Experimental probes of emergent symmetries in the quantum Hall system

TL;DR

This paper reviews experimental evidence for emergent holomorphic modular symmetry in quantum Hall systems, showing that low-temperature data align with theoretical predictions and exploring dualities extending the symmetry.

Contribution

It demonstrates the agreement of experimental quantum critical points with $ ext{Gamma}_0(2)$ symmetry predictions and introduces extended dualities involving non-holomorphic particle-hole symmetry.

Findings

Quantum critical points match $ ext{Gamma}_0(2)$ predictions at low temperatures.

Experimental data supports duality relations derived from emergent modular symmetry.

Deviations at higher temperatures indicate experiments are not in the quantum critical regime.

Abstract

Experiments studying renormalization group flows in the quantum Hall system provide significant evidence for the existence of an emergent holomorphic modular symmetry $\Gamma_0(2)$. We briefly review this evidence and show that, for the lowest temperatures, the experimental determination of the position of the quantum critical points agrees to the parts \emph{per mille} level with the prediction from $\Gamma_0(2)$. We present evidence that experiments giving results that deviate substantially from the symmetry predictions are not cold enough to be in the quantum critical domain. We show how the modular symmetry extended by a non-holomorphic particle-hole duality leads to an extensive web of dualities related to those in plateau-insulator transitions, and we derive a formula relating dual pairs $(B,B_d)$ of magnetic field strengths across any transition. The experimental data obtained for the transition studied so far is in excellent agreement with the duality relations following from this emergent symmetry, and rule out the duality rule derived from the ``law of corresponding states". Comparing these generalized duality predictions with future experiments on other transitions should provide stringent tests of modular duality deep in the non-linear domain far from the quantum critical points.