Fractal energy carpets in non-Hermitian Hofstadter quantum mechanics

We study the non-Hermitian Hofstadter dynamics of a quantum particle with biased motion on a square lattice in the background of a magnetic field. We show that in quasimomentum space, the energy spectrum is an overlap of infinitely many inequivalent fractals. The energy levels in each fractal are space-filling curves with Hausdorff dimension 2. The band structure of the spectrum is similar to a fractal spider web in contrast to the Hofstadter butterfly for unbiased motion.

Figure 1

Biased motion in a magnetic field.

Figure 2

The energy spectrum $E_q$ for odd $q=7$ (top) and even $q=8$ (bottom) as functions of the quasimomentum $k_y$ for periodic (solid lines) and antiperiodic (dashed lines) boundary conditions on the $x$ axis. The $q$ energy branches are marked by different colors.

Figure 3

The band energy spectrum (the horizontal axis) with respect to the minimal value of the normalized magnetic flux ${\beta }^{}=1/q\in [0,1]$ with $q=1,2,...,100$ (the vertical axis) for periodic boundary conditions.

Figure 4

The band energy spectrum (the horizontal axis) with respect to the value of the normalized magnetic flux $\beta =p/q\in [0,1]$ with $q=1,2,...,70$ and $p=1,...,q$ (the vertical axis) for the periodic (the red lines) and antiperiodic (the blue lines) boundary conditions. The darkness (lightness) of the lines visualize high (low) values of the density of states (13).

Figure 5

Top: the density of states (13) as a function of the energy $E$ for $q=2$ (the blue solid line), $q=4$ (the orange long-dashed line), and $q=8$ (the violet short-dashed line) levels. Bottom: the energy levels as a function of the quasimomentum $k_y$.

Figure 6

The energy spectra (9) are shown as functions of the quasi-momentum $k_y$ at (normalized) magnetic fields $\beta =1/q$ for $q=m^n$ series with (a) $m=2, n=1,\cdots ,6$; (b) $m=3, n=1,\cdots ,4$; (c) $m=5, n=1,\cdots ,3$.

Figure 7

Zooming into the fractal structure of the energy levels of the non-Hermitian Hofstadter model for (normalized) magnetic fields $\beta =1/q$ with $q=2^n$ and increasing $n$.

Figure 8

The flattened energy spectra (47) vs the quasimomentum $k_y$ for $q=m^n$ series with (a) $m=2, n=1,\cdots ,6$; (b) $m=3, n=1,\cdots ,4$; and (c) $m=5, n=1,\cdots ,3$. These plots correspond the original energy spectra shown in Fig. 6.

Figure 9

The blue horizontal lines: The arccosine (47) of the energy levels (9) for the series $q=m^n$ at fixed power $n$ and for (a) $m=2$, (b) $m=3$, and (c) $m=5$. The shadowed regions and the arrows show the self-similarity features.

Figure 10

The real and imaginary parts of the energy spectrum for $q=8$ plotted against the quasimomentum $k_y$. The horizontal blue lines represent the real spectrum ($\mathrm{Im}E=0$) while the vertical red-orange lines correspond to the complex spectrum with a nonzero imaginary part ($\mathrm{Im}E\ne 0$). A projection of the energy spectrum on the bottom of the plot is also presented for convenience (gray color).