Dielectric breakdown in a Mott insulator: Many-body Schwinger-Landau-Zener mechanism studied with a generalized Bethe ansatz

The nonadiabatic quantum tunneling picture, which may be called the many-body Schwinger-Landau-Zener mechanism, for the dielectric breakdown of Mott insulators in strong electric fields is studied in the one-dimensional Hubbard model. The tunneling probability is calculated by a method due to Dykhne-Davis-Pechukas with an analytical continuation of the Bethe-ansatz solution for excited states to a non-Hermitian case. A remarkable agreement with the time-dependent density-matrix renormalization-group result is obtained.

Figure 1

(Color online) Many-body energy levels against the complex AB flux $\Phi$ for a finite half-filled 1D Hubbard model $\left(L=10,N_{\uparrow }=N_{\downarrow }=5,U=0.5\right)$. Only charge excitations are plotted. Quantum tunneling occurs between the ground state (labeled as $n=0$) and a low-lying excited state $\left(n=1\right)$ as the flux $\Phi \left(t\right)=Ft$ increases on the real axis while the tunneling is absent for the states plotted as dashed lines. The wavy lines starting from the singular points $(\times )$ at $\Phi \left(t^{\ast }\right)$ represent the branch cuts for different Riemann surfaces, along which the solutions $n=0$ and $n=1$ are connected. In the DDP approach, the tunneling factor is calculated from the dynamical phase associated with adiabatic time evolution (DDP path) that encircles a gap-closing point at $\Phi \left(t^{\ast }\right)$ on the complex $\Phi$ plane.

Figure 2

(Color online) Schematic configurations (displayed here for $U=4.0$) of the charge rapidities for the lowest charge-excited state $|{\psi }_1\left(i\Psi \right)⟩$ (a) for $\Psi =0$ and (b) for a finite $\Psi$. $\mathcal{C}$ corresponds to the ground-state continuum with occupied states (green circles, printed gray). In the excited state, two holes $k_l$ and $k_m$ (open circles) near $\pi +ib$ appear in the continuum while two rapidities ${\kappa }_1$ and ${\kappa }_2$ outside the continuum $\mathcal{C}$ are occupied. The surface represents the real part of the excitation energy $\text{Re}\epsilon \left(k\right)$ [plotted here for $\text{Re}\epsilon \left(k\right)\ge 0$ and $\text{Im}k>0$] which gives the energies of holes at $k_l,k_m$. At $\Psi =0$, ${\kappa }_1,{\kappa }_2$ sit on the $\text{Re}\epsilon \left(k\right)=0$ curve.

Figure 3

(Color online) (a) The threshold field strength $F$ against $U$ obtained by the present DDP formalism [solid line; Eq. (14)]. Inset: The energy difference between the ground state and the excited state against $b$ for various values of $U$. (b) The decay rate $\Gamma$ of the ground state against the electric field $F$ obtained by the DDP formalism [solid line; Eq. (15)]. In (a) and (b), the symbols represent the time-dependent DMRG result (Ref. 5).