Reservoir-induced Thouless pumping and symmetry-protected topological order in open quantum chains

Abstract

We introduce a classification scheme for 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 the topological properties of non-interacting and interacting closed and open systems as well 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 the analogous unitary system, the Rice-Mele model, symmetries enforce topological properties which lead to a non-trivial winding of the geometric Zak phase upon cyclic variations of model parameters. Associated with this is a winding of the many-body polarization, corresponding to a quantized transport in the bulk (Thouless pump). We here show that in the open system, where the Zak phase looses its meaning, the same symmetries enforce a winding of the generalized manybody polarization. This winding is shown to be robust against Hamiltonian perturbations as well as homogeneous dephasing and particle losses.

Topological bands for ultracold atoms

“…The concept of topological pumping can also be extended to the case of open systems, as shown by Linzner et al (2016) and Hu et al (2017). Using a proper engineering of the Liouvillian (108) of a fermionic 1D chain, Linzner et al (2016) could generalize the result of Sec. IV.C.1 for the Rice-Mele model, and prove the quantization of the variation of the many-body polarization after a closed loop in parameter space.…”

Uhlmann number in translational invariant systems

“…On the other hand, the fate of these topological ordered phases remains unclear, when a mixed state is the faithful description of the quantum system, either because of thermal equilibrium, or due to out-ofequilibrium conditions. Over the last few years, different attempts have been done to reconcile the above topological criteria with a mixed state configuration [16][17][18][19][20][21][22][23][24][25][26]. The recent success of the Uhlmann approach [27] in describing the topology of 1D Fermionic systems [18,19], remains in higher dimensions [20] not as straightforward [21].…”

Topological invariants in dissipative extensions of the Su-Schrieffer-Heeger model

“…(7c) is very similar to the one of the non-Hermitian Bloch Hamiltonian in Eq. (9). In case of unbroken PT symmetry, that is if all eigenvalues of H eff are real-valued, the complex Zak phase which is picked up by the mth Bloch band, described by the left and right eigenvectors χ m | and |φ m of the Bloch Hamiltonian associated with eigenvalue E m , is defined as [36,38]…”

“…In analogy with the line of argument of Hatsugai [39] for Hermitian systems we use the quantized real part of the complex Zak phase as topological invariant to characterize topological phases in the periodic lattice system described by Eq. (9). The quantization of the real part of the complex Zak phase is ensured by PT symmetry, and thus the corresponding topological phases are protected by PT symmetry (see App.…”