Mathematical aspects of the Kubo formula for electrical conductivity with dissipation

Alexander B. Watson; Dionisios Margetis; Mitchell Luskin

doi:10.1007/s13160-023-00613-7

Mathematical aspects of the Kubo formula for electrical conductivity with dissipation

Alexander B. Watson, Dionisios Margetis, Mitchell Luskin

Mathematics

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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.

Original language	English (US)
Pages (from-to)	1765-1795
Number of pages	31
Journal	Japan Journal of Industrial and Applied Mathematics
Volume	40
Issue number	3
DOIs	https://doi.org/10.1007/s13160-023-00613-7
State	Published - Sep 2023

Bibliographical note

Publisher Copyright:
© 2023, The JJIAM Publishing Committee and Springer Nature Japan KK, part of Springer Nature.

Keywords

Dissipation
Electrical conductivity
Kubo formula
Linear-response
Trace formula
von Neumann equation

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T1 - Mathematical aspects of the Kubo formula for electrical conductivity with dissipation

AU - Watson, Alexander B.

AU - Margetis, Dionisios

AU - Luskin, Mitchell

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KW - Electrical conductivity

KW - Kubo formula

KW - Linear-response

KW - Trace formula

KW - von Neumann equation

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Mathematical aspects of the Kubo formula for electrical conductivity with dissipation

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