Competing color superconductivity and color Kondo effect in quark matter

TL;DR

The paper introduces an integrable 1+1D SU(3) toy model that captures the competition between bulk color superconductivity (mass gap $m$) and the color Kondo effect (boundary coupling $J$) due to a heavy quark impurity. Using Bethe Ansatz, the authors derive exact boundary Bethe equations and an RG-invariant parameter $d$ to classify infrared boundary phases, revealing three distinct regimes: a Kondo phase with complete multi-particle screening and a Kondo scale $T_K$, a Yu–Shiba–Rusinov (YSR) phase where a boundary-bound state screens the impurity, and an unscreened phase with a residual local moment. The work shows how dynamically generated bulk properties interplay with boundary impurities, producing a rich spectrum of boundary states and transitions, including a first-order transition at $oldsymbol{ ext{δ}}=rac{5}{4}$ in the YSR regime. These findings provide insights into impurity physics under extreme QCD conditions and may guide future explorations in cold-atom emulations and dense astrophysical environments.

Abstract

The competition between bulk color superconductivity and the localized screening of a heavy quark impurity, analogous to the Kondo effect, leads to a rich spectrum of phenomena in dense quark matter. We investigate this competition at the edge of a superconducting quark bulk, where both the superconducting gap and the Kondo scale are dynamically generated in a tractable toy model. Utilizing the exact Bethe Ansatz method, we elucidate the resulting boundary physics. We identify distinct regimes characterized by either multi-particle Kondo screening or an unscreened local moment. Crucially, we also uncover a novel intermediate phase featuring impurity screening through a single-particle bound state formed within the superconducting gap. The toy model presented in this work highlights the complex interplay between dynamically generated bulk properties and boundary impurities in extreme QCD environments, offering potential insights into phenomena occurring in heavy-ion collisions and compact stars.

Figures (9)

• Figure 1: Phase diagram of QCD in temperature and chemical potential phase showing the hadronic, quark-gluon plasma, and color superconducting phase.
• Figure 2: Cartoon depiction of the model Eq. \ref{['modelham']} where left and right moving chiral fermions with marginal bulk attractive current-current interaction $g$ and open boundary conditions are coupled to an SU(3) moment at the boundary with Kondo coupling $J$.
• Figure 3: The weak-coupling RG flow diagram for Hamiltonian Eq. \ref{['modelham']} is given by two RG equations Eq. \ref{['ggflow']} and Eq. \ref{['jjflow']}. The flow is away from the non-interacting point $J=0=g$. The magenta curves denote the Kondo phase, where the boundary coupling flows to strong coupling, and the impurity is screened by a multiparticle Kondo cloud. The cyan lines indicate the YSR phase, where the impurity is screened by a single-particle bound mode. Finally, the orange lines depict the unscreened regime, where the boundary coupling flows to zero, leaving the impurity unscreened. The dashed black curves demarcate the three boundary phases of the model. The dashed gray curve within the cyan YSR region indicates the locus where the energy of the boundary bound state $E_{\delta}$ vanishes, corresponding to the level crossing at $\delta = 5/4$ shown in Fig. \ref{['fig:Eng-bm-delta']}. Across this line, the bound-state energy changes sign, separating the screened ($E_{\delta}<0$) and unscreened ($E_{\delta}>0$) regimes within the YSR phase.
• Figure 4: Schematic phase diagram of the SU(3)–Gross–Neveu model with a massive quark impurity at one edge. The low-lying spectrum is plotted as a function of the RG-invariant $d$, which, when imaginary, is written $d = i\delta$. The YSR state exists only for $\frac{1}{2} < \delta < 2$, and the magnitude of its energy is shown by the solid magenta line. A first-order quantum phase transition between the screened $(p,q)=(0,0)$ and unscreened $(p,q)=(1,0)$ impurity configurations occurs at $\delta = \frac{5}{4}$. For $\delta > 2$, the impurity remains unscreened, and no YSR states screening the impurity are present. In the Kondo regime, $\delta < \frac{1}{2}$ (i.e. $d\in\mathbb{R}$), many-body screening generates a Kondo scale $T_K$. The shaded orange region denotes the continuum of bulk excitations above the mass gap $m$. The two faint vertical guides at $\delta=1$ and $\delta=\tfrac{3}{2}$ delimit the parametric range in which a midgap antiquark excitation carrying triality 2 exists with its energy in range $\frac{m}{2}<E_\delta<m$, whereas for $\delta>\frac{3}{2}$ its energy is $E(\theta)=m\cosh\left(\frac{2\pi}{3}\theta\right)$ such that it merges into the continuum of excitations $E>m$.
• Figure 5: Weight diagrams of SU(3) multiplets plotted in the $(Y, I_3)$ plane. In the nested Bethe Ansatz description, each irreducible representation is labeled by a Dynkin label $(p, q)$. For the class of fundamental excitations formed by symmetric combinations of holes, the integers $p$ and $q$ count the number of holes in the Bethe root distributions at each nesting rank: $p$ counts holes in the rank-1 equations and $q$ in the rank-2 equations. For example, (a) the meson octet corresponds to $(1,1)$, with one hole in each level; (b) the baryon decuplet corresponds to $(3,0)$, with three holes in the first level and none in the second; and (c) the antibaryon multiplet corresponds to $(0,3)$. The Bethe Ansatz ground state is the SU(3) singlet $(0,0)$, in which all Bethe roots are real and both levels are fully filled with no holes. This hole-counting interpretation applies only to symmetric excitations. More generally, antisymmetric or mixed-symmetry states—such as those arising in the tensor product decomposition of multiple fundamental excitations—require the inclusion of string solutions, i.e., complex bound states of Bethe roots.