Universal Edge Transport in Interacting Hall Systems

Abstract

We study the edge transport properties of 2d interacting Hall systems, displaying singlemode chiral edge currents. For this class of many-body lattice models, including for instance the interacting Haldane model, we prove the quantization of the edge charge conductance and the bulk-edge correspondence. Instead, the edge Drude weight and the edge susceptibility are interaction-dependent; nevertheless, they satisfy exact universal scaling relations, in agreement with the chiral Luttinger liquid theory. Moreover, charge and spin excitations differ in their velocities, giving rise to the spin-charge separation phenomenon. The analysis is based on exact renormalization group methods, and on a combination of lattice and emergent Ward identities. The invariance of the emergent chiral anomaly under the renormalization group flow plays a crucial role in the proof. phases [35]. We prove the exact quantization of the edge conductance, for weak interactions: all interaction corrections cancel out. Combined with [27] and with the noninteracting bulk-edge correspondence [39,51,16], this result provides the first proof of the bulk-edge correspondence for an interacting many-body quantum system. Moreover, we also consider the edge Drude weight and the edge susceptibility, both for charge and spin degrees of freedom; we find explicit expressions for these quantities, which turn out to be nonuniversal in the coupling strength. Nevertheless, the Drude weight D and the susceptibility κ satisfy the universal scaling relation D " κv 2 c , as in the Luttinger model. Finally, we compute the two-point function, and we show that it exhibits spin-charge separation.Notice that our analysis does not extend in a straightforward way to the case of multi-edge currents. The reason being the scattering between different edge modes. In the renormalization group terminology, the edge states scattering is a marginal process; our method allows to control the scattering between edge states with the same velocity, thanks to the comparison with the chiral Luttinger model (see below), but does not allow to control the scattering of edge states with different velocities. We leave the generalization to multi-edge channels Hall systems as a very interesting open problem, on which we plan to come back in the future.The paper is organized as follows. In Section 2 we introduce the class of interacting lattice models we will consider, and we define bulk and edge transport coefficients, in the linear response regime. In Section 3 we recall some known facts about noninteracting Hall systems. Then, in Section 4 we present our main result, Theorem 4.1. In the rest of the paper, we discuss the proof of Theorem 4.1. In Section 5 we introduce a functional integral representation for fermionic lattice models, and in particular in Section 5.2 we derive a rigorous relationship between the model of interest and an interacting one-dimensional quantum field theory. This result actually applies to models with a general number of edge states. Starting from Section 6, we restr...

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