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arXiv:1504.03729 · Physical Review B

Logarithmic correlations in quantum Hall plateau transitions

Romain Vasseur

Published Jul 15, 2015• 5 citations
arXivHTMLPDF

Abstract

The critical behavior of quantum Hall transitions in two-dimensional disordered electronic systems can be described by a class of non-unitary conformal field theories with logarithmic correlations. Using the replica trick and the underlying symmetries of these quantum critical points, the author shows how to construct non-perturbatively disorder-averaged observables in terms of Green's functions that scale logarithmically at criticality.

DOI
10.1103/physrevb.92.014205

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The critical behavior of quantum Hall transitions in two-dimensional disordered electronic systems can be described by a class of non-unitary conformal field theories with logarithmic correlations. Using the replica trick and the underlying symmetries of these quantum critical points, the author shows how to construct non-perturbatively disorder-averaged observables in terms of Green's functions that scale logarithmically at criticality.
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Conformal invariance often accompanies criticality in Hermitian systems. However, its fate in non-Hermitian settings is less clear, especially near exceptional points where the Hamiltonian becomes non-diagonalizable. Here we investigate whether a 1+1-dimensional gapless non-Hermitian system can admit a conformal description, focusing on a PT-symmetric free-fermion field theory. Working in the biorthogonal formalism, we identify the conformal structure of this theory by constructing a traceless energy-momentum tensor whose Fourier modes generate a Virasoro algebra with central charge c=−2c=-2c=−2. This yields a non-Hermitian, biorthogonal realization of a logarithmic conformal field theory, in which correlation functions exhibit logarithmic scaling and the spectrum forms Virasoro staggered modules that are characterized by universal indecomposability parameters. We further present a microscopic construction and show how the same conformal data (with finite-size corrections) can be extracted from the lattice model at exceptional-point criticality, thereby supporting the field-theory prediction.
The fermion disorder operator has been shown to reveal the entanglement information in 1D Luttinger liquids and 2D free and interacting Fermi and non-Fermi liquids emerging at quantum critical points (QCPs) [W. Jiang et al., arXiv:2209.07103]. Here we study, by means of large-scale quantum Monte Carlo simulation, the scaling behavior of the disorder operator in correlated Dirac systems. We first demonstrate the logarithmic scaling behavior of the disorder operator at the Gross-Neveu (GN) chiral Ising and Heisenberg QCPs, where consistent conformal field theory (CFT) content of the GN-QCP in its coefficient is found. Then we study a 2D monopole-free deconfined quantum critical point (DQCP) realized between a quantum-spin Hall insulator and a superconductor. Our data point to negative values of the logarithmic coefficients such that the DQCP does not correspond to a unitary CFT. Density matrix renormalization group calculations of the disorder operator on a 1D DQCP model also detect emergent continuous symmetries.
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We construct an effective-field theory for the quantum Hall system which embodies both localization and fractional statistics. The latter involves a Chern-Simons interaction, while the former involves a generalization of conventional localization theory. The theory is invariant under ``complexified'' duality transformations of the conductivities which appear as effective parameters of the model. By exploiting these parameter space symmetries, as well as the conformal symmetry which appears at renormalization-group fixed points, we are able to extract a precise prediction for the whole scaling diagram. It exhibits both fractional and integer phases, the exact location of all fixed points, and universal scaling exponents. With a plausible identification of the universality class of the theory in the replica limit, the value of the critical exponent for the delocalization transition between plateaus in the Hall conductivity is found to be 7/3, in apparent agreement with available scaling experiments.
We study the spin quantum Hall effect and transitions between Hall plateaus in quasi-two-dimensional network models consisting of several coupled layers. Systems exhibiting the spin quantum Hall effect belong to class C in the symmetry classification for Anderson localization, and for network models in this class there is an established mapping between the quantum problem and a classical one involving random walks. This mapping permits numerical studies of plateau transitions in much larger samples than for other symmetry classes, and we use it to examine localization in systems consisting of nnn weakly coupled layers. Standard scaling ideas lead one to expect nnn distinct plateau transitions, but in the case of the unitary symmetry class this conclusion has been questioned. Focusing on a two-layer model, we demonstrate that there are two separate plateau transitions, with the same critical properties as in a single-layer model, even for very weak interlayer coupling.