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.
Highlights
One of the remarkable features of the Integer Quantum Hall Effect (QHE) is the impressive precision of the quantization of the plateaus observed in the experiments
[12,24,30], or on the properties of anomalies [15], offered an alternative view on the QHE: they indicated that quantization should persists in the presence of many body interaction, but such conclusions were based on manipulations of divergent series, or of effective actions arising in a formal scaling limit
In Lemma 3.4, by differentiating the Ward Identities associated with the continuity equation, and by combining the result with the Schwinger–Dyson equation, we show that the Euclidean Hall conductivity of HR is constant in U, provided that ξ(U, mR,−, φ), δ(U, mR,−, φ) are differentiable in U and that the Fourier transform of the Euclidean correlation functions is smooth in the momenta, for any fixed mR,− = 0
Summary
One of the remarkable features of the Integer Quantum Hall Effect (QHE) is the impressive precision of the quantization of the plateaus observed in the experiments. The bulk-edge correspondence for a class of weakly interacting fermionic systems displaying single-mode chiral edge currents was proved [2] Given these results on the quantization of the Hall conductivity in weakly interacting systems (i.e., with interaction strength smaller than the gap), one naturally wonders what happens for stronger interactions. The interaction may be marginal, as in the case of the anisotropic Hofstadter model, recently considered in [33]: in this case, the gaps with integer label are stable, but new gaps corresponding to fractional labels are expected to open It would be, very interesting to further investigate such cases, where fractional Hall conductances may potentially appear, as well as to include disorder effects, which are essential for the very existence of Hall plateaus.
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