Breakdown of an Electric-Field Driven System: A Mapping to a Quantum Walk

Quantum transport properties of electron systems driven by strong electric fields are studied by mapping the Landau-Zener transition dynamics to a quantum walk on a semi-infinite one-dimensional lattice with a reflecting boundary, where the sites correspond to energy levels and the boundary the ground state. Quantum interference induces a distribution localized around the ground state, and a delocalization transition occurs when the electric field is increased, which describes the dielectric breakdown in the original electron system.

Figure 1

(a) Idealized energy levels of an electron system on a ring plotted against the AB flux $\varphi =FLt/h$, which increases linearly with time after the electric field $F$ is turned on at $t=0$. Two paths (A and B) for the Landau-Zener transition among neighboring levels are shown. (b) A mapping to a quantum walk, where the energy levels are mapped to sites of the qubits and the ground state to a reflecting boundary.

Figure 2

Time evolution of $|{\psi }_{L,R}(n,\tau ){|}^2$ with the initial condition ${\psi }_L(0,0)=1$. Here we set $p=0.2<p_c={\sin }^2\theta$, and only the amplitudes at even $n$ are displayed.

Figure 3

Total momentum $J_{\mathrm{edge}}$ and energy $E_{\mathrm{edge}}$ contributed by the edge component [Eq. (8)] against the electric field $F$ for $\theta =\pi /4$. The upper figure depicts the weight of the edge state.