Berry connection induced anomalous wave-packet dynamics in non-Hermitian systems

Berry phases strongly affect the properties of crystalline materials, giving rise to modifications of the semiclassical equations of motion that govern wave-packet dynamics. In non-Hermitian systems, generalizations of the Berry connection have been analyzed to characterize the topology of these systems. While the topological classification of non-Hermitian systems is being developed, little attention has been paid to the impact of the new geometric phases on dynamics and transport. In this work, we derive the full set of semiclassical equations of motion for wave-packet dynamics in a system governed by a non-Hermitian Hamiltonian, including corrections induced by the Berry connection. We show that non-Hermiticity is manifested in the anomalous weight rate and velocity terms that are present already in one-dimensional systems, in marked distinction from the Hermitian case. We express the anomalous weight and velocity in terms of the Berry connections defined in the space of left and right eigenstates and compare the analytical results with numerical lattice simulations. Our work specifies the conditions for observing the anomalous contributions to the semiclassical dynamics and thereby paves the way to their experimental detection, which should be within immediate reach in currently available metamaterials.

Figure 1

Energy dispersion of the non-Hermitian SSH model (38) for the parameter choices specified in Table 1. Panels (a)–(d) show the real (dark blue lines) and imaginary parts (olive lines) of the energies as a function of the crystal momentum $k$. The dashed lines denote the bands we use to construct wave packets whose dynamics we simulate numerically. Depending on $k$, the energies of these bands may have the largest imaginary part of both bands (green shaded regions) or the smallest imaginary part (red shaded regions). Panels (e)–(h) show the imaginary part of the energies as a function of their real part (gray lines). The red and green dots mark the energies at $k=0$ and $k=\pi$, respectively, and the red arrows point from $k=0$ towards positive $k>0$. The parameters are chosen such that the models (I), (II), and (IV) have a line gap and (III) has a point gap. Model (II) is at a fine-tuned point where ${\varepsilon }_k={\varepsilon }_{k+\pi }$ coincide, hence each of the two band contours in panel (f) flattens to an arc.

Figure 2

Time evolution of the normalized weight difference $(N_{+}-N_{-})/(N_{+}+N_{-})$ for the non-Hermitian SSH model with the parameters chosen as specified in Table 1 [SSH (I)] and two different initial momenta $k_i=k_c{|}_{t=0}$: (a) $k_i=\pi /2$ with $\mathrm{Im}{\varepsilon }_{k_i}>0$, (b) $k_i=3\pi /2$ with $\mathrm{Im}{\varepsilon }_{k_i}<0$. The different colors denote different strengths of the electric field. In all panels, we use periodic boundary conditions with $L=780$ sites, a wave-packet width $\sigma =0.0067$, and discrete time steps with $t_{n+1}-t_n=0.0168/{\varepsilon }_0$.

Figure 3

Time evolution of the anomalous weight rate, normalized by the electric field, for the different parameter regimes summarized in Table 1. We extract the anomalous weight rate by subtracting $2\mathrm{Im}{\varepsilon }_k$ from $\dot{N}/N$ and plot the result divided by the strength of the electric field. In panels (a)–(d), we show the resulting anomalous weight rate at fixed wave-packet width ($\sigma =0.0067$) for various strengths of the electric field (with the different panels showing different parameter choices), whereas we compare different wave-packet widths at a fixed electric field ($eE=0.0135{\varepsilon }_0$) in panels (e)–(h). The red dashed lines show the analytic expectation based on Eq. (27) with $eE=0.0135{\varepsilon }_0$ (the expectation for larger fields is slightly shifted since the central momentum at $t$ depends on the field strength). All quantities are evaluated at $t=12.57/{\varepsilon }_0$. We use $L=780$ sites with periodic boundary conditions and discrete time steps with $t_{n+1}-t_n=0.0109/{\varepsilon }_0$ for SSH (III) and $t_{n+1}-t_n=0.0168/{\varepsilon }_0$ for the other parameter choices.

Figure 4

Time evolution of the anomalous velocity for the different parameter regimes summarized in Table 1. We extract the anomalous velocity by subtracting ${\partial }_k{\varepsilon }_k{|}_{k_c}$ from ${\dot{r}}_c$ and plot the result divided by the strength of the electric field. In panels (a)–(d), we show the resulting anomalous velocity at fixed wave-packet width ($\sigma =0.0067$) for various strengths of the electric field (with the different panels showing different parameter choices), whereas we compare different wave-packet widths at a fixed electric field ($eE=0.0135{\varepsilon }_0$) in panels (e)–(h). The red dashed lines show the analytic expectation based on Eq. (28) with $eE=0.0135{\varepsilon }_0$ (the expectation for larger fields is slightly shifted since the central momentum at $t$ depends on the field strength). As in Fig. 3, all quantities are evaluated at $t=12.57/{\varepsilon }_0$, and we use $L=780$ sites with periodic boundary conditions and discrete time steps with $t_{n+1}-t_n=0.0109/{\varepsilon }_0$ for SSH (III) and $t_{n+1}-t_n=0.0168/{\varepsilon }_0$ for the other parameter choices.

Figure 5

The shift of a wave packet's central momentum due to its nonzero width in momentum space. (a) Without an electric field, the absolute value of the weights $w_k\left(t\right)$ increases or decreases in time, depending on the sign of the imaginary part of the energies [cf. panel (b)]. A wave packet initialized at $k_c{|}_{t=0}=0$ (black line) loses weight for $k<0$ (red shaded area) and gains weight for $k>0$ (green shaded area) when evolving in time, such that its shape and central momentum changes (gray line). (c) We show ${\partial }_t{k}_c{|}_{t=0}$ for different widths $\sigma$ that collapse onto the same line when rescaling it by $2{\sigma }^2$. The rescaled value equals ${\partial }_k\mathrm{Im}{\varepsilon }_k$ (gray dashed line); cf. Eq. (B6) and note that $\mathrm{Im}{\varepsilon }_{k_c}=0$. In all simulations, we employ the SSH model [Eq. (38)] with the parameters specified in Table 1 [SSH (I)]. We use $L=600$ lattice sites and periodic boundary conditions.