Study of Scaling in the Fractional Quantum Hall Effect

Abstract

In the composite fermion model of the fractional quantum Hall e ect, composite fermions experience, in addition to the usual potential disorder, also a magnetic ux disorder. Motivated by this, we investigate the localization properties of a single fermion in two dimensions, moving in the presence of both potential and magnetic ux disorders, but with a non-zero average magnetic eld. It is found that the exponent characterizing the divergence of the localization length is not changed upon the addition of the ux disorder, provided it is not too large, suggesting that the transitions between fractionally quantized Hall plateaus belong to the same universality class as those between the integrally quantized ones. PACS numbers: 73.40.Hm In the absence of a magnetic eld, an arbitrary weak random potential localizes all states of an electron in two dimensions 1]. In contrast, existence of extended states, separated by bands of localized states, is crucial for the explanation of the quantum Hall e ect 2]. Numerical studies 3,4] have made a convincing case that the localization length of an electron, con ned to two dimensions and exposed to a strong magnetic eld, diverges at a single energy near the center of each Landau level, and the divergence is characterized by an exponent := jE E c j ;(1) where 7=3, independent of the Landau level (LL) index. No analytical derivation of the value of the exponent yet exists. Experimentally, the width of the transition region (e.g., the width of the conductance peak) in the integer quantum Hall regime (IQHE) regime has been found to vanish in a range of temperature as T , where 0:42, independent of the LL in question 5]. This is consistent with the theoretical value of in a variable range hopping theory 6], which obtains = 1= (in other words, writing = 1=z , this work shows that the dynamical exponent z is equal to unity for the Coulomb interaction). Direct measurements of the localization length are also in agreement with7=3 7]. At rst glance, the nature of localization would appear to be much more complex in the fractional quantum Hall e ect (FQHE) regime, since interactions are an intimate part of the physics here and cannot be neglected. However, in the composite fermion model 8], the essential part of the interelectron interaction is incorporated through the formation of composite fermions, which themselves can be treated as non-interacting to a good approximation. An e ective single-particle description of the strongly correlated electron state thus becomes possible in terms of composite fermions. This paper studies the scaling of the localization length in the FQHE regime in the framework of a non-interacting composite fermion model. While the dynamics of composite fermions is similar to that of electrons in a weaker magnetic eld, an essential di erence is that the composite fermions also experience a random ux disorder in addition to the usual random potential disorder 9{11]. The main conclusion of the paper is that the exponent is the same for non-interacting composite fer...

Signatures of spin in theν=1∕3fractional quantum Hall effect

“…The latter is due to fluctuations in the density which, via the flux attachment, produce fluctuations in the effective magnetic field felt by CF. It has been shown 27 that the scaling law ͑3͒ holds for CF as for electrons. The main difference is in the scaling of ⌫ due to the two scattering mechanisms.…”