Heavy-quark spin polarization induced by the Kondo effect in a magnetic field

We propose a new mechanism of the heavy-quark spin polarization (HQSP) in quark matter induced by the Kondo effect under an external magnetic field. The Kondo effect is caused by a condensate between a heavy and a light quark called the Kondo condensate leading to a mixing of the heavy- and light-quark spins. Thus, the HQSP is driven through the Kondo effect from light quarks coupling with the magnetic field in quark matter. For demonstration, we employ the Nambu-Jona-Lasinio type model under a magnetic field and investigate the HQSP within the linear response theory with vertex corrections required by the $U(1{)}_{\mathrm{EM}}$ electromagnetic gauge invariance. As a result, we find that the HQSP arises significantly with the appearance of the Kondo effect. Our findings are testable in future sign-problem-free lattice simulations.

Figure 1

A schematic picture of the HQSP induced by the Kondo effect under a magnetic field. The HQSP is absent without the Kondo effect in a heavy-quark mass limit $m_Q\to \infty$ (left), while it significantly emerges in the presence of the Kondo effect (right). In this figure, we have assumed that the electric charge of light quarks is positive.

Figure 2

A schematic picture of the dispersion relations of q.p., $p_0=E_{\mathit{p}}^{+}$ (red solid line); q.h., $p_0=E_{\mathit{p}}^{-}$ (blue dashed line); and a.p., $p_0=E_{\mathit{p}}^{\mathrm{a}}$ (purple dotted line).

Figure 3

The Feynman diagrams contributing to the HQSP. Blobs represent the corrected vertices required by $U(1{)}_{\mathrm{EM}}$ gauge invariance. The diagrams (i) and (ii) correspond to the intraband processes, while (iii) and (iv) the interband ones.

Figure 4

The $\mu$ dependence of the HQSP response function in the dynamical limit. In this figure, ${\tilde{\mathrm{\Pi }}}_{0H}^{\mathrm{dyn}}$ (dashed purple line) and ${\tilde{\mathrm{\Pi }}}_H^{\mathrm{dyn}}$ (solid blue line) with ${\widehat{e}}_Q={\widehat{e}}_q=+\frac{2}{3}$ are plotted.

Figure 5

The $\mu$ dependence of the HQSP response function in the dynamical limit. In this figure, ${\tilde{\mathrm{\Pi }}}_{0H}^{\mathrm{dyn}}$ (dashed purple line) and ${\tilde{\mathrm{\Pi }}}_H^{\mathrm{dyn}}$ (solid red line) with ${\widehat{e}}_Q=+\frac{2}{3},{\widehat{e}}_q=-\frac{1}{3}$ are plotted.

Figure 6

The $q_0$ dependence of the HQSP response function for vanishing $\mathit{q}$ at $\mu =0.5\mathrm{GeV}$. In this figure, ${\tilde{\mathrm{\Pi }}}_{0H}$ (dashed purple line) and ${\tilde{\mathrm{\Pi }}}_H(q_0,\mathbf{0})$ (solid blue line) with ${\widehat{e}}_Q={\widehat{e}}_q=+\frac{2}{3}$ are plotted.

Figure 7

The $q_0$ dependence of the HQSP response function for vanishing $\mathit{q}$ at $\mu =0.5\mathrm{GeV}$. In this figure, ${\tilde{\mathrm{\Pi }}}_{0H}$ (dashed purple line) and ${\tilde{\mathrm{\Pi }}}_H(q_0,\mathbf{0})$ (solid red line) with ${\widehat{e}}_Q=+\frac{2}{3},{\widehat{e}}_q=-\frac{1}{3}$ are plotted.

Figure 8

The $\mu$ dependence of the HQSP response function in the static limit. In this figure, ${\tilde{\mathrm{\Pi }}}_{0H}^{\mathrm{sta}}$ (dashed purple line) and ${\tilde{\mathrm{\Pi }}}_H^{\mathrm{sta}}$ (solid blue line) with ${\widehat{e}}_Q={\widehat{e}}_q=+\frac{2}{3}$ are plotted.

Figure 9

The $\mu$ dependence of the HQSP response function in the static limit. In this figure, ${\tilde{\mathrm{\Pi }}}_{0H}^{\mathrm{sta}}$ (dashed purple line) and ${\tilde{\mathrm{\Pi }}}_H^{\mathrm{sta}}$ (solid red line) with ${\widehat{e}}_Q=+\frac{2}{3},{\widehat{e}}_q=-\frac{1}{3}$ are plotted.

Figure 10

The $q$ dependence of the HQSP response function for vanishing $q_0$ at $\mu =0.5\mathrm{GeV}$ ($q\equiv |\mathit{q}|$). In this figure, ${\tilde{\mathrm{\Pi }}}_{0H}$ (dashed purple line) and ${\tilde{\mathrm{\Pi }}}_H(q_0,\mathbf{0})$ (solid blue line) with ${\widehat{e}}_Q={\widehat{e}}_q=+\frac{2}{3}$ are plotted.

Figure 11

The $q$ dependence of the HQSP response function for vanishing $q_0$ at $\mu =0.5\mathrm{GeV}$ ($q\equiv |\mathit{q}|$). In this figure, ${\tilde{\mathrm{\Pi }}}_{0H}$ (dashed purple line) and ${\tilde{\mathrm{\Pi }}}_H(q_0,\mathbf{0})$ (solid red line) with ${\widehat{e}}_Q=+\frac{2}{3},{\widehat{e}}_q=-\frac{1}{3}$ are plotted.

Figure 12

The HQSP response function induced by the Kondo effect and that by the ZI as functions of $T$. For details, see the text.