[Paper Review] Mathematical aspects of the Kubo formula for electrical conductivity with dissipation

Alexander Watson, Dionisios Margetis+1 more|arXiv (Cornell University)|Apr 9, 2023

Quantum and electron transport phenomena4 citations

TL;DR

This paper presents a systematic formal derivation of the Kubo formula for electrical conductivity in quantum systems with dissipation, modeling scattering via Poisson-distributed events. It derives a general expression for conductivity that reduces to well-known forms—such as Drude and periodic tight-binding models—providing a unified framework for analyzing electronic transport in materials like graphene and free electrons.

ABSTRACT

In this expository article, we present a systematic formal derivation of the Kubo formula for the linear-response current due to a time-harmonic electric field applied to non-interacting, spinless charged particles in a finite volume in the quantum setting. We model dissipation in a transparent way by assuming a sequence of scattering events occurring at random-time intervals modeled by a Poisson distribution. By taking the large-volume limit, we derive special cases of the formula for free electrons, continuum and tight-binding periodic systems, and the nearest-neighbor tight-binding model of graphene. We present the analogous formalism with dissipation to derive the Drude conductivity of classical free particles.

Motivation & Objective

• To provide a systematic formal derivation of the Kubo formula for linear-response electrical conductivity in non-interacting, spinless quantum particles under time-harmonic electric fields.
• To incorporate dissipation in a transparent way by modeling scattering events as Poisson processes with inverse mean scattering time Γ.
• To derive the large-volume limit of the Kubo formula, yielding special cases for free electrons, continuum periodic potentials, tight-binding models, and graphene.
• To establish the classical Drude conductivity as a limiting case of the quantum formalism, validating consistency across regimes.
• To offer a unified mathematical framework that connects diverse electronic transport models through a single, generalizable formula.

Proposed method

• Models dissipation via a sequence of random scattering events with exponentially distributed intervals, parameterized by the inverse mean scattering time Γ.
• Derives the general Kubo formula in the form $\sigma_{lm}(\omega) = -\frac{e^2}{\hbar^2} \tilde{\operatorname{Tr}}\left\{ (\partial_l H)(\mathcal{L}_H - i\omega + \Gamma)^{-1} \partial_m \Phi(H) \right\} $, where $\tilde{\operatorname{Tr}}$ is the trace density per unit volume.
• Applies the formalism to derive specific results for free particles, continuum periodic potentials, tight-binding limits, and the nearest-neighbor graphene model.
• Uses time-ordered expectation values and stochastic averaging over Poisson-distributed scattering times to compute the current response.
• Treats the classical limit by deriving the Drude conductivity from the same framework, showing consistency with known results.
• Employs the Liouvillian $\mathcal{L}_H = \frac{i}{\hbar}[H, \cdot]$ and momentum-space derivations $\partial_l = -i[X_l, \cdot]$ to express the formula in momentum and Fourier domains.

Experimental results

Research questions

• RQ1 How can the Kubo formula for electrical conductivity be formally derived in the presence of dissipation modeled by Poisson-distributed scattering events?
• RQ2 What are the limiting forms of the general Kubo formula in the cases of free electrons, periodic potentials, and tight-binding models?
• RQ3 How does the quantum Kubo formula reduce to the classical Drude model under appropriate limits?
• RQ4 What is the role of the inverse scattering time Γ in modifying the frequency-dependent conductivity?
• RQ5 Can the general formula consistently reproduce known results such as the Drude peak and interband contributions in periodic systems?

Key findings

• The general Kubo formula $\sigma_{lm}(\omega) = -\frac{e^2}{\hbar^2} \tilde{\operatorname{Tr}}\left\{ (\partial_l H)(\mathcal{L}_H - i\omega + \Gamma)^{-1} \partial_m \Phi(H) \right\} $ is derived for non-interacting, spinless particles under time-harmonic fields with dissipation.
• For free electrons, the formula yields the Drude conductivity $\sigma(\omega) = \frac{e^2 \tilde{N}}{m(-i\omega)} \left( \frac{1}{\Gamma} - \frac{1}{\Gamma - i\omega} \right) $, matching the classical result in the limit of small frequency.
• In the tight-binding limit of periodic systems, the formula reduces to standard forms used in band structure calculations, including interband contributions.
• For graphene in the nearest-neighbor tight-binding model, the formalism reproduces known conductivity behavior, including the universal value in the clean limit.
• The classical Drude conductivity is recovered in the large-volume, weak-field, and classical-limit regimes, confirming consistency with established physics.
• The derivation shows that the inclusion of Γ via Poissonian scattering leads to a natural damping of the response function, resolving divergences in the zero-frequency limit.