# Spin current Kondo effect in frustrated Kondo systems

Jiangfan Wang  Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Science, Beijing 100190, China  Yi-
feng Yang<EMAIL_ADDRESS>Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Science, Beijing 100190,
China  School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100190, China  Songshan Lake Materials Laboratory, Dongguan, Guangdong
523808, China

###### Abstract

Magnetic frustrations can enhance quantum zero-point motion in spin systems
and lead to exotic topological insulating states. When coupled to mobile
electrons, they may lead to unusual non-Fermi liquid or metallic spin liquid
states whose nature has not been well explored. Here, we propose a spin
current Kondo mechanism underlying a series of non-Fermi liquid phases on the
border of Kondo and magnetic phases in a frustrated three-impurity Kondo
model. This mechanism is confirmed by renormalization group analysis and
describes movable Kondo singlets called “holons” induced by an effective
coupling between the spin current of conduction electrons and the vector
chirality of localized spins. Similar mechanism may widely exist in all
frustrated Kondo systems and be detected through spin-current noise
measurements.

##  I Introduction

Kondo interactions between localized spins and mobile electrons underly many
exotic quantum many-body states in condensed matter physics review_si2010 ;
YangPNAS2017 . Geometric frustration adds additional richness to this problem
coleman2010frustration . In frustrated Kondo lattice systems, experiments have
uncovered rich phase diagrams with unusual non-Fermi liquid (NFL) or metallic
spin liquid phases zhao2019CePdAl ; Canfield2011YbAgGe ; Nakatsuji2006PrIrO ;
Kim2013YbPtPb . The nature of these exotic phases is still under debate
coleman2010frustration ; zhao2019CePdAl ; Wang2020 . A minimum model for
studying the interplay between geometric frustrations and the many-body Kondo
coupling is the three impurity Kondo model (3IKM) Ferrero2007 ; Konig2020 ;
Coleman2021Hund ; Kudasov2002 ; Paul1996 ; Ingersent2005 ; Lazarovits2005 .

Geometric frustrations typically lead to static or fluctuating noncolinear or
noncoplanar configurations measured by the spin vector chirality
($\bm{S}_{i}\times\bm{S}_{j}$) or scalar chirality
($\bm{S}_{i}\times\bm{S}_{j})\cdot\bm{S}_{k}$ Wen1988chiral ; Grohol2005 ;
McCulloch2008 ; Machida2010PrIrO . When coupled to mobile electrons, the spin
chirality may give rise to anomalous electron transport properties restricted
by parity ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries. For
example, a classical spin trimer with nonzero vector chirality may cause spin
Hall effect Ishizuka2019 , while that with nonzero scalar chirality can induce
anomalous Hall effect Ishizuka2019 ; Ishizuka2018 . Similarly, in a Mott
insulator Batista2008  or a nanoscale conducting ring coupled to ferromagnetic
leads Tatara2003 , scalar chirality can induce circular electric currents. The
correlation between spin vector chirality and electron spin current may be
enhanced by the Kondo effect YDWang2020 . But how their strong coupling may
affect the many-body ground state has not been explored.

In this work, we study the 3IKM with a dynamical large-$N$ Schwinger boson
approach Yashar1D ; Komijani2019 ; wang2019quantum ; Wang2020 , combined with
renormalization group (RG) analysis. We found that the spin vector chirality
can induce an effective spin current Kondo effect (SCK) and causes a series of
intermediate NFL phases featured with a fractional holon phase shift ($\pi/3$
or $2\pi/3$) due to partial Kondo screening. These phases emerge on the border
of magnetic and Kondo phases and may be viewed as a quantum superposition of
local Kondo singlets on different impurity sites. Evidently, a prerequisite of
the SCK is the existence of electron spin or charge currents. Previous studies
on 3IKM usually assumed independent electron baths Ferrero2007 ; Konig2020 ;
Coleman2021Hund  and thus forbid such current flow. Our work is based on an
improved treatment of the nonlocal spin current interaction in a shared-bath
model and the NFL ground states may have a lower energy due to the transfer of
Kondo singlets between impurity sites Kudasov2002 . In fact, numerical RG
analyses have previously discovered stable NFL fixed points within the shared-
bath 3IKM Paul1996 ; Ingersent2005 ; Lazarovits2005  but not in the
independent-bath model when $C_{3}$ symmetry is present Ferrero2007 . The SCK
mechanism may also be responsible for the metallic spin liquid in the
frustrated Kondo lattice.

##  II Model and Methods

We start with the following Hamiltonian of 3IKM:

$H=\sum_{{\bm{p}}a\alpha}\epsilon_{\bm{p}}c_{{\bm{p}}a\alpha}^{\dagger}c_{{\bm{p}}a\alpha}+J_{K}\sum_{i=1}^{3}\bm{s}_{i}\cdot\bm{S}_{i}+J_{H}\sum_{\left\langle
ij\right\rangle}\bm{S}_{i}\cdot\bm{S}_{j},$ (1)

where $a$ and $\alpha$ denote the channel (orbital) and spin indices of
conduction electrons, $i$ labels the three vertices of an equilateral triangle
with side length $|{\bm{r}}_{i}-{\bm{r}}_{j}|=R$ (see Figure 1(a)). The
electron dispersion is chosen as $\epsilon_{\bm{p}}={\bm{p}}^{2}/2\pi-1$, with
$|\bm{p}|\leq 2\sqrt{\pi}$ and half-band width $D=1$, to ensure the $C_{3}$
symmetry of the Hamiltonian and simplify the calculations.
$\bm{s}_{i}=\frac{1}{2}\sum_{a\alpha\beta}c_{ia\alpha}^{\dagger}\bm{\sigma}_{\alpha\beta}c_{ia\beta}$
is the electron spin density at ${\bm{r}}_{i}$, and $\bm{S}_{i}$ is the local
spin. The Heisenberg term describes antiferromagnetic exchange interaction
between local spins and is assumed to be independent of $R$.

Figure 1: (Color online) The three-impurity Kondo model (3IKM) and its phase
diagram. (a) Schematic picture of 3IKM, showing three antiferromagnetically
coupled impurity spins (red bold arrows) sitting on the vertices of an
equilateral triangle with side length $R$ and Kondo coupled to a two-
dimensional electron bath with a parabolic dispersion. (b) The large-$N$ phase
diagram of 3IKM in terms of $R$ and $T_{K}/J_{H}$. The black curves with data
points are the phase boundaries determined by the jump of holon phase shift,
$\delta_{\chi}$. The number in each phase denotes the value
$\delta_{\chi}/\pi$. The intensity plot with a red-blue color scale shows the
correlation between the conduction electron spin current $\mathbf{j}_{ij}^{s}$
and the impurity spin vector chirality
$\mathbf{J}_{ij}^{S}\equiv\mathbf{S}_{i}\times\mathbf{S}_{j}$. (c) Schematic
pictures illustrating the antiferromagnetic (left) and ferromagnetic (right)
spin current correlations. The symbol $\otimes$ and $\odot$ denote two
opposite spin polarizations of the spin current.

The Schwinger boson representation has been widely used in studying frustrated
spin systems Read1991SpN ; Wang2006PSG ; FlintSpN2009 . In this
representation, the local spin operator is written as
$\bm{S}_{i}=\frac{1}{2}\sum_{\alpha\beta}b_{i\alpha}^{\dagger}\bm{\sigma}_{\alpha\beta}b_{i\beta}$,
with a constraint fixing the number of bosons at each site, $n_{b}(i)=2S$,
where $S$ is the spin size. Using a mean-field like decomposition, the
Heisenberg term becomes

$J_{H}\bm{S}_{i}\cdot\bm{S}_{j}\rightarrow\left(\tilde{\Delta}_{ij}^{\dagger}\Delta_{ij}-\tilde{\Gamma}_{ij}^{\dagger}\Gamma_{ij}\right)+H.c.+\frac{4(|\Delta_{ij}|^{2}-|\Gamma_{ij}|^{2})}{J_{H}},$
(2)

where
$\tilde{\Delta}_{ij}^{\dagger}\equiv\sum_{\alpha}\text{sgn}(\alpha)b_{i\alpha}^{\dagger}b_{j,-\alpha}^{\dagger}$
creates a singlet valence bond, and
$\tilde{\Gamma}_{ij}\equiv\sum_{\alpha}b_{i\alpha}^{\dagger}b_{j\alpha}$ is
the boson hopping term. $\Delta_{ij}$ and $\Gamma_{ij}$ are two auxiliary
fields in the functional integral, with mean-field values
$\Delta_{ij}=-J_{H}\left\langle\tilde{\Delta}_{ij}\right\rangle/4$ and
$\Gamma_{ij}=-J_{H}\left\langle\tilde{\Gamma}_{ij}\right\rangle/4$. In the
mean-field theory, the constraint is enforced on average by adding a lagrange
multiplier term $\sum_{i}\lambda_{i}(n_{b}(i)-2S)$ to the Hamiltonian. Without
Kondo coupling ($J_{K}=0$), the ground state of the Heisenberg model has a
total effective spin $S_{tot}=1/2$ for half-integer spins and $S_{tot}=0$ for
integer spins. The former becomes Kondo screened immediately by turning on a
small $J_{K}$, while the latter remains unscreened within a finite range of
$J_{K}$.

In the limit of large $J_{K}$, the physics is dominated by the Kondo term,
which can be decoupled by introducing a spinless fermionic holon field
$\chi_{ia}$ that creates a Kondo quasi-bound state Wang2020 ,

$J_{K}\bm{s}_{i}\cdot\bm{S}_{i}\rightarrow\sum_{a\alpha}\frac{1}{\sqrt{2}}b_{i\alpha}^{\dagger}c_{ia\alpha}\chi_{ia}+H.c.+\sum_{a}\frac{\chi_{ia}^{\dagger}\chi_{ia}}{J_{K}}.$
(3)

The effective onsite energy of holon decreases as one increases $J_{K}$, and
becomes negative at large $J_{K}$. It is then energetically favorable to bind
a spinon and a conduction hole into a holon, forming a Kondo singlet below the
characteristic Kondo temperature $T_{K}=De^{-2D/J_{K}}$. For $T_{K}/J_{H}\gg
1$, all impurities are Kondo screened, so that each impurity site is occupied
by a holon with positive electric charge. The charge conservation ensures
equal numbers of negative charges being released into the electron Fermi sea,
leading to a Kondo resonance peak in the Kondo impurity model, or a large
electron Fermi surface in the lattice model Coleman2005sum ; Wang2020 .

To study the physics at intermediate $T_{K}/J_{H}$, we perform a large-$N$
calculation, with $\alpha=\pm 1,\cdots,\pm N/2$, $a=1,\cdots,K$, and
$\kappa\equiv K/N=2S/N$ fixed ($\kappa=0.2$ in this paper). Here, we focus on
the uniform real solution, $\Delta_{ij}=\Delta$, $\Gamma_{ij}=\Gamma$ and
$\lambda_{i}=\lambda$. The $C_{3}$ symmetry allows us to take Fourier
transform along the triangle,
$b_{j\alpha}=\frac{1}{\sqrt{3}}\sum_{h}b_{h\alpha}e^{i2\pi hj/3}$ (same for
$\chi$ and $c$ fields), where $h=0,\pm 1$ is the “helicity” number. The
large-$N$ solution is obtained by solving the following self-consistent
equations (see Appendix A1):

$\displaystyle\Sigma_{b}(h,i\nu_{n})=$
$\displaystyle-\frac{\kappa}{3\beta}\sum_{h^{\prime}l}G_{\chi}(h-h^{\prime},i\nu_{n}-i\omega_{l})g_{c}(h^{\prime},i\omega_{l}),$
(4a) $\displaystyle\Sigma_{\chi}(h,i\omega_{n})=$
$\displaystyle\frac{\kappa}{3\beta}\sum_{h^{\prime}l}G_{b}(h+h^{\prime},i\omega_{n}+i\omega_{l})g_{c}(h^{\prime},i\omega_{l}),$
(4b)

where

$\displaystyle g_{c}(h,i\omega_{n})=$
$\displaystyle\frac{1}{\mathcal{V}}\sum_{\bm{p}}\frac{1+\frac{2}{3}\sum_{j}\cos\left(\frac{2\pi
h}{3}+\bm{p}\cdot(\bm{r}_{j}-\bm{r}_{j+1})\right)}{i\omega_{n}-\epsilon_{\bm{p}}},$
(5a) $\displaystyle G_{b}(h,i\nu_{n})=$
$\displaystyle\frac{1}{\gamma_{b}(h,i\nu_{n})-3\Delta^{2}\delta_{|h|,1}/\gamma_{b}(-h,-i\nu_{n})},$
(5b) $\displaystyle G_{\chi}(h,i\omega_{n})=$
$\displaystyle\frac{1}{-1/J_{K}-\Sigma_{\chi}(h,i\omega_{n})},$ (5c)

are the Green’s functions of conduction electrons, spinons and holons,
respectively. Here $\gamma_{b}(h,i\nu_{n})=i\nu_{n}-\lambda+2\Gamma\cos(2\pi
h/3)-\Sigma_{b}(h,i\nu_{n})$, and $i\nu_{n}$ ($i\omega_{n}$) is the bosonic
(fermionic) Matsubara frequency. $\Delta$, $\Gamma$ and $\lambda$ are treated
as variational parameters. The Green’s functions are symmetric under inversion
of helicity $h\rightarrow-h$, which is ensured by the real $\Gamma$ and the
symmetric dispersion $\epsilon_{\bm{p}}$. In the limit of
$R=|\bm{r}_{i}-\bm{r}_{j}|\rightarrow\infty$, both $g_{c}$ and $G_{\chi}$
become independent of $h$, and our model reduces to the independent-bath
model. In the limit $R=0$, one has
$g_{c}(h,\omega)=g_{c}(0,\omega)\delta_{h0}$ and only the $h=0$ electron
states are coupled to the impurities. In this paper, we focus on the
intermediate range $1\leq R\leq 5$ that is physically relevant for real
materials.

##  III Results

###  III.1 Phase diagram

The zero temperature phase diagram in terms of $T_{K}/J_{H}$ and $R$ is shown
in Figure 1(b), containing a series of intermediate regions determined by the
jump of the holon phase shift $\delta_{\chi}$. At small $T_{K}/J_{H}$ and for
integer spins, the three local spins form a total singlet with
$\delta_{\chi}/\pi=0$. At large $T_{K}/J_{H}$, the spins are fully Kondo
screened, with a maximal phase shift $\delta_{\chi}/\pi=1$. At intermediate
$T_{K}/J_{H}$, two different phases with $\delta_{\chi}/\pi=1/3$ and $2/3$
occur alternately with increasing $R$. The non-integer $\delta_{\chi}/\pi$
indicates that one or two of the spins are Kondo screened on average.

The holon phase shift is defined as
$\delta_{\chi}\equiv\sum_{h}\text{Im}\ln(-G_{\chi}^{-1}(h,0))/3$. At zero
temperature, the quantity
$3\delta_{\chi}/\pi=\sum_{h}\theta(-\varepsilon_{h}^{\chi})$ ($\theta(x)$ is
the step function) simply counts the number of holon levels below the Fermi
energy. The holon energy level, $\varepsilon_{h}^{\chi}$, is determined by the
equation $1/J_{K}+\Sigma_{\chi}^{\prime}(h,\varepsilon_{h}^{\chi})=0$. Figure
2(a) shows its evolution with $T_{K}/J_{H}$ at three different $R$.
$\varepsilon_{h}^{\chi}$ evolves from above the Fermi level at small
$T_{K}/J_{H}$, to below the Fermi level at large $T_{K}/J_{H}$. For $R=1,2$,
$\varepsilon_{0}^{\chi}$ and $\varepsilon_{\pm 1}^{\chi}$ are well separated,
leaving a finite range of $T_{K}/J_{H}$ where $\varepsilon_{0}^{\chi}$ and
$\varepsilon_{\pm 1}^{\chi}$ are on the opposite sides of the Fermi level.
However for $R=1.68$, $\varepsilon_{0}^{\chi}$ and $\varepsilon_{\pm
1}^{\chi}$ are nearly degenerate and cross the Fermi level at the same
$T_{K}/J_{H}$.

Therefore, $\delta_{\chi}/\pi$ consists of several plateaus ($0,1/3,2/3,1$)
separated by jumps at the phase boundaries, as shown in Figure 2(b). Due to
the global U(1) symmetry, there is a Ward identity,
$\sum_{h}\delta_{c}(h)=3\delta_{\chi}/N$, relating the electron phase shift,
$\delta_{c}(h)\equiv-\text{Im}\ln[1-g_{c}(h,0^{+})\Sigma_{c}(h,0^{+})]$, to
the holon phase shift Coleman2005sum . Based on the Friedel sum rule, there
will be a Kondo resonance peak containing $\Delta
n_{c}=\sum_{h\alpha}\delta_{c}(h)/\pi=3\delta_{\chi}/\pi$ additional electrons
per channel, indicating $\Delta n_{c}$ local spins being “delocalized”.

Figure 2: (Color online) Holon phase shift and spin current correlations. (a)
Evolution of the holon energy levels with $T_{K}/J_{H}$ for $R=1$, $2$ and
$1.68$. (b-d) The holon phase shift, spin current correlation, and local Kondo
correlation as functions of $T_{K}/J_{H}$. Data for $R=1$, $2$ and $1.68$ are
shown with red, green and blue points, respectively.

The holon levels seem to be “pinned” at the Fermi energy at the phase
boundaries, consistent with previous studies on the Kondo lattice Komijani2019
; wang2019quantum . Here, well defined holon levels (with sharp quasiparticle
peaks) only appear inside the energy range $|\omega|<D_{\chi}\sim 10^{-3}D$
around the Fermi level, consistent with the single ion Kondo temperature
$T_{K}/D=e^{-2D/J_{K}}\approx 0.0067$ that we fixed in our calculations.

###  III.2 Spin current Kondo effect

The intermediate phases are featured with an enhanced spin current
correlation,
$\left\langle\mathbf{j}_{ij}^{s}\cdot\mathbf{J}_{ij}^{S}\right\rangle_{D_{\chi}}$,
as shown by the color-coded intensity plot of Figure 1(b). The correlation has
a negative (positive) sign in the $\delta_{\chi}/\pi=1/3$ ($2/3$) phase,
corresponding to an AFM (FM) SCK effect as illustrated in Figure 1(c). Here
the electron spin current is defined as,
$\bm{j}_{ij}^{s}=ic_{j\alpha}^{\dagger}\bm{\sigma}_{\alpha\beta}c_{i\beta}/2+h.c.$,
and the impurity spin current is
$\bm{J}_{ij}^{S}\equiv\bm{S}_{i}\times\bm{S}_{j}$. The identification of
vector chirality as a spin current is supported by the equation of motion of
the Heisenberg model, $d\bm{S}_{j}/dt=J_{H}\sum_{i}\bm{S}_{i}\times\bm{S}_{j}$
Bruno2005 ; Katsura2005 . The subscript “$D_{\chi}$” denotes an average over
the coherent part of the holon spectrum that lies within the range
$|\omega|\leq D_{\chi}$ around the Fermi level. Physically, this accounts for
the retardation of the SCK effect due to the “slowly-moving” holons.

The SCK effect arises because only $\varepsilon_{0}^{\chi}$ or
$\varepsilon_{\pm 1}^{\chi}$ are occupied in the intermediate phases. This
means that a holon can be thermally excited to a different level, producing a
net “moment” (helicity) along the triangle. These moving holons (Kondo
singlets) can mediate a nonlocal Kondo scattering process described by
$c_{ja\alpha}^{\dagger}c_{ia\beta}b_{i\beta}^{\dagger}b_{j\alpha}$ Wang2020 ,
which is directly related to the spin current correlation via the relation
$i\bm{\sigma}_{\alpha\beta}\cdot(\bm{S}_{i}\times\bm{S}_{j})=(b_{j\beta}^{\dagger}\tilde{\Gamma}_{ij}b_{i\alpha}-b_{i\beta}^{\dagger}\tilde{\Gamma}_{ji}b_{j\alpha})/2$.

More details on the evolution of
$\left\langle\mathbf{j}_{ij}^{s}\cdot\mathbf{J}_{ij}^{S}\right\rangle_{D_{\chi}}$
with $T_{K}/J_{H}$ are shown in Figure. 2(c) for different $R$. We see it is
substantially enhanced at intermediate $T_{K}/J_{H}$ for $R=1,2$, but with
opposite signs, and strongly suppressed for $R=1.68$. As discussed in Appendix
A2, the main contribution to the spin current correlation is proportional to
$n_{\pm 1}^{\chi}-n_{0}^{\chi}$, where $n_{h}^{\chi}$ is the holon “occupation
number”. For $n_{0}^{\chi}>0$ and $n_{\pm 1}^{\chi}=0$, one has
$\delta_{\chi}/\pi=1/3$ and the AFM SCK effect, while for $n_{0}^{\chi}=0$ and
$n_{\pm 1}^{\chi}>0$, one has $\delta_{\chi}/\pi=2/3$ and the FM SCK effect.
When both $n_{0}^{\chi}$ and $n_{\pm 1}^{\chi}$ are zero
($\delta_{\chi}/\pi=0$) or finite ($\delta_{\chi}/\pi=1$), the SCK effect is
suppressed. In fact, the Pauli exclusion principle forbids holons to move when
all the holon levels are occupied, or equivalently, when all the spins are
fully Kondo screened. This is exactly what happens in the independent-bath
model, where $n_{0}^{\chi}=n_{\pm 1}^{\chi}$ always holds.

The local Kondo correlation can be calculated in a similar way, leading to
$\left\langle\mathbf{s}_{i}\cdot\mathbf{S}_{i}\right\rangle=C-\frac{1}{3J_{K}^{2}}(n_{0}^{\chi}+n_{1}^{\chi}+n_{-1}^{\chi})$,
where $C$ represents contributions from incoherent part of the holon spectrum.
As shown in Figure 2(d), the Kondo correlation is always negative
(antiferromagnetic) with an absolute value increasing monotonically with
$T_{K}/J_{H}$.

###  III.3 Spin correlation

The SCK effect is a combined effect of the Kondo and inter-site magnetic
correlations. It requires not only movable holons, but also movable spinons.
The spin current correlation is proportional to the spinon hopping amplitude
$\Gamma$. As shown in Figure 3(a), $\Gamma$ decreases monotonically with
increasing $T_{K}/J_{H}$, and vanishes deep inside the Kondo phase. This is
another reason why SCK effect is suppressed for large $T_{K}/J_{H}$. The
spinon pairing amplitude $\Delta$ has very similar behaviors to $\Gamma$. They
are related to the impurity spin correlation through
$\left\langle\bm{S}_{i}\cdot\bm{S}_{i+1}\right\rangle=(\Gamma^{2}-\Delta^{2})/(4J_{H}^{2})$.
As shown in Figure 3(b), it is always negative with an absolute value
decreasing monotonically with increasing $T_{K}/J_{H}$. Direct comparison of
Figure 2(d) and Figure 3(b) reveals complementary behavior of the Kondo and
spin correlations, showing strong competition between the Kondo effect and the
magnetic interaction. Both stay relatively “constant” within each phase but
change rapidly near the phase boundaries, as is seen from the peaks of the
derivative of $\left\langle\bm{S}_{i}\cdot\bm{S}_{i+1}\right\rangle$ with
respect to $T_{K}/J_{H}$ (inset of Figure 3(b)).

Figure 3: (Color online) Spin correlations and spinon density of states. (a)
The spinon hopping amplitude as a function of $T_{K}/J_{H}$ for $R=1$, $2$ and
$1.68$. (b) The spin correlation function
$\left\langle\bm{S}_{i}\cdot\bm{S}_{j}\right\rangle$ as a function of
$T_{K}/J_{H}$ for $R=1$, $2$ and $1.68$. The inset shows its derivative with
respect to $T_{K}/J_{H}$ for $R=1$ and $2$. (c,d) Intensity plots of the
spinon’s density of states, $-\sum_{h}G_{b}^{\prime\prime}(h,\omega)/(3\pi)$,
for $R=1.68$ and $R=2$, showing their evolutions with $T_{K}/J_{H}$.

The boundary of the intermediate phases is marked with gapless holon and
spinon excitations. This is seen from the intensity plots of spinon’s density
of states (DOS), $-\sum_{h}G_{b}^{\prime\prime}(h,\omega)/(3\pi)$, shown in
Figure 3(c) and Figure 3(d) for $R=1.68$ and $2$, respectively. The gapless
fractional excitations are expected to cause quantum critical behaviors
Komijani2019 ; Wang2020 . Away from the phase boundaries, both spinon and
holon develop gaps in their spectra. Here, the intermediate phase has a small
holon gap due to the discrete nature of the holon levels, which is expected to
be closed in the lattice version causing a dispersive holon band and a hidden
holon Fermi surface Wang2020 .

###  III.4 Renormalization group analysis

To clarify the origin of the SCK, we perform a simplified renormalization
group (RG) analysis and integrate out the scattering processes involving high
energy electrons or holes Anderson1970 . This gives a low temperature
effective Hamiltonian with an emergent “strongly coupled” SCK term, suggesting
that the SCK effect is energetically favored.

The renormalized Hamiltonian with a reduced electron band width
$D^{\prime}=D-\delta D$ has a modified spin-electron scattering term
$H_{K}^{\prime}=H_{K}+\Delta H_{K}$, with

$\displaystyle\Delta H_{K}=$ $\displaystyle-\frac{J_{K}^{2}\rho\delta
D}{4D}\sum_{ij}c_{ia\alpha}^{\dagger}c_{ja\beta}\sigma_{\alpha\gamma}^{\mu}\sigma_{\gamma\beta}^{\nu}$
$\displaystyle\times\left(\sum_{\bm{q}^{+}}e^{i\bm{q}\cdot\bm{R}_{ij}}S_{i}^{\mu}S_{j}^{\nu}-\sum_{\bm{q}^{-}}e^{i\bm{q}\cdot\bm{R}_{ij}}S_{j}^{\nu}S_{i}^{\mu}\right).$
(6)

Here $\mu,\nu$ denote the spatial components,
$\bm{R}_{ij}=\bm{r}_{i}-\bm{r}_{j}$, and $\bm{q}^{\pm}$ satisfies
$D^{\prime}<\epsilon_{\bm{q}^{+}}<D$, $-D<\epsilon_{\bm{q}^{-}}<-D^{\prime}$,
respectively. The summation over $i,j$ contains two parts. For $i=j$, Eq. (6)
gives the usual correction to the Kondo term, $\frac{J_{K}^{2}\rho\delta
D}{2D}\sum_{i}c_{ia\alpha}^{\dagger}c_{ia\beta}\bm{\sigma}_{\alpha\beta}\cdot\bm{S}_{i}$.
For $i\neq j$, it gives rise to a nonlocal scattering term,

$J_{K}^{\prime}\sum_{\left\langle
ij\right\rangle}\left[\bm{j}^{s}_{ij}\cdot\bm{J}_{ij}^{S}-K_{ij}\bm{S}_{i}\cdot\bm{S}_{j}\right],$
(7)

where $K_{ij}=c_{ia\alpha}^{\dagger}c_{ja\alpha}+h.c.$ and
$J_{K}^{\prime}=\frac{J_{K}^{2}\rho\delta
D}{2D}[J_{0}(q_{+}R)-J_{0}(q_{-}R)]$, with $J_{0}(x)$ the Bessel function and
$q_{\pm}=\sqrt{2\pi(1\pm D)}$. The scalar and vector products of spins occur
simultaneously due to the identity
$\sigma_{\alpha\gamma}^{\mu}\sigma_{\gamma\beta}^{\nu}=\delta_{\alpha\beta}\delta_{\mu\nu}+i\epsilon_{\mu\nu\eta}\sigma_{\alpha\beta}^{\eta}$.
Once averaged over the conduction electron sea, the second term contributes a
part of the well-known Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which
competes with the Kondo effect to govern the fundamental physics of a Kondo
lattice or multi-impurity Kondo system. The spin current interaction term is
seen to emerge on the same level of the RKKY interaction but was often ignored
in the literature. As we have seen, it is exactly this term that is
responsible for the intermediate phases.

For simplicity, instead of performing a complete RG study of the 3IKM, we are
interested here in the flow of $J_{K}^{\prime}$ in different regions of the
phase diagram. This is done via the Schrieffer-Wolff transformation of a
general Hamiltonian containing both the local and nonlocal interaction terms
(see Appendix A3). The RG equation for $g_{K}^{\prime}\equiv
J_{K}^{\prime}\rho$ reads:

$\frac{dg_{K}^{\prime}}{dl}=\beta(g_{K}^{\prime})=\frac{f_{R}(l)}{2}g_{K}^{2}+2g_{K}g_{K}^{\prime}+O(g_{K}^{4}),$
(8)

where $g_{K}\equiv J_{K}\rho$, $l\equiv-\ln D$ and $f_{R}(l)\equiv
J_{0}(q_{+}(l)R)-J_{0}(q_{-}(l)R)$. Therefore, $g_{K}^{\prime}$ strongly
depends on the fixed point $g^{*}$ that $g_{K}$ flows to at low temperature.
In the RKKY dominated region (the magnetic phase), $g_{K}$ is suppressed and
$g^{*}=0$ at zero temperature Nejati2017 . This leads to a vanished
$\beta(g_{K}^{\prime})$ in the right hand of Eq. (8), so that $g_{K}^{\prime}$
becomes marginal in this phase. In the fully Kondo screened phase,
$g^{*}=\infty$, and the perturbative RG breaks down at low energy scale.
However, both the RKKY and SCK terms are expected to be suppressed due to the
fully quenched local spins. Between these two extremes, one expects $g_{K}$ to
flow to some stable (NFL) or unstable (quantum critical point) intermediate
fixed point $0<g^{*}<\infty$. This indeed happens for the quantum critical
point of the two-impurity Kondo model Mitchell2012 , the Anderson or Kondo
impurity model with a pseudogapped fermion bath Ingersent1998RG , the Bose-
Fermi Kondo model Si2002BFKM , and the stable NFL fixed point of 3IKM Paul1996
; Lazarovits2005 .

Figure 4: (Color online) Renormalization group flow. (a) The RG flow of the
coupling constant $g_{K}^{\prime}$ for different $R$, assuming that the local
Kondo coupling $g_{K}$ flows to a fixed point $g^{*}=0.5$. The initial value
is $g_{K}=0.2$, corresponding to $J_{K}=0.4$ in the large-$N$ calculations.
(b) $g_{K}^{\prime}$ as a function of $R$ at a fixed scale $l=10$ with
$g^{*}=0.5$. (c) Intensity plot of
$\text{sgn}(g_{K}^{\prime})\ln(|g_{K}^{\prime}|+1)$ on the $R$-$g^{*}$ plane
at the same scale $l=10$, showing the sign change of $g_{K}^{\prime}$ with
varying $R$.

The behaviors of $g_{K}^{\prime}$ may be illustrated in the intermediate
coupling regime $0<g^{*}<1$, where the perturbative RG can be qualitatively
trusted. For clarity, we assume a simplest flow equation for the Kondo
coupling, $dg_{K}/dl=g_{K}^{2}-g_{K}^{3}/g^{*}$, that gives such a fixed point
Nozieres1980 . Figure 4(a) compares the RG flow of $g_{K}$ and
$g_{K}^{\prime}$ for different $R$ with $g^{*}=0.5$. One finds that
$g_{K}^{\prime}$ grows rapidly to strong coupling for $R=1$ and $2.2$, but
remains zero for $R\approx 1.66$. This is because the initial flow, namely the
sign of $g_{K}^{\prime}$, is controlled by the first term of Eq. (8) that
depends on $R$, while its exponential growth at large $l$ is caused by the
second term of Eq. (8). As shown in Figure 4(b), $g_{K}^{\prime}$ at a fixed
scale $l=10$ (corresponding to a small energy scale $T\sim e^{-10}D\sim
0.01T_{K}$) oscillates as a function of $R$ between positive and negative, and
stays zero for a series of nodes $R_{n}\approx 1.66,2.84,4.05,\cdots$. The
value of $R_{n}$ is nearly independent of $g^{*}$, as can be seen from the
intensity plot on the $R$-$g^{*}$ plane shown in Figure 4(c). As $g^{*}$
approaches zero, $g_{K}^{\prime}$ also becomes negligible. The nodes $R_{n}$
and the associated sign change of $g_{K}^{\prime}$ resembles qualitatively to
the large-$N$ phase diagram shown in Figure 1(b). Therefore, the RG analysis
supports that the strong coupling SCK effect is indeed a characteristic
feature of the intermediate NFL phases and may be crucial in turning an
unstable fixed point into a stable one in the 3IK model.

##  IV Discussion and conclusion

We have theoretically studied the phase diagram of 3IKM with a single shared
electron bath using the dynamical large-$N$ Schwinger boson approach, focusing
on the emergent NFL state and its microscopic origin. Combined with RG
analysis, we show that the intermediate phase is featured with a strong SCK
effect, a nonlocal Kondo scattering effect mediated by itinerant or movable
Kondo singlets (holons). We noticed that Eq. (7) can also arise in the two-
impurity Kondo model (2IKM), and was demonstrated to destabilize the “Varma-
Jones” critical point into a smooth crossover Logan2011 . In this case,
applying our Schwinger boson approach does not yield an intermediate ground
state Coleman-SWB-2006 . This is because that the SCK correlation is
proportional to the spinon hopping amplitude $\Gamma$ which is absent in the
2IKM. Thus, frustrations are crucial for stabilizing such NFL states.

In our calculations, we have focused on the ground state solutions that keep
both $\mathcal{P}$ and $\mathcal{T}$ symmetries. Due to the strong SCK
coupling, it is likely that the intermediate ground states spontaneously break
these symmetries, leading to nonzero
$\left\langle\bm{S}_{i}\times\bm{S}_{j}\right\rangle$ and
$\left\langle(\bm{S}_{i}\times\bm{S}_{j})\cdot\bm{S}_{k}\right\rangle$. This
can also be realized by adding a Dzyaloshinskii-Moriya interaction
Moriya1960DM  or a Zeeman term to the Hamiltonian. A mean-field description of
such states assumes a complex spinon hopping $\Gamma=|\Gamma|e^{i\phi/3}$,
here $\phi$ is the flux through the triangle related to the scalar chirality
via
$\left\langle\bm{S}_{1}\cdot(\bm{S}_{2}\times\bm{S}_{3})\right\rangle=\frac{N^{3}|\Gamma|^{3}}{2J_{H}^{3}}\sin(\phi)$.
A persistent circular current $\left\langle
J_{\bigtriangleup}\right\rangle=-2\text{Im}\sum_{j\alpha}\left\langle
c_{j\alpha}^{\dagger}c_{j+1,\alpha}\right\rangle$ can be induced due to the
asymmetry between $h=1$ (clockwise) and $h=-1$ (counterclockwise) electron
states. In fact, an additional term
$g_{C}J_{\bigtriangleup}\bm{S}_{1}\cdot(\bm{S}_{2}\times\bm{S}_{3})$ with
$g_{C}\propto g_{K}g_{K}^{\prime}$ will appear in the RG calculation due to
the interference between local and nonlocal interactions. Therefore, within
the intermediate phases, an applied external current will be deflected by the
persistent circular current, leading to an anomalous Hall effect. Similar
mechanism was studied in Ref. Ishizuka2018 ; Ishizuka2019  using transport
theories with the assumption of classical spins.

In reality, $J_{H}$ may be given by the RKKY interaction and oscillate with
$R$. Then the observed phase diagram might be distorted. In particular, for a
ferromagnetic spin interaction ($J_{H}<0$), there will be no frustration and
intermediate phases Coleman2021Hund , and the three impurity spins will align
and form an effective large spin to be fully screened by conduction electrons.
Nevertheless, we believe that the SCK effect is a widely existing property in
frustrated Kondo systems and may be responsible for the metallic spin liquid
state in frustrated Kondo lattice zhao2019CePdAl ; Canfield2011YbAgGe ;
Nakatsuji2006PrIrO ; Kim2013YbPtPb ; Wang2020 . One of its consequences is the
suppression or enhancement of the thermal spin-current noise,
$\left\langle(\mathbf{j}^{s}+\mathbf{J}^{S})^{2}\right\rangle=\left\langle(\mathbf{j}^{s})^{2}\right\rangle+\left\langle(\mathbf{J}^{S})^{2}\right\rangle+2\left\langle\mathbf{j}^{s}\cdot\mathbf{J}^{S}\right\rangle$,
which can be transformed into charge noise by the inverse spin Hall effect and
is experimentally measurable Kamra2014SHN . The spin-current noise should
therefore be reduced (enhanced) in the presence of AFM (FM) SCK effect. Our
prediction may be verified in future experiment on frustrated Kondo lattice
systems.

This work was supported by the National Key R&D Program of MOST of China
(Grant No. 2017YFA0303103), the National Natural Science Foundation of China
(NSFC Grant No. 12174429, No. 11774401, No. 11974397), the Strategic Priority
Research Program of the Chinese Academy of Sciences (Grant No. XDB33010100),
and the Youth Innovation Promotion Association of CAS.

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## Appendix A

###  A.1  Action and self-consistent equations

With the uniform real mean-field assumption, the action of 3IKM is written as:

$\displaystyle S=$ $\displaystyle
S_{c0}+\int_{\tau}\left\\{\sum_{h\alpha}b_{h\alpha}^{\dagger}\left(\partial_{\tau}+\lambda-2\Gamma\cos\frac{2\pi
h}{3}\right)b_{h\alpha}-6S\beta\lambda\right.$
$\displaystyle+i\Delta\sum_{h\alpha}\text{sgn}(\alpha)b_{h\alpha}b_{-h,-\alpha}\sin\frac{2\pi
h}{3}+c.c.+\frac{3\beta N}{J_{H}}(\Delta^{2}-\Gamma^{2})$
$\displaystyle+\left.\frac{1}{\sqrt{3N}}\sum_{hh^{\prime}a\alpha}b_{h\alpha}^{\dagger}c_{h^{\prime}a\alpha}\chi_{h-h^{\prime},a}+c.c.+\sum_{ha}\frac{|\chi_{ha}|^{2}}{J_{K}}\right\\},$
(9)

where
$S_{c0}=-\sum_{hna\alpha}c_{hna\alpha}^{\dagger}g_{c}^{-1}(h,i\omega_{n})c_{hna\alpha}$
describes the free conduction electrons. The only vertex is the spinon-
electron-holon three point vertex, which gives rise to the form of self-
energies in Eq. (4) at large-$N$. The Green’s function for each field can be
conveniently derived from the action. The three mean-field variables $\Delta$,
$\Gamma$ and $\lambda$ are determined through the minimization equations
$\partial\ln Z/\partial\Delta=\partial\ln Z/\partial\Gamma=\partial\ln
Z/\partial\lambda=0$, where $Z=\int\mathcal{D}[b,c,\chi]\exp[-S]$ is the
partition function. This gives

$\displaystyle\kappa=$
$\displaystyle-\frac{1}{3\beta}\sum_{hn}G_{b}(h,i\nu_{n})$ (10a)
$\displaystyle\frac{\Gamma}{J_{H}}=$
$\displaystyle\frac{1}{3\beta}\sum_{hn}G_{b}(h,i\nu_{n})\cos\frac{2\pi h}{3}$
(10b) $\displaystyle\frac{\Delta}{J_{H}}=$
$\displaystyle\frac{1}{3\beta}\sum_{hn}F_{b}(h,i\nu_{n})i\sin\frac{2\pi
h}{3},$ (10c)

where $F_{b}(h,i\nu_{n})\equiv-\text{sgn}(\alpha)\left\langle
b_{hn\alpha}b_{-h,-n,-\alpha}\right\rangle$ is the anomalous Green’s function
of spinon.

###  A.2  Spin current correlation function

Using the SU(2) Schwinger boson representation, we can write the spin current
correlation as

$\displaystyle\left\langle\mathbf{j}_{ij}^{s}\cdot\mathbf{J}_{ij}^{S}\right\rangle=$
$\displaystyle-\frac{1}{4}\sum_{a\alpha\beta\gamma}\left(\left\langle
c_{ja\alpha}^{\dagger}c_{ia\beta}b_{i\beta}^{\dagger}b_{i\gamma}b_{j\gamma}^{\dagger}b_{j\alpha}\right\rangle\right.$
$\displaystyle\left.-\left\langle
c_{ja\alpha}^{\dagger}c_{ia\beta}b_{j\beta}^{\dagger}b_{j\gamma}b_{i\gamma}^{\dagger}b_{i\alpha}\right\rangle\right)+c.c..$
(11)

We then calculate the above expression in the large-$N$ limit. The leading
order contraction of the six-body average contains the term
$\sum_{\gamma}\left\langle
b_{i\gamma}b_{j\gamma}^{\dagger}\right\rangle=\sum_{\gamma}\left\langle
b_{j\gamma}b_{i\gamma}^{\dagger}\right\rangle=-\frac{N\Gamma}{J_{H}}$. The
leading order Feynman diagrams of the remaining four-body averages have order
of magnitudes $\sum_{a\alpha\beta}\left\langle
c_{ja\alpha}^{\dagger}c_{ia\beta}b_{i\beta}^{\dagger}b_{j\alpha}\right\rangle\sim
O(NK)$ and $\sum_{a\alpha\beta}\left\langle
c_{ja\alpha}^{\dagger}c_{ia\beta}b_{j\beta}^{\dagger}b_{i\alpha}\right\rangle\sim
O(K)$, respectively. The latter is sub-leading due to an additional
$\delta$-function, thus can be neglected. The former is calculated as

$\displaystyle\sum_{a\alpha\beta}\left\langle
c_{ja\alpha}^{\dagger}(\tau)c_{ia\beta}(\tau^{\prime})b_{i\beta}^{\dagger}(\tau^{\prime})b_{j\alpha}(\tau)\right\rangle$
$\displaystyle=$
$\displaystyle-\frac{NK}{3\beta}\sum_{hn}\Sigma_{\chi}(h,i\omega_{n})^{2}G_{\chi}(h,i\omega_{n})e^{i2\pi
h(j-i)/3}e^{i\omega_{n}(\tau^{\prime}-\tau)}$ $\displaystyle=$
$\displaystyle\frac{NK}{3}\int_{-\infty}^{\infty}\frac{d\omega}{\pi}\frac{e^{\omega(\tau^{\prime}-\tau)}}{e^{\omega/T}+1}\left[\frac{G_{\chi}^{\prime\prime}(0,\omega)-G_{\chi}^{\prime\prime}(1,\omega)}{J_{K}^{2}}\right.$
$\displaystyle\left.-\Sigma_{\chi}^{\prime\prime}(0,\omega)+\Sigma_{\chi}^{\prime\prime}(1,\omega)\right].$
(12)

Instead of equal-time correlation, here we take $\tau^{\prime}-\tau\sim
D_{\chi}^{-1}$ to account for the time needed for the heavy holons to
propagate from site $i$ to site $j$. At zero temperature $T=0$, the
exponential factor effectively reduces the range of frequency integral to
$-D_{\chi}\leq\omega\leq 0$. Practically, we calculate the following quantity:

$\displaystyle\left\langle\mathbf{j}_{ij}^{s}\cdot\mathbf{J}_{ij}^{S}\right\rangle_{D_{\chi}}\equiv$
$\displaystyle\frac{2\Gamma}{3J_{H}D_{\chi}}\int_{-D_{\chi}}^{0}\frac{d\omega}{\pi}\left[\frac{G_{\chi}^{\prime\prime}(0,\omega)-G_{\chi}^{\prime\prime}(1,\omega)}{J_{K}^{2}}\right.$
$\displaystyle\left.-\Sigma_{\chi}^{\prime\prime}(0,\omega)+\Sigma_{\chi}^{\prime\prime}(1,\omega)\right],$
(13)

where we have multiplied the right hand side of Eq. (11) with a factor
$4/(N^{2}KD_{\chi})$ to give an average-over-$D_{\chi}$ result with order
$O(1)$ at large-$N$. We take $D_{\chi}=0.001$ to account for the holon
spectrum near the Fermi energy, containing only well-defined quasiparticles
with negligible imaginary part of self-energies. In this way, Eq. (13) is
dominated by the term proportional to $n_{1}^{\chi}-n_{0}^{\chi}$, where
$n_{h}^{\chi}=-\int_{-D_{\chi}}^{0}\frac{d\omega}{\pi}G_{\chi}^{\prime\prime}(0,\omega)$
is the holon “occupation number”.

###  A.3  Derivation of RG equations

To derive the RG equation of $J_{K}^{\prime}$ to the order $O(J_{K}^{3})$, we
start with the following general Hamiltonian:

$\displaystyle H(D)=$ $\displaystyle
H_{c0}(D)+J_{K}\sum_{i}\bm{s}_{i}\cdot\bm{S}_{i}$ (14) $\displaystyle+$
$\displaystyle J_{K}^{\prime}\sum_{\left\langle
ij\right\rangle}\left(\bm{j}_{ij}^{s}\cdot\bm{J}_{ij}^{S}-K_{ij}\bm{S}_{i}\cdot\bm{S}_{j}\right)+J_{H}^{\prime}\sum_{\left\langle
ij\right\rangle}\bm{S}_{i}\cdot\bm{S}_{j},$

where $H_{c0}(D)$ is the free conduction electron term, and $D$ is the half
band width of conduction band. The last term arises when integrating out the
high energy conduction holes and contributes a major part of the RKKY
interaction, $J_{RKKY}=J_{H}^{\prime}-J_{K}^{\prime}\left\langle
K_{ij}\right\rangle$. In the second-order perturbation, it does not enter the
RG flow of $J_{K}^{\prime}$ and is therefore dropped in the following
analysis. The renormalized Hamiltonian with a reduced half band width
$D^{\prime}=D-\delta D$ can be obtained using the standard Schrieffer-Wolff
transformation and yields

$\displaystyle H(D^{\prime})=$ $\displaystyle
H_{c0}(D^{\prime})+J_{K}(D^{\prime})\sum_{i}\bm{s}_{i}\cdot\bm{S}_{i}$
$\displaystyle+J_{K}^{\prime}(D^{\prime})\sum_{\left\langle
ij\right\rangle}\left(\bm{j}_{ij}^{s}\cdot\bm{J}_{ij}^{S}-K_{ij}\bm{S}_{i}\cdot\bm{S}_{j}\right)$
$\displaystyle+J_{C}(D^{\prime})J_{\bigtriangleup}\bm{S}_{1}\cdot(\bm{S}_{2}\times\bm{S}_{3})+\cdots$
(15)

where $``\cdots"$ contains higher order scattering terms and pure spin
interaction terms. The renormalized coupling constants in the above equation
read

$\displaystyle J_{K}(D^{\prime})=$ $\displaystyle J_{K}+J_{K}^{2}\rho\delta\ln
D,$ (16a) $\displaystyle J_{K}^{\prime}(D^{\prime})=$ $\displaystyle
J_{K}^{\prime}+\left[2J_{K}J_{K}^{\prime}+\frac{J_{K}^{2}}{2}f_{R}(D)\right]\rho\delta\ln
D,$ (16b) $\displaystyle J_{C}(D^{\prime})=$
$\displaystyle-\frac{J_{K}J_{K}^{\prime}}{2}f_{R}(D)\rho\delta\ln D,$ (16c)

where $f_{R}(D)\equiv J_{0}(q_{+}(D)R)-J_{0}(q_{-}(D)R)$. By defining
$g_{K}\equiv J_{K}\rho$, $g_{K}^{\prime}\equiv J_{K}^{\prime}\rho$,
$g_{C}\equiv J_{C}\rho$ and $l\equiv-\ln D$, the above equations can be
rewritten as

$\displaystyle\frac{\delta g_{K}}{\delta l}=$ $\displaystyle g_{K}^{2},$ (17a)
$\displaystyle\frac{\delta g_{K}^{\prime}}{\delta l}=$
$\displaystyle\frac{g_{K}^{2}}{2}f_{R}(l)+2g_{K}g_{K}^{\prime},$ (17b)
$\displaystyle\frac{\delta g_{C}}{\delta l}=$
$\displaystyle\frac{g_{K}g_{K}^{\prime}}{2}f_{R}(l).$ (17c)

The Schrieffer-Wolff transformation only gives an order $O(J_{K}^{2})$ beta
function for $g_{K}$. However, the beta functions for $g_{K}^{\prime}$ and
$g_{C}$ are of order $O(J_{K}^{3})$, since $g_{K}^{\prime}\sim O(J_{K}^{2})$.
At this order, the coupling constant $g_{C}$ does not enter the other RG
equations, and is completely determined by the flows of $g_{K}$ and
$g_{K}^{\prime}$.