Heavy-quark spin polarization induced by the Kondo effect in a magnetic field

The Kondo effect is known as one of the important phenomena in the quantum many-body system Kondo (1964); Hewson (1993). This effect arises when heavy impurities exist in matter coupling with itinerant fermions through non-Abelian interactions such as the spin exchange. The Kondo effect changes transport properties significantly; e.g., it turns the electrical resistivity from decrement to increment with the temperature lowered.

Recently, it has been discussed that the Kondo effect persists universally in high-energy physics as well, although the effect was originally discovered in condensed-matter physics. For example, in the context of quantum chromodynamics (QCD), the effect is driven when heavy quarks ($c,b$) exist as an impurity in dense quark matter formed by light quarks ($u,d$) Yasui and Sudoh (2013); Hattori et al. (2015). This is particularly referred to as the QCD Kondo effect. In this case, the non-Abelian interaction is supplied by $SU(N_{c})$ color exchange ($N_{c}=3$ is the number of colors). The Kondo effect is expected to arise also in the hadronic phase at lower density. In fact, it was shown that the Kondo effect arises in nuclear matter when $\Sigma_{c}$ and $\Sigma^{*}_{c}$ baryons or $\bar{D}$ and $\bar{D}^{*}$ mesons exist as impurities, where the spin and/or isospin exchange serves as the non-Abelian interaction Yasui and Sudoh (2013, 2017); Yasui (2016); Yasui and Miyamoto (2019). In addition, many works on the Kondo effect in a relativistic system have been done in the literatures Ozaki et al. (2016); Yasui et al. (2019); Yasui (2017); Kanazawa and Uchino (2016); Kimura and Ozaki (2017); Yasui et al. (2017); Suzuki et al. (2017); Yasui and Ozaki (2017); Kimura and Ozaki (2019); Fariello et al. (2019); Hattori et al. (2019); Suenaga et al. (2020a, b); Kanazawa (2020); Araki et al. (2021a, b); Kimura (2021); Suenaga et al. (2021); Ishikawa et al. (2021). In this way, the interdisciplinary study bridging condensed-matter physics and QCD/hadron physics focused on the Kondo effect is attracting attention.

In a field-theoretical treatment, the Kondo effect is caused by a condensate (a hybridization) formed by a light itinerant fermion and a heavy impurity called Kondo condensate. This condensate not only provides a gap of the itinerant fermions near the Fermi level but also causes a mixing between the itinerant fermion and the heavy impurity as induced by an $s$-$d$ interaction in the Anderson model Anderson (1961). In quark matter, due to the mixing, in Ref. Suenaga et al. (2021), the magnetically induced axial current of light quarks in quark matter, namely, the chiral separation effect (CSE) Son and Zhitnitsky (2004); Metlitski and Zhitnitsky (2005); Newman and Son (2006), was found to be enhanced under the Kondo effect. In solid states, in Ref. Araki et al. (2021b), the magnetic responses of a Dirac and nonrelativistic bands with their hybridizations were investigated, showing that the spin-orbit crossed part of the magnetic susceptibility of the Dirac (nonrelativistic) band is enhanced (suppressed) due to the presence of the hybridizations. Those findings show that the Kondo condensate (hybridization) has a great impact on magnetic responses of the fermions.

In this paper, we propose a new mechanism of the heavy-quark spin polarization (HQSP) induced by the Kondo effect under a magnetic field. When the Kondo effect is absent, the HQSP does not arise significantly, since the Zeeman interaction of a heavy quark is of order $1/m_{Q}$ ($m_{Q}$ is the mass of a heavy quark) and is suppressed for sufficiently large $m_{Q}$. In particular, it vanishes at $m_{Q}\to\infty$. On the other hand, when the Kondo effect is present, the HQSP can arise without $1/m_{Q}$ suppression. The Kondo condensate correlates the heavy-quark spin with the light-quark spin coupling to the magnetic field in a medium. In other words, the condensate converts the magnetic response of the light-quark spin into that of heavy-quark spin. Schematically this effect is depicted in Fig. 1. This new mechanism of the HQSP is expected to be testable in future sign-problem-free lattice simulations.

This paper is organized as follows. In Sec. II, we introduce the model for demonstration and derive the Green’s function of the fermions. In Sec. III, we show the analytic evaluation for the HQSP within the linear response theory with the vertex corrections. In Sec. IV, we study the HQSP response function in the so-called dynamical and static limits in detail, toward a better understanding of the HQSP in the timelike and spacelike momentum regions. Referring to it, in Sec. V, we present numerical results of the HQSP in both the momentum regions. In Sec. VI, we compare the HQSP induced by the Kondo effect with that by the ordinary Zeeman effect, and in Sec. VII, we conclude the present work.

Here, we present our effective model to study the HQSP induced by the Kondo effect under a magnetic field. One of the most useful models for the demonstration is the Nambu-Jona-Lasinio (NJL) type model including a four-point interaction between a light and a heavy quark Yasui et al. (2019, 2017). Namely, we start with the following Lagrangian:

	$\displaystyle{\cal L}$	$\displaystyle=$	$\displaystyle\bar{\psi}(i{\ooalign{\hfil/\hfil\crcr$D$}}+\mu\gamma_{0})\psi+\Psi_{v}^{\dagger}iD_{0}\Psi_{v}-\lambda(\Psi^{\dagger}_{v}\Psi_{v}-n_{Q})$		(4)
			$\displaystyle-g\Big{[}(\bar{\psi}\Psi_{v})(\Psi^{\dagger}_{v}\psi)+(\bar{\psi}\gamma^{i}\Psi_{v})(\Psi_{v}^{\dagger}\gamma^{i}\psi)\Big{]}\ .$

In this Lagrangian, $\psi$ is the light quark described by the ordinary Dirac theory, while $\Psi_{v}$ is the heavy one described by the heavy-quark effective theory (HQET) Eichten and Hill (1990); Georgi (1990); Neubert (1994); Manohar and Wise (2000). Thus, $\Psi_{v}$ is defined by $\Psi_{v}={\rm e}^{im_{Q}t}\frac{1+\gamma_{0}}{2}\Psi$, with $\Psi$ being the relativistic heavy fermion in Dirac theory, and $\mu$ is the chemical potential of the light quark included to access finite density. It should be noted that $\lambda$ is a Lagrange multiplier included to impose a condition $\Psi_{v}^{\dagger}\Psi_{v}=n_{Q}$, with $n_{Q}$ a space-averaged heavy-quark density. In the following analysis, we examine the HQSP for the $\lambda=0$ case as a clear demonstration. The common coupling $g$ for the scalar and vector interactions is derived by the Fierz transformation from the one-gluon exchange interaction Ebert et al. (1995). In Eq. (4) the covariant derivatives are defined by

	$\displaystyle D_{\mu}\psi$	$\displaystyle=$	$\displaystyle(\partial_{\mu}+ie_{q}A_{\mu})\psi\ ,$
	$\displaystyle D_{0}\Psi_{v}$	$\displaystyle=$	$\displaystyle(\partial_{0}+ie_{Q}A_{0})\Psi_{v}\ ,$		(5)

where the electromagnetic gauge field $A^{\mu}=(A^{0},{\bm{A}})$ is introduced to incorporate interactions with the magnetic field. The coupling constants $e_{q}$ and $e_{Q}$ are electric charges of the light and heavy quarks, respectively. We take $e_{q}=+\frac{2}{3}e$ for the $u$ quark, $e_{q}=-\frac{1}{3}e$ for the $d$ quark, and $e_{Q}=+\frac{2}{3}e$ for the $c$ quark, with the elementary charge $e$.

For the Lagrangian (4), we introduce the Kondo condensate made of the light and heavy quarks to describe the ground state governed by the Kondo effect. For this purpose, we make use of the mean field approximation for $\bar{\psi}\Psi_{v}$ and $\bar{\psi}\gamma^{i}\Psi_{v}$ by rewriting the Lagrangian (4) as

	$\displaystyle{\cal L}$	$\displaystyle=$	$\displaystyle\bar{\psi}(i{\ooalign{\hfil/\hfil\crcr$D$}}+\mu\gamma_{0})\psi+\Psi_{v}^{\dagger}iD_{0}\Psi_{v}$		(9)
			$\displaystyle-g\Big{[}\langle\bar{\psi}\Psi_{v}\rangle\Psi_{v}^{\dagger}\psi+\langle\bar{\psi}\gamma^{i}\Psi_{v}\rangle\Psi_{v}^{\dagger}\gamma^{i}\psi+({\rm h.c.})\Big{]}$
			$\displaystyle+g\Big{[}\langle\bar{\psi}\Psi_{v}\rangle\langle\Psi_{v}^{\dagger}\psi\rangle+\langle\bar{\psi}\gamma^{i}\Psi_{v}\rangle\langle\Psi_{v}^{\dagger}\gamma^{i}\psi\rangle\Big{]}\ ,$

with $\lambda=0$ taken. In the previous works Yasui et al. (2019, 2017), it was found that the reasonable ground state is realized by assuming the so-called hedgehog ansatz provided by

$\displaystyle\langle\bar{\psi}\Psi_{v}\rangle=\frac{1}{g}\Delta\ ,\ \ \langle\bar{\psi}\gamma^{i}\Psi_{v}\rangle=\frac{1}{g}\Delta\hat{p}^{i}\ ,$

(10)

with $\hat{p}^{i}\equiv p^{i}/|{\bm{p}}|$ defined in momentum space, where ${\bm{p}}$ is identical to the momentum of the heavy and light quarks.¹¹1Thus, $\hat{p}^{i}$ in Eq. (10) is not used to minimize the thermodynamic potential. In Eq. (10), $\Delta$ serves as the Kondo condensate made of a light quark and a heavy quark. The value of $\Delta$ should be determined by a variational method on the free energy of the system. Despite the violation of $O(3)$ rotational invariance due to the magnetic field, we employ the ansatz (10) as an appropriate configuration of the Kondo effect since we make use of the linear response theory for a weak magnetic field.

The nontrivial phase characterized by $\Delta\neq 0$ is called the Kondo phase, whereas the trivial phase by $\Delta=0$ is called the normal phase.²²2Due to the ansatz (10), only positive-energy parts of light and heavy quarks are correlated to form the Kondo condensate, as is explicitly shown in Eq. (24). Such a condensate is reasonable since the heavy quark carries only positive-energy parts due to its heavy mass, and for the light quark, the Fermi sea is sufficiently formed where antiparticle contributions are rather suppressed. Therefore, the Kondo phase with the ansatz (10) is expected to be one of the thermodynamically preferred phases Yasui et al. (2017).

From the Lagrangian (9) together with the ansatz (10), the inverse of Green’s function of the fermions in the absence of a gauge field is read as

$\displaystyle i{\cal G}_{0}^{-1}(p_{0},{\bm{p}})=\left(\begin{array}[]{ccc}p_{0}+\mu&-{\bm{p}}\cdot{\bm{\sigma}}&\Delta^{*}\\ {\bm{p}}\cdot{\bm{\sigma}}&-p_{0}-\mu&-\Delta^{*}\hat{\bm{p}}\cdot{\bm{\sigma}}\\ \Delta&\Delta\hat{\bm{p}}\cdot{\bm{\sigma}}&p_{0}\\ \end{array}\right)\ .$

(14)

This Green’s function is a $6\times 6$ matrix since the light and heavy quarks are four-component and two-component spinors, respectively. The inverse of Eq. (14) yields the Green’s function which is of the form Suenaga et al. (2021)

$\displaystyle\tilde{\cal G}_{0}(p_{0},{\bm{p}})=\left(\begin{array}[]{cc}\tilde{\cal G}_{0}^{\bar{\psi}\psi}(p_{0},{\bm{p}})&\tilde{\cal G}_{0}^{\bar{\psi}\Psi_{v}}(p_{0},{\bm{p}})\\ \tilde{\cal G}_{0}^{{\Psi}_{v}^{\dagger}\psi}(p_{0},{\bm{p}})&\tilde{\cal G}_{0}^{{\Psi}_{v}^{\dagger}\Psi_{v}}(p_{0},{\bm{p}})\\ \end{array}\right)\ ,$

(17)

with the elements

	$\displaystyle\tilde{\cal G}_{0}^{\bar{\psi}\psi}(p_{0},{\bm{p}})$	$\displaystyle=$	$\displaystyle i\left[\frac{U_{+}({\bm{p}})}{p_{0}-E_{\bm{p}}^{+}}+\frac{U_{-}({\bm{p}})}{p_{0}-E_{\bm{p}}^{-}}\right]\Lambda_{\rm p}$
			$\displaystyle+i\frac{U_{\rm a}({\bm{p}})}{p_{0}-{E}_{\bm{p}}^{\rm a}}\Lambda_{\rm a}\,,$
	$\displaystyle\tilde{\cal G}_{0}^{\bar{\psi}\Psi_{v}}(p_{0},{\bm{p}})$	$\displaystyle=$	$\displaystyle i\left[\frac{V^{*}_{+}({\bm{p}})}{p_{0}-E_{\bm{p}}^{+}}+\frac{V^{*}_{-}({\bm{p}})}{p_{0}-E_{\bm{p}}^{-}}\right]\Lambda_{{\rm p}{\rm H}}\ ,$
	$\displaystyle\tilde{\cal G}_{0}^{{\Psi}_{v}^{\dagger}\psi}(p_{0},{\bm{p}})$	$\displaystyle=$	$\displaystyle i\left[\frac{V_{+}({\bm{p}})}{p_{0}-E_{\bm{p}}^{+}}+\frac{V_{-}({\bm{p}})}{p_{0}-E_{\bm{p}}^{-}}\right]\Lambda_{{\rm H}{\rm p}}\ ,$
	$\displaystyle\tilde{\cal G}_{0}^{{\Psi}_{v}^{\dagger}\Psi_{v}}(p_{0},{\bm{p}})$	$\displaystyle=$	$\displaystyle i\left[\frac{W_{+}({\bm{p}})}{p_{0}-E_{\bm{p}}^{+}}+\frac{W_{-}({\bm{p}})}{p_{0}-E_{\bm{p}}^{-}}\right]{\bm{1}}\ .$		(18)

The matrices $\Lambda_{\rm p}$ and $\Lambda_{\rm a}$ are the projection operators for positive- and negative-energy states of the light quark defined by

$\displaystyle\Lambda_{\rm p}\equiv\frac{1+\hat{\bm{p}}\cdot{\bm{\alpha}}}{2}\gamma_{0}\ ,\ \ \Lambda_{\rm a}\equiv\frac{1-\hat{\bm{p}}\cdot{\bm{\alpha}}}{2}\gamma_{0}\ ,$

(19)

with ${\bm{\alpha}}=\gamma_{0}{\bm{\gamma}}$, respectively, and ${\bm{1}}$ in Eq. (18) is a $2\times 2$ unit matrix. Similarly, $\Lambda_{\rm pH}$ and $\Lambda_{\rm Hp}$ are the operators mixing the positive-energy states of light and heavy quarks defined by

$\displaystyle\Lambda_{{\rm p}{\rm H}}\equiv\left(\begin{array}[]{c}1\\ \hat{\bm{p}}\cdot{\bm{\sigma}}\\ \end{array}\right)\ ,\ \ \Lambda_{{\rm H}{\rm p}}\equiv\left(\begin{array}[]{cc}1&-\hat{\bm{p}}\cdot{\bm{\sigma}}\\ \end{array}\right)\ ,$

(23)

respectively. The dispersion relations $E_{\bm{p}}^{+}$, $E_{\bm{p}}^{-}$, and $E_{\bm{p}}^{\rm a}$ are given by

	$\displaystyle E_{\bm{p}}^{+}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(|{\bm{p}}|-\mu+\sqrt{(|{\bm{p}}|-\mu)^{2}+8|\Delta|^{2}}\right)\ ,$
	$\displaystyle E_{\bm{p}}^{-}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(|{\bm{p}}|-\mu-\sqrt{(|{\bm{p}}|-\mu)^{2}+8|\Delta|^{2}}\right)\ ,$
	$\displaystyle E^{\rm a}_{\bm{p}}$	$\displaystyle=$	$\displaystyle-|{\bm{p}}|-\mu\ .$		(24)

In the following analysis, the modes carrying the dispersions $E_{\bm{p}}^{+}$, $E_{\bm{p}}^{-}$, and $E_{\bm{p}}^{\rm a}$ are referred to as the Kondo quasiparticle (“q.p.”), the Kondo quasihole (“q.h.”), and the light antiparticle (“a.p.”), respectively. The factors $U({\bm{p}})$’s, $V({\bm{p}})$’s, and $W({\bm{p}})$’s in Eq. (18) are

	$\displaystyle U_{+}({\bm{p}})=\frac{2(|\Delta|^{2}+|{\bm{p}}|E_{\bm{p}}^{+})}{(E_{\bm{p}}^{-}-E_{\bm{p}}^{+})(E_{\bm{p}}^{\rm a}-E_{\bm{p}}^{+})}\ ,$
	$\displaystyle U_{-}({\bm{p}})=\frac{2(|\Delta|^{2}+|{\bm{p}}|E_{\bm{p}}^{-})}{(E_{\bm{p}}^{+}-E_{\bm{p}}^{-})(E_{\bm{p}}^{\rm a}-E_{\bm{p}}^{-})}\ ,$
	$\displaystyle U_{\rm a}({\bm{p}})=1\ ,$
	$\displaystyle V_{+}({\bm{p}})=\frac{\Delta}{E_{\bm{p}}^{-}-E_{\bm{p}}^{+}}\ ,\ \ V_{-}({\bm{p}})=\frac{\Delta}{E_{\bm{p}}^{+}-E_{\bm{p}}^{-}}\ ,$
	$\displaystyle W_{+}({\bm{p}})=\frac{E_{\bm{p}}^{+}-|{\bm{p}}|+\mu}{E_{\bm{p}}^{+}-E_{\bm{p}}^{-}}\ ,\ \ W_{-}({\bm{p}})=\frac{E_{\bm{p}}^{-}-|{\bm{p}}|+\mu}{E_{\bm{p}}^{-}-E_{\bm{p}}^{+}}\ .$
			(25)

The dispersion relations (24) and the factors (25) clearly show that only the positive-energy component of light quarks couples to the Kondo condensate. It should be noted that $U_{\pm}({\bm{p}})$ and $W_{\pm}({\bm{p}})$ satisfy $U_{+}({\bm{p}})+U_{-}({\bm{p}})=1$ and $W_{+}({\bm{p}})+W_{-}({\bm{p}})=1$, respectively.

A schematic picture of the dispersion relations of q.p. (red solid line), q.h. (blue dashed line), and a.p. (purple dotted line) is shown in Fig. 2. This figure shows that q.p. always lives above the Fermi level, while q.h. and a.p. are below the Fermi level. The energy gap between q.p. and q.h. at the identical momentum ${\bm{p}}$ satisfies

	$\displaystyle\delta E$	$\displaystyle\equiv$	$\displaystyle E_{\bm{p}}^{+}-E_{\bm{p}}^{-}$		(26)
		$\displaystyle=$	$\displaystyle\sqrt{(|{\bm{p}}|-\mu)^{2}+8|\Delta|^{2}}$
		$\displaystyle\geq$	$\displaystyle\sqrt{8|\Delta|^{2}}\ ,$

where the minimum value is given at $|{\bm{p}}|=\mu$. In other words, the threshold energy of pair creation or pair annihilation of q.p. $+$ q.h. at rest frame is found to be $\delta E_{\rm min.}=\sqrt{8|\Delta|^{2}}$ as indicated in Fig. 2.

From the Lagrangian (9) with the ansatz (10) inserted, the thermodynamic potential in the absence of the gauge field is given by

	$\displaystyle\Omega/V$	$\displaystyle=$	$\displaystyle-N_{c}\sum_{\zeta=+,-,{\rm a}}\int^{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\Bigg{[}{E}_{\bm{p}}^{\zeta}+\frac{2}{\beta}{\rm ln}\left(1+{\rm e}^{-\beta{E}^{\zeta}_{\bm{p}}}\right)\Bigg{]}$		(27)
			$\displaystyle+\frac{2}{g}|\Delta|^{2}\ ,$

with the three-dimensional volume and the inverse temperature, $V$ and $\beta=1/T$, respectively. The cutoff parameter $\Lambda$ is included to regularize the integral, and $N_{c}$ is the number of colors which is taken to be $N_{c}=3$. By taking a derivative of the thermodynamic potential (27) with respect to $\Delta$, the gap equation that the Kondo condensate satisfies is derived as

$\displaystyle 2N_{c}\int^{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\frac{f_{F}(E_{\bm{p}}^{+})-f_{F}(E_{\bm{p}}^{-})}{E_{\bm{p}}^{+}-E_{\bm{p}}^{-}}+\frac{1}{g}=0\ ,$

(28)

with the Fermi-Dirac distribution function $f_{F}(\epsilon)=1/({\rm e}^{\beta\epsilon}+1)$. At zero temperature, $f_{F}(E_{\bm{p}}^{+})-f_{F}(E_{\bm{p}}^{-})\to-1$, and hence an approximate solution of Eq. (28) at $T=0$ can be found as Yasui et al. (2019, 2017)

$\displaystyle\Delta\sim\Lambda{\rm exp}\left(-\frac{1}{N_{c}g\rho(\mu)}\right)\ ,$

(29)

with the density of states of a massless quark at the Fermi level, $\rho(\mu)=\frac{\mu^{2}}{2\pi^{2}}$, under assumptions of $\Delta\ll\mu,\Lambda$ and $N_{c}g\rho(\mu)\ll 1$. The solution (29) shows that the Kondo condensate $\Delta$ significantly appears in cold dense quark matter. The integration in Eq. (28) is mostly governed by modes near the Fermi sphere for the noninteracting case $|{\bm{p}}|\sim\mu$. This fact allows us to estimate the temperature dependence of $\Delta$ in an easy way. Namely, the factor by distribution functions $f_{F}(E_{\bm{p}}^{+})-f_{F}(E_{\bm{p}}^{-})$ at the Fermi sphere turns into

$\displaystyle\Big{[}f_{F}(E_{\bm{p}}^{+})-f_{F}(E_{\bm{p}}^{-})\Big{]}_{|{\bm{p}}|\sim\mu}\sim 2f_{F}\big{(}\sqrt{2}|\Delta|\big{)}-1\ .$

(30)

Here, if $T\ll\sqrt{2}\Delta$, then thermal fluctuations are rather suppressed, leading to

$\displaystyle\Big{[}f_{F}(E_{\bm{p}}^{+})-f_{F}(E_{\bm{p}}^{-})\Big{]}_{|{\bm{p}}|\sim\mu}\to-1\ ,$

(31)

which is the same value as that at $T=0$. In other words, because of the gapped nature of q.p. and q.h., at lower temperature, the magnitude of $\Delta$ does not change significantly from that at zero temperature. Numerically such a tendency was found in Ref. Suzuki et al. (2017).

We employ the gap equation (28) to determine the size of the Kondo condensate in the following analysis, since the magnetic field is treated perturbatively based on the linear response theory.

In this section, we give analytic evaluation of the HQSP induced by the Kondo effect under a magnetic field with vertex corrections required by the $U(1)_{\rm EM}$ gauge symmetry.