Kondo effect driven by chirality imbalance

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.


I. INTRODUCTION
The Kondo effect [1][2][3][4][5] is known as a phenomenon which occurs in metal including heavy impurities. It leads to drastic modifications of the transport properties of conducting (or itinerant) electrons at low temperature. While, in the conventional case, itinerant electrons are treated as nonrelativistic fermions, recent studies show that the Kondo effect can be realized also in systems with relativistic fermions.
In relativistic massless fermions, one of the interesting characteristics is their chirality i.e. the left-handed and * suenaga@mail.ccnu.edu.cn † k.suzuki.2010@th.phys.titech.ac.jp ‡ araki.yasufumi@jaea.go.jp § yasuis@keio.jp right-handed degrees of freedoms. In this paper, we propose a novel type of Kondo effect: the Kondo effect driven by a chirality imbalance (or chiral chemical potential µ 5 ). This is similar to the "usual" Kondo effect induced on the Fermi surface but slightly different in the sense that it occurs even at zero chemical potential, µ = 0. We particularly study the Kondo effect at finite µ 5 perturbatively and non-perturbatively: The former is accomplished by the renormalization group (RG) analysis at one loop, and the latter is by the mean-field analysis.
To investigate systems with µ 5 will give a motivation for Monte Carlo (lattice) simulations of strongly correlated quantum systems such as the Kondo effect and quark-gluon dynamics, which is one of the promising tools to nonperturbatively study them. While Monte Carlo simulations with a finite chemical potential µ suffer from the sign problem, at finite chiral chemical potential µ 5 , the sign problem is absent [36] (also see Refs. [37][38][39][40][41]). Therefore, when the Kondo effects are induced by finite µ 5 , we expect that Monte Carlo simulations with µ 5 would be promising for measuring the Kondo effect.
Our analyses can also be extended to Dirac/Weyl semimetals with energy splitting among Dirac/Weyl cones, such as in Weyl semimetals with broken inversion symmetry [42][43][44]. Such an effect may also be reproduced by Zeeman splitting of spin-degenerate Dirac cones in topological Dirac semimetals [45], such as Cd 3 As 2 [46,47]. This paper is organized as follows. In Sec. II, we consider the Kondo effect at finite µ 5 from an effective Lagrangian and a perturbation calculation. In Sec. III, to study the Kondo effect in the nonperturbative region, we formalize a mean field approach, and show the phase diagram of the Kondo effect on the plane of temperature and µ 5 . Section IV is devoted to our conclusion and outlook. In this section, we show the emergence of the Kondo effect at finite µ 5 within a perturbative scheme, which can be signaled by existence of a Landau pole in the renormalization group (RG) flow for the effective coupling between a light fermion and a heavy fermion [48]. 1 We start our discussion by the following Lagrangian to describe a scattering between a light fermion and a heavy fermion: in which ψ and Ψ denote the light fermion and heavy fermion fields, respectively. µ 5 is the chiral chemical potential and M Q is the heavy fermion mass whose value is significantly larger than the typical scale of the theory. t a with an index a = 1, . . . , N 2 − 1 is the generator of the SU (N ) group characterizing a non-Abelian interaction. In terms of the interaction manner between the light fermion and the heavy fermion, we have employed a vector-type contact interaction. 2 G > 0 is the coupling constant. We notice that, in this section, we introduce the heavy fermion field (Ψ) as a Dirac spinor which includes an anti-particle as well as a particle com-respectively, which are diagrammatically indicated in Fig. 1. u(p) and U (q) are the Dirac wavefunctions for the light and heavy fermions, respectively, with p = p i (p f ) and q = q i (q f ) the initial (final) momenta. In Eq. (4), we have employed the imaginary-time formalism to take into account the finite temperature effect, so that the propagators S l (k) and S h (k) take the form of and and∆ where γ ≡ (γ 1 , γ 2 , γ 3 ) is the spatial components of the Dirac gamma matrices. In these expressions, P ± = (1 ± γ 5 )/2 is the right-handed or left-handed projection operator, and the Matsubara frequency is ω n = (2n + 1)πT (n = 0, ±1, ±2, · · · ). The detailed calculation of the one loops in Eqs. (5) and (6) within the imaginary-time formalism is provided in Appendix A.
Before showing the results of Eq.
(2), we notice some important points about the fermion wavefunctions u(p) (ū(p)) or U (q) (Ū (q)). First, in terms of the light fermion wavefunction, it is useful to separate the light fermion transition part in Eq. (2) into the right-handed and lefthanded ones by defining u R = P + u and u L = P − u, since the Lagrangian (1) preserves the axial current.
Next, in terms of the heavy fermion wavefunction, as is well known, the free Dirac spinor can be decomposed into with U ± (q) ≡ Λ ± U (q), by defining the projection operator with respect to the positive-energy (+) and negativeenergy (−) solutions of the Dirac equation: with q 0 = | q| 2 + M 2 Q . When we measure the energy of the fermion from M Q as in the non-relativistic system, Tree-level amplitude .
One-loop amplitude . i.e., by shifting the energy of the positive-energy and negative-energy components commonly as q 0 → q 0 −M Q , we need to cost at least 2M Q for the excitation of the negative-energy component, which can be ignored in the limit of M Q → ∞. Therefore, when we consider such a situation, we can drop U − (q) in Eq. (11), and replace U (q) by U + (q). By taking the above arguments into account, the treelevel amplitude in Eq. (3) can be reduced to in which we have used a fact ofŪ (14) in the limit of M Q → ∞ with the initial-and final-state light fermions inhabiting the "Fermi surface", i.e. the initial-and finial-state light fermions satisfy the kinematics of (p 0 , | p|) = (0, µ 5 ) for the right-handed fermion while (p 0 , | p|) = (2µ 5 , µ 5 ) for the left-handed fermion (p µ stands for p µ i and p µ f collectively), due to the Dirac equation.f β (E) is the Fermi distribution function,f β (E) = 1/(e βE + 1) with inverse temperature β = 1/T , and ρ 0 = µ 2 5 /(2π 2 ) is the density of state on the Fermi surface.
From the above considerations, it turns out that Eqs. (13) and (14) lead to the RG equation [48] as for the coupling G(Λ) of only the right-handed fermion, where the effective coupling G(Λ) depends on the energy scale Λ measured from the Fermi surface. Alternatively, The initial values areḠ0 = 3 atΛ0 = 0.2.
the RG equation (15) can be converted into the dimensionless one as by definingΛ = Λ/µ 5 ,Ḡ(Λ) = G(Λ)µ 2 5 , andβ = βµ 5 (T = T /µ 5 ). We comment that Eq. (16) is reduced to the simple form,Λ The resulting RG flow of the dimensionless couplingḠ with N = 3 is shown in Fig. 2. In this plot, the results withT = 0,T = 0.02, andT = 0.2 are shown. As an example, the initial values are taken to beḠ 0 ≡Ḡ(Λ 0 ) = 3 at the initial high-energy scale Λ 0 = 0.2. The results clearly show the logarithmic divergences at lower-energy scales and the emergence of the Landau poles at the energy scaleΛ =Λ K lower thanΛ 0 (or temperature), implying the appearance of the Kondo effect. We callΛ K the Kondo scale. This behavior is easily understood by the fact that the right-hand side of Eq. (16) is always negative. It is important to note that the Kondo scale is generated dynamically through the quantum processes accompanying the non-Abelian interaction 3 . The existence of the Kondo scale is more clearly confirmed in the case of zero temperature (T = 0). In fact, from Eq. (17), we obtain the analytic form of the solution as leading toΛ The last inequality indicates that the Kondo scale is the low-energy scale, so that it is exponentially smaller than the high-energy scaleΛ 0 . At finite temperature, we notice that, as the temperature becomes higher, the value ofΛ K becomes smaller. Thus, this behavior implies the suppression of the Kondo effect by finite temperature effects.

III. MEAN-FIELD APPROACH
At the low-energy scale below the Kondo scale, we need to describe the Kondo effect in a nonperturbative way. For this purpose, we adopt a mean-field approach describing a mixing between a light relativistic fermion and a heavy fermion based on the treatment in Refs. [12,16].

A. Mean-field Lagrangian
For the light relativistic fermions, we use the one-flavor light-fermion field ψ with a chemical potential µ and a chiral chemical potential µ 5 . 4 For the heavy fermions, we use a redefined field based on the so-called heavy-quark effective theory [49,50] (see Refs. [51,52] for reviews): (1, 0) are the mass and four-velocity of the heavy fermion at rest (the rest frame), respectively. After this redefinition, only the positive-energy component of the original Dirac spinor of the heavy-fermion field survives by the projection operator 1 2 (1 + γ 0 ). The original mass M Q is subtracted by the factor e iM Q v·x .
As a result, the effective Lagrangian is given by where the SU (N ) non-Abelian interaction term is a fourpoint vertex, andG is the coupling constant in the interaction between a light fermion and a heavy particle. 5 λ and n Q are the Lagrange multiplier and heavyparticle density, respectively, for the constraint condi- [12,16]. 6 Thus, the number density of heavy particles are controlled by the Lagrange multiplier method, so that one needs not to regard λ as a chemical potential of heavy particles. 7 As a mean-field approximation, we assume the following form of the condensate, which is the so-called Kondo condensate [12,16]: wherep ≡ p/p (p ≡ | p|) is the unit vector for the threedimensional momentum p. 8 The angle brackets O denote the vacuum expectation value for an operator O.
Note that ∆ R(L) is a complex number, which indicates the mixing between the light fermion and the heavy particle. Thus, |∆ R(L) | gives the absolute value of the Kondo condensate. From Eq. (20), as a result, the mean-field Lagrangian is written as where φ ≡ (ψ t , (Ψ pos v ) t ) contains the six components with the Dirac four-spinor of the light-fermion field ψ and the positive-energy projected components (two-spinor) of the heavy-particle field Ψ t v ≡ ((Ψ pos v ) t , 0). The factor 2 in front of |∆ R(L) | 2 comes from the ansatz (21) and (22). The inverse propagator of φ is given by 3 If there is no non-Abelian interaction (or the generator t a ) in the Lagrangian (1), all the logarithmic divergences from M (1) in Eqs. (14) are canceled, and hence the Kondo scale disappears. 4 The one flavor is a simplified setup, but we can easily extend our formalism to multi-flavor fermions, ψ ≡ (ψ t 1 , ψ t 2 , · · · , ψ t N f ) [12,16]. 5 Note that the four-point interaction in Eq. (20) can be obtained by the Fiertz transformation from Eq. (1). See e.g., Refs. [12,16]. Using the projection operators for the chirality of the light fermions, ψ R = 1+γ 5 2 ψ and ψ L = 1−γ 5 2 ψ, we can easily check the chiral symmetry for the four-point interaction terms: |ψΨv| 2 + |ψiγ 5 Ψv| 2 + |ψ γΨv| 2 + |ψγ 5 γΨv| 2 = 2 |ψ R Ψv| 2 + |ψ L Ψv| 2 + |ψ R γΨv| 2 + |ψ L γΨv| 2 . 6 Notice thatΨv = Ψ † v in the rest frame. 7 The Kondo effect for a single heavy particle within the same mean-field ansatz is formalized in Ref. [13]. 8 The momentum dependence in Eq. (22) is called the hedgehog solution. We assumed the scalar and hedgehog condensate have the same value of ∆ R(L) .
in the standard representation of the Dirac matrices.

B. Dispersion relations
By solving det[G(p 0 , p) −1 ] = 0, we obtain the six energy-momentum dispersion relations with µ R,L ≡ µ ± µ 5 . The four modes, E R± and E L± , are the mixing modes (quasiparticles) between the light fermion and the heavy particle, which are induced by the nonzero value of the Kondo condensate ∆ R(L) . On the other hand,Ẽ R andẼ L are the decoupling anti-particle modes. The obtained dispersion realtions and the wave functions lead to the quasiparticle fermions, but they preserve the topological properties for the original massless Dirac fermions, where the Berry's curvature induces the monopoles in momentum space [16].
A schematic figure of these dispersion relations is shown in Fig. 3. Among them, the quasiparticles with E R− and E L− are essential for the Kondo effect because the Kondo condensate is induced by the occupation of quasiparticles under E(p) = 0.

C. Thermodynamic potential
From the modes in Eqs. (25)-(28), the thermodynamic potential at finite temperature T is obtained as Ω(T, µ, µ 5 , λ; ∆ R(L) ) = N Λcut 0 f (T, µ, µ 5 , λ; p) p 2 dp 2π 2 where Λ cut is an ultraviolet cutoff parameter of the momentum integral, and the integrand is From the minimization condition of Eq. (29) or the gap equation ∂Ω/∂∆ R = ∂Ω/∂∆ L = 0, we can determine ∆ R(L) in a self-consistent way. In this model setting, the free parameters areG and Λ cut (and N ), and they can be tuned for a specific system, as it will be explained later.

D. Numerical results
The Kondo condensate ∆ R as a function of µ 5 > 0 is plotted in Fig. 4. Here we use, for example,G = 2/Λ 2 cut and 4/Λ 2 cut at N = 3. We find that ∆ R is enhanced as µ 5 increases. This behavior indicates that the (relativistic) Kondo effect is induced by finite µ 5 . This is consistent with the result from the perturbative analysis in Sec. II. We emphasize that the usual (nonrelativistic and relativistic) Kondo effects occur at finite µ, but the Kondo effect at finite µ 5 appears even when µ = 0. This is a unique property of relativistic fermions composing matter including impurities. Such Kondo effects can be realized in relativistic-fermion matter, i.e. Weyl/Dirac metal/semimetals and quark matter.
Furthermore, within our model, the phase transition along the µ 5 axis is a crossover. This is the same result as the transition along the µ axis is also a crossover (see Appendix B).
Within our parameters, we numerically find that, for µ 5 > 0, the Kondo effect is dominated by the righthanded condensate ∆ R , and the value of the left-handed condensate ∆ L is almost zero. On the other hand, in the case of µ 5 < 0, ∆ L dominates the Kondo effect.
For a typical parameter in the QCD Kondo effect, we apply the coupling constant,G = G c , where G c ≡ 2/Λ 2 cut and Λ cut = 0.65 GeV, and the number of the colors is N = 3. These parameters are the same as those used in the Nambu-Jona-Lasinio model with a four-point interaction between a light quark and a light antiquark [53]. When we use G c , we find ∆ R = 7.9 MeV at µ 5 = 0.5 GeV. If we use a stronger coupling constant, the Kondo effect is increasingly enhanced, as shown by the blue curve in Fig. 4. Note that, if we extrapolate the results to 0.75 µ 5 /Λ cut , then we find a sudden decrease of ∆ R , but this behavior is an artifact from the cutoff Λ cut in our model.
We comment the possible setup on lattice QCD simulations. At finite µ, the Monte-Carlo simulations suffer from the sign problem, so that it is difficult to measure the QCD Kondo effect (by finite µ) by using lattice simulations. On the other hand, at finite µ 5 , we can escape from the sign problem [36][37][38][39][40][41], and the QCD Kondo effect (by finite µ 5 ) will be observed.
Finally, we give a discussion on the temperature dependence of ∆ R at finite µ 5 . In Fig. 5, we show ∆ R on the T -µ 5 plane. We observe that, when a finite T is switched on, the value of ∆ R decreases: the Kondo effect is suppressed by finite-temperature effects, which is again consistent with the perturbative analysis in Sec. II. While the transition along the µ 5 axis at T = 0 is a crossover, the transition at finite T > 0 is a second order.

IV. CONCLUSION AND OUTLOOK
In this paper, we proposed the Kondo effect driven by a chirality imbalance (or chiral chemical potential µ 5 ) from the point of view of the two theoretical approaches. Using the perturbative approach, we found the infrared divergence of scattering amplitude as a signal of the Kondo effect. Using the mean-field approach, we found that the Kondo condensate is enhanced by finite µ 5 . These are universal properties in relativistic-fermion matter with heavy impurities and a chirality imbalance, which can be attributed to the enhancement of the density of states at the Fermi surface. Our findings generalize the analysis of the Kondo effect in Dirac/Weyl electron systems with an energy splitting among Dirac cones [26], involving various types of SU (N ) exchange interactions, such as spin, isospin, and color. The interplay effect between the exchange interaction and particular spin-orbit coupling in crystalline electron systems, such as topological Dirac semimetal Cd 3 As 2 , is left for further analysis.
As a topics not covered in the present study, we comment that the response to magnetic and electric fields would be interesting. For example, when µ 5 is coupled to a magnetic field, an electric current can be induced, which is the so-called chiral magnetic effect [36,54]. The correlation between the chiral transport phenomena and the Kondo effects will be worth to be studied. See for example the discussion of the transport coefficients in the Kondo effect in relativistic fermion gas [18].
In the context of QCD, lattice simulations at finite µ 5 evade from the sign problem [36][37][38][39][40][41], so that we can numerically measure the QCD Kondo effects in a fully nonperturbative way. The ground state of QCD in the lowtemperature and/or low-chemical potential region is the chiral-symmetry breaking phase characterized by the chiral condensate, and the ground state in the high-chemical potential region is expected to be the color superconducting phase characterized by diquark condensate. These condensates could exclude the Kondo condensate [14,17] or might induce a "coexistence" phase with two order parameters [17]. The topological properties of the QCD Kondo effect is also an interesting issue [16]. However, the conclusion from the effective models depends on the coupling constants of the interactions, and in the future it should be checked based on QCD.
In particular, the properties of chiral condensates at finite µ 5 have been studied from chiral effective models [36,, Schwinger-Dyson equations [76,77], and lattice QCD simulations [39][40][41]. One of the characteristic properties is the catalysis effect of the chiral symmetry breaking by finite µ 5 . Therefore, in matter with a chirality imbalance and impurities, the two catalysis effects of the chiral symmetry breaking and Kondo effect could be correlated.
In addition, in two-(or multi-) component fermion systems, the situation including an imbalance between the chemical potentials of different fermions would be also important. In QCD, the isospin chemical potential µ I , an imbalance between up-and down-quark chemical poten-tials, is realized in neutron-rich nuclei and neutron stars, and lattice QCD simulations are also applicable [78][79][80][81]. For a similar external parameter to µ 5 , the effects from the chiral isospin chemical potential µ I5 could be also interesting [71,72,[82][83][84][85][86]. Appendix A: Matsubara summation in Eqs. (5) and (6).
In this appendix, we show a detailed calculation of Matsubara summation in the one-loop amplitudes in Eqs. (5) and (6).
Within the imaginary-time formalism, Eqs. (5) and (6) are rewritten to and respectively, where the Matsubara Green's functions for the light and heavy fermions are given bỹ and∆ with the Matsubara frequency ω n = (2n + 1)πT (n = 0, ±1, ±2, · · · ). Therefore, apart from the spinor and SU (N ) non-Abelian algebras, we need to calculate and for the evaluation of M (1a) and M (1b) . First, let us demonstrate a detailed calculation of I 1 . The three-momentum integral in Eq. (A5) is performed by the conventional procedure as in the vacuum, such that we show only the zeroth components of the momentum or coordinate space explicitly below. The inverse Fourier transformations of the Matsubara Green's func-tions∆ l i(ω n − i 5 µ 5 ) and∆ h (iω n ) given in Eqs. (A3) and (A4) can be defined bỹ Then, by making use of the Poisson summation formula we get the following equation: The Matsubara Green's function∆ l(h) (τ ) is defined by an analytic continuation of the greater Green's function Here, we remind that the Fourier transformation of the greater Green's function S > l(h) (t) can be expressed as [87] S > l(h) (t) = in whichf β (k 0 ) is the Fermi distribution function, f β (k 0 ) = 1/(e βk0 + 1) (β = 1/T ), andρ l(h) is the spectral functionρ Then, we find that Eq. (A9) can be rewritten to . By performing the analytic continuations of iω qi → q 0 i + i , iω pi → p 0 i + i , and replacing the energy of the external heavy fermion by its mass as q 0 → M Q , together with the M Q → ∞ limit, we find that Eq. (A14) is reduced to withΛ + ≡ lim M Q →∞ (/ q + M Q )/(2M Q ) = (1 + γ 0 )/2. Hence, by combining the three-momentum integral, finally we can evaluate I 1 in Eq. (A5) as Note that we are interested in only the real part of the amplitude, so that the imaginary parts have been omitted.
the second line, we find where we have replaced the density of states by that on the Fermi surface, ρ 0 = µ 2 5 /(2π 2 ) , since this study is based on the assumption that µ 5 is sufficiently large.
Appendix B: Mean-field approach for Kondo effect at finite µ In this appendix, in order to compare the Kondo effects at finite µ and µ 5 , we show the phase diagram at finite µ using the same formalism as those in the main text. In the upper panel of Fig. 6, we show the µ-µ 5 phase diagram of ∆ R . In the region with large µ and/or µ 5 , we find the appearance of the Kondo phase with nonzero ∆ R . The transitions along either the µ axis or the µ 5 axis are still a crossover. Note that, in the region with large µ+µ 5 , ∆ R is suddenly suppressed and becomes zero, but this behavior is an artifact from the ultraviolet cutoff, as mentioned in the main text. Therefore, we cannot conclude the true physics in this region, which is beyond the scope of this model. As shown in the lower panel of Fig. 6, in the region with large µ but small µ 5 , we find that ∆ L is also enhanced. This behavior indicates that the "usual" Kondo effect induced by only finite chemical potential is realized, where both the right-handed and left-handed condensates contribute to the Kondo effect (namely, ∆ R ≈ ∆ L ). [2] A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, 1993).