primitive vectors ${\mathit{e}}_1$ and ${\mathit{e}}_2$. Energy dispersions $E(\mathit{k})/t$ for (b) $t^{\prime }=0$, (c) $t^{\prime }/t=0.2$, and (d) $t^{\prime }/t=0.4$.

Figure 2

Density of states for the model with $t^{\prime }/t=0.4$ depicted in the inset is plotted against the spatial correlation length of the random component of the magnetic field $\eta$ for a uniform magnetic field $\varphi /{\varphi }_0=1/100$ and an amplitude of the random magnetic field $\sigma /{\varphi }_0=0.0029$. The result is an average over $5\times {10}^3$ samples with the system size $30a\times 30a$.