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

Marcello Porta, Harman Preet Singh

TL;DR

The paper provides a rigorous foundation for linear response in gapless quantum systems by proving the zero-temperature, infinite-volume validity of the Kubo formula for 1d gapless fermions and for edge modes of 2d topological insulators under weak, slowly varying perturbations. The authors develop an Euler-like scaling framework and reformulate the real-time Duhamel series in terms of imaginary-time Euclidean correlations, enabling uniform control of higher-order terms via loop cancellations tied to emergent chiral symmetry. A central outcome is an explicit leading-order expression for the density and current response, governed solely by the Fermi velocities, with subleading corrections vanishing in the scaling limit; this yields the quantization of edge conductance from quantum dynamics. The approach blends rigorous Duhamel/Wick-rotation techniques with careful estimates of auxiliary dynamics and uses bosonization-inspired cancellations to simplify the perturbation theory without assuming integrability, offering a pathway to extend to weakly interacting gapless systems and to derive generalized hydrodynamic descriptions.

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.

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

TL;DR

The paper provides a rigorous foundation for linear response in gapless quantum systems by proving the zero-temperature, infinite-volume validity of the Kubo formula for 1d gapless fermions and for edge modes of 2d topological insulators under weak, slowly varying perturbations. The authors develop an Euler-like scaling framework and reformulate the real-time Duhamel series in terms of imaginary-time Euclidean correlations, enabling uniform control of higher-order terms via loop cancellations tied to emergent chiral symmetry. A central outcome is an explicit leading-order expression for the density and current response, governed solely by the Fermi velocities, with subleading corrections vanishing in the scaling limit; this yields the quantization of edge conductance from quantum dynamics. The approach blends rigorous Duhamel/Wick-rotation techniques with careful estimates of auxiliary dynamics and uses bosonization-inspired cancellations to simplify the perturbation theory without assuming integrability, offering a pathway to extend to weakly interacting gapless systems and to derive generalized hydrodynamic descriptions.

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 quantum Hall systems from quantum dynamics.

Paper Structure

This paper contains 27 sections, 17 theorems, 298 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Data availability statement.
  4. Conflict of interest.
  5. Lattice fermions
  6. Hamiltonian and Gibbs state
  7. Dynamics and linear response
  8. Perturbing the system.
  9. Density and current operators.
  10. Imaginary time evolution.
  11. Euclidean correlation functions and Wick rotation.
  12. Large scale response of $1d$ systems
  13. Proof of Theorem \ref{['thm:main1d']}
  14. Auxiliary dynamics
  15. Duhamel series of the auxiliary dynamics
  16. ...and 12 more sections

Key Result

Proposition 2.3

Let $\tau_{t}(\mathcal{A})$ be the Heisenberg evolution of the observable $\mathcal{A}$: The following identity holds true: For $\eta > 0$, the sum in the right-hand side is absolutely convergent.

Figures (3)

  • Figure 1: Qualitative representation of the rescaled local potential entering the definition of $\mathcal{P}$, Eq. (\ref{['eq:Pdef']}).
  • Figure 2: Typical form for the spectrum of $H$. The purple region corresponds to the "bulk spectrum". The red curves are the edge modes localized at $x_{2} = 0$, while the dotted curves are the edge modes localized at $x_{2} = L-1$.
  • Figure 3: Representation of the lattice $\Gamma_{L}$: the red semi-circle is the typical support of the perturbation $\theta \mu(\theta x)$, whereas $1\ll \ell \ll 1/\theta$ is the length of the fiducial line entering the definition of the edge 2-current $j_{\nu,x}^{\ell}$.

Theorems & Definitions (38)

  • Remark 2.2
  • Proposition 2.3: Duhamel series
  • proof
  • Remark 2.4
  • Proposition 2.5: Wick rotation
  • Theorem 3.1: Response of one-dimensional systems
  • Corollary 3.2
  • Remark 3.3
  • Proposition 4.1: Approximation by the auxiliary dynamics
  • Remark 4.2
  • ...and 28 more