Topology and stability of the Kondo phase in quark matter

We investigate properties of the ground state of a light-quark matter with heavy-quark impurities. This system exhibits the “QCD Kondo effect” where the interaction strength between a light quark near the Fermi surface and a heavy quark increases with decreasing energy of the light quark toward the Fermi energy and diverges at some scale near the Fermi energy, called the Kondo scale. Around and below the Kondo scale, we must treat the dynamics nonperturbatively. As a typical nonperturbative method to treat the strong coupling regime, we adopt a mean-field approach where we introduce a condensate, the Kondo condensate, representing a mixing between a light quark and a heavy quark, and determine the ground state in the presence of the Kondo condensate. We show that the ground state is a topologically nontrivial state and the heavy-quark spin forms the hedgehog configuration in the momentum space. We can define the Berry phase for the ground-state wave function in the momentum space, which is associated with a monopole at the position of a heavy quark. We also investigate fluctuations around the mean field in the random-phase approximation and show the existence of (excitonlike) collective excitations made of a hole $h$ of a light quark and a heavy quark $Q$.

Figure 1

Schematic figures of the energy-momentum dispersion relations in the presence of the Kondo condensate: $N_f=1$ (top panel) and $N_f\ge 2$ (bottom panel). The units are arbitrary, but we fixed here $\mu =0.5$ and $\lambda =0$. The origin of the vertical axis corresponds to the Fermi surface. The thick solid lines are the occupied states below the Fermi sphere. The dashed line for $N_f=1$ means the dispersion of a free light quark at finite density. The numbers in the square brackets indicate the number of degeneracy for each of spin up and down. The arrows stand for the excitations from the quark in the ground state (cf. Sec. 4a).

Figure 2

Plots of ${\overrightarrow{S}}_{E_p^{-}}^{(\gamma )}$ (left: $\gamma =+1$ and right: $\gamma =-1$) in momentum space [Eq. (64)].

Figure 3

Effective heavy-quark spins $S_{\pm }(p)$ as functions of $p$ [Eq. (65)]. The solid (dashed) line corresponds to $S_{-}(p)$ [$S_{+}(p)$]. We used typical values of parameters: $\mu =0.5\mathrm{GeV}$, $\lambda =0$, $|\mathrm{\Delta }|=0.01\mathrm{GeV}$.

Figure 4

The fraction of the light-quark component $n_{\pm }(p)$ [Eq. (68)] (top panel) and the effective heavy-quark spin $S_{\pm }(p)$ [Eq. (65)] (bottom panel) as functions of energy $E$ for $\lambda =0$.

Figure 5

A one-loop diagram of the self-energy $\frac{1}{i}\mathrm{\Pi }(k_0,\overrightarrow{k})$. Solid lines are quark propagators $G$ in the presence of the Kondo condensate. We denote $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\ell }^{(i)}$ and $\overline{\mathrm{\Gamma }}={\overline{\mathrm{\Gamma }}}_{\ell }^{(i)}$ for $\ell =\{\mathrm{S},\mathrm{V},\mathrm{P},\mathrm{A}\}$ and $i=1,\dots ,N_f$.

Figure 6

Numerical results of ${\omega }_k(=k_0)$ as functions of $k=|\overrightarrow{k}|$ for $h_{i}Q$ modes ($J^P=0^{\pm },1^{\pm }$) with $i=1$ (upper panel) and $i=2,\dots ,N_f$ (lower panel). The dashed lines are thresholds for free $h_i$ and $Q$, and the upper gray regions are the scattering states.

Figure 7

Schematic pictures of the particle-hole excitations from the state inside the Fermi sphere ($E_p<0$; thick curves) to the state outside the Fermi sphere ($E_p>0$; thin curves). See the text for more explanations.

Figure 8

Plots of the thermodynamic potentials as functions of $|\mathrm{\Delta }|$. The dashed line is for ${\mathrm{\Omega }}^{(\mathrm{i})}$, and the solid line is for ${\mathrm{\Omega }}^{(\mathrm{ii})}$. The parameters we used are $g_c{\mathrm{\Lambda }}^2=1$, $N_c=3$, $\mu =0.5\mathrm{GeV}$, $\mathrm{\Lambda }=1\mathrm{GeV}$, and $T=0$.