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

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.

Figure 1

(a) Open-system analog of the Rice-Mele model: One-dimensional fermion or spin chain with alternating pairwise coupling to two different Markovian reservoirs (Lindblad generators $L_j^A$ and $L_j^B)$. The unit cell labeled by “$j$” contains a left “$L$” (blue) and a right “$R$” (green) lattice site. (b) The many-body polarization shows a nontrivial winding, here in the fermionic model (see text), upon cyclic variations of parameters of the Liouvillian. On the axes with inversion symmetry (blue and green) the change in polarization is strictly quantized $\mathrm{\Delta }P=0,\pm 1/2$. On the special path along the periphery (red), the steady state is pure.

Figure 2

Many-body polarization under slow variation of the parameters evaluated along a circular path (red) as indicated in (a) with radius $R=\sqrt{{\varepsilon }^2+{\lambda }^2}=1$. Different perturbations are considered numerically, using TEBD (solid) for the spin chain $(L=16$ unit cells) or exact diagonalization for the four-site periodic fermionic chain $(L=2$ unit cells) (dashed): (b) spatially random magnetic field in $z$ direction $V/\mathrm{\Gamma }=1$ (disorder average consisting of 50 samples). (c) local dephasing with homogeneous losses and pumping both with rates ${\gamma }_{\perp }/\mathrm{\Gamma }=0.5$, where ${\gamma }_{\parallel }={\gamma }_{\perp }$.

Figure 3

(a) purity spectrum of fermionic model. (b) Dominant values of the spectrum $\sigma \left({\rho }_{\mathrm{ss}}\right)$ of the steady-state density matrix of the spin model for additional homogeneous dephasing ${\gamma }_{\perp }=0.5\mathrm{\Gamma }$ at $\varepsilon =0$ obtained from exact diagonalization of a four-site problem.