We introduce a classification of symmetry-protected topological phases applicable to stationary states of open systems based on a generalization of the many-body polarization. The polarization can be used to probe topological properties of noninteracting and interacting closed and open systems and remains a meaningful quantity even in the presence of moderate particle-number fluctuations. As examples, we discuss two open-system versions of a topological Thouless pump in the steady state of one-dimensional lattices driven by Markovian reservoirs. In an analogous unitary system, the Su-Shrieffer-Heeger model, symmetries enforce a quantization of the geometric Zak phase, which acts as a topological invariant. Introducing a further degree of freedom, a nontrivial winding of the phase can be observed upon cyclic variations of parameters. Associated with this is a quantization or, respectively, a winding of the polarization corresponding to a quantized transport (Thouless pump). We here show that in the open system, where the Zak phase loses its meaning, the same symmetries enforce a quantization and more generally a winding of the generalized many-body polarization. These features are shown to be robust against Hamiltonian perturbations as well as homogeneous dephasing and particle losses.
By means of time-dependent density-matrix renormalization-group calculations, we study topological quantum pumping in a strongly interacting system. The system under consideration is described by the Hamiltonian of a one-dimensional extended Bose-Hubbard model in the presence of a correlated hopping which breaks lattice inversion symmetry. This model has been predicted to support topological pumping [E. Berg, M. Levin, and E. Altman, Phys. Rev. Lett. 106, 110405 (2011)]. The pumped charge is quantized and of a topological nature. We provide a detailed analysis of the finite-size scaling behavior of the pumped charge and its deviations from the quantized value. Furthermore, we also analyze the nonadiabatic corrections due to the finite frequency of the modulation. We consider two configurations: a closed ring where the time dependence of the parameter induces a circulating current and a finite open-ended chain where particles are dragged from one edge to the opposite edge, due to the pumping mechanism induced by the bulk.
Quantized charge pumping in quantum Hall systems is a bulk topological property, typically manifesting on gapless boundaries as spectral flow through chiral edge modes upon flux insertion. When the boundary is anomaly-free, it can be symmetrically gapped by symmetric mass generation (SMG) interactions. However, the effects on quantized charge pumping at an interacting gapped boundary are unclear. We address this using density-matrix renormalization group simulations of a four-leg ladder quasi-(2+1)d realization of the (1+1)d 3-4-5-0 lattice chiral fermion model with six-fermion interactions at one edge. We find gapless modes near the interacting edge for both weak and strong couplings, but the edge can be in a fully gapped SMG phase at intermediate coupling. We also identify spontaneous breaking of the model's
Z2 symmetry above a critical interaction strength. Across all three boundary phases, we find quantized charge pumping with the same pumped charge per flux quantum. The spatial distribution of the pumped charge varies between phases. Besides charge pumping, we study how a single injected fermion propagates under open boundary conditions. In the SMG phase, an interaction-induced flavor-mixing current develops at the endpoints of the gapped edge, which is absent in the weak- and strong-coupling limits. This current reveals a charge-redistribution mechanism unique to the SMG phase. Additionally, we propose an analogy between charge pumping in this model and a topological leptogenesis scenario in the (3+1)d Standard Model. The anomaly-free chiral U(1) symmetries of the 3-4-5-0 model play a role analogous to the baryon-minus-lepton symmetry of the Standard Model. In both settings, charge pumping can be associated with a continuous deformation along a classically defined but quantum-anomalous direction, while a discrete subgroup remains an exact quantum symmetry.