PHYSICAL REVIEW RESEARCH 2, 023312 (2020) Kondo effect driven by chirality imbalance Daiki Suenaga,1,* Kei Suzuki,2,† Yasufumi Araki,2,‡ and Shigehiro Yasui3,§ 1Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China 2Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai 319-1195, Japan 3Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan (Received 10 January 2020; accepted 5 May 2020; published 9 June 2020) We propose a novel mechanism of the Kondo effect driven by a chirality imbalance (or chiral chemical potential) of relativistic light fermions. This effect is realized by the mixing between a right- or left-handed fermion and a heavy impurity in the chirality imbalanced matter even at zero density. This is different from the usual Kondo effect induced by finite density. We derive the Kondo effect from both a perturbative calculation and a mean-field approach. We also discuss the temperature dependence of the Kondo effect. The Kondo effect at nonzero chiral chemical potential can be tested by future lattice simulations. DOI: 10.1103/PhysRevResearch.2.023312 I. INTRODUCTION handed degrees of freedoms. In this paper, we propose a novel type of Kondo effect: the Kondo effect driven by a The Kondo effect [1–5] is known as a phenomenon which chirality imbalance (or chiral chemical potential μ ). This occurs in metal including heavy impurities. It leads to drastic 5 is similar to the “usual” Kondo effect induced on the Fermi modifications of the transport properties of conducting (or surface but slightly different in the sense that it occurs even itinerant) electrons at low temperature. While, in the conven- at zero chemical potential, μ = 0. We particularly study the tional case, itinerant electrons are treated as nonrelativistic Kondo effect at finite μ perturbatively and nonperturbatively: fermions, recent studies show that the Kondo effect can be 5 The former is accomplished by the renormalization group realized also in systems with relativistic fermions. (RG) analysis at one loop, and the latter is by the mean-field One example of the Kondo effect realized in the relativistic analysis. In the present work, we do not take into account the system is the isospin Kondo effect. This effect can be induced effects from interactions between two light fermions such as near the Fermi surface of nucleons with a heavy hadron chiral condensate in order to focus on the Kondo effect in a such as (∗) or D¯ (∗) existing as an impurity, where the non- c transparent way. Abelian SU(2) interaction between the light nucleon and the To investigate systems with μ will give a motivation heavy hadron is supplied by an isospin exchange [6–9]. In 5 for Monte Carlo (lattice) simulations of strongly correlated the context of quantum chromodynamics (QCD) in which quantum systems such as the Kondo effect and quark-gluon the non-Abelian SU(3) interaction is governed by the color dynamics, which is one of the promising tools to nonper- interaction mediated by gluons, the so-called QCD Kondo turbatively study them. While Monte Carlo simulations with effect which may be realized in quark matter composed of up a finite chemical potential μ suffer from the sign problem, and down (and also often strange) quarks with a heavy (charm at finite chiral chemical potential μ , the sign problem is or bottom) quark, has been studied in the literatures [6,10–22]. 5 absent [36] (also see Refs. [37–41]). Therefore, when the Moreover, in the context of electron systems with Dirac or Kondo effects are induced by finite μ , we expect that Monte Weyl dispersion in solid states, called Dirac or Weyl semimet- 5 Carlo simulations with μ would be promising for measuring als, it has been seen that the vanishing density of states 5 the Kondo effect. around the Dirac or Weyl points leads to an anomalous Kondo In the context of QCD, a chirality imbalance might be screening behavior, distinct from normal metals [14,23–35]. realized in the heavy-ion collision (HIC) experiments. Argu- In relativistic massless fermions, one of the interesting ments on its possibility have a long history [42], and there characteristics is their chirality, i.e., the left-handed and right- are some scenarios leading to local parity violation, such as the sphaleron transition [43–45], the parallel color electric *[email protected] and magnetic fields (or the Glasma) [44,46], and disoriented †[email protected] pseudoscalar condensates [47,48]. At early state of HIC, ‡[email protected] heavy quarks are also produced by hard processes mediated 1 §[email protected] by gluons from nucleon-nucleon scattering at high energy. 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) 1These heavy quarks can play a role of heavy impurities without and the published article’s title, journal citation, and DOI. satisfying any chemical equilibrium conditions, when we focus on 2643-1564/2020/2(2)/023312(13) 023312-1 Published by the American Physical Society SUENAGA, SUZUKI, ARAKI, AND YASUI PHYSICAL REVIEW RESEARCH 2, 023312 (2020) Thus HIC is expected to be a possible environment to study The scattering amplitude between the light fermion and the the Kondo effect with chirality imbalance.2 heavy fermion up to one loop is of the form Our analyses can also be extended to Dirac or Weyl M = M(0) + M(1), semimetals with energy splitting among Dirac or Weyl cones (2) in electronic band structure, such as in Weyl semimetals with where M(0) and M(1) are the amplitude at tree level and at broken inversion symmetry [49–51]. Such an effect may be one-loop level, respectively. Explicitly, M(0) and M(1) are reproduced as well by Zeeman splitting of spin-degenerate obtained as Dirac cones in topological Dirac semimetals [52], such as M(0) = aγ μ ¯ aγ Cd3As2 [53,54]. Gu¯(p f )t u(pi )U (q f )t μU (qi )(3) This paper is organized as follows. In Sec. II, we consider and the Kondo effect at finite μ5 from an effective Lagrangian and a perturbation calculation. In Sec. III, to study the Kondo M(1) = M(1a) + M(1b), (4) effect in the nonperturbative region, we formalize a mean field approach, and show the phase diagram of the Kondo effect on with μ the plane of temperature and 5. Section IV is devoted to our d3k M(1a) =−G2T u¯(p )t aγ μS (k)t bγ ν u(p ) conclusion and outlook. (2π )3 f l i n × ¯ aγ − + bγ , II. PERTURBATIVE APPROACH U (q f )t μSh(qi k pi )t νU (qi ) (5) In this section, we show the emergence of the Kondo and μ effect at finite 5 within a perturbative scheme, which can be d3k M(1b) =− 2 aγ μ bγ ν signaled by existence of a Landau pole in the renormalization G T u¯(p f )t Sl (k)t u(pi ) (2π )3 group (RG) flow for the effective coupling between a light n fermion and a heavy fermion [55].3 b a We start our discussion by the following Lagrangian to ×U¯ (q f )t γν Sh(qi + k − p f )t γμU (qi ), (6) describe a scattering between a light fermion and a heavy respectively, which are diagrammatically indicated in Fig. 1. fermion: u(p) and U (q) are the Dirac wave functions for the light and / / heavy fermions, respectively, with p = pi (p f ) and q = qi (q f ) L = ψ¯ (i∂ + μ5γ0γ5)ψ + ¯ (i∂ − MQ ) the initial (final) momenta. In Eq. (4), we have employed a μ a + G(ψ¯t γ ψ)(¯ t γμ) , (1) the imaginary-time formalism to take into account the finite ψ temperature effect, so that the propagators Sl (k) and Sh(k)take in which and denote the light-fermion and heavy-fermion the form of fields, respectively. μ5 is the chiral chemical potential and MQ = ˜ ω − μ is the heavy-fermion mass whose value is significantly larger Sl (k) P 5 l (i( n i 5 5)) (7) a = than the typical scale of the theory. t with an index a 5=± 1,...,N2 − 1 is the generator of the SU(N) group charac- and terizing a non-Abelian interaction. In terms of the interaction manner between the light fermion and the heavy fermion, we Sh(k) = ˜ h(iωn ), (8) have employed a vector-type contact interaction.4 G > 0isthe coupling constant. We notice that, in this section, we introduce with the heavy-fermion field () as a Dirac spinor which includes i(−ωn + i 5μ5)γ0 + k · γ an antiparticle as well as a particle component. However, later, ˜ l (i(ωn − i 5μ5)) =− (9) (ω − i μ )2 +|k|2 we will take a limit of MQ →∞to describe the emergence of n 5 5 the Kondo effect in more transparent way. and −iωnγ0 + k · γ − MQ ˜ h(iωn ) =− , (10) ω2 +||2 + 2 n k MQ the short time scale in which the weak decay of the heavy quarks where γ ≡ (γ 1,γ2,γ3) is the spatial components of the Dirac does not take place. The Kondo effect can evolve regardless of any P± = ± γ / chemical equilibriums on the heavy quarks. gamma matrices. In these expressions, (1 5) 2isthe 2 right-handed or left-handed projection operator, and the Mat- The realization of chiral chemical potential large enough in ex- ω = + π = , ± , ± ,..