Quantization of the Interacting Hall Conductivity in the Critical Regime

Abstract

The Haldane model is a paradigmatic 2d lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a 'topological' phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current-current correlations. Our method provides a full construction of the critical curves, which are modified ('dressed') by the electron-electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods. arXiv:1803.11213v1 [cond-mat.str-el]

Anomaly Non-renormalization in Interacting Weyl Semimetals

“…We also setĴ j, p ≡Ĵ j, p (0). A straightforward computation gives, letting 31) whereĴ j (k, p) should be understood as a 2 × 2 matrix, and…”

“…In connection with their emergent QFT description, the transport coefficients of these systems are characterized by remarkable universality properties: examples include the optical conductivity of graphene [60] and the Hall conductance [11,66]. A mathematical proof of universality in the presence of interactions has been established in [6,12,13,26,[28][29][30][31]39,57]. More recently, 3D Dirac systems have been experimentally realized [16,46,47,62], following the theoretical predictions of [63] and [24,59,65], see [9,38] for reviews; these semimetals are dubbed 'Dirac' or 'Weyl' semimetals, depending on whether the Fermi points coincide or are at distinct locations in the Brillouin zone.…”

A Supersymmetric Hierarchical Model for Weakly Disordered 3d Semimetals

“…as x − y → ∞. (1.6) It turns out that the reduced dimensionality of the Fermi surface allows to use RG methods to construct the low/zero temperature interacting Gibbs state of the model, both in two (corresponding to graphene-like systems) and in three dimensions, and to prove universality results for transport coefficients; see [45][46][47][48][49]60,61]. We refer the reader to [58] for a review of recent applications of rigorous RG methods interacting condensed matter systems.…”

Universal Edge Transport in Interacting Hall Systems

“…The proof of [27] works provided the interaction strength is smaller than the gap in the single-particle spectrum. More recently, the strategy of [27] has been improved in [28,29], to prove quantization of the Hall conductivity arbitrarily close to criticality, for models displaying conical intersections in the spectrum at the critical point, like the Haldane-Hubbard model. The same methods can be used to prove the universality of the longitudinal conductivity for interacting graphene-like models, [26], whose spectrum is gapless.…”

“…The same methods can be used to prove the universality of the longitudinal conductivity for interacting graphene-like models, [26], whose spectrum is gapless. Thus, the main advantage of the field-theoretic methods of [26,28,29] is that they allow to prove universality of transport coefficients without assuming the presence of a gap in the spectrum.…”