Quantum field theoretical structure of electrical conductivity of cold and dense fermionic matter in the presence of a magnetic field

We have gone through a detailed calculation of the two-point correlation function of vector currents at finite density and magnetic field by employing the real time formalism of finite temperature field theory and Schwinger’s proper-time formalism. With respect to the direction of the external magnetic field, the parallel and perpendicular components of electric conductivity for the degenerate relativistic fermionic matter are obtained from the zero-momentum limit of the current-current correlator, using the Kubo formula. Our quantum-field theoretical expressions and numerical estimations are compared with those obtained from the relaxation-time approximation methods of kinetic theory and its Landau quantized extension, which can be called classical and quantum results, respectively. All the results are merged in the classical domain i.e., the high-density and low-density magnetic field region, but in the remaining (quantum) domain, quantum results carry a quantized information like the Shubnikov-de Haas oscillation along the density and magnetic field axes. We have obtained a completely new quantum-field theoretical expression for perpendicular conductivity of degenerate relativistic fermionic matter. Interestingly, our quantum field theoretical calculation provides a new mathematical form of the cyclotron frequency with respect to its classical definition, which might require more future research to interpret the phenomena.

Figure 1

The symmetric Schwinger-Keldysh contour $C$ in the complex time ($\tau$) plane used in the RTF with $t_0\to \infty$ and inverse temperature $\beta =1/T$. The two horizontal segments of the contour are labeled as ‘(1)’ and ‘(2)’, respectively.

Figure 2

The variation of $l_{\max }$ as a function of (a) a magnetic field for different values of chemical potential, and (b) the chemical potential for different values of magnetic field.

Figure 3

The variation of CM and QM/QFT values of ${\sigma }_{\parallel }/({\tau }_c{\mu }^2)$ as a function of (a) the magnetic field for different values of $\mu$, and (b) fermion chemical potential for different values of $eB$.

Figure 4

The variation of the CM, QM, and QFT estimation of ${\sigma }_{\perp }/({\tau }_c{\mu }^2)$ as a function of (a) magnetic field for $\mu =300\mathrm{MeV}$, (b) magnetic field for $\mu =400\mathrm{MeV}$, (c) fermion chemical potential for $eB=0.05{\mathrm{GeV}}^2$, and (d) fermion chemical potential for $eB=0.10{\mathrm{GeV}}^2$. The corresponding $eB=0$ graphs are also shown in subfigures (c) and (d) for comparison.

Figure 5

The variation of ${\sigma }_{\perp }^{\mathrm{QFT}}/{\sigma }_{\perp }^{\mathrm{QM}}$ as a function of (a) magnetic field for different values of $\mu$, and (b) fermion chemical potential for different values of $eB$.

Figure 6

The variation of the ratio ${\tau }_B/{\tilde{\tau }}_B$ as a function of (a) magnetic field for different values of $\mu$, and (b) fermion chemical potential for different values of $eB$.