Detecting Topological Transitions in Two Dimensions by Hamiltonian Evolution

We show that the evolution of two-component particles governed by a two-dimensional spin-orbit lattice Hamiltonian can reveal transitions between topological phases. A kink in the mean width of the particle distribution signals the closing of the band gap, a prerequisite for a quantum phase transition between topological phases. Furthermore, for realistic and experimentally motivated Hamiltonians, the density profile in topologically nontrivial phases displays characteristic rings in the vicinity of the origin that are absent in trivial phases. The results are expected to have an immediate application to systems of ultracold atoms and photonic lattices.

Figure 1

(a) The $m-t_2$ phase diagram for the spin-orbit Hamiltonian defined by Eq. (1). Different phases are characterized by the Chern numbers $C$. The red dots correspond to critical values of $t_2$ and $m$ along the $m=-4t_1$ and $t_2=-t_1$ dashed lines, respectively. The behavior of (b) $\partial s/\partial t_2$ [(c) $\partial s/\partial m$] along the appropriate direction is shown as a function of $t_2$ ($m$). The red vertical lines represent the phase boundaries, and we set $t=\hslash =t_1=1$.

Figure 2

Particle density distributions $p=⟨\mathit{r}|\psi (t)⟩{|}^2$. Results for $m=-20t_1$ and a $599\times 599$ lattice are shown (a)–(c) for $t_2=4.5t_1$ and $t=28\hslash /t_1$ ($C=0$), and (d)–(f) for $t_2=5.5t_1$ and $t=23\hslash /t_1$ ($C=1$). Raw data are presented in (a) and (d); smoothed densities in (b) and (e) are obtained by convolving with the function $e^{-(k_x^2+k_y^2)/{\gamma }^2}$, $\gamma =12.45$, representing the finite resolution of experimental imaging and initial state preparation. Slices through the centers of the particle densities (the blue dashed lines) are shown in (c) and (f). Images (b) and (e) are frames from movies included in the Supplemental Material.