Correlation-assisted quantized charge pumping

We investigate charge pumping in the vicinity of order-obstructed topological phases, i.e., symmetry-protected topological phases masked by spontaneous symmetry breaking in the presence of strong correlations. To explore this, we study a prototypical Su-Schrieffer-Heeger model with finite-range interaction that gives rise to orbital charge density wave order and characterize the impact of this order on the model's topological properties. In the ordered phase, where the many-body topological invariant loses quantization, we find that not only is quantized charge pumping still possible, but it is even assisted by the collective nature of the orbital charge density wave order. Remarkably, we show that the Thouless pump scenario may be used to uncover the underlying topology of order-obstructed phases.

Figure 1

Schematic of correlation-assisted charge pumping. (a) Sketch of the correlation-assisted pump cycle in parameter space (solid line) and the effective parameter cycle in the absence of correlations (dashed line). Blue and orange backgrounds in the phase diagram denote the orbital character of the sublattice ordering (sublattices A and B, respectively). (b) Sketch of the order parameter over two correlation-assisted pump cycles which do not (top) and do (bottom) pump nonzero charge. Both cycles enter and exit the ordered phase, but only in the bottom cycle, corresponding to the cycle in (a), does the ordering change orbital character, leading to nonzero pumped charge.

Figure 2

(a) Phase diagram for the interacting SSH model with $d=0.4, \tau =1.0$. TI: topological insulator, BI: band insulator, OOBI: order-obstructed band insulator, OOTI: order-obstructed topological insulator. Red triangles denote mean-field theory results for the BI $\to$ TI phase boundary. Black squares denote finite-size scaling calculations from fitting the orbital CDW phase transition to the Ising universality class. The blue line delineates regions of single (right) and double (left) ground state degeneracy for OBC systems. Green crosses and the solid green line indicate the polarization phase boundary. Simulations were performed on systems with up to $L=100$ unit cells. Gray shading in the center represents the region of uncertainty given the accessible system sizes. (b) Typical band structures in each phase. Color signifies the orbital character with respect to the BI basis, and the level of transparency represents strength of band mixing.

Figure 3

Fractionalization in the symmetry-broken phase, with $J=0.1, d=0.4, \tau =1.0$. (a) Resta polarization $P$, computed using DMRG, and the Zak phase, computed from DMRG and MFT for $L=256$ unit cells. In the infinite $V$ limit in the symmetry-broken phase, $\mathcal{Z}\to 0$, and $P\to \pm q{a}_0/4$. (b) System-size dependence of the polarization. The polarization curve converges with system size but does not obey finite-size scaling. $P$ is in units of $q{a}_0/2$.

Figure 4

Correlation-assisted Thouless pump cycles. Left: sketch of the cycle in the phase diagram. Middle: order parameter along the cycle. Right: polarization along the cycle. Pump cycles within (a) OOBI and (b) OOTI phases. In (c) and (d), the system enters and exits the ordered phase. (d) successfully transports charge, but (c) does not. Shaded blue (orange) regions denote orbital ordering with occupation localized on sublattice $\mathrm{a}$ ($\mathrm{b}$), across which the order parameter attains peak magnitude in the thermodynamic limit. In numerics, (d) transports charge $\mathrm{\Delta }Q=-2.010\left(4\right)$, which is asymptotically quantized to $\mathrm{\Delta }Q\to 2$ in the thermodynamic limit. Each cycle was performed by adiabatically connecting 100 independent ground state calculations for systems of $L=100$ unit cells. Specific parameter values used to generate these results are detailed in Table I in the Supplemental Material [46].