Large Scale Response of Gapless $1d$ and Quasi-$1d$ Systems

TL;DR

This paper rigorously analyzes the transport properties of gapless 1D and quasi-1D quantum systems under slow, weak perturbations, proving the validity of the Kubo formula and deriving explicit response functions.

Contribution

It provides a rigorous proof of the Kubo formula for a class of perturbations in gapless 1D systems, connecting response functions to lattice conservation laws and quantized conductance.

Findings

Validation of Kubo formula in the zero temperature, infinite volume limit.

Explicit form of the leading response function derived.

Proof of quantized edge conductance in 2D quantum Hall systems.

Abstract

We consider the transport properties of non-interacting, gapless one-dimensional quantum systems and of the edge modes of two-dimensional topological insulators, in the presence of time-dependent perturbations. We prove the validity of Kubo formula, in the zero temperature and infinite volume limit, for a class of perturbations that are weak and slowly varying in space and in time, in an Euler-like scaling. The proof relies on the representation of the real time Duhamel series in imaginary time, which allows to prove its convergence uniformly in the scaling parameter and in the size of system, at low temperatures. Furthermore, it allows to exploit a suitable cancellation for the scaling limit of the model, related to the emergent anomalous chiral gauge symmetry of relativistic one-dimensional fermions. The cancellation implies that, as the temperature and the scaling parameter are sent to zero, the linear response is the only contribution to the full response of the system. The explicit form of the leading contribution to the response function is determined by lattice conservation laws. In particular, the method allows to prove the quantization of the edge conductance of $2d$ quantum Hall systems from quantum dynamics.