Kubo formula for spin hydrodynamics: spin chemical potential as leading order in gradient expansion

TL;DR

This paper develops a first-order dissipative spin hydrodynamic framework with the spin chemical potential as the leading gradient term, analyzing its consistency and novel effects like entropy production and cross-diffusion.

Contribution

It introduces a new spin hydrodynamics model with the spin chemical potential as leading order, and demonstrates its consistency via entropy current and Kubo formalism.

Findings

Entropy production from spin-orbit coupling affects fluid frame equivalence.

Identification of cross-diffusion effects involving spin and charge gradients.

Equivalence of two spin tensor decomposition methods via Kubo relations.

Abstract

We present a first-order dissipative spin hydrodynamic framework, where the spin chemical potential $\omega^{\mu\nu}$ is treated as the leading term in the hydrodynamic gradient expansion, i.e., $\omega^{\mu\nu}\sim \mathcal{O}(1)$. We argue that for the consistency of the theoretical framework, the energy-momentum tensor needs to be symmetric at least up to order $\mathcal{O}(\partial)$. We consider the phenomenological form of the spin tensor, where it is anti-symmetric in the last two indices only. A comprehensive analysis of spin hydrodynamics is conducted using both macroscopic entropy current analysis and microscopic Kubo formalism, establishing consistency between the two approaches. A key finding is the entropy production resulting from spin-orbit coupling, which alters the traditional equivalence between the Landau and Eckart fluid frames. Additionally, we identify cross-diffusion effects, where vector dissipative currents are influenced by gradients of both spin chemical potential and chemical potential corresponding to the conserved charge through off-diagonal transport coefficients. Two distinct methods for decomposing the spin tensor are proposed, and their equivalence is demonstrated through Kubo relations.