Non-Hermiticity from non-reciprocal hoppings has been shown recently to demonstrate the non-Hermitian skin effect (NHSE) under open boundary conditions (OBCs). Here we study the interplay of this effect and the Anderson localization in a non-reciprocal quasiperiodic lattice, dubbed nonreciprocal Aubry-André model, and a rescaled transition point is exactly proved. The non-reciprocity can induce not only the NHSE, but also the asymmetry in localized states with two Lyapunov exponents for both sides. Meanwhile, this transition is also topological, characterized by a winding number associated with the complex eigenenergies under periodic boundary conditions (PBCs), establishing a bulk-bulk correspondence. This interplay can be realized by an elaborately designed electronic circuit with only linear passive RLC devices instead of elusive non-reciprocal ones, where the transport of a continuous wave undergoes a transition between insulating and amplifying. This initiative scheme can be immediately applied in experiments to other non-reciprocal models, and will definitely inspires the study of interplay of NHSEs and more other quantum/topological phenomena.

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We show how to define a dynamical topological invariant for general one-dimensional topological systems after a quantum quench. Focusing on two-band topological insulators, we demonstrate that the reduced momentum-time manifold can be viewed as a series of submanifold S 2 , and thus we are able to define a dynamical topological invariant on each of the sphere. We also unveil the intrinsic relation between the dynamical topological invariant and the difference of topological invariant of the initial and final static Hamiltonian. By considering some concrete examples, we illustrate the calculation of the dynamical topological invariant and its geometrical meaning explicitly.Introduction.-In the last decade, the study of topological quantum matter is one of the most attractive topic in condensed matter physics [1][2][3][4][5], and our knowledge of topological properties for various quantum systems has been widely expanded. In contrast to equilibrium systems, what we know about the topological quantum matter out of equilibrium is quite rare [6]. The topology properties far from equilibrium have been studied in different ways, such as the dynamics of edge states [7][8][9], dynamical quantum phase transition [10][11][12], Floquet topological states [13][14][15], etc. The rapid development of cold atom experiments provides a powerful tool to study the dynamics far from equilibrium [16][17][18][19], and the evolution of a quantum state can be visualized with the method of Bloch state tomography [20][21][22].