Detecting Bulk Topology of Quadrupolar Phase from Quench Dynamics

Direct measurement of a bulk topological observable in topological phase of matter has been a long-standing issue. Recently, detection of bulk topology through quench dynamics has attracted growing interests. Here, we propose that topological characters of a quantum quadrupole insulator can be read out by quench dynamics. Specifically, we introduce a quantity, a quadrupole moment weighted by the eigenvalues of the chiral operator, which takes zero for the trivial phase and finite for the quadrupolar topological phase. By utilizing an efficient numerical method to track the unitary time evolution, we elucidate that the quantity we propose indeed serves as an indicator of topological character for both noninteracting and interacting cases. The robustness against disorders is also demonstrated.

Figure 1

Schematics of the BBH model. The unit cell is indicated by a yellow shade, and green circles schematically represent the initial positions of the two particles. Note that $\pi$ flux threads each square plaquette.

Figure 2

Schematic figures of two decoupled limits. The panel (a) corresponds to $t_b=0$, i.e., the topologically trivial case, and (b) corresponds to $t_a=0$, i.e., the topologically nontrivial case.

Figure 3

The MCQM for the clean system (i.e., $W=0$) with (a) $t_a=-0.3$, $t_b=-1.0$ and (b) $t_a=-1.0$, $t_b=-0.3$. (c) The time-averaged MCQM, ${\overline{\mathcal{C}}}_q$, as a function of $|t_a|/|t_b|$. The average is taken over $t\in [0,50]$, and the parameters in the actual simulations are set such that $\max \{|t_a|,|t_b|\}=1$.

Figure 4

The MCQM for the disordered system with (a) $t_a=-0.3$, $t_b=-1.0$, $W=0.2$ and (b) $t_a=-0.3$, $t_b=-1.0$, $W=1.0$. The error bars are represented by the shades.