Cyclotron quantization and mirror-time transition on nonreciprocal lattices

Unidirectional transport and localized cyclotron motion are two opposite physical phenomena. Here, we study the interplay effects between them on nonreciprocal lattices subject to a magnetic field. We show that, in the long-wavelength limit, the trajectories of the wave packets always form closed orbits in four-dimensional complex space. Therefore, the semiclassical quantization rules persist despite the nonreciprocity, which preserves real Landau levels. We predict a different type of non-Hermitian spectral transition induced by the spontaneous breaking of the combined mirror-time reversal ($\mathcal{MT}$) symmetry, which generally exists in such systems. An order parameter is proposed to describe the $\mathcal{MT}$ phase transition, not only to determine the $\mathcal{MT}$ phase boundary but also to quantify the degree of $\mathcal{MT}$-symmetry breaking. Such an order parameter can be generally applied to all types of non-Hermitian phase transitions.

Figure 1

Schematic illustration of (a) nonreciprocal Harper-Hofstadter model and (b) nonreciprocal honeycomb lattice model with unequal hopping strengths $t\pm {\delta }_x$ in the $x$ direction and equal hopping $t$ in others. The gray dashed box in (b) denotes the unit cell composed of two sites $A,B$ and ${\mathit{a}}_{1,2}$ are the unit vectors. (c) Low-energy parabolic dispersion with the linear imaginary part corresponding to the square lattice model in (a). (d) Dirac cone dispersion and its imaginary part corresponding to the honeycomb lattice in (b). The signs of the imaginary part of the energy coincide with those of the velocity. The contours in (c) and (d) are the closed orbits that satisfy the Onsager-Lifshitz quantization rule. (e) Semiclassical cyclotron motion of charged particles in a magnetic field. (f) Semiclassical picture of the $\mathcal{MT}$ symmetry, in which the successive actions of the $\mathcal{MT}$ operation ($y\to -y,v_x\to -v_x$) and the time evolution $U\left(t\right)$ leave the state unchanged.

Figure 2

Energy spectra under the $x,y$-PBC calculated in momentum space for (a) ${\delta }_x=0$ and (b) ${\delta }_x=0.2$. (c) Complex energy spectra for different $B$ marked in (b). (d) Complex energy spectra as a function of ${\delta }_x$ with $B=0.05$. In all figures, $t=0.5$.

Figure 3

Projections of the semiclassical trajectories in the real and imaginary $x-y$ planes. (a) Closed orbits in the long-wavelength limit with $x\left(0\right)=1, y\left(0\right)=0.75, p_x\left(0\right)=0, p_y\left(0\right)=0$ and (b) open trajectories beyond this limit with $x\left(0\right)=2, y\left(0\right)=2.5, p_x\left(0\right)=0.4, p_y\left(0\right)=0$. Other parameters are $B=0.26, {\delta }_x=0.1$.

Figure 4

Order parameter $d_{\text{HS}}$ as a function of ${\delta }_x$ and ${\gamma }_B$ for the nonreciprocal square lattice with (a) $B=0$ and (b) $B=0.02$. Critical points in (a) mark the real-to-complex spectral transition and the phase boundary defined by the $d_{\text{HS}}$ contours is fitted by exponential functions. Other parameters are $M=50, t=0.5$.