An RG potential for the quantum Hall effects

Abstract

The phenomenological analysis of fully spin-polarized quantum Hall systems, based on holomorphic modular symmetries of the renormalization group (RG) flow, is generalized to more complicated situations where the spin or other "flavors" of charge carriers are relevant, and where the symmetry is different. We make the simplest possible ansatz for a family of RG potentials that can interpolate between these symmetries. It is parametrized by a single number $a$ and we show that this suffices to account for almost all scaling data obtained to date. The potential is always symmetric under the main congruence group at level two, and when $a$ takes certain values this symmetry is enhanced to one of the maximal subgroups of the modular group. We compute the covariant RG $\beta$-function, which is a holomorphic vector field derived from the potential, and compare the geometry of this gradient flow with available temperature driven scaling data. The value of $a$ is determined from experiment by finding the location of a quantum critical point, i.e., an unstable zero of the $\beta$-function given by a saddle point of the RG potential. The data are consistent with $a\in \mathbb{R}$, which together with the symmetry leads to a generalized semi-circle law.

10 figures, sligthly updated discussion and refs, accepted for PRB

References in corpus (15)

• Boron nitride substrates for high-quality graphene electronics
• Landau Level Splitting in Graphene in High Magnetic Fields
• Multicomponent fractional quantum Hall effect in graphene
• Spontaneous Symmetry Breaking and Quantum Hall Effect in Graphene
• Conformal Invariance and Shape-Dependent Conductance of Graphene Samples
• Graphene, universality of the quantum Hall effect and redefinition of the SI system
• Metal to insulator transition on the N = 0 Landau level in graphene
• The Quantum Hall Effect in Graphene: Emergent Modular Symmetry and the Semi-circle Law
• Geometric scaling in the quantum Hall system
• Modular symmetry and temperature flow of conductivities in quantum Hall systems with varying Zeeman energy
• An experimental study on $\Gamma$(2) modular symmetry in the quantum Hall system with a small spin-splitting
• Duality, the Semi-Circle Law and Quantum Hall Bilayers
• Experimental probes of emergent symmetries in the quantum Hall system
• Holomorphic and anti-holomorphic conductivity flows in the quantum Hall effect
• On the origin of duality in the quantum Hall system

Cited by in corpus (2)

• Probing temperature-driven flow lines in a gated two-dimensional electron gas with tunable spin-splitting
• Elliptic mirror of the quantum Hall effect