We use linear response theory to derive both the non-dissipative and dissipative effects of spin polarization for massive and massless interacting spin 1/2 particles in a relativistic fluid. We list and classify all the possible contributions up to first order in gradients of hydrodynamic fields including the axial chemical potential and the spin potential, and we obtain the corresponding Kubo formulas. We find that all the possible dissipative contributions, except those coming from the gradients of spin potential, require a chiral imbalance or parity violating interactions. In a fluid with chiral imbalance we find a chiral version of the spin Hall effect, i.e. a spin polarization is induced by the gradients of temperature and of axial chemical potential in the direction orthogonal to the momentum of the particle and to the gradients. Moreover, we identify several other new non-dissipative contributions that are not present for free fields.

Romania

Matteo Buzzegoli

Kubo formulas for spin polarization in dissipative relativistic spin hydrodynamics: a first-order gradient expansion approach

Victor E. Ambrus

France

Maxim N. Chernodub

Corrigendum to “Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism” [Physics Letters B 855 (2024) 138757/PLB 138795]

The strongly interacting matter created in relativistic heavy-ion collisions possesses several conserved quantum numbers, such as baryon number, strangeness, and electric charge. The diffusion process of these charges can be characterized by a diffusion matrix that describes the mutual influence of the diffusion of various charges. We derive the Kubo relations for evaluating diffusion coefficients as elements of a diffusion matrix. We further demonstrate that in the weak coupling limit, the diffusion matrix elements obtained through Kubo relations reduce to those obtained from kinetic theory with an appropriate identification of the relaxation times. We illustrate this evaluation in a toy model of two interacting scalar fields with two conserved charges.

India

Sourav Dey

Poland

Amaresh Jaiswal

Hiranmaya Mishra

Diffusion coefficient matrix for multiple conserved charges: a Kubo approach

The electrical conductivity of hot and dense quark matter is calculated using the 2-flavour gauged Nambu–Jona–Lasinio (NJL) model in the presence of a chiral imbalance quantified in terms of a chiral chemical potential (CCP). To this end, the in-medium spectral function corresponding to the vector current correlator is evaluated employing the real time formulation of finite temperature field theory. Taking the long wavelength limit of the spectral function we extract the electrical conductivity using the Green–Kubo relation. The thermal widths of the quarks/antiquarks that appear in the expression of electrical conductivity are calculated by considering the $$2\rightarrow 2$$ scattering in the NJL model. The scattering amplitudes containing the polarization functions of the mesonic modes in the scalar and pseudoscalar channels are also evaluated by considering finite value of CCP. We find that the ratio of electrical conductivity to temperature has significant dependence on CCP especially in the low temperature region.

Snigdha Ghosh

Nilanjan Chaudhuri

Sourav Sarkar

Pradip Roy

Effect of chiral imbalance on the electrical conductivity of hot and dense quark matter using Green–Kubo method within the 2-flavour gauged NJL model

We discuss the imaginary-time formalism for field theories in thermal equilibrium in uniformly accelerating frames. We show that under a Wick rotation of Minkowski spacetime, the Rindler event horizon shrinks to a point in a two-dimensional subspace tangential to the acceleration direction and the imaginary time. We demonstrate that the accelerated version of the Kubo-Martin-Schwinger (KMS) condition implies an identification of all spacetime points related by integer-multiple rotations in the tangential subspace about this Euclidean Rindler event-horizon point, with the rotational quanta defined by the thermal acceleration, <math><mi>α</mi><mo>=</mo><mi>a</mi><mo>/</mo><mi>T</mi></math>. In the Wick-rotated Rindler hyperbolic coordinates, the KMS relations reduce to standard (anti-)periodic boundary conditions in terms of the imaginary proper time (rapidity) coordinate. Our findings pave the way to study, using first-principle lattice simulations, the Hawking-Unruh radiation in geometries with event horizons, phase transitions in accelerating Early Universe and early stages of quark-gluon plasma created in relativistic heavy-ion collisions.

Victor E. Ambruş

Acceleration as a circular motion along an imaginary circle: Kubo-Martin-Schwinger condition for accelerating field theories in imaginary-time formalism

We have explored the multicomponent structure of electrical conductivity of relativistic Fermionic and bosonic fluid in the presence of a magnetic field by using the Kubo approach. This is done by explicitly evaluating the thermomagnetic vector current spectral functions using the real time formalism of finite temperature field theory and the Schwinger proper time formalism. In the absence of a magnetic field, the one-loop diagramatic representation of the Kubo expression of any transport coefficients is exactly the same with a relaxation time approximation (RTA)-based expression, but this equality does not hold for a finite magnetic field picture due to a lack of proper implementation of quantum effect in a latter approach. We have shown this discrepancy for a particular transport coefficient—electrical conductivity, whose starting point in the Kubo approach will be an electromagnetic current-current correlator and its one-loop skeleton diagram carrying two scalar/Dirac propagators for a scalar/Dirac fluid. Through a numerical comparison between RTA and Kubo expressions of conductivity components (parallel and perpendicular), we have attempted to interpret a detailed quantum field theoretical effect, contained by the Kubo expression but not by the RTA expression. In a classical RTA expression we get a magnetic field independent parallel conductivity due to zero Lorentz force but in the field theoretical Kubo expression, it decreases and increases with the magnetic field for a scalar and Dirac medium, respectively, due to the Landau quantization effect. This parallel component of conductivity can be interpreted as a zero momentum limit of quantum fluctuation with the same Landau level internal lines, while for a perpendicular component of conductivity, fluctuation with Landau level differences <math><mo>±</mo><mn>1</mn></math> are noticed, which might be a new realization of transportation in a field theoretical sector.

Sarthak Satapathy

Sabyasachi Ghosh

Kubo estimation of the electrical conductivity for a hot relativistic fluid in the presence of a magnetic field

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.

China

Jin Hu

Kubo formulae for first-order spin hydrodynamics

The expressions of the shear viscosity and the bulk viscosity components in the presence of an arbitrary external magnetic field for a system of hot charged scalar bosons (spin 0) as well as for a system of hot charged Dirac fermions (spin <math><mfrac><mn>1</mn><mn>2</mn></mfrac></math>) have been derived by employing the one-loop Kubo formalism. This is done by explicitly evaluating the thermomagnetic spectral functions of the energy-momentum tensors using the real time formalism of finite temperature field theory and the Schwinger proper time formalism. In the present work, a rich quantum field theoretical structure in the expressions of the viscous coefficients in nonzero magnetic field are found, which are different from their respective expressions obtained earlier via kinetic-theory-based calculations; though, in the absence of a magnetic field, the one-loop Kubo and the kinetic-theory-based expressions for the viscosities are known to be identical. We have identified that Kubo and kinetic-theory-based results of viscosity components follow a similar kind of temperature and magnetic field dependency. The relaxation time and the synchrotron frequency in the kinetic theory formalism are realized to be connected, respectively, with the thermal width of propagator and the transitions among the Landau levels of the charged particles in the Kubo formalism. We believe that the connection of the latter quantities is quite new and probably the present work is the first time addressing this interpretation along with the new expressions of viscosity components, not seen in existing works.

One-loop Kubo estimations of the shear and bulk viscous coefficients for hot and magnetized bosonic and fermionic systems

We compute thermal one-point functions in Maxwell’s theory sourced by vorticity for the zilch and its higher spin extensions via the Kubo formalism. This leads to a generalization of the recent results of [M. N. Chernodub, A. Cortijo, and K. Landsteiner, Phys. Rev. D 98, 065016 (2018)] to any spin, and their value suggests a relation with possible anomalies for the higher spin tower of currents.

Spain

Christian Copetti

Jorge Fernández-Pendás

Higher spin vortical zilches from Kubo formulae

Uncharged relativistic fluids in 3+1 dimensions have three independent thermodynamic transport coefficients at second order in the derivative expansion. Fluids with a single global U(1) current have nine, out of which seven are parity preserving. We derive the Kubo formulas for all nine thermodynamic transport coefficients in terms of equilibrium correlation functions of the energy-momentum tensor and the current. All parity-preserving coefficients can be expressed in terms of two-point functions in flat space without external sources, while the parity-violating coefficients require three-point functions. We use the Kubo formulas to compute the thermodynamic coefficients in several examples of free field theories.