Dynamical Buildup of a Quantized Hall Response from Nontopological States

We consider a two-dimensional system initialized in a topologically trivial state before its Hamiltonian is ramped through a phase transition into a Chern insulator regime. This scenario is motivated by current experiments with ultracold atomic gases aimed at realizing time-dependent dynamics in topological insulators. Our main findings are twofold. First, considering coherent dynamics, the nonequilibrium Hall response is found to approach a topologically quantized time-averaged value in the limit of slow but nonadiabatic parameter ramps, even though the Chern number of the state remains trivial. Second, adding dephasing, the destruction of quantum coherence is found to stabilize this Hall response, while the Chern number generically becomes undefined. We provide a geometric picture of this phenomenology in terms of the time-dependent Berry curvature.

Figure 1

(a) Parameter ramp in the system Hamiltonian from the nontopological to the topological regime through a phase transition. The insets show cuts of the band structure along the $k_x$ axis ($k_y=0$), illustrating (i) the initial state, (ii) the creation of excitations near the energy gap closing point, and (iii) the nonequilibrium state after the transition. (b),(c) Nonequilibrium Hall response ${\mathrm{\Sigma }}_{xy}(t)$ [see Eq. (4)] and quasistatic ansatz ${\tilde{\mathrm{\Sigma }}}_{xy}(t)$ [see Eq. (5)], with (b) coherent and (c) dephasing dynamics for the dephasing rate ${\gamma }_k=0.15$, for the ramp $m(t)=m_i+(m_f-m_i)[1-\exp (-vt)]$, $m_i=-2.7$, $m_f=-1.0$, $v=0.1$ of the Hamiltonian (1). In (b), ${\mathcal{C}}_s$ of the pure state [see Eq. (2)] trivially equals ${\tilde{\mathrm{\Sigma }}}_{xy}$, and the dashed horizontal line denotes the long-time average of ${\mathrm{\Sigma }}_{xy}$. System size: $120\times 120$ sites in all simulations. The finite size [27] causes a small deviation of ${\mathcal{C}}_s$ from zero in (b).

Figure 2

Berry curvature ${\mathcal{F}}_k$. (a),(b) Lower band ${\mathcal{F}}_k$ for (a) the initial Hamiltonian $H(m_i)$ and (b) the final Hamiltonian $H(m_f)$. (c) ${\mathcal{F}}_k(t)$ for the coherently evolved state at $t>t_c$. (d) Discountinous ${\mathcal{F}}_k$ for the dephased steady state. Corresponding weighted curvature ${\sqrt{p}}_k{\mathcal{F}}_k$ [the integrand of Eq. (5)] is shown in panel (e). (f) $\mathcal{F}(k_x,0)$ of the coherently evolved states at $t>t_c$, for $v=0.1$ and $v=0.5$, respectively. The simulations are done with a local adaptive method in momentum space resolving system sizes of up to $1795\times 1795$ sites. Ramp velocity $v=0.5$ (c)–(e), $m_i=-2.5$, $m_f=-1$.

Figure 3

(a) Bloch vector ${\widehat{n}}_k$ of the coherently evolving state at $t\gg t_c$. Arrows depict the in-plane configuration $({\widehat{n}}_k^x,{\widehat{n}}_k^y)$, whereas $n_k^z$ is indicated with color. (b) Occupation in the eigenbasis of $H(m_f)$ parametrized by ${\tilde{n}}_k^z$. Contour ${\mathrm{\Gamma }}_p$ (closed white curve) defined by ${\tilde{n}}_k^z=0$, i.e., at an equal weight superposition of the upper and lower band. (c) ${\widehat{n}}_z(k_x,0)$ as a smooth function of $k_x$ for $k_y=0$ for the coherently time-evolved state (blue solid) and the ground state of $H(m_f)$ (blue dashed). (d) ${\widehat{n}}_k^z(k_x,0)$ for the dephased steady state, which exhibits a discontinuous jump around the purity gap closing point. $m_i=-2.5$, $m_f=-1$, $v=0.5$ in all plots. ${\gamma }_k=0.5$ in (d). The system size is $120\times 120$ sites in all plots.