Multi-channel Luttinger Liquids at the Edge of Quantum Hall Systems

Abstract

We consider the edge transport properties of a generic class of interacting quantum Hall systems on a cylinder, in the infinite volume and zero temperature limit. We prove that the large-scale behavior of the edge correlation functions is effectively described by the multi-channel Luttinger model. In particular, we prove that the edge conductance is universal, and equal to the sum of the chiralities of the non-interacting edge modes. The proof is based on rigorous renormalization group methods, that allow to fully take into account the effect of backscattering at the edge. Universality arises as a consequence of the integrability of the emergent multi-channel Luttinger liquid combined with lattice Ward identities for the microscopic 2d theory.

Anomaly Non‐Renormalization, Lattice QFT, and Universality of Transport Coefficients

“…[71], extending a previous study in the continuum in ref. [76], shows that the rcc corresponding to the fermion-boson interaction is essentially not flowing e h ≈ e in contrast with Equation (88). A lowest order computation of the conductivity is in ref.…”

“…Similar ideas have been used to establish the bulk‐edge correspondence in Hall insulators in presence of interaction. [ 88 ] Even when at the edge of Hall insulators there are fermions with both chiralities, which interact with short range potential, the conductivity is non‐renormalized.…”

Adiabatic Evolution of Low-Temperature Many-Body Systems

“…The proof is based on a rigorous Wick rotation, which makes it possible to rewrite the Duhamel expansion for the quantum evolution of the system in terms of time-ordered, Euclidean (or imaginary-time) connected correlation functions. Previously, this idea has been used to rigorously study the linear and quadratic response in a number of interacting gapped or gapless systems [ 5 , 26 , 29 , 42 ]. Here, we extend this strategy at all orders in the Duhamel expansion for the time-evolution of the state, and we use it to prove convergence of the Duhamel series for the real-time dynamics.…”

“…Specifically, the combination of cluster expansion with rigorous renormalization group recently allowed to study the low temperature properties of a wide class of interacting gapless systems, and in particular to access their transport coefficients defined in the framework of linear response. Among the recent works, we mention the construction of the ground state of the two-dimensional Hubbard model on the honeycomb lattice [ 24 ] and the proof of universality of the longitudinal conductivity of graphene [ 25 ]; the construction of the topological phase diagram of the Haldane-Hubbard model [ 27 , 28 ]; the proof of the non-renormalization for the chiral anomaly of Weyl semimetals [ 29 ]; the proof of Luttinger liquid behavior for interacting edge modes of two-dimensional topological insulators and the proof of universality of edge conductance [ 5 , 41 , 42 ]. It would be very interesting to prove the validity of linear response in the setting considered in these works, starting from many-body quantum dynamics.…”