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Competing color superconductivity and color Kondo effect in quark matter

Pradip Kattel ^*, Abay Zhakenov, and Natan Andrei

Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA

^*Contact author: pradip.kattel@rutgers.edu

Phys. Rev. D 112, 094012 – Published 10 November, 2025

DOI: https://doi.org/10.1103/q6kz-2vl2

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 multiparticle 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 quantum chromodynamics environments, offering potential insights into phenomena occurring in heavy-ion collisions and compact stars.

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References (86)

Ian J. R. Aitchison and Anthony J. G. Hey, Gauge Theories in Particle Physics: A Practical Introduction, -2 Volume Set (Taylor & Francis, London, 2012).
David Bailin and Alexander Love, Introduction to Gauge Field Theory Revised Edition (Taylor & Francis, London, 1993).
William Marciano and Heinz Pagels, Quantum chromodynamics, Phys. Rep. 36, 137 (1978).
Walter Greiner, Stefan Schramm, and Eckart Stein, Quantum Chromodynamics (Springer Science & Business Media, New York, 2007).
Szabolcs Borsányi, Zoltan Fodor, Christian Hoelbling, Sandor D. Katz, Stefan Krieg, and Kalman K. Szabo, Full result for the QCD equation of state with $2 + 1$ flavors, Phys. Lett. B 730, 99 (2014).
Alexei Bazavov, Tanmoy Bhattacharya, Carleton DeTar, H.-T. Ding, Steven Gottlieb, Rajan Gupta, P. Hegde, U. M. Heller, Frithjof Karsch, Edwin Laermann et al., Equation of state in ( $2 + 1$ )-flavor QCD, Phys. Rev. D 90, 094503 (2014).
Kohsuke Yagi, Tetsuo Hatsuda, and Yasuo Miake, Quark-Gluon Plasma: From Big Bang to Little Bang, Vol. 23 (Cambridge University Press, Cambridge, England, 2005).
Sourav Sarkar, Helmut Satz, and Bikash Sinha, The Physics of the Quark-Gluon Plasma: Introductory Lectures, Vol. 785 (Springer, New York, 2009).
Jean Letessier and Johann Rafelski, Hadrons and Quark–Gluon Plasma (Cambridge University Press, Cambridge, England, 2002).
Roman Pasechnik and Michal Šumbera, Phenomenological review on quark–gluon plasma: Concepts vs. observations, Universe 3, 7 (2017).
Jürgen Berges, Michal P. Heller, Aleksas Mazeliauskas, and Raju Venugopalan, QCD thermalization: Ab initio approaches and interdisciplinary connections, Rev. Mod. Phys. 93, 035003 (2021).
Aleksi Kurkela, Eduardo S. Fraga, Jürgen Schaffner-Bielich, and Aleksi Vuorinen, Constraining neutron star matter with quantum chromodynamics, Astrophys. J. 789, 127 (2014).
Krishna Rajagopal and Frank Wilczek, The condensed matter physics of QCD, in At The Frontier of Particle Physics (World Scientific, Singapore, 2001), pp. 2061–2151, 10.1142/9789812810458_0043.
Mei Huang, Color superconductivity at moderate baryon density, Int. J. Mod. Phys. E 14, 675 (2005).
Larry McLerran and Robert D. Pisarski, Phases of dense quarks at large Nc, Nucl. Phys. A796, 83 (2007).
Larry McLerran and Sanjay Reddy, Quarkyonic matter and neutron stars, Phys. Rev. Lett. 122, 122701 (2019).
Gert Aarts, Introductory lectures on lattice QCD at nonzero baryon number, J. Phys. Conf. Ser. 706, 022004 (2016).
Mark Alford, Color-superconducting quark matter, Annu. Rev. Nucl. Part. Sci. 51, 131 (2001).
Mark G. Alford, Andreas Schmitt, Krishna Rajagopal, and Thomas Schäfer, Color superconductivity in dense quark matter, Rev. Mod. Phys. 80, 1455 (2008).
Hai cang Ren, Color superconductivity in a dense quark matter, arXiv:hep-ph/0307125.
Igor A. Shovkovy, Two lectures on color superconductivity*, Found. Phys. 35, 1309 (2004).
Mark G. Alford and Sanjay Reddy, Compact stars with color superconducting quark matter, Phys. Rev. D 67, 074024 (2002).
Mark G. Alford, Jeffrey A. Bowers, and Krishna Rajagopal, Color superconductivity in compact stars, J. Phys. G 27, 541 (2000).
Boris Tomasik, Ultrarelativistic heavy-ion collisions: Exploring the phase diagram of QCD, arXiv:nucl-th/0610042.
Michael Buballa, NJL-model analysis of dense quark matter, Phys. Rep. 407, 205 (2005).
Hosein Gholami, Marco Hofmann, and Michael Buballa, Renormalization-group consistent treatment of color superconductivity in the NJL model, Phys. Rev. D 111, 014006 (2025).
Axel de la Macorra et al., Cosmology and quantum field theory: A study of the Nambu-Jona-Lasinio model, arXiv:1511.06210.
Hosein Gholami, Lennart Kurth, Ugo Mire, Michael Buballa, and Bernd-Jochen Schaefer, Renormalizing the quark-meson-diquark model, arXiv:2505.22542.
Koichi Hattori, Kazunori Itakura, Sho Ozaki, and Shigehiro Yasui, QCD Kondo effect: Quark matter with heavy-flavor impurities, Phys. Rev. D 92, 065003 (2015).
Shigehiro Yasui, Kei Suzuki, and Kazunori Itakura, Kondo phase diagram of quark matter, Nucl. Phys. A983, 90 (2019).
R. Fariello, Juan C. Macías, and F. S. Navarra, The QCD Kondo phase in quark stars, arXiv:1901.01623.
Prabal Adhikari, Martin Ammon, Sidney S. Avancini, Alejandro Ayala, Aritra Bandyopadhyay, David Blaschke, Fabio L Braghin, Pavel Buividovich, Rafael P. Cardoso, Casey Cartwright et al., Strongly interacting matter in extreme magnetic fields, Prog. Part. Nucl. Phys. 146, 104199 (2026).
Daiki Suenaga, Kei Suzuki, and Shigehiro Yasui, QCD Kondo excitons, Phys. Rev. Res. 2, 023066 (2020).
Jun Kondo, The Physics of Dilute Magnetic Alloys (Cambridge University Press, Cambridge, England, 2012).
Alexander Cyril Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, England, 1993), 10031, 1.
Natan Andrei, K. Furuya, and J. H. Lowenstein, Solution of the Kondo problem, Rev. Mod. Phys. 55, 331 (1983).
A. M. Tsvelick and P. B. Wiegmann, Exact results in the theory of magnetic alloys, Adv. Phys. 32, 453 (1983).
Ian Affleck and Andreas W. W. Ludwig, The Kondo effect, conformal field theory and fusion rules, Nucl. Phys. B352, 849 (1991).
Shigehiro Yasui, Kei Suzuki, and Kazunori Itakura, Topology and stability of the Kondo phase in quark matter, Phys. Rev. D 96, 014016 (2017).
Koichi Hattori, Xu-Guang Huang, and Robert D. Pisarski, Emergent QCD Kondo effect in two-flavor color superconducting phase, Phys. Rev. D 99, 094044 (2019).
The QCD Lagrangian is $L_{QCD} = - \frac{1}{4} F_{μ ν}^{a} F^{a μ ν} + {\bar{ψ}}_{i} (i γ^{μ} D_{μ} - m_{i}) ψ_{i}$ (46)where the field strength tensor $F_{μ ν}^{a}$ is defined as $F_{μ ν}^{a} = \partial_{μ} A_{ν}^{a} - \partial_{ν} A_{μ}^{a} + g f^{a b c} A_{μ}^{b} A_{ν}^{c}$ , and the covariant derivative $D_{μ}$ is $D_{μ} = \partial_{μ} - i g τ^{a} A_{μ}^{a}$ , with $A_{μ}^{a}$ being the gluon fields, $f^{a b c}$ the structure constants of the SU(3) gauge group, $ψ_{i}$ the quark fields with flavor index $i$ , $m_{i}$ is the mass of the $i$ -th quark. and $τ^{a}$ are the generators of the SU(3) group that acts on the 3-dimensional color space of quarks.
Hiroaki Kohyama, Possible interpretation on the origin of four-fermion interaction, arXiv:1601.07832.
Pradip Kattel, Abay Zhakenov, and Natan Andrei, Thermodynamics in a split Hilbert space: Quantum impurity at the edge of a one-dimensional superconductor, arXiv:2508.19330.
Taro Kimura and Sho Ozaki, Conformal field theory analysis of the QCD Kondo effect, Phys. Rev. D 99, 014040 (2019).
Olivier Parcollet, Antoine Georges, Gabriel Kotliar, and Anirvan Sengupta, Overscreened multichannel SU(n) Kondo model: Large-n solution and conformal field theory, Phys. Rev. B 58, 3794 (1998).
Andreas W. W. Ludwig and Ian Affleck, Exact conformal-field-theory results on the multichannel Kondo effect: Asymptotic three-dimensional space-and time-dependent multipoint and many-particle Green’s functions, Nucl. Phys. B428, 545 (1994).
David J. Gross and Andre Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10, 3235 (1974).
N. Andrei and J. Ho Lowenstein, Derivation of the chiral gross-neveu spectrum for arbitrary SU(n) symmetry, Phys. Lett. B 90, 106 (1980).
Using the SU(3) Fierz identity $(τ^{a})_{i j} (τ^{a})_{k l} = \frac{2}{3} (S_{s})_{i k} (S_{s})_{l j} - \frac{4}{3} (A_{a})_{i k} (A_{a})_{l j}$ , one obtains $(ψ_{L}^{†} τ^{a} ψ_{L}) (ψ_{R}^{†} τ^{a} ψ_{R}) = \frac{2}{3} (ψ_{L}^{†} S_{s} ψ_{R}^{†}) (ψ_{R} S_{s} ψ_{L}) - \frac{4}{3} (ψ_{L}^{†} A_{a} ψ_{R}^{†}) (ψ_{R} A_{a} ψ_{L})$ , showing that the antisymmetric color $\bar{3}$ channel is attractive. In $1 + 1$ dimensions, this attraction generates a finite mass gap and algebraically decaying pairing correlations rather than a true condensate.
Andrés Jerez, Natan Andrei, and Gergely Zaránd, Solution of the multichannel Coqblin-Schrieffer impurity model and application to multilevel systems, Phys. Rev. B 58, 3814 (1998).
Y. Luh, Bound state in superconductors with paramagnetic impurities, Acta Phys. Sin. 21, 75 (1965).
Hiroyuki Shiba, Classical spins in superconductors, Prog. Theor. Phys. 40, 435 (1968).
A. I. Rusinov, Superconductivity near a paramagnetic impurity, ZhETF Pis. Red. 9, 146 (1969) [JETP Lett. 9, 85 (1969)], http://jetpletters.ru/ps/1353/article_20434.shtml.
Akio Sakurai, Comments on superconductors with magnetic impurities, Prog. Theor. Phys. 44, 1472 (1970).
K. J. Franke, G. Schulze, and J. I. Pascual, Competition of superconducting phenomena and Kondo screening at the nanoscale, Science 332, 940 (2011).
Alex Zazunov, Stephan Plugge, and Reinhold Egger, Fermi-liquid approach for superconducting Kondo problems, Phys. Rev. Lett. 121, 207701 (2018).
Toshio Aoi and Yoshika Masuda, Magnetic impurity effects in Kondo superconductor, La3Al(Ce). J. Phys. Soc. Jpn. 37, 673 (1974).
E. Müller-Hartmann and J. Zittartz, Kondo effect in superconductors, Phys. Rev. Lett. 26, 428 (1971).
Tamifusa Matsuura, The effects of impurities on superconductors with Kondo effect, Prog. Theor. Phys. 57, 1823 (1977).
Fumihiko Takano and Seitaro Matayoshi, Kondo effect in superconductor, Prog. Theor. Phys. 41, 45 (1969).
J. C. Cuevas, A. Levy Yeyati, and A. Martín-Rodero, Kondo effect in normal-superconductor quantum dots, Phys. Rev. B 63, 094515 (2001).
John Bardeen, Leon N. Cooper, and J. Robert Schrieffer, Microscopic theory of superconductivity, Phys. Rev. 106, 162 (1957).
Pradip Kattel, Abay Zhakenov, and Natan Andrei, Overscreened spin- $\frac{1}{2}$ Kondo impurity and Shiba state at the edge of a one-dimensional spin-1 superconducting wire, Phys. Rev. B 112, 085103 (2025).
Parameshwar R. Pasnoori, Colin Rylands, and Natan Andrei, Kondo impurity at the edge of a superconducting wire, Phys. Rev. Res. 2, 013006 (2020).
Parameshwar R. Pasnoori, Natan Andrei, Colin Rylands, and Patrick Azaria, Rise and fall of Yu-Shiba-Rusinov bound states in charge-conserving $s$ -wave one-dimensional superconductors, Phys. Rev. B 105, 174517 (2022).
Adriano Amaricci, Andrea Richaud, Massimo Capone, Nelson Darkwah Oppong, and Francesco Scazza, Engineering the Kondo impurity problem with alkaline-earth atom arrays, Phys. Rev. A 112, 043301 (2025).
Shintaro Taie, Ryo Yamazaki, Seiji Sugawa, and Yoshio Takahashi, Realization of $SU (2) \times SU (6)$ system of fermions in a cold atomic gas, Phys. Rev. Lett. 105, 190401 (2010).
F. Scazza, C. Hofrichter, M. Höfer, Piere C. De Groot, Immanuel Bloch, and Simon Fölling, Observation of two-orbital spin-exchange interactions with ultracold SU(N)-symmetric fermions, Nat. Phys. 10, 779 (2014).
Guido Pagano, Marco Mancini, Giulio Cappellini, Patrizia Lombardi, Florian Schäfer, Hui Hu, Xia-Ji Liu, Jacopo Catani, Chiara Sias, Massimo Inguscio, and Leonardo Fallani, A one-dimensional liquid of fermions with tunable spin, Nat. Phys. 10, 198 (2014).
We also stress that the running coupling constant in the Bethe Ansatz formalism is nonanalytically related to the running coupling constant in standard UV cutoffs. While this implies that the renormalization group (RG) flows derived in the two frameworks are parametrized differently, we emphasize that the beta function computed from the Bethe Ansatz matches the perturbative QFT result in the limit of small bare coupling, thereby confirming consistency between the two approaches in the weak-coupling regime. (See Ref. [36] for a detailed discussion of the two cutoff schemes.).
Hans Bethe, Zur theorie der metalle: I. Eigenwerte und eigenfunktionen der linearen atomkette, Z. Phys. 71, 205 (1931).
Chen-Ning Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19, 1312 (1967).
M. Gaudin, Un systeme a une dimension de fermions en interaction, Phys. Lett. 24A, 55 (1967).
H. Bergknoff and H. B. Thacker, Structure and solution of the massive thirring model, Phys. Rev. D 19, 3666 (1979).
P. Zinn-Justin and N. Andrei, The generalized multi-channel Kondo model: Thermodynamics and fusion equations, Nucl. Phys. B528, 648 (1998).
We include the factor $\frac{\sqrt{3}}{2}$ up front because, in an $S U (N)$ theory, the mass of a rank- $r$ hole is generally given by $m_{r} = m \sin (\frac{π r}{N}) .$ as shown in Ref. [48].
Pradip Kattel, Parameshwar R. Pasnoori, J. H. Pixley, and Natan Andrei, Edge modes and boundary impurities in the anisotropic Heisenberg spin chain, Phys. Rev. B 111, 174430 (2025).
Takuya Kanazawa and Shun Uchino, Overscreened Kondo effect,(color) superconductivity, and Shiba states in Dirac metals and quark matter, Phys. Rev. D 94, 114005 (2016).
Virgil V. Baran and Jens Paaske, Yu-Shiba-Rusinov states in color superconducting quark matter, arXiv:2502.15930.
Yupeng Wang, Exact solution of the open Heisenberg chain with two impurities, Phys. Rev. B 56, 14045 (1997).
Pradip Kattel, Parameshwar R. Pasnoori, J. H. Pixley, Patrick Azaria, and Natan Andrei, Kondo effect in the isotropic Heisenberg spin chain, Phys. Rev. B 109, 174416 (2024).
Colin Rylands, Exact boundary modes in an interacting quantum wire, Phys. Rev. B 101, 085133 (2020).
A. Kapustin and S. Skorik, Surface excitations and surface energy of the antiferromagnetic XXZ chain by the Bethe Ansatz approach, J. Phys. A 29, 1629 (1996).
C. Destri and J. H. Lowenstein, Analysis of the Bethe-Ansatz equations of the chiral-invariant Gross-Neveu model, Nucl. Phys. B205, 369 (1982).
Yupeng Wang, Wen-Li Yang, Junpeng Cao, and Kangjie Shi, Off-Diagonal Bethe Ansatz for Exactly Solvable Models (Springer, New York, 2015).
The full expression for the $β$ –function of the boundary coupling $J$ , Eq. (e7), is derived exactly within the Bethe–Ansatz cutoff scheme and matches the expression obtained via perturbative calculations in the universal regime. The pole at $32 g^{2} = 9$ is outside the universal regime, similar to the pole in the Bethe-Ansatz coupling $c = 2 J / (1 - J^{2})$ in the standard Kondo model (see Ref. [36] for a detailed discussion about the difference between the two cutoff schemes). In other words, the reliable domain of this expression is limited to $g^{2} ≪ 1$ and $J^{2} ≪ 1$ , where it correctly reproduces the weak–coupling flow, while the exact Bethe-Ansatz solution reveals a smooth and analytic crossover from the UV to the IR fixed point with no actual singularities.

Phys. Rev. D 112, 094012– Published 10 November, 2025

Vol. 112, Iss. 9 — 1 November 2025

Received 9 September 2025
Accepted 21 October 2025

DOI: https://doi.org/10.1103/q6kz-2vl2

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Competing color superconductivity and color Kondo effect in quark matter

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