Band structures under non-Hermitian periodic potentials: Connecting nearly-free and bi-orthogonal tight-binding models

We explore band structures of one-dimensional open systems described by periodic non-Hermitian operators, based on continuum models and tight-binding models. We show that imaginary scalar potentials do not open band gaps but instead lead to the formation of exceptional points as long as the strength of the potential does not exceed a threshold value, which is in contrast to closed systems where real potentials open a gap with infinitesimally small strength. The imaginary vector potentials hinder the separation of low-energy bands because of the lifting of degeneracy in the free system. In addition, we construct tight-binding models through bi-orthogonal Wannier functions based on Bloch wavefunctions of the non-Hermitian operator and its Hermitian conjugate. We show that the bi-orthogonal tight-binding model well reproduces the dispersion relations of the continuum model when the complex scalar potential is sufficiently large.

Figure 1

Eigenvalues in the complex plane and dispersion relations with $A=0$ and $V\left(x\right)=csin\left(2\pi x\right)$: (a) $c=0$, (b) $c=5i$, (c) $c=20i$, (d) $c=30i$, and (e) $c=80i$. Green lines are obtained from the continuum model by solving Eq. (6), and blue dashed lines in the center and bottom rows of (c)–(e) represent dispersion relations of first and second bands obtained from the bi-orthogonal tight-binding model discussed in Sec. 4. In the bottom of (b)–(e), only $\mathrm{Im}\left[{\varepsilon }_1\left(k\right)\right]$ and $\mathrm{Im}\left[{\varepsilon }_2\left(k\right)\right]$ are shown but $\mathrm{Im}\left[{\varepsilon }_3\left(k\right)\right]=0$ is not shown to avoid the figures becoming too crowded, while the third band is also shown in the top and center figures of (b)–(d). In (d) and (e), $\mathrm{Re}\left[{\varepsilon }_1\left(k\right)\right]$ and $\mathrm{Re}\left[{\varepsilon }_2\left(k\right)\right]$ take the same value in the whole Brillouin zone.

Figure 2

The gap sizes ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ as functions of the strength of the scalar potential for $A=0$. The potentials are [(a), (b)] $V\left(x\right)=csin\left(2\pi x\right)$ with purely imaginary $c$ and (c) $V\left(x\right)=bcos\left(2\pi x\right)+csin\left(2\pi x\right)$ with $b=20$ and purely imaginary $c$. Solid (red and blue) lines show numerical results and dashed (green) lines correspond to the results from the perturbation analysis: (a) ${\mathrm{\Delta }}_1=c$, (b) ${\mathrm{\Delta }}_2={\left|c\right|}^2/8{\pi }^2$, and (c) ${\mathrm{\Delta }}_1=\sqrt{\pm 2b\left|\delta c\right|}$. In (c), red and blue lines correspond to $\mathrm{Re}\left({\mathrm{\Delta }}_1\right)$ and $\mathrm{Im}\left({\mathrm{\Delta }}_1\right)$, respectively.

Figure 3

Eigenvalues in the complex plane and dispersion relations when $A=0$ with $V\left(x\right)=csin\left(2\pi x\right)$ when (a) $c=5+20i$ and (b) $c=20+80i$. The green lines are calculated from the continuous model, and the blue dashed lines are the results from tight-binding approximation.

Figure 4

The dispersion relation when $A=0$ and $V\left(x\right)=bcos\left(2\pi x\right)+csin\left(2\pi x\right)$ with $b=20$ and $c=25i$.

Figure 5

Eigenvalues in the complex plane and dispersion relations when $A=i$ and $V\left(x\right)=csin\left(2\pi x\right)$ with (a) $c=0$, (b) $c=30i$, (c) $c=40i$, and (d) $c=80i$. The green lines are calculated from the continuous model, and the blue dashed lines are the results from tight-binding approximation.

Figure 6

Bloch wavefunctions at $k=0, {\psi }_{k=0}^n\left(x\right)$, under $V\left(x\right)=csin\left(2\pi x\right)$ when (a) $A=0, c=200i$; (b) $A=0, c=20+80i$; (c) $A=0, c=40+80i$; and (d) $A=1, c=80i$. Red squares and blue asterisks respectively correspond to first and second bands $n=1,2$. Green solid lines and black dashed lines are imaginary parts and real parts (if nonzero) of $V\left(x\right)$, respectively.

Figure 7

Wannier functions $w_1^0\left(x\right)$ (blue stars) and $w_2^0\left(x\right)$ (red squares) when $A=0$ and $V\left(x\right)=csin\left(2\pi x\right)$ with (a) $c=200i$ and (b) $c=20i$. While the Wannier function in (a) is obtained from an individual band, that in (b) is based on mixing of the two bands as explained in the main text. The green lines show the imaginary potentials $\mathrm{Im}\left[V\right(x\left)\right]$.

Figure 8

Schematic pictures which show the tight-binding lattices corresponding to our model for various regimes (a) when two bands are separated and $A=0$, (b) when two bands are not separated and $A=0$, and (c) when two bands are separated and $A\ne 0$.

Figure 9

The ratio of hopping amplitudes as functions of the strength of the potential when $A=0$ and $V\left(x\right)=csin\left(2\pi x\right)$. In (a), where $c$ is imaginary, blue solid squares and purple open circles respectively correspond to $|t_{11}^2/t_{11}^1|$ and $|t_{12}^0/t_{11}^0|$. Three dashed lines show parameters used in Figs. 1, 1, and 1. In (b), where $c$ is complex and $\mathrm{Re}\left(c\right)=d$ and $\mathrm{Im}\left(c\right)=4d$, blue squares and red circles respectively show $|t_{11}^2/t_{11}^1|$ and $|t_{22}^2/t_{22}^1|$, as functions of $d$. The left and right dashed lines correspond to parameters in Figs. 3 and 3.