Kubo formula for spin hydrodynamics: Spin chemical potential as the leading order term in a gradient expansion

Abstract

We present a first-order dissipative spin hydrodynamic framework, where the spin chemical potential 𝜔𝜇⁢𝜈 is treated as the leading term in the hydrodynamic gradient expansion, i.e., 𝜔𝜇⁢𝜈 ∼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 𝒪⁡(∂) . We consider the phenomenological form of the spin tensor, where it is antisymmetric 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.