Extended flat band, entanglement, and topological properties in a Creutz ladder

In this work, we study the entanglement and topological properties of an extended flat-band Creutz ladder by considering a compacted localized state (CLS). Based on the CLS picture, we find a multiple flat-band extension from the conventional two flat-band Creutz ladder. A simple vertical interchain coupling leads to a four complete flat-band system and creates an additive $\pi$-flux pattern on the Creutz ladder. Interestingly, the strong coupling induces a topological phase transition where the distribution of CLSs is modified: upper and lower flat-band CLSs are paired up. This pairing leads to the destruction of the CLS entanglement and, hence, to a vanishing edge mode (i.e., the breakdown of the nontrivial topological phase). Finally, we study the localization dynamics induced by the presence of complete flat bands in this extended flat-band system.

Figure 1

(a) A conventional Creutz ladder. (b) Additive $\pi$ flux induced by vertical interchain coupling. (c) Extended unit cell for the four-flat-band system. (d) Bulk-band structure of $H_{\mathrm{Cd}}\left(k\right)$ for ${\mathrm{\Delta }}_0=t_0$.

Figure 2

(a) Two types of ground states described by the CLS. The blue vertical solid line represents the entanglement cut. Under a periodic boundary condition, the two lines divide the periodic system into two subsystems. (b) Entanglement entropy of the half-filled ground state under varying ${\mathrm{\Delta }}_0$. (c) Entanglement spectrum of the half filled ground state under varying ${\mathrm{\Delta }}_0$. (d) Scaling behavior of the entanglement spectrum under varying subsystem size $\ell$. In all cases, the ladder length is $L=50$, and $t_0=1$.

Figure 3

Energy spectrum for open boundary condition: (a) ${\mathrm{\Delta }}_0<2t_0$, (b) ${\mathrm{\Delta }}_0>2t_0$. The red and green dots represents the left and right localized edge modes. These modes have nonzero energy. (c) Band crossing of the second and third bands triggers a topological phase transition at half filling. Each bulk band is characterized by a Zak phase. (d) Zak phase for lower-half and upper-half states.

Figure 4

Return probability in a single particle dynamics. For any value of ${\mathrm{\Delta }}_0$, the RP does not decay. $t_0=1$.

Figure 5

Energy spectrum: (a) Periodic boundary condition result with ${\mathrm{\Delta }}_0=1$. (b) Open boundary condition result with ${\mathrm{\Delta }}_0=0.5$. The red line is doubly degenerate, indicating the presence of both left- and right-localized edge modes. For both cases, $t_1=t_0=1$.

Figure 6

Single lower flat-band CLS.

Figure 7

Single pairing state of the lower and upper flat-band CLSs.