Massless Dirac Fermions on a Space‐Time Lattice with a Topologically Protected Dirac Cone

Abstract

The symmetries that protect massless Dirac fermions from a gap opening may become ineffective if the Dirac equation is discretized in space and time, either because of scattering between multiple Dirac cones in the Brillouin zone (fermion doubling) or because of singularities at zone boundaries. Here an implementation of Dirac fermions on a space-time lattice that removes both obstructions is introduced. The quasi-energy band structure has a tangent dispersion with a single Dirac cone that cannot be gapped without breaking both time-reversal and chiral symmetries. It is shown that this topological protection is absent in the familiar single-cone discretization with a linear sawtooth dispersion, as a consequence of the fact that there the time-evolution operator is discontinuous at Brillouin zone boundaries.

Tangent Fermions: Dirac or Majorana Fermions on a Lattice Without Fermion Doubling

“…Tangent fermions play a unique role in these materials, they represent the only class of 2D lattice fermions with a topologically protected Dirac cone. [ 32 ] What distinguishes them from slac fermions is that the dispersion can be regularized on a space‐time lattice, producing a smooth energy–momentum relation across the entire Brillouin zone, of the form $$\tan ^2 (\varepsilon /2) =\tan ^2 (k_x/2) +\tan ^2 (k_y/2)$$ in dimensionless units. The sawtooth dispersion of slac fermions, in contrast, retains singularities at Brillouin zone boundaries when time and space are both discretized (see Figure 12 ).…”

“…In ref. [32] the evolution equation (Equation ( 28)) was used to calculate the time dependence of a state Ψ(x, y, t) incident along the x-axis on a rectangular barrier (height V 0 ). The initial state is a Gaussian wave packet, Ψ(x, y, 0) = (4𝜋w 2 ) −1∕2 e ik 0 x e −(x 2 +y 2 )∕2w 2…”

“…Klein tunneling of tangent fermions was studied in ref. [36], based on a space‐time lattice generalization [ 32 ] of the generalized eigenproblem (Equation (22)). The stationary equation $${\cal H}\Psi =E{\cal P}\Psi$$ can be converted into a time‐dependent equation (time step $$\delta t$$) upon substitution of Ψ on the left‐hand‐side by $$\tfrac{1}{2}[\Psi (t+\delta t)+\Psi (t)]$$, and of $$E\Psi$$ on the right‐hand‐side by $$(i\hbar /\delta t)[\Psi (t+\delta t)-\Psi (t)]$$.…”

“…This topological protection is lost on the lattice for each of the discretized Hamiltonians discussed in the previous section—except for one: Tangent fermions retain a topologically protected Dirac cone. [ 32 ]…”

“…Following ref. [32] we apply the staggered potential $$V(x)=V\cos (\pi x/a)$$, switching from $$+ V$$ to $$-V$$ between even and odd‐numbered lattice sites. This potential couples the states at k and $$k+\pi /a$$, as described by the Hamiltonian $$$$\begin{equation} H_{V}(k)=\def\eqcellsep{&}\begin{pmatrix} H(k)&V/2\\ V/2&H(k+\pi /a) \end{pmatrix} \end{equation}$$$$The Brillouin zone is halved to $$|k|<\pi /2a$$, with the band structure [ 33 ] shown in Figure 4 .…”