Wide-range-tunable Dirac-cone band structure in a chiral-time-symmetric non-Hermitian system

We establish a connection between an arbitrary Hermitian tight-binding model with chiral ($\mathcal{C}$) symmetry and its non-Hermitian counterpart with chiral-time ($\mathcal{CT}$) symmetry. We show that such a non-Hermitian Hamiltonian is pseudo-Hermitian. The eigenvalues and eigenvectors of the non-Hermitian Hamiltonian can be easily obtained from those of its parent Hermitian Hamiltonian. It provides a way to generate a class of non-Hermitian models with a tunable full real band structure by means of additional imaginary potentials. We also present an illustrative example that could achieve a cone structure from the energy band of a two-layer Hermitian square lattice model.

Figure 1

Schematics for the system with ${\mathrm{\Lambda }}_{\mathrm{A}}={\mathrm{\Lambda }}_{\mathrm{B}}=4$ to illustrate the connection between the systems of Eqs. (1) and (15). (a) A bipartite lattice consists of sublattices A and B, which are connected by bond $J_{ij}$ which is across the $i\mathrm{th}$ site in sublattice A and the $j\mathrm{th}$ site in sublattice B. In the absence of imaginary potentials, i.e., $\gamma =0$, it has $\mathcal{C}$ symmetry, which ensures that the system has the spectrum $\pm {\varepsilon }_0\left(n\right)$ ($n=1,2,3,4$). In the presence of imaginary potentials, it has $\mathcal{CT}$ symmetry and becomes a pseudo-Hermitian system. (b) An ensemble of noninteracting half-spins in a complex external magnetic field. It is an equivalent system of (a) when the local magnetic field for spin $n$ has the form $\vec{B}\left(n\right)$.

Figure 2

(a) Schematic illustration of a bilayer square lattice with staggered imaginary potentials. The two sublattices are denoted by A (red) and B (green), respectively. The intra- and interlayer hopping strengths are $J$ and $T$, respectively. For $\gamma =0$, the system has both $\mathcal{C}$ and $\mathcal{T}$ symmetries, while nonzero $\gamma$ breaks the $\mathcal{C}$ symmetry but maintains the $\mathcal{CT}$ symmetry. The additional staggered imaginary potentials make the simple lattice have a tunable cone band structure.

Figure 3

Three-dimensional plots of band structures of bilayer square lattice with periodic boundary condition. Staggered imaginary potentials $\pm i\gamma$ are applied throughout the lattice and here we set $\lambda =\pm$. The parameters are (a) $T=5J,\gamma =0$; (b) $T=5J,\gamma =0.98J$; (c) $T=5J,\gamma =J.$ Here the key difference between the cases for (a) and (c) is that the dispersion relation in the bottom of band is quadratic for (a) but linear (cone) for (c). The band structure in case (a) is trivial in the context that Dirac cone in lattice system has been shown to exhibit some novel features. Although a Hermitian system can support Dirac dispersion (e.g., honeycomb lattice), the speed of electron (slope of the cone) is not tunable.

Figure 4

DOS per unit cell as a function of energy (in units of $J$) computed from the energy dispersion Eq. (30) with several typical values of $\gamma =0$ (blue), $0.98J$ (red), $J$ (black). And here we set $T=5J$. Also shown is a zoom-in of the densities of states close to the zero-energy point, which can be approximated by $D\left(\varepsilon \right)\propto \left|\varepsilon \right|$. The approximate expression in Eq. (36) is also plotted (solid green) as comparison.