Criticality of Two-Dimensional Disordered Dirac Fermions in the Unitary Class and Universality of the Integer Quantum Hall Transition

Abstract

Two-dimensional (2D) Dirac fermions are a central paradigm of modern condensed matter physics, describing low-energy excitations in graphene, in certain classes of superconductors, and on surfaces of 3D topological insulators. At zero energy E ¼ 0, Dirac fermions with mass m are band insulators, with the Chern number jumping by unity at m ¼ 0. This observation lead Ludwig et al. [Phys. Rev. B 50, 7526 (1994)] to conjecture that the transition in 2D disordered Dirac fermions (DDF) and the integer quantum Hall transition (IQHT) are controlled by the same fixed point and possess the same universal critical properties. Given the far-reaching implications for the emerging field of the quantum anomalous Hall effect, modern condensed matter physics, and our general understanding of disordered critical points, it is surprising that this conjecture has never been tested numerically. Here, we report the results of extensive numerics on the phase diagram and criticality of 2D DDF in the unitary class. We find a critical line at m ¼ 0, with an energy-dependent localization length exponent. At large energies, our results for the DDF are consistent with state-of-the-art numerical results ν IQH ¼ 2.56-2.62 from models of the IQHT. At E ¼ 0, however, we obtain ν 0 ¼ 2.30-2.36 incompatible with ν IQH . This result challenges conjectured relations between different models of the IQHT, and several interpretations are discussed.

Fractional Chern insulators with a non-Landau level continuum limit

“…We also demonstrated how this is reflected in their finite-wavevector electromagnetic responses to an applied electric field. We introduced a transfer matrix and studied the electronic delocalization transition in the lowest level of the zeroquadratic model with quenched onsite disorder and found a localization-length critical exponent of 2.57(3), which is in close agreement that of the disordered Hofstadter model 2.58(3) [50], and in concurrence with localizationlength exponents for Landau levels and the IQHE [44][45][46][47][48][49][50][51][52][53][54][55]. This provides evidence that supports a broad universality of the IQHE localization-delocalization critical exponent independent of specific model parameters.…”

“…The value of this critical exponent has drawn much attention, with experimental values of approximately ν = 2.38 [42,43]. Theoretical values are less clear, with recent values between 2.37 and 2.62 being reported for different models [44][45][46][47][48][49][50][51][52][53][54][55]. A recent large-scale numerical study of the disordered Hofstadter model using the recursive Green's function method found ν = 2.58(3) [50].…”

Composite fermion nonlinear sigma models

“…The numerically calculated localization length critical exponent ν calc (varying from about 2.48-2.62 with the specific lattice model and calculation details [8][9][10][11][12][13][14][15][16]) do not appear to lie within the error bars of the measured exponent ν ≈ 2.38 [17,18]. (Some recent works are critical of these theoretical results: inclusion of additional types of disorder may be relevant [19] and/or the critical scaling regime may require significantly larger systems sizes [20]; see also [21,22].) Furthermore, the difference of the calculated and measured dynamical critical exponents z calc − z ≈ 1 [23].…”

Numerical evidence for marginal scaling at the integer quantum Hall transition

“…In a recent study, a collaboration involving the present authors obtained ν ≈ 2.33 (3) in models of disordered Dirac fermions at the nodal point energy. 19 Such disordered Dirac fermions were conjectured before to be in the IQHT universality class. 20 To sum up, the IQHT sets itself apart from other Anderson transitions in two ways: (i) A significant apparent variability of numerical estimates of ν across different models assumed to be in the same universality class; (ii) A very small (and possibly vanishing 10,14,15 ) leading irrelevant exponent |y|.…”

“…We also show that the proposed scaling is consistently found for a disordered 2d Dirac model. 19 Tuning away from criticality in Sec. V, we demonstrate how the marginal flow equations can mimic relevant scaling with an effective exponent ν eff , offering a new perspective on the variability of numerically determined ν discussed above.…”