Thermoelectric coefficients of two-flavor quark matter from the Kubo formalism

TL;DR

The study addresses thermoelectric transport in hot, dense two-flavor quark matter using the Kubo formalism within the Nambu–Jona-Lasinio model. Transport coefficients—the electrical conductivity $\sigma$, Seebeck coefficient $Q$, and modified thermal conductivity $\tilde{\kappa}$—are computed from retarded two-point correlators of currents and heat flux, with quark spectral functions derived from one-meson-exchange diagrams above the Mott transition and evaluated in a $1/N_c$ expansion. The results show that $Q$ and the Thomson coefficient $\rho$ increase approximately linearly with temperature and decrease with chemical potential, with divergences as $\mu \to 0$, and they provide rough estimates of electric fields generated by thermal gradients in heavy-ion collisions. This work offers a quantum-statistical framework for thermoelectric effects in strongly interacting QCD matter, with implications for electromagnetic-field evolution and charge transport in quark-gluon plasma environments.

Abstract

The hot quark matter created in heavy-ion collision experiments can exhibit strong temperature and chemical-potential gradients, which in turn can generate electric fields through thermoelectric effects. In this work, we investigate two relevant thermoelectric coefficients -- the thermopower (Seebeck coefficient) and the Thomson coefficient -- of two-flavor quark matter using the Kubo formalism and the Nambu--Jona-Lasinio model as an effective description of dense, finite-temperature QCD. The required two-point equilibrium correlation functions are evaluated using the Matsubara formalism of thermal field theory, applying a 1/Nc expansion to the relevant multi-loop Feynman diagrams. We employ previously derived quark spectral functions obtained from one--meson-exchange diagrams above the Mott transition temperature. Our numerical results show that both thermoelectric coefficients increase approximately linearly with temperature and decrease with increasing chemical potential. We also estimate the magnitude of the electric fields that can be generated in heavy-ion collisions by thermal gradients via the Seebeck effect.

Figures (7)

• Figure 1: Contributions to a generic two-point correlation function from ${\cal O}(N_c^1)$ (first and second lines) and ${\cal O}(N_c^0)$ (the third line) diagrams up to the first-order terms with respect to the coupling constant $G$ associated with a pair of $\Gamma=\Gamma^0_{s/ps}$ matrices. The straight lines represent dressed quark propagators.
• Figure 2: The Dyson-Schwinger equation for the constituent quark mass. The dashed and solid lines stand for the bare and dressed quark propagators, respectively, and the vertex $\Gamma=1$. The wavy line represents the four-fermion coupling $G$.
• Figure 3: The Bethe-Salpeter equation for mesonic modes. The double lines represent the dressed meson propagators, the solid lines stand for the dressed quark propagators, and the vertex assumes the values $\Gamma^0_{s}=1$ for $\sigma$ meson and $\Gamma^0_{ps}=i\bm\tau\gamma_5$ for pions.
• Figure 4: The phase diagram of strongly interacting quark matter at the leading order of $1/N_c$ expansion. The shaded area shows the region where our approximations are applicable. The inner bound is the Mott temperature $T_{\rm M}$ (in the case of broken chiral symmetry) or $T_{\rm M 0}\equiv T_c$ (in the case where chiral symmetry is intact), and the upper bound is $T_{\rm max}$ above which no mesonic modes are found.
• Figure 5: Dependence of three Lorentz components of the quark spectral functions $a=-mA_s$ (solid lines), $b=-\varepsilon A_0$ (dash-dotted lines) and $c=-pA_v$ (dashed lines) on the quark momentum at $\mu=0$ for $T=0.22$ GeV (a) and $T=0.3$ GeV (b). These spectral functions are shown at three energies $\varepsilon_1 = 0.1$, $\varepsilon_2 = 0.3$, and $\varepsilon_3 = 0.5$ GeV, as indicated in the plot.