Search for long-living topological solutions of the nonlinear φ⁴ field theory

We look for long-living topological solutions of classical nonlinear $(1+1)-$ dimensional $\varphi^4$ field theory. To that effect we use the well-known cut-and-match method. In this framework, new long-living states are obtained in both topological sectors. In particular, in one case a highly excited state of a kink is found. We discover several ways of energy reset. In addition to the expected emission wave packets (with small amplitude), for some selected initial conditions the production of kink-antikink pairs results in a large energy reset. Also, the topological number of a kink in the central region changes in the contrast of saving full topological number. At lower excitation energies there is a long-living excited vibrational state of the kink; this phenomenon is the final stage of all considered initial states. Over time this excited state of the kink changes to a well-known linearized solution - a discrete kink excitation mode. This method yields a qualitatively new way to describe the large-amplitude bion, which was detected earlier in the kink-scattering processes in the nontopological sector.Comment: 13 pages, 11 figure

Visualizing the connection between edge states and the mobility edge in adiabatic and nonadiabatic topological charge transport

The ability to pump quantized amounts of charge is one of the hallmarks of topological materials. An archetypical example is Laughlin's gauge argument for transporting an integer number of electrons between the edges of a quantum Hall cylinder upon insertion of a magnetic flux quantum. This is mathematically equivalent to the equally famous suggestion of Thouless that an integer number of electrons is pumped between two ends of a one-dimensional quantum wire upon sliding a charge-density wave over a single wavelength. We use the correspondence between these descriptions to visualize the detailed dynamics of the electron flow during a single pumping cycle, which is difficult to do directly in the quantum Hall setup because of the gauge freedom inherent in its description. We find a close correspondence between topological edge states and the mobility edges in charge-density wave, quantum Hall, and other topological systems. We illustrate this connection by describing an alternative, nonadiabatic mode of topological transport that displaces precisely the opposite amount of charge compared to the adiabatic pump. We discuss possible experimental realizations in the context of ultracold atoms and photonic waveguide experiments

Visualizing topological transport

We present a mathematically simple procedure for explaining and visualizing the dynamics of quantized transport in topological insulators. The procedure serves to illustrate and clarify the dynamics of topological transport in general, but for the sake of concreteness, it is phrased here in terms of electron transport in a charge-ordered chain, which may be mapped exactly onto transport between edge channels in the Integer Quantum Hall Effect. This approach has the advantage that it allows a direct visualization of the real-space and real-time evolution of the electronic charges throughout the topological pumping cycle, thus demystifying how charge flows between remote edges separated by an insulating bulk, why the amount of transported charge is given by a topological invariant, and how continuous driving yields a discrete, quantized amount of transported charge

Visualizing topological transport

We present a mathematically simple procedure for explaining and visualizing the dynamics of quantized transport in topological insulators. The procedure serves to illustrate and clarify the dynamics of topological transport in general, but for the sake of concreteness, it is phrased here in terms of electron transport in a charge-ordered chain, which may be mapped exactly onto transport between edge channels in the Integer Quantum Hall Effect. This approach has the advantage that it allows a direct visualization of the real-space and real-time evolution of the electronic charges throughout the topological pumping cycle, thus demystifying how charge flows between remote edges separated by an insulating bulk, why the amount of transported charge is given by a topological invariant, and how continuous driving yields a discrete, quantized amount of transported charge