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arXiv:2101.08440 · Physical review. D/Physical review. D.

Kubo formulae for first-order spin hydrodynamics

Jin Chun Hu

Published Jun 21, 2021• 34 citations
arXivHTMLPDF

Abstract

We derive Kubo formulae for first-order spin hydrodynamics based on nonequilibrium statistical operators method. In first-order spin hydrodynamics, there are two new transport coefficients besides the ordinary ones appearing in first-order viscous hydrodynamics. They emerge due to the incorporation of the spin degree of freedom into fluids and the spin-orbital coupling. Zubarev's nonequilibrium statistical operator method can be well applied to investigate these quantum effects in fluids. The Kubo formulae, based on the method of nonequilibrium statistical operators, are related to equilibrium (imaginary-time) infrared Green's functions, and all the transport coefficients can be determined when the microscopic theory is specified.

DOI
10.1103/physrevd.103.116015

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Linear response theory relates hydrodynamic transport coefficients to equilibrium retarded correlation functions of the stress-energy tensor and global symmetry currents in terms of Kubo formulas. Some of these transport coefficients are non-dissipative and affect the fluid dynamics at equilibrium. We present an algebraic framework for deriving Kubo formulas for such thermal transport coefficients by using the equilibrium partition function. We use the framework to derive Kubo formulas for all such transport coefficients of superfluids, as well as to rederive Kubo formulas for various normal fluid systems.
We address the question of whether transport coefficients obtained from a unitary closed system setting, i.e., the standard equilibrium Green-Kubo formula, are the same as the ones obtained from a weakly driven nonequilibrium steady-state calculation. We first derive a nonequilibrium Kubo-like expression for the steady-state diffusion constant expressed as a time integral of either a current or a conserved density nonequilibrium correlation function. This expression has certain advantages over the equilibrium Green-Kubo formula, but it is not clear if it gives the same value of the diffusion constant. We then rigorously show that if the unitary dynamics is diffusive, the nonequilibrium formula indeed gives exactly the same transport coefficient. The form of finite-size correction is also predicted. Theoretical results are verified by an explicit calculation of the diffusion constant in several interacting many-body models.
Abstract Hydrodynamics is a powerful emergent theory for the large-scale behaviours in many-body systems, quantum or classical. It is a gradient series expansion, where different orders of spatial derivatives provide an effective description on different length scales. We report the first fully general derivation of third-order, or ‘dispersive’, terms in the hydrodynamic expansion. Our derivation is based on general principles of statistical mechanics, along with the assumption that the complete set of local and quasi-local conserved densities constitutes a good set of emergent degrees of freedom. We obtain fully general Kubo-like expressions for the associated hydrodynamic coefficients (also known as Burnett coefficients), and we determine their exact form in quantum integrable models, introducing in this way purely quantum higher-order terms into generalised hydrodynamics. We emphasise the importance of hydrodynamic gauge fixing at diffusive order, where we claim that it is parity-time-reversal, and not time-reversal, invariance that is at the source of Einstein’s relation, Onsager’s reciprocal relations, the Kubo formula and entropy production. At higher hydrodynamic orders we introduce a more general, nth order ‘symmetric’ gauge, which we show implies the validity of the higher-order hydrodynamic description.
At second order in gradients, conformal relativistic hydrodynamics depends on the viscosity $\ensuremath{\eta}$ and on five additional ``second-order'' hydrodynamical coefficients ${\ensuremath{\tau}}_{\ensuremath{\Pi}}$, $\ensuremath{\kappa}$, ${\ensuremath{\lambda}}_{1}$, ${\ensuremath{\lambda}}_{2}$, and ${\ensuremath{\lambda}}_{3}$. We derive Kubo relations for these coefficients, relating them to equilibrium, fully retarded three-point correlation functions of the stress tensor. We show that the coefficient ${\ensuremath{\lambda}}_{3}$ can be evaluated directly by Euclidean means and does not in general vanish.