Numerical evidence for marginal scaling at the integer quantum Hall transition

Abstract

The integer quantum Hall transition (IQHT) is one of the most mysterious members of the family of Anderson transitions. Since the 1980s, the scaling flow close to the critical fixed point in the parameter plane spanned by the longitudinal and Hall conductivities has been studied vigorously both by experiments and with numerical simulations. Despite all efforts, it is notoriously difficult to pin down the precise values of critical exponents, which seem to vary with model details and thus challenge the principle of universality. Recently, M. Zirnbauer [Nucl. Phys. B 941, 458 (2019)] has conjectured a conformal field theory for the transition, in which linear terms in the beta-functions vanish, leading to a very slow flow in the fixed point's vicinity which we term marginal scaling. In this work, we provide numerical evidence for such a scenario by using extensive simulations of various network models of the IQHT at unprecedented length scales. At criticality, we confirm the marginal scaling of the longitudinal conductivity towards the conjectured fixed-point value σ * xx = 2/π. Away from criticality we describe a mechanism that could account for the emergence of an effective critical exponents ν eff , which is necessarily dependent on the parameters of the model. We confirm this idea by exact numerical determination of ν eff in suitably chosen models.

Localization of Dirac Fermions in Finite-Temperature Gauge Theory

“…A better understanding of the field theory describing the critical point was obtained only recently, in terms of a conformal field theory deformed only by marginal perturbations, that emerge from the spontaneous breaking of the replica (super)symmetry of the relevant nonlinear sigma model [159]. This proposal is quantitatively supported by numerical results, and can explain the apparent numerical discrepancies [158]. A transition between localized and delocalized modes was observed in the two-dimensional unitary Anderson model [160].…”

“…While numerical evidence qualitatively supported this idea [157], significant quantitative discrepancies between different microscopic models were observed (see Ref. [158] for a summary), in contrast with the expected universality of the transition. A better understanding of the field theory describing the critical point was obtained only recently, in terms of a conformal field theory deformed only by marginal perturbations, that emerge from the spontaneous breaking of the replica (super)symmetry of the relevant nonlinear sigma model [159].…”

How spectrum-wide quantum criticality protects surface states of topological superconductors from Anderson localization: Quantum Hall plateau transitions (almost) all the way down

“…The longitudinal conductivity is known to be σ xx IQHPT 0.58 ± 0.02 [58] from numerical Kubo computations. Excitingly, the IQHPT transition is conjectured to itself be described by a class AIII n = 4 WZNW CFT [90,91]. If true, this would correspond to a generalized version of Haldane's conjecture for one-dimensional halfinteger spin quantum antiferromagnets.…”

Marginal CFT perturbations at the integer quantum Hall transition

“…Based on that belief, many authors have calculated the critical indices ν and y by numerical simulation of various models -we will not go into any details here but simply offer an incomplete list of selected references: [16,30,2,26,35,28]. A short synopsis is that the published results for y lie in the range of −1 < y ≤ 0 and for ν between 2.34 and 2.61 (be informed, however, that a very recent study of a two-channel network model by Dresselhaus, Sbierski, and Gruzberg [8] finds values for ν as large as 3.4 and even 3.9).…”

“…So far, we have not used any property of O(w, w) other than locality. Now, we recall the specific form (39) of O(0) (using invariance under spatial translations to set w = w = 0 for notational simplicity) and compute the left-hand side of (42) from the expression (8) for the current J:…”