Abstract
The new states of matter and concomitant quantum critical phenomena re- vealed by the quantum Hall effect appear to be accompanied by an emergent modular symmetry. The extreme rigidity of this infinite symmetry makes it easy to falsify, but two decades of experiments have failed to do so, and the location of quantum critical points predicted by the symmetry is in increasingly accurate agreement with scaling ex- periments. The symmetry severely constrains the structure of the effective quantum field theory that encodes the low energy limit of quantum electrodynamics of 10 10 charges in two dirty dimensions. If this is a non-linear σ-model the target space is a torus, rather than the more familiar sphere. One of the simplest toroidal models gives a critical (cor- relation length) exponent that agrees with the value obtained from numerical simulations of the quantum Hall effect.
Highlights
The new states of matter and concomitant quantum critical phenomena revealed by the quantum Hall effect appear to be accompanied by an emergent modular symmetry
One of the earliest and most successful methods developed to circumvent this problem used to be called phenomenology, a precursor of what today is called effective field theory (EFT) [5]. The essence of this idea is to use some of the global properties of the theory — in the case of low energy quantum chromodynamics (QCD) the geometry of flavour symmetries that are observed in the hadronic spectrum
In our discussion above of infinite symmetries we immediately focused on holomorphic Möbius functions γ(z) =/(cz + d) ∈ C, both because these are intimately related to the number theory apparent in the quantum Hall effect (QHE), and because they carry the simplest representations of modular groups, which are the prototypes of these symmetries
Summary
A fact that never fails to baffle our students is that a charge (Q) moving in a magnetic field (B) feels a force that is orthogonal to both the field and the velocity (v) of the charge. Measured in the fundamental unit of resistance RK = h/e2 = 25.812807557(18) kΩ ≈ 26 kΩ (h is Planck’s constant, and e is the electron charge), the steps observed by von Klitzing are quantized at inverse integer values (IQHE): RH = 1/p [RK] (p ∈ Z) These integers can be understood as “spectroscopy" of Landau levels, but the second big surprise is how accurate this quantisation is, essentially exact (parts per billion) with current technology. The fractional quantum Hall effect (FQHE) was totally unexpected and heralded a brave new world of quantum weirdness The importance of these discoveries was immediately recognised by the community, who rewarded von Klitzing, Laughlin, Störmer and Tsui with Nobel Prizes in 1985 and 1998. Three decades after their discovery we are still struggling to understand these new phases of condensed matter, the quantum phase transitions separating them, and the effective quantum field theory needed to understand the scaling, universality and critical properties of these systems
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