Yu-Shiba-Rusinov states in color superconducting quark matter

TL;DR

This work investigates impurity-induced subgap states in dense quark matter where color superconductivity competes with the QCD Kondo effect for a single heavy-quark impurity. It extends the surrogate model solver (SMS) to a color-superconducting Kondo setting by representing the bulk with a small set of effective levels whose g-function matches the Matsubara dependence of the impurity–bulk hybridization, enabling efficient computation of subgap spectra. The authors find two phase transitions as the impurity coupling grows: an initial transition to impurity color screening by one of four light-quark species, followed by a transition to a fully overscreened state involving all species; two-quark or three-quark screening phases appear only as subgap excitations. The SMS framework converges rapidly (even with L̃ = 1) and remains tractable as additional surrogate levels are included, offering a practical route to exploring more realistic QCD scenarios such as SU(3) color and density-dependent couplings.

Abstract

The high-density region of the QCD phase diagram displays an intricate competition between color superconductivity and the QCD Kondo effect due to color exchange in quark matter containing a single heavy quark impurity. We explore the characteristic impurity-induced superconducting subgap states arising in such systems by generalizing the surrogate model solver, recently considered in the context of condensed matter physics [Phys. Rev. B 108, L220506 (2023)]. The method consists of approximating the full superconducting bulk by only a small number of effective levels whose parameters are chosen so as to best reproduce the Matsubara frequency dependence of the impurity-bulk hybridization function. We numerically solve a surrogate QCD Kondo model describing a quantum impurity color-exchange coupled to a two-color two-flavor superconducting bulk. The results directly indicate the presence of multiple phase transitions as the coupling of the impurity to the bulk is increased, due to the interplay between various overscreened states. The methods introduced here are straightforward enough to be extended to more realistic QCD scenarios.

Figures (5)

• Figure 1: Energy difference $E_{o}-E_e$ (in units of $\Delta$) between the lowest even- and odd-fermion-parity states of the surrogate Kondo models for $\tilde{L}=1,2,3$ vs the dimensionless color-exchange coupling $\mathcal{J}$.
• Figure 2: Energies (relative to the ground state) of the lowest-lying states in each sector of definite SC quasiparticle number $N_{\text{qp}}=0,...,4$ vs the dimensionless color-exchange coupling $\mathcal{J}$, as obtained by the $\tilde{L}=1$ (ZBW) model. Continuous (dashed) lines denote states with even (odd) fermion parity).
• Figure 3: Color pseudo-spin properties of the lowest $T^z_{\text{tot}}=1/2$, even fermion-parity states vs the dimensionless color-exchange coupling $\mathcal{J}$, for the surrogate Kondo models $\tilde{L}=1,2,3$. (a) Average impurity color pseudo-spin $\langle T_{\text{imp}}^z\rangle$. (b) Impurity-bulk color pseudo-spin correlation function $\langle \vec{T}_{\text{imp}}\cdot \vec{T}_{\text{SC}}\rangle$, with gridlines at $0, -1, -1.5$. (c) Total color pseudo-spin squared $\langle \vec{T}_{\text{tot}}^2\rangle$, with gridlines at $0.75$ and $3.75$.
• Figure 4: Color pseudo-spin properties of the lowest $T^z_{\text{tot}}=0$, odd fermion-parity states vs the dimensionless color-exchange coupling $\mathcal{J}$, for the surrogate Kondo models $\tilde{L}=1,2,3$. (a) Impurity-bulk color pseudo-spin correlation function $\langle \vec{T}_{\text{imp}}\cdot \vec{T}_{\text{SC}}\rangle$, with gridlines at $-0.75$ and $-1.25$. (c) Total color pseudo-spin squared $\langle \vec{T}_{\text{tot}}^2\rangle$, with gridlines at $0$ and $2$.
• Figure 5: Energy difference $E_{o}-E_e$ (in units of $\Delta$) between the lowest even- and odd-fermion-parity states of the surrogate Kondo models of Eq. (\ref{['htilde0']}) for $\tilde{L}=1,..,6$ vs the dimensionless exchange coupling $\mathcal{J}$.