[Paper Review] Spin Polarization Induced by Inhomogeneous Dynamical Condensate

Ziyue Wang, Pengfei Zhuang|arXiv (Cornell University)|Jan 3, 2021

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper investigates spin polarization in quark matter induced by an inhomogeneous dynamical chiral condensate within a kinetic theory framework based on the Nambu–Jona-Lasinio model. It shows that even without collision terms, an initially unpolarized system can develop spin polarization due to the condensate's contribution to the energy-momentum and angular momentum tensors, and that the global equilibrium spin distribution remains stable under inhomogeneous mass, relaxing the Killing condition on thermal vorticity.

ABSTRACT

The role of dynamical chiral condensate in spin polarization is investigated in a kinetic theory framework. Transport equations for quark matter are derived in the mean-field approximation for the Nambu--Jona-Lasinio model. The dynamical condensate carries part of the energy momentum tensor (EMT) and the angular momentum tensor (AMT), the conservation of EMT and AMT can be proved from the kinetic equations as required by the symmetry. The transport equations of vector and axial-vector components are derived taking the spin decomposition as well as semi-classical expansion. Inhomogeneous mass introduces novel effect at $\mathcal{O}(\hbar)$, for an initially unpolarized system, spin polarization can be generated from the dynamical chiral condensate, even without the collision term. The stable spin distribution function is found to be robust, namely in the case with non-trivial dynamical mass, the spin polarization is still enslaved by the thermal vorticity, while the Killing condition can be loosened.

Motivation & Objective

• To investigate the role of dynamical chiral condensate in inducing spin polarization in quark matter under non-equilibrium conditions.
• To derive transport equations for vector and axial-vector components in the presence of inhomogeneous dynamical mass using semi-classical expansion.
• To examine whether spin polarization can emerge without collision terms, relying solely on mean-field dynamics.
• To determine the stability of the global equilibrium spin distribution when the thermal vorticity condition is relaxed due to inhomogeneous condensates.

Proposed method

• Derives transport equations for quark matter using the Wigner function approach and mean-field approximation in the Nambu–Jona-Lasinio model.
• Applies semi-classical expansion to obtain classical and first-order transport equations for vector and axial-vector components of the spin distribution.
• Considers the dynamical chiral condensate as a source of inhomogeneous mass that contributes to the energy-momentum and angular momentum tensors.
• Uses the Keldysh formalism to derive collisionless spin transport equations and evaluates self-energy contributions via the NJL model.
• Proves conservation of energy-momentum and angular momentum tensors from the kinetic equations, ensuring consistency with symmetries.
• Analyzes the spin distribution function in the presence of inhomogeneous condensates, focusing on the stability of the equilibrium solution.

Experimental results

Research questions

• RQ1 Can spin polarization be generated in an initially unpolarized system without collision terms, solely through the dynamics of an inhomogeneous chiral condensate?
• RQ2 How does the inclusion of a dynamical, spatially varying mass affect the conservation laws of energy-momentum and angular momentum in kinetic theory?
• RQ3 To what extent is the global equilibrium spin distribution, typically tied to thermal vorticity, still valid when the Killing condition is relaxed due to inhomogeneous condensates?
• RQ4 What is the role of the condensate in mediating the conversion between orbital and spin angular momentum in a collisionless regime?

Key findings

• Spin polarization can be induced in an initially unpolarized system via the inhomogeneous dynamical chiral condensate even in the absence of collision terms.
• The dynamical condensate contributes to both the energy-momentum and angular momentum tensors, and its dynamics preserve the conservation of these tensors as required by symmetry.
• The stable solution for the spin distribution function remains robust under inhomogeneous mass, maintaining alignment with thermal vorticity despite the relaxation of the Killing condition.
• The condensate's spatial variation introduces novel effects at O(ħ), enabling spin generation through mean-field dynamics alone.
• The equivalence of the transport equations is proven by showing that momentum integrals involving the condensate and Wigner function yield vanishing contributions to angular momentum, preserving total conservation.
• The results demonstrate that the global equilibrium spin distribution is not disrupted by inhomogeneous condensates, indicating a high degree of stability in the spin transport framework.