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arXiv:1712.04942 · Physical review. B./Physical review. B

Non-Abelian bosonization and modular transformation approach to superuniversality

Aaron Hui, Eun-Ah Kim, Michael Mulligan

Published Mar 20, 2019• 24 citations
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Abstract

Quantum Hall inter-plateau transitions are physical exemplars of quantum phase transitions. Near each of these transitions, the measured electrical conductivity scales with the same correlation length and dynamical critical exponents, i.e., the critical points are superuniversal. In apparent contradiction to these experiments, prior theoretical studies of quantum Hall phase transitions within the framework of Abelian Chern-Simons theory coupled to matter found correlation length exponents that depend on the value of the quantum critical Hall conductivity. Here, we use non-Abelian bosonization and modular transformations to theoretically study the phenomenon of superuniversality. Specifically, we introduce a new effective theory that has an emergent $U(N)$ gauge symmetry with any $N>1$ for a quantum phase transition between an integer quantum Hall state and an insulator. We then use modular transformations to generate from this theory effective descriptions for transitions between a large class of fractional quantum Hall states whose quasiparticle excitations have Abelian statistics. We find the correlation length and dynamical critical exponents are independent of the particular transition within a controlled 't Hooft large $N$ expansion, i.e., superuniversal! We argue that this superuniversality could survive away from this controlled large $N$ limit using recent duality conjectures.

DOI
10.1103/physrevb.99.125135

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We explore the physics of the quantum Hall effect using the Haldane mapping of dimerized SU(N+M)\mathrm{SU}(N+M)SU(N+M) spin chains, the large NNN expansion, and the density matrix renormalization group technique. We show that while the transition is first order for N+M>2N+M>2N+M>2, the system at zero temperature nevertheless displays a continuously diverging length scale $\ensuremath{\xi}$ (correlation length). The numerical results for (N,M)=(3,1),(2,2),(5,1)(N,M)=(3,1),\phantom{\rule{4pt}{0ex}}(2,2),\phantom{\rule{4pt}{0ex}}(5,1)(N,M)=(3,1),(2,2),(5,1), and (7,1)(7,1)(7,1) indicate that $\ensuremath{\xi}$ is a directly observable physical quantity, namely the spatial width of the edge states. We relate the physical observables of the quantum spin chain to those of the quantum Hall system (and, hence, the $\ensuremath{\vartheta}$ vacuum concept in quantum field theory). Our numerical investigations provide strong evidence for the conjecture of superuniversality which says that the dimerized spin chain quite generally displays all the basic features of the quantum Hall effect, independent of the specific values of NNN and MMM. For the cases at hand we show that the singularity structure of the quantum Hall plateau transitions involves a universal function with two scale parameters that may in general depend on NNN and MMM. This includes not only the Hall conductance but also the ground state energy as well as the correlation length $\ensuremath{\xi}$ with varying values of $\ensuremath{\vartheta}\ensuremath{\sim}\ensuremath{\pi}$.
We suggest that a unified description of the integer and fractional phases of the quantum Hall system may be possible if the scaling diagram of transport coefficients is invariant under linear fractional (modular) transformations. In this model the hierarchy of states, as well as the observed universality of critical exponents, are consequences of a discrete SL(2,openZ) symmetry acting on the parameter space of an effective quantum-field theory. Available scaling data on the position of delocalization fixed points in the integer case and the position of mobility fixed points in the fractional case agree with the model within experimental accuracy.