Non-adiabatic effect on Laughlin’s argument of the quantum Hall effect

We have numerically studied a non-adiabatic charge transport in the quantum Hall system pumped by a magnetic flux, as one of the simplest theoretical realizations of nonadiabatic Thouless pumping. In the adiabatic limit, a pumped charge is quantized, known as Laughlin’s argument in a cylindrical lattice. In a uniform electric field, we obtained a formula connecting quantized pumping in the adiabatic limit and no-pumping in the sudden limit. The intermediate region between the two limits is determined by the Landau gap. A randomness or impurity effect is also discussed. In the paper by Laughlin[1], the quantum Hall system on a cylinder with two edges penetrated by an Aharonov-Bohm (AB) flux Φ is considered, where the flux changes from 0 to a flux quantum Φ0 = h/e adiabatically. The change of the AB flux enforces electrons to move form one edge to the other edge. In the adiabatic limit, net charge of transported electron is quantized due to a request from the gauge transformation, which is known as Laughlin’s argument in a cylindrical lattice. This topological aspect is the key feature of quantization of the Hall conductivity[2–4]. Although Laughlin’s argument gives a theoretical explanation for the quantum Hall effect, quantized charge transport is realized in various situations as Thouless pumping[5]. Thouless argued that an electron system in time dependent potential such as right moving potential U(x, t) = sin (2π(x/L− t/T )) with period T and L can pump a quantized charge in analogy with water pumping by Archimedean screw[6]. Recently, in mesoscopic system, electron pumping by adiabatic change of a cyclic potential U(x, t) has attracted much attention both experimentally[7] and theoretically[6]. Not only adiabatic pumping but also non-adiabatic pumping is also important and realized easily in experimental situation[8], which needs a extension of original Thouless pumping theoretically[9]. To increase net current induced by successive pumping, non-adiabatic pumping is thought to be more efficient because fast pumping transports more electrons[10]. In this paper, going back to the cylindrical system of Laughlin’s argument, we change the AB flux Φ non-adiabatically, i.e., non-adiabatic effect on edge-state pumping. For this purpose, we introduce a time-dependent flux Φ(t) = Φ0t/T with the period T . This system is one of the simplest theoretical realizations of non-adiabatic Thouless pumping. In addition, we shall study a square lattice penetrated by the flux Φ without boundary as shown in Fig. 1. There is no edge state at Φ = 0 and edge states induced by Φ. One of our motivations is to study how a edge state goes through Φ = Φ0 because the edge state comes across the second