On the self-similarity in quantum Hall systems

Abstract

Abstract. -The Hall-resistance curve of a two-dimensional electron system in the presence of a strong perpendicular magnetic field is an example of self-similarity. It reveals plateaus at low temperatures and has a fractal structure. We show that this fractal structure emerges naturally in the Hamiltonian formulation of composite fermions. After a set of transformations on the electronic model, we show that the model, which describes interacting composite fermions in a partially filled energy level, is self-similar. This mathematical property allows for the construction of a basis of higher generations of composite fermions. The collective-excitation dispersion of the recently observed 4/11 fractional-quantum-Hall state is discussed within the present formalism.A branch of a snow flake observed under a microscope appears to be almost the same as the macroscopic flake. This is one of the most prominent examples of self-similarity and fractal structure in nature [1]. Another example of this phenomenon is the curve of the transverse Hall resistance of a two-dimensional (2D) electron system in the presence of a strong perpendicular magnetic field [2]: at low temperatures, the primary sequence of plateaus in the Hall resistance, which occur at specific values of the magnetic field, repeats itself qualitatively in a different field range (see Fig. 1). The phenomenon of plateau formation in the Hall resistance is known as the quantum Hall effect. One distinguishes the integral and the fractional quantum Hall effect (IQHE and FQHE). The IQHE [3] is found at lower magnetic fields B and is a manifestation of the quantisation of the electron energy, with charge −e, into equidistant energy levels, called Landau levels (LLs). The level degeneracy is characterised by the flux density n B = eB/h, and the filling of the levels is given by ν = n el /n B , where n el is the electron density. When n LLs are completely filled (ν = n), one finds the IQHE: when the magnetic field is lowered, the number of states per level is reduced, and some electrons are therefore promoted to a higher LL. These electrons become localised due to residual impurities in the sample and thus do not contribute to the electrical transport. As a consequence, the Hall resistance is not sensitive to the change in the magnetic field and remains at its original quantised value R H = h/e 2 n. This gives rise to a plateau when the Hall resistance is plotted as a function of B. The FQHE is found at fractional values of the filling factor, i.e. when a LL -mostly the lowest -is only partially filled [4]. This effect is due to the Coulomb interaction between the electrons, c EDP Sciences

Quantum Hall system in Tao-Thouless limit

“…5 is a fractal, self-similar, one: enlarging any region of ν reproduces the original figure [71,72]. This fractal structure is connected to the hierarchy construction of fractional QH states.…”

Design and characterization of electrons in a fractal geometry

“…Fractals have been investigated in a wide variety of research areas, ranging from polymers 5 , porous systems 6 , electrical storage 7 and stretchable electronics 8 down to molecular 5 , 9 – 11 and plasmonic 12 fractals. On the quantum level, fractal properties emerge in the behavior of electrons under perpendicular magnetic fields, for example in the Hofstadter butterfly 13 and in quantum Hall resistivity 14 , 15 . In addition, a multi-fractal behavior has been observed for the wave functions at the transition from a localized to delocalized regime in disordered electronic systems 16 – 18 .…”

Transition from Free to Interacting Composite Fermions away fromν=1/3

“…The experimental evidence for such higher order states bring into question the validity of the weakly interacting CF framework [4]. While these higher order fractions were actually predicted in the earlier hierarchical model [5,6], it was argued that the CF model may account for these new fractions when CF residual interactions are incorporated beyond the mean-field level [7,8,9,10,11,12,13,14].…”