# Quasi-bound state in the $\bar{K}NNN$ system

N.V. Shevchenko<EMAIL_ADDRESS>Nuclear Physics Institute, 25068 Řež,
Czech Republic

###### Abstract

The paper is devoted to the $\bar{K}NNN$ system, which is an exotic system
consisting of an antikaon and three nucleons. Dynamically exact four-body
Faddeev-type equations were solved and characteristics of the quasi-bound
state in the system were evaluated. Three antikaon-nucleon and three nucleon-
nucleon potentials were used, so the dependence of the four-body pole
positions on the two-body interaction models was studied. The resulting
binding energies $B^{\rm Chiral}_{K^{-}ppn}\sim 30.5-34.5$ MeV obtained with
chirally motivated and $B^{\rm SIDD}_{K^{-}ppn}\sim 46.4-52.0$ MeV obtained
with phenomenological antikaon-nucleon potentials are close to those obtained
for the $K^{-}pp$ system with the same $\bar{K}N$ and $NN$ potentials, while
the four-body widths $\Gamma_{K^{-}ppn}\sim 38.2-50.9$ MeV are smaller.

few-body physics, antikaon-nucleon systems, four-body Faddeev equations

##  I Introduction

Attractive nature of $\bar{K}N$ interaction lead to suggestions, that quasi-
bound states can exist in few-body systems consisting of antikaons and
nucleons [1]. In particular, a deep and relatively narrow quasi-bound state
was predicted in the lightest three-body $\bar{K}NN$ system [2]. Many
theoretical calculations of the system were performed after that using
different methods and inputs. All of them agree, that the quasi-bound state
really exists in spin-zero state of $\bar{K}NN$, usually denoted as $K^{-}pp$,
but predict quite different binding energies and widths of the state.

The experimental situation is unsettled as well: several candidates for the
$K^{-}pp$ state were reported by different experiments [3, 4, 5], while other
experiments left the matter unsettled [6, 7]. However, the measured binding
energies and especially decay widths of the state differ from each other and
are far from all theoretical predictions. The most recent results by J-PARK
E15 experiment [8, 9] for the binding energy are comparable to some
theoretical predictions, but the width of the $K^{-}pp$ quasi-bound state is
still too large.

In recent years we performed a series of calculations of different states of
the three-body $\bar{K}NN$ and $\bar{K}\bar{K}N$ systems [10], using
dynamically exact Faddeev-type equations in AGS form with coupled $\bar{K}NN$
and $\pi\Sigma N$ channels. In particular, we evaluated $K^{-}pp$ quasi-bound
state binding energy and width using three different models of $\bar{K}N$
interaction. The same was done for the $\bar{K}\bar{K}N$ system. We also
demonstrated, that there is no quasi-bound states, caused by pure strong
interactions, in spin one state of $\bar{K}NN$ system, which is $K^{-}np$. In
addition, we calculated near-threshold amplitudes of $K^{-}$ elastic
scattering on deuteron. Finally, we evaluated $1s$ level shift in kaonic
deuterium, which is an atomic state, caused by presence of the strong
$\bar{K}N$ interaction in comparison to the pure Coulomb state.

Four-body $\bar{K}NNN$ system is another system with strangeness, which could
lead more light on the question of antikaon-nucleon interaction. Some
theoretical calculations of the quasi-bound state were already performed [1,
11, 12, 13], but more accurate calculations are needed. We calculated binding
energy and width of the state using four-body Faddeev-type equations by
Grassberger and Sandhas [14]. Only these dynamically exact equations in
momentum representation can treat energy-dependent $\bar{K}N$ potentials,
necessary for the this system, exactly. We used our two-body antikaon-nucleon
potentials, constructed for the three-body AGS calculations [10], as the
input.

##  II Four-body Faddeev-type equations

The three-body Faddeev-type equations in Alt-Grassberger-Sandhas form

$U_{\alpha\beta}(z)=(1-\delta_{\alpha\beta})G^{-1}_{0}(z)+\sum_{\gamma=1}^{3}(1-\delta_{\alpha\gamma})T_{\gamma}(z)G_{0}(z)U_{\gamma\beta}(z)$
(1)

define the three-body transition operators $U_{\alpha\beta}(z)$, which
describe process $\beta+(\alpha\gamma)\to\alpha+(\beta\gamma)$. The $G_{0}(z)$
in Eq. (1) is three-body Green function, Faddeev partition indices
$\alpha,\beta=1,2,3$ simultaneously define a particle ($\alpha$) and the
remained pair ($\beta\gamma$), $\alpha\neq\beta\neq\gamma$. The operator
$T_{\alpha}(z)$ is a two-body $T$-matrix, describing interaction in the
($\beta\gamma$) pair.

A separable potential $V_{\alpha}$ leading to a separable $T$-matrix

$V_{\alpha}=\lambda_{\alpha}|g_{\alpha}\rangle\langle g_{\alpha}|\to
T_{\alpha}(z)=|g_{\alpha}\rangle\tau_{\alpha}(z)\langle g_{\alpha}|$ (2)

allows to write the three-body AGS equations in the form

$X_{\alpha\beta}(z)=Z_{\alpha\beta}(z)+\sum_{\gamma=1}^{3}Z_{\alpha\gamma}(z)\tau_{\gamma}(z)X_{\gamma\beta}(z)$
(3)

with new transition $X_{\alpha\beta}$ and kernel $Z_{\alpha\beta}$ operators,
defined by

$\displaystyle X_{\alpha\beta}(z)$ $\displaystyle=$ $\displaystyle\langle
g_{\alpha}|G_{0}(z)U_{\alpha\beta}(z)G_{0}(z)|g_{\beta}\rangle,$ (4)
$\displaystyle Z_{\alpha\beta}(z)$ $\displaystyle=$
$\displaystyle(1-\delta_{\alpha\beta})\langle
g_{\alpha}|G_{0}(z)|g_{\beta}\rangle.$ (5)

Here for simplicity the one-term separable potentials Eq.(2) used, while in
general $V_{\alpha}$ can consist of $N$ terms.

The four-body Faddeev-type Grassberger-Sandhas equations were derived in
Ref.[14]

$\displaystyle U_{\alpha\beta}^{\sigma\rho}(z)$ $\displaystyle=$
$\displaystyle(1-\delta_{\sigma\rho})\delta_{\alpha\beta}G_{0}^{-1}(z)T_{\alpha}^{-1}(z)G_{0}^{-1}(z)+$
(6)
$\displaystyle\qquad\sum_{\tau,\gamma}(1-\delta_{\sigma\tau})U_{\alpha\gamma}^{\tau}G_{0}(z)T_{\gamma}(z)G_{0}(z)\,U_{\gamma\beta}^{\tau\rho}$

In addition to the free Green function $G_{0}(z)$, which now acts in four-body
space, and the two-body $T$-matrix $T_{\alpha}(z)$, three-body
$U_{\alpha\beta}^{\tau}(z)$ and four-body $U_{\alpha\beta}^{\sigma\rho}(z)$
operators enter the system Eq.(6). The high indices $\sigma,\rho,\tau$ define
a partition, which could be $3+1$ or $2+2$ type, while the low indices
$\alpha,\beta$ define two-body subsystems of the particular three-body
subsystem, denoted by the high index.

If the separable potentials Eq. (2), leading to the corresponding separable
$T$-matrices, is used, the system Eq. (6) can be rewritten in the same way, as
the three-body one. The new system of equations

$\bar{U}^{\sigma\rho}_{\alpha\beta}(z)=(1-\delta_{\sigma\rho})(\bar{G_{0}}^{-1})_{\alpha\beta}(z)+\sum_{\tau,\gamma,\delta}(1-\delta_{\sigma\tau})\bar{T}^{\tau}_{\alpha\gamma}(z)(\bar{G_{0}})_{\gamma\delta}(z)\bar{U}^{\tau\rho}_{\delta\beta}(z)$
(7)

contains new operators

$\displaystyle\bar{U}^{\sigma\rho}_{\alpha\beta}(z)=\langle
g_{\alpha}|G_{0}(z)U^{\sigma\rho}_{\alpha\beta}(z)G_{0}(z)|g_{\beta}\rangle,$
(8) $\displaystyle\bar{T}^{\tau}_{\alpha\beta}(z)=\langle
g_{\alpha}|G_{0}(z)U^{\tau}_{\alpha\beta}(z)G_{0}(z)|g_{\beta}\rangle,$ (9)
$\displaystyle(\bar{G_{0}})_{\alpha\beta}(z)=\delta_{\alpha\beta}\tau_{\alpha}(z).$
(10)

It is seen that the four-body system with separable potentials Eq. (7) looks
similarly to Eq. (1), which describes a three-body system with arbitrary
potentials. This analogy can be used for further modification of the
equations. Namely, if the three-body $T$-matrices
$\bar{T}^{\tau}_{\alpha\beta}(z)$ in Eq. (7) are presented in a separable form

$\bar{T}^{\tau}_{\alpha\beta}(z)=|\bar{g}^{\tau}_{\alpha}\rangle\bar{\tau}^{\tau}_{\alpha\beta}(z)\langle\bar{g}^{\tau}_{\beta}|,$
(11)

the four-body equations Eq. (7) can be rewritten as

$\bar{X}^{\sigma\rho}_{\alpha\beta}(z)=\bar{Z}^{\sigma\rho}_{\alpha\beta}(z)+\sum_{\tau,\gamma,\delta}\bar{Z}^{\sigma\tau}_{\alpha\gamma}(z)\bar{\tau}^{\tau}_{\gamma\delta}(z)\bar{X}^{\tau\rho}_{\delta\beta}(z)$
(12)

with new transition $\bar{X}^{\sigma\rho}_{\alpha\beta}$ and kernel
$\bar{Z}^{\sigma\rho}_{\alpha\beta}$ operators, defined by

$\displaystyle\bar{X}^{\sigma\rho}_{\alpha\beta}(z)$ $\displaystyle=$
$\displaystyle\langle\bar{g}^{\sigma}_{\alpha}|(\bar{G_{0}})_{\alpha\alpha}(z)\bar{U}^{\sigma\rho}_{\alpha\beta}(z)(\bar{G_{0}})_{\beta\beta}(z)|\bar{g}^{\rho}_{\beta}\rangle,$
(13) $\displaystyle\bar{Z}^{\sigma\rho}_{\alpha\beta}(z)$ $\displaystyle=$
$\displaystyle(1-\delta_{\sigma\rho})\langle\bar{g}^{\sigma}_{\alpha}|(\bar{G_{0}})_{\alpha\beta}(z)|\bar{g}^{\rho}_{\beta}\rangle.$
(14)

We solved the four-body equations Eq. (12) with separable two-body
$T$-matrices being an input and three-body $T$-matrices
$\bar{T}_{\alpha\beta}^{\tau}$ represented in separable form.

##  III Separable three-body amplitudes

We used $\bar{K}N$ and $NN$ potentials, which are separable ones by
construction. Therefore, separable versions of three-body and 2+2 amplitudes,
entering the equations (12), should be constructed. These amplitudes are
described by three-body AGS equations Eq.(3), which being written in momentum
basis for $s$-wave interactions have a form

$X_{\alpha\beta}(p,p^{\prime};z)=Z_{\alpha\beta}(p,p^{\prime};z)+\sum_{\gamma=1}^{3}4\pi\int_{0}^{\infty}Z_{\alpha\gamma}(p,p^{\prime\prime};z)\,\tau_{\gamma}(p^{\prime\prime};z)\,X_{\gamma\beta}(p^{\prime\prime},p^{\prime};z)p^{\prime\prime
2}dp^{\prime\prime}.$ (15)

Here $p,p^{\prime}$ and $z$ are relative momenta and three-body energy. It is
possible to evaluate eigenvalues $\lambda_{n}$ and eigenfunctions
$g_{n\alpha}(p;z)$ of the system Eq.(15) from

$g_{n\alpha}(p;z)=\frac{1}{\lambda_{n}}\,\sum_{\gamma=1}^{3}4\pi\int_{0}^{\infty}Z_{\alpha\gamma}(p,p^{\prime};z)\,\tau_{\gamma}(p^{\prime};z)\,g_{n\gamma}(p^{\prime};z)p^{\prime
2}dp^{\prime}$ (16)

with normalization condition

$\sum_{\gamma=1}^{3}4\pi\int_{0}^{\infty}g_{n\gamma}(p^{\prime};z)\,\tau_{\gamma}(p^{\prime};z)\,g_{n^{\prime}\gamma}(p^{\prime};z)p^{\prime
2}dp^{\prime}=-\delta_{nn^{\prime}}.$ (17)

Knowledge of the eigenvalues and eigenfunction allows us to write down
Hilbert-Schmidt expansion (HSE) of the kernel functions $Z_{\alpha\beta}$:

$Z_{\alpha\beta}^{\rm
HSE}(p,p^{\prime};z)=-\sum_{n=1}^{\infty}\lambda_{n}g_{n\alpha}(p;z)g_{n\beta}(p^{\prime};z),$
(18)

which leads to the separable three-body amplitude

$X_{\alpha\beta}^{\rm
HSE}(p,p^{\prime};z)=-\sum_{n=1}^{\infty}\frac{\lambda_{n}}{1-\lambda_{n}}g_{n\alpha}(p;z)g_{n\beta}(p^{\prime};z).$
(19)

Since the kernel function $Z_{\alpha\beta}$ and three-body amplitude
$X_{\alpha\beta}$ are energy-dependent functions, being off energy shell in
the four-body equations Eq.(12), it is necessary to solve eigenequations
Eq.(16) with normalization condition Eq.(17) for every value of the three-body
energy during the four-body calculations. It is timely consuming work. Instead
of this we used Energy Dependent Pole Expansion/Approximation (EDPE/EDPA)
method, suggested in [17] specially for the four-body GS equations. It needs
solution of the eigenequations Eq.(16) only once, for a fixed energy $z_{\rm
fix}$. Usually it is chosen to be the binding energy $z_{\rm fix}=E_{B}$ if a
bound state in the system exists or $z_{\rm fix}=0$ if not. After that energy
dependent form-factors

$g_{n\alpha}(p;z)=\sum_{\gamma=1}^{3}4\pi\int_{0}^{\infty}Z_{\alpha\gamma}(p,p^{\prime};z)\,\tau_{\gamma}(p^{\prime};z_{\rm
fix})\,g_{n\gamma}(p^{\prime};z_{\rm fix})p^{\prime 2}dp^{\prime}$ (20)

and propagators

$\displaystyle\left(\Theta(z)\right)^{-1}_{mn}\,=\,\sum_{\gamma=1}^{3}4\pi\int_{0}^{\infty}g_{m\gamma}(p^{\prime};z)\,\tau_{\gamma}(p^{\prime};z_{\rm
fix})\,g_{n\gamma}(p^{\prime};z_{\rm fix})p^{\prime 2}dp^{\prime}$ (21)
$\displaystyle-\sum_{\gamma=1}^{3}4\pi\int_{0}^{\infty}g_{m\gamma}(p^{\prime};z)\,\tau_{\gamma}(p^{\prime};z)\,g_{n\gamma}(p^{\prime};z)p^{\prime
2}dp^{\prime}$

are calculated. Finally, the separable three-body amplitude can be written in
a form

$X_{\alpha\beta}^{\rm
EDPE}(p,p^{\prime};z)=\sum_{m,n=1}^{\infty}g_{m\alpha}(p;z)\,\Theta_{mn}(z)\,g_{n\beta}(p^{\prime};z).$
(22)

If only one term is taken in the sums in Eq.(22), the Energy Dependent Pole
Expansion turns into Energy Dependent Pole Approximation. It is seen, that
EDPE method needs only one solution of the eigenvalue equations Eq.(16) and
calculations of the integrals Eqs.(20,21) after that. According to the
authors, the method converges faster than Hilbert-Schmidt expansion, it is
accurate already with one term.

Three-body form-factors $g_{\alpha}$ in Eqs.(20,21,22) are denoted as
$\bar{g}^{\rho}_{\alpha}$ in the four-body equations, while three-body energy-
dependent functions $\Theta(z)$ are denoted as
$\bar{\tau}^{\rho}_{\alpha\beta}$ (four-body equations are written down for
EDPA, i.e. with only one term is taken in Eq.(22), $m=n=1$).

We tried different versions of numerical treatment of Eqs.(16,17) and
Eqs.(20,21,22). The best result was obtained when the eigenequations Eq.(16)
with normalization condition Eq.(17) were solved using such number of
integration points, which is enough for exact calculations of the
corresponding binding energy of the three-body subsystem. For this sake 20
point is enough. The whole set of the eigenvalues and the corresponding
eigenfunctions (in this case 20 eigenfunctions for each of 20 eigenvalues) was
evaluated. We used ZGEEV subroutine from Intel oneAPI Math Kernel Library -
Fortran, which computes the eigenvalues and, optionally, the left and/or right
eigenvectors for an n-by-n complex nonsymmetric matrix. After that the first
term in Eq.(22) with $m=n=1$ gives the correct binding energy (more exactly,
$\Theta_{11}(z){\longrightarrow\atop{z\to E_{B}}}\infty$) since we strictly
set the first eigenvalue $\lambda_{1}=1$ (of cause, when binding energy in the
system exists). We checked how the amplitudes converge, and found that,
really, the first term already gives the result close to the original value of
$X_{\alpha\beta}(p,p^{\prime};z)$ at the discrete set of momenta, used in
Eqs.(16,17).

##  IV Four-body equations for the $\bar{K}NNN$ system

Two types of partitions for a four-body system: $3+1$ and $2+2$, - for the
$\bar{K}NNN$ system are: $|\bar{K}+(NNN)\rangle$, $|N+(\bar{K}NN)\rangle$ and
$|(\bar{K}N)+(NN)\rangle$. We started by writing down the system Eq.(12) for
$18$ channels considering three nucleons as nonidentical particles. Four-body
asymptotic states are denoted by ${}^{\sigma}_{\alpha}$ indices with
$\sigma=1,2,3$ stands for $|\bar{K}+(NNN)\rangle$, $|N+(\bar{K}NN)\rangle$ and
$|(\bar{K}N)+(NN)\rangle$ partitions, correspondingly, and $\alpha=$
$N_{i}N_{j}$ or $\bar{K}N_{i}$ ($i,j=1,2,3$, $i\neq j$) denotes the pair in
the two- or three-body subsystem:

$\displaystyle\bar{g}^{1}_{N_{i}N_{j}}:|\bar{K}+(N_{1}+N_{2}N_{3})\rangle,|\bar{K}+(N_{2}+N_{3}N_{1})\rangle,|\bar{K}+(N_{3}+N_{1}N_{2})\rangle,$
$\displaystyle\bar{g}^{2}_{N_{i}N_{j}}:|N_{1}+(\bar{K}+N_{2}N_{3})\rangle,|N_{2}+(\bar{K}+N_{3}N_{1})\rangle,|N_{3}+(\bar{K}+N_{1}N_{2})\rangle,$
$\displaystyle\bar{g}^{2}_{\bar{K}N_{i}}:|N_{1}+(N_{2}+\bar{K}N_{3})\rangle,|N_{2}+(N_{3}+\bar{K}N_{1})\rangle,|N_{3}+(N_{1}+\bar{K}N_{2})\rangle,$
$\displaystyle\quad\quad\quad\\!|N_{1}+(N_{3}+\bar{K}N_{2})\rangle,|N_{2}+(N_{1}+\bar{K}N_{3})\rangle,|N_{3}+(N_{2}+\bar{K}N_{1})\rangle,$
$\displaystyle\bar{g}^{3}_{N_{i}N_{j}}:|(N_{2}N_{3})+(\bar{K}+N_{1})\rangle,|(N_{3}N_{1})+(\bar{K}+N_{2})\rangle,|(N_{1}N_{2})+(\bar{K}+N_{3})\rangle,$
$\displaystyle\bar{g}^{3}_{\bar{K}N_{i}}:|(\bar{K}N_{1})+(N_{2}+N_{3})\rangle,|(\bar{K}N_{2})+(N_{3}+N_{1})\rangle,|(\bar{K}N_{3})+(N_{1}+N_{2})\rangle$

After antisymmetrization, necessary for a system with identical fermions, only
five states, plotted in Fig.1, remains.

Figure 1: States of the four-body system equations.

The kernel functions $\bar{Z}^{\sigma\rho}_{\alpha}$ of the system of four-
body equations Eq.(12) can be seen in Fig.2
($\bar{Z}^{\sigma\rho}_{\alpha\beta}$ carrys only one bottom index due to
$\delta_{\alpha\beta}$ function in it’s definition).

Figure 2: Kernel functions $\bar{Z}^{\sigma\rho}_{\alpha}$ of the four-body
system equations.

Looking for a quasi-bound state needs solution of the homogeneous system of
equations, which can be written in a matrix form

$\hat{X}=\hat{Z}\,\hat{\tau}\,\hat{X}$ (24)

with

$\bar{X}^{\rho}_{\alpha}=\left(\begin{tabular}[]{c}$\bar{X}^{1}_{NN}$\\\
$\bar{X}^{2}_{NN}$\\\ $\bar{X}^{2}_{\bar{K}N}$\\\ $\bar{X}^{3}_{NN}$\\\
$\bar{X}^{3}_{\bar{K}N}$\\\
\end{tabular}\right),\qquad\qquad\bar{Z}^{\sigma\rho}_{\alpha}=\left(\begin{tabular}[]{ccccc}0&$\bar{Z}^{12}_{NN}$&0&$\bar{Z}^{13}_{NN}$&0\\\
$\bar{Z}^{21}_{NN}$&0&0&$\bar{Z}^{23}_{NN}$&0\\\
0&0&$\bar{Z}^{22}_{\bar{K}N}$&0&$\bar{Z}^{23}_{\bar{K}N}$\\\
$\bar{Z}^{31}_{NN}$&$\bar{Z}^{32}_{NN}$&0&0&0\\\
0&0&$\bar{Z}^{32}_{\bar{K}N}$&0&0\\\ \end{tabular}\right),$ (25)
$\bar{\tau}^{\rho}_{\alpha\beta}=\left(\begin{tabular}[]{ccccc}$\bar{\tau}^{1}_{NN,NN}$&0&0&0&0\\\
0&$\bar{\tau}^{2}_{NN,NN}$&$\bar{\tau}^{2}_{NN,\bar{K}N}$&0&0\\\
0&$\bar{\tau}^{2}_{\bar{K}N,NN}$&$\bar{\tau}^{2}_{\bar{K}N,\bar{K}N}$&0&0\\\
0&0&0&$\bar{\tau}^{3}_{NN,NN}$&$\bar{\tau}^{3}_{NN,\bar{K}N}$\\\
0&0&0&$\bar{\tau}^{3}_{\bar{K}N,NN}$&$\bar{\tau}^{3}_{\bar{K}N,\bar{K}N}$\\\
\end{tabular}\right).$ (26)

However, our $\bar{K}N$ and $NN$ potentials, which we use for the $\bar{K}NNN$
system calculations, are isospin- and spin-dependent interaction models, in
addition $V_{NN}$ is a two-term potential. Due to this elements of the
matrices $\bar{Z}^{\sigma\rho}_{\alpha}$ in Eq.(25) and
$\bar{\tau}^{\rho}_{\alpha\beta}$ in Eq.(26), entering the antisymmetrized
equations Eq.(24), are matrices themselves containing elements with additional
indices
$\bar{Z}^{\sigma\rho(m_{3},n_{3})}_{{\alpha(m_{2},n_{2});}\atop{(i,ss^{\prime})}}$
and
$\bar{\tau}^{\sigma(m_{3})}_{{\alpha\beta(m_{2},m_{2}^{\prime});}\atop{(ii^{\prime},ss^{\prime})}}$.
Particular forms of nine elements-matrices $\bar{Z}^{\sigma\rho}_{\alpha}$ of
Eq.(25) and nine elements-matrices $\tau^{\rho}_{\alpha\beta}$ of Eq.(26) are
presented in the Appendix. The additional indices $m_{3},n_{3}$ denote number
of a separable term of the three-body or $2+2$ amplitudes (at the first step
only one separable term was used for the three-body $\bar{K}NN$, $NNN$ and 2+2
$\bar{K}N+NN$ amplitudes in Eq. (22), so that $m_{3}=n_{3}=1$). Separable
indices $m_{2},n_{2}$ of the two-body subsystems (i.e. potentials) are
$m_{2}=1$ for $V_{\bar{K}N}$ and $m_{2}=1,2$ for $V_{NN}$. The remained
indices $i,i^{\prime}$ and $s,s^{\prime}$ are two-body isospins and spins,
correspondingly.

The unknown four-body amplitudes in Eq.(25) have a general form
$\bar{X}^{\sigma\rho(m_{3},n_{3})}_{\alpha\beta(m_{2},n_{2});(ii^{\prime},ss^{\prime})}$.
Finally, the system to be solved consists of $18$ coupled equations.

##  V Two-body interactions and three-body subsystems

###  V.1 Two-body input: $\bar{K}N$ and $NN$ potentials

Both $\bar{K}N$ and $NN$ potentials, which we used, are separable isospin- and
spin-dependent ones in $s$-wave. Three our separable antikaon-nucleon
potentials were constructed for our three-body calculations of the $\bar{K}NN$
and $\bar{K}\bar{K}N$ systems [10]. They are: two phenomenological potentials
with coupled $\bar{K}N-\pi\Sigma$ channels, having one- $V_{\bar{K}N}^{\rm
1,SIDD}$ or two-pole $V_{\bar{K}N}^{\rm 2,SIDD}$ structure of the
$\Lambda(1405)$ resonance [18] and a chirally motivated model
$V_{\bar{K}N}^{\rm Chiral}$ with coupled $\bar{K}N-\pi\Sigma-\pi\Lambda$
channels and the two-pole structure [19]. All three potentials describe low-
energy $K^{-}p$ scattering, namely: elastic $K^{-}p\to K^{-}p$ and inelastic
$K^{-}p\to MB$ cross-sections and threshold branching ratios
$\gamma,R_{c},R_{n}$. They also reproduce $1s$ level shift of kaonic hydrogen
caused by the strong $\bar{K}N$ interaction in comparison to the pure Coulomb
level, measured by SIDDHARTA experiment [20]: $\Delta_{1s}^{SIDD}=-283\pm
36\pm 6$ eV, and it’s width $\Gamma_{1s}^{SIDD}=541\pm 89\pm 22$ eV. All the
experimental data are described by three our potentials with equally high
accuracy. In addition, elastic $\pi\Sigma$ cross-sections with isospin
$I_{\pi\Sigma}=0$, provided by all three potentials, have a bump in a region
of the $\Lambda(1405)$ resonance (according to PDG [21]:
$M_{\Lambda(1405)}^{PDG}=1405.1^{+1.3}_{-1.0}$ MeV,
$\Gamma_{\Lambda(1405)}^{PDG}=50.5\pm 2.0$ MeV). The poles corresponding to
the $\Lambda(1405)$ resonance are situated at

$\displaystyle z_{\Lambda(1405)-1}^{\rm 1,SIDD}$ $\displaystyle=$
$\displaystyle 1426-i\,48{\,\rm MeV}$ (27) $\displaystyle
z_{\Lambda(1405)-1}^{\rm 2,SIDD}$ $\displaystyle=$ $\displaystyle
1414-i\,58{\,\rm MeV},\quad z_{\Lambda(1405)-2}^{\rm 2,SIDD}=1386-i\,104{\,\rm
MeV}$ (28)

for the phenomenological potentials with one- and two-pole structure,
correspondingly [19], and at

$z_{\Lambda(1405)-1}^{\rm Chiral}=1417-i\,33{\,\rm MeV},\quad
z_{\Lambda(1405)-2}^{\rm Chiral}=1406-i\,89{\,\rm MeV}$ (29)

for the chirally motivated potential [22].

These three antikaon-nucleon potentials with coupled $\bar{K}N-\pi\Sigma$
channels were used in their original form in three-body AGS equations [10]
with coupled $\bar{K}NN-\pi\Sigma N$ three-body channels. By this the channel
coupling was taken into account in a direct way. The four-body GS equations
Eq.(12) are too complicated for introducing additional particle channels and
performing coupled-channel calculations. Due to this we used the exact optical
versions of our $\bar{K}N$ potentials [23], which are one-channel
$V_{\bar{K}N(-\pi\Sigma-\pi\Lambda)}$. They have exactly the same elastic part
as the potential with coupled channels, while all in-elasticity is taken into
account in an energy-dependent imaginary part of the potential. It was
demonstrated in our three-body calculations [23, 24], that such exact optical
potentials give quite accurate results for $K^{-}d$ scattering length or
quasi-bound state position and width of the $K^{-}pp$ system in comparison
with the results obtained with the coupled-channel form of interaction models.
Due to this we assume that it is a good approximation for the four-body
calculations as well.

Nucleon-nucleon potentials $V^{\rm TSA-A}_{NN}$ and $V^{\rm TSA-B}_{NN}$ used
in our three-body calculations [10] were used here together with the new
version of the two-term separable $NN$ potential $V^{\rm TSN}_{NN}$, described
in [24]. All three potentials reproduce Argonne v18 $NN$ phase shifts at low
energies up to $500$ MeV with change of sign, which means they are repulsive
at short distances. They give proper singlet and triplet $NN$ scattering
lengths and deuteron binding energy. The new nucleon-nucleon potential $V^{\rm
TSN}_{NN}$ reproduces Argonne v18 $NN$ phase shifts of $pp$ scattering
slightly better than the previously used ones, its parameters are more natural
then those of the older $V_{NN}^{\rm TSA}$.

In principle, Coulomb interaction also should be included in the equations,
but we are interested in the quasi-bound state caused mainly by strong
potentials. We assume that in such a state Coulomb interaction plays a minor
role and can be neglected.

###  V.2 $3+1$ and $2+2$ partitions

We investigated $\bar{K}NNN$ system with the lowest value of the four-body
isospin $I^{(4)}=0$ and spin $S^{(4)}$ one half, which can be denoted as
$K^{-}ppn$ or $\bar{K}^{0}nnp$. The total angular momentum is zero, all two-
body interactions are $s$-wave ones. For the $\bar{K}NNN$ system with these
quantum numbers the following three-body subsystems contribute:

  * •

$\bar{K}NN$ with isospin $I^{(3)}=1/2$ and spin $S^{(3)}=0$ ($K^{-}pp$) or
spin $S^{(3)}=1$ ($K^{-}np$)

  * •

$NNN$ with isospin $I^{(3)}=1/2$ and spin $S^{(3)}=1/2$ (3H or 3He)

together with the $2+2$ partition

  * •

$\bar{K}N+NN$ with isospin $I^{(4)}=0$ and spin $S^{(4)}=1/2$.

The corresponding three-body amplitudes is not an input, they are calculated
during the four-body calculations.

Table 1: Dependence of the binding energy $B_{K^{-}pp}^{\rm Opt}$ (MeV) and width $\Gamma_{K^{-}pp}^{\rm Opt}$ (MeV) of the quasi-bound state in the $K^{-}pp-\bar{K}^{0}np$ subsystem ($\bar{K}NN,S^{(3)}=0$) on three $\bar{K}N$ and three $NN$ interaction models. Calculations were performed using the exact optical $\bar{K}N$ potentials. The binding energy is counted from the threshold energy of the $K^{-}pp$ system $z_{th,{\rm K^{-}pp}}=m_{\bar{K}}+2\,m_{N}$ $=2373.485$ MeV. | $V_{NN}^{\rm TSA-A}$ | $V_{NN}^{\rm TSA-B}$ | $V_{NN}^{\rm TSN}$
---|---|---|---
| $B_{K^{-}pp}^{\rm Opt}$ | $\Gamma_{K^{-}pp}^{\rm Opt}$ | $B_{K^{-}pp}^{\rm Opt}$ | $\Gamma_{K^{-}pp}^{\rm Opt}$ | $B_{K^{-}pp}^{\rm Opt}$ | $\Gamma_{K^{-}pp}^{\rm Opt}$
$V_{\bar{K}N}^{\rm 1,SIDD}$ | $55.4$ | $60.9$ | $54.3$ | $60.8$ | $53.3$ | $64.7$
$V_{\bar{K}N}^{\rm 2,SIDD}$ | $48.2$ | $46.2$ | $47.5$ | $45.9$ | $46.7$ | $48.4$
$V_{\bar{K}N}^{\rm Chiral}$ | $31.9$ | $42.2$ | $33.2$ | $48.7$ | $29.9$ | $48.2$

The three-body $\bar{K}NN$ system with different quantum numbers was studied
in our previous works. In particular, quasi-bound state pole positions and
widths in the $K^{-}pp$ system ($\bar{K}NN$ with isospin $I^{(3)}=1/2$ and
spin $S^{(3)}=0$) were calculated in [22] with older $V_{NN}^{\rm TSA-B}$
nucleon-nucleon potential. Recently the calculations were repeated with the
new $V_{NN}^{\rm TSN}$ [24], the results can be found in Table 4 of the paper.
Binding energies $B_{K^{-}pp,\rm Opt}$ and widths $\Gamma_{K^{-}pp,\rm Opt}$
calculated using the exact optical potentials are shown in Table 1 for three
antikaon-nucleon $V_{\bar{K}N}^{\rm 1,SIDD}$, $V_{\bar{K}N}^{\rm 2,SIDD}$,
$V_{\bar{K}N}^{\rm Chiral}$ and three nucleon-nucleon $V_{NN}^{\rm TSA-A}$,
$V_{NN}^{\rm TSA-B}$, $V_{NN}^{\rm TSN}$ potentials. It is seen that the new
$NN$ potential changed quasi-bound state positions in $K^{-}pp$ system by few
MeV.

Table 2: Dependence of the binding energy $B_{K^{-}np}^{\rm Opt}$ (MeV) and width $\Gamma_{K^{-}np}^{\rm Opt}$ (MeV) of the quasi-bound state in the $K^{-}np-\bar{K}^{0}nn$ subsystem ($\bar{K}NN,S^{(3)}=1$) on hree $\bar{K}N$ and three $NN$ interaction models. Calculations were performed using the exact optical $\bar{K}N$ potentials. The energy is counted from the $K^{-}d$ threshold $z_{th,{\rm K^{-}d}}=m_{\bar{K}}+2m_{N}+E_{\rm deu}$ $=2371.26$ MeV. | $V_{NN}^{\rm TSA-A}$ | $V_{NN}^{\rm TSA-B}$ | $V_{NN}^{\rm TSN}$
---|---|---|---
| $B_{K^{-}np}^{\rm Opt}$ | $\Gamma_{K^{-}np}^{\rm Opt}$ | $B_{K^{-}np}^{\rm Opt}$ | $\Gamma_{K^{-}np}^{\rm Opt}$ | $B_{K^{-}np}^{\rm Opt}$ | $\Gamma_{K^{-}np}^{\rm Opt}$
$V_{\bar{K}N}^{\rm 1,SIDD}$ | $1.6$ | $70.1$ | $0.8$ | $67.6$ | $1.9$ | $68.7$
$V_{\bar{K}N}^{\rm 2,SIDD}$ | $5.2$ | $63.7$ | $5.0$ | $61.4$ | $5.6$ | $62.7$
$V_{\bar{K}N}^{\rm Chiral}$ | $2.6$ | $46.4$ | $2.4$ | $53.1$ | $2.3$ | $45.5$

No quasi-bound states similar to that one in $K^{-}pp$ were found in the
$K^{-}np$ system, which is $\bar{K}NN$ with isospin $I^{(3)}=1/2$ and spin
$S^{(3)}=1$ in our previous calculations [19]. However, new nucleon-nucleon
potential $V^{\rm TSN}_{NN}$ changed the picture: the quasi-bound state caused
purely by strong interactions can exist in the $K^{-}np$ system [23] in
addition to the atomic kaonic deuterium. The binding energies $B_{K^{-}np,\rm
Opt}$ and width $\Gamma_{K^{-}np,\rm Opt}$ evaluated with the exact optical
versions of the $V_{\bar{K}N}^{\rm 1,SIDD}$, $V_{\bar{K}N}^{\rm 2,SIDD}$,
$V_{\bar{K}N}^{\rm Chiral}$ potentials and three nucleon-nucleon $V_{NN}^{\rm
TSA-A}$, $V_{NN}^{\rm TSA-B}$, $V_{NN}^{\rm TSN}$ potentials can be seen in
Table 2. The binding energy of the state is so close to the $K^{-}d$
threshold, that relatively weak $NN$ interaction, playing in the case of
strong quasi-bound state secondary role, can resolve the question of existence
of the state.

The binding energies and widths of the quasi-bound states of the $\bar{K}NN$
systems with spin zero and one were calculated using the three-body AGS
equations. Details of three-body calculations can be found in [23, 25]. The
codes for numerical solution of the three-body AGS equations for the
$\bar{K}NN$ systems were than modified in order to construct separable
versions of the three-body amplitudes, as described in Section III.

Table 3: Dependence of the binding energy $B_{NNN}$ (MeV) and width $\Gamma_{NNN}$ (MeV) of the quasi-bound state in the $NNN$ subsystem ($S^{(3)}=1/2$) on three $NN$ models. The binding energy is counted from the threshold energy of the $NNN$ system $z_{th,{\rm NNN}}=3\,m_{N}$ $=2816.76$ MeV. $V_{NN}^{\rm TSA-A}$ | $V_{NN}^{\rm TSA-B}$ | $V_{NN}^{\rm TSN}$
---|---|---
$9.03$ | $9.04$ | $9.52$

The three-body AGS equations Eq.(3) were written and numerically solved for
the three-nucleon system $NNN$ with three $NN$ potentials $V_{NN}^{\rm
TSA-A}$, $V_{NN}^{\rm TSA-B}$, $V_{NN}^{\rm TSN}$ as an input. The calculated
binding energies of the system are equal for both 3H and 3He nuclei since
Coulomb interaction was not taken into account. They are presented in Table 3.
The resulting energies are larger than known values for binding energy of
triton $8.4820$ MeV and helium-3 $7.7181$ MeV nucleus. Such overestimation is
typical for separable $NN$ potentials. The numerical code was afterwards
changed for construction of separable version of the $NNN$ amplitude.

Table 4: Dependence of the binding energy $B_{K^{-}p+np}^{\rm Opt}$ (MeV) and width $\Gamma_{K^{-}p+np}^{\rm Opt}$ (MeV) of the quasi-bound state in the $K^{-}p+np$ partition ($\bar{K}N+NN,S^{(4)}=1/2$) on three $\bar{K}N$ and three $NN$ interaction models. Calculations were performed using the exact optical $\bar{K}N$ potentials. The binding energy is counted from the threshold energy of the $K^{-}npp$ system $z_{th,{\rm K^{-}npp}}=m_{\bar{K}}+3\,m_{N}$ $=3312.405$ MeV. | $V_{NN}^{\rm TSA-A}$ | $V_{NN}^{\rm TSA-B}$ | $V_{NN}^{\rm TSN}$
---|---|---|---
| $B_{K^{-}p+np}^{\rm Opt}$ | $\Gamma_{K^{-}p+np}^{\rm Opt}$ | $B_{K^{-}p+np}^{\rm Opt}$ | $\Gamma_{K^{-}p+np}^{\rm Opt}$ | $B_{K^{-}p+np}^{\rm Opt}$ | $\Gamma_{K^{-}p+np}^{\rm Opt}$
$V_{\bar{K}N}^{\rm 1,SIDD}$ | $20.0$ | $83.6$ | $19.1$ | $81.1$ | $20.3$ | $82.2$
$V_{\bar{K}N}^{\rm 2,SIDD}$ | $25.4$ | $71.3$ | $24.0$ | $70.8$ | $25.1$ | $70.4$
$V_{\bar{K}N}^{\rm Chiral}$ | $18.8$ | $54.5$ | $20.9$ | $58.5$ | $17.9$ | $53.7$

Finally, the partition of the $2+2$ type $\bar{K}N+NN$ is a system with two
non-interacting pairs of particles. It is a special part for the four-body
Faddeev-type equations and not a simple sum of two-body $\bar{K}N$ and $NN$
amplitudes. $\bar{K}N+NN$ partition is described by a three-body system of AGS
equations, therefore, it is a ”three-body” amplitude. Only $\bar{K}N+NN$
partition with spin one $NN$ pair has a quasi-bound state, the binding
energies and widths are shown in Table 4. The separable version of the
corresponding $\bar{K}N+NN$ amplitude was constructed in similar way as in the
case of $\bar{K}NN$ and $NNN$ three-body subsystems.

##  VI Results and discussion

We solved the four-body GS equations for the $K^{-}ppn-\bar{K}^{0}nnp$ system
with the antikaon-nucleon and nucleon-nucleon potentials described above. As
the first step we used only one separable term in separable representation of
the three-body $\bar{K}NN$, $NNN$ and ”three-body” $\bar{K}N+NN$ amplitudes
Eq.(22), so that we used EDPA method. The binding energies $B_{K^{-}ppn}$ and
widths $\Gamma_{K^{-}ppn}$ of the $\bar{K}NNN$ system evaluated using three
exact optical versions of the antikaon-nucleon potentials and three nucleon-
nucleon interaction models are presented in Table 5.

It is seen that the binding energy $B_{K^{-}ppn}$ and width
$\Gamma_{K^{-}ppn}$ of the four-body quasi-body state strongly depend on the
model of antikaon-nucleon interaction. In fact, it is a property of all three-
body $\bar{K}NN$ and $\bar{K}\bar{K}N$ systems as well. Phenomenological
$\bar{K}N(-\pi\Sigma)$ potentials give comparable binding energies, while the
chirally motivated potential led to much more shallow state. As for the
widths, the models of antikaon-nucleon interaction having two-pole structure
of $\Lambda(1405)$ resonance, which are the phenomenological
$V_{\bar{K}N}^{\rm 2,SIDD}$ and chiral $V_{\bar{K}N}^{\rm Chiral}$, have very
close widths, while one-pole $V_{\bar{K}N}^{\rm 1,SIDD}$ potential lead to
much larger width.

Table 5: Dependence of the binding energy $B_{K^{-}ppn}$ (MeV) and width $\Gamma_{K^{-}ppn}$ (MeV) of the quasi-bound state in the $K^{-}ppn-\bar{K}^{0}nnp$ system on three $\bar{K}N$ and $NN$ interaction models.  | $V_{NN}^{\rm TSA-A}$ | $V_{NN}^{\rm TSA-B}$ | $V_{NN}^{\rm TSN}$ | Other results
---|---|---|---|---
| $B_{K^{-}ppn}$ | $\Gamma_{K^{-}ppn}$ | $B_{K^{-}ppn}$ | $\Gamma_{K^{-}ppn}$ | $B_{K^{-}ppn}$ | $\Gamma_{K^{-}ppn}$ | $B_{K^{-}ppn}$ | $\Gamma_{K^{-}ppn}$
$V_{\bar{K}N}^{\rm 1,SIDD}$ | $52.0$ | $50.4$ | $50.3$ | $49.6$ | $51.2$ | $50.8$ |  |
$V_{\bar{K}N}^{\rm 2,SIDD}$ | $47.0$ | $39.6$ | $46.4$ | $38.2$ | $46.4$ | $39.9$ |  |
$V_{\bar{K}N}^{\rm Chiral}$ | $32.6$ | $39.7$ | $34.5$ | $50.9$ | $30.5$ | $42.8$ |  |
AY [1] |  |  |  |  |  |  | 108.0 | 20.0
BGL [11] |  |  |  |  |  |  | 29.3 | 32.9
OHHMH [12]: |  |  |  |  |  |  |  |
$V_{\bar{K}N}^{\rm Kyoto-I}$ |  |  |  |  |  |  | 45.3 | 25.5
$V_{\bar{K}N}^{\rm Kyoto-II}$ |  |  |  |  |  |  | 49.7 | 69.4
$V_{\bar{K}N}^{\rm AY}$ |  |  |  |  |  |  | 72.6 | 78.6
ME [13]: |  |  |  |  |  |  |  |
$V_{\bar{K}N}^{\rm 1,SIDD}$ |  |  |  |  |  |  | 73.5 | 22.0
$V_{\bar{K}N}^{\rm 2,SIDD}$ |  |  |  |  |  |  | 58.5 | 27.0
$V_{\bar{K}N}^{\rm IKS\,chiral}$ |  |  |  |  |  |  | 41.4 | 31.5

As for the dependence of the obtained results on the models of $NN$
interactions, it seem they play more visible role than in the case of
$\bar{K}NN$ system. Nucleon-nucleon interaction plays visible role together
with the chirally motivated antikaon nucleon potential, while the results
obtained with phenomenological $\bar{K}N$ interaction models depend on the
$V_{NN}$ slightly.

Comparing the four-body binding energies and widths in Table 5 with those
obtained for the three-body spin zero $\bar{K}NN$ system usually denoted as
$K^{-}pp$ Table 1, it is seen that the binding energies remained almost the
same or become slightly smaller after adding one neutron, while the widths
become much smaller. So that the additional to $K^{-}pp$ neutron changes the
binding energy of the system slightly, but it ”tightenes” the quasi-bound
state.

Obviously, the four-body binding energies are much larger and the
corresponding widths are much smaller than those for the other, spin one state
of the $\bar{K}NN$ state ($K^{-}np$).

Results for the binding energies and widths of other authors are also
presented in Table 5. The largest binding energy and smaller width were
predicted in [1] by using many-body G-matrix formalism for the few-body system
and an antikaon-nucleon model, which does not reproduce actual experimental
data on $K^{-}p$ scattering and kaonic hydrogen.

The authors of variational calculation [11] used an energy dependent chiral
$\bar{K}N$ potential, which, however, was calculated at set of fixed energies.
Also ithe maginary part of the quasi-bound state was calculated approximately.
The result of [11] is not far from our binding energy and width evaluated with
our chirally motivated antikaon-nucleon potential, taken into account exactly.

Variational calculations were also performed in [12] with energy dependent
Kyoto and energy independent AY [26] $\bar{K}N$ potentials. The authors also
had to fix the antikaon-nucleon energy in the Kyoto potential, and they did it
in two ways, denoting the corresponding models as Kyoto-I and Kyoto-II. Huge
difference between the widths obtained with these two Kyoto potentials shows
that chosen procedure is not quite reliable. Besides, the binding energy and
width of the four-body quasi-bound state, calculated in [12] with AY potnetial
[26], differs drastically from the original AY results [1].

The most intriguing is the difference between our results and those of [13]
since the authors solved the same four-body Faddeev-type equations, moreover,
they used our phenomenological $V_{\bar{K}N}^{\rm 1,SIDD}$ and
$V_{\bar{K}N}^{\rm 2,SIDD}$ potentials. Nucleon-nucleon PEST $V_{NN}$
potential, which is a separable version of Paris nucleon-nucleon potential,
was used there. But we do not think, that different from our’s $NN$ potential
could change the results so drastically, by $\sim 20$ MeV, for both: binding
energy (make it much larger) and width (make it much smaller).

One of the possible reasons of differences could follow from the way of
separable version of the three-body and $2+2$ amplitudes construction. The
formulas for the EDPE energy dependent form-factors and propagators in [13]
differ from our Eqs.(20,21) by $1/\lambda$ factor on the right-hand side for
the first and by $m\leftrightarrow n$ in the second formula. We checked the
formulas by comparing the approximate $Z_{\alpha\beta}(p,p^{\prime};z)$ at
fixed momenta and energy with the original ones, and obtained agreement up to
$\sim 10$ significant digits while the whole number of separable terms (which
is 20 in our case) is used.

Besides, setting the binding energy for the $(\bar{K}N)_{I=0}+NN$ to the
energy of the quasi-bound $\bar{K}N$ state ($\Lambda(1405)$ resonance), made
in [13], is not quite understandable. The matter is the total four-body
isospin $I^{(4)}=0$ means that the $NN$ subsystem should also have $I_{NN}=0$,
which is a deuteron. Due to this, the total binding energy of this partition
should include both two-body energies: $B_{\Lambda(1405)}$ and $B_{\rm deu}$.
In fact, the energy of the $\bar{K}N+NN$ ”three-body” system with isospins of
both pairs equal to zero differs from simple sum of the two two-body energies,
see Table 4, it comes from solution of a three-body equation. As for another
state of the $\bar{K}N+NN$ partition, it does not have bound states since
$\bar{K}N$ and $NN$ in this case have two-body isospin $I^{(2)}=1$ and none of
them is bound. Setting the energy in this case to deuteron binding energy, as
is done in [13], is quite strange.

Our very preliminary results for the pole position of the quasi-bound state in
the $K^{-}ppn-\bar{K}^{0}nnp$ system calculated with two phenomenological
potentials were published in [27]. They differ from those presented in Table 5
drastically: binding energies are much larger while the width are smaller. The
reason is the different procedure of the separable representation of the
three- and ”three”-body amplitudes. In contrast to the present usage of the
Intel MKL library subroutine, we used some hand made procedure for eigenvalues
and eigenfunctions evaluation. Such difference in the procedure also can
explain differences between our present results and those of [13].

Finally, the binding energy the three-body $K^{-}pp$ quasi-bound state
evaluated in [13] with our phenomenological $V_{\bar{K}N}^{\rm 1,SIDD}$
potential differ from our result by $\sim 6$ MeV and almost coinsides with the
energy calculated with $V_{\bar{K}N}^{\rm 2,SIDD}$. Our three-body binding
energies are sufficiently different for every of the three nucleon-nucleon
potentials, see Table 1. Keeping this closeness of the three-body results in
mind, it is hard to understand large difference ($15$ MeV) between the
corresponding four-body results in [13].

##  VII Conclusion

The four-body Faddeev-type GS equations for search of the quasi-bound state in
the $\bar{K}NNN$ system were written down and solved. The binding energies
$B^{\rm Chiral}_{K^{-}ppn}\sim 30.5-34.5$ MeV obtained with chirally motivated
and $B^{\rm SIDD}_{K^{-}ppn}\sim 46.4-52.0$ MeV obtained with phenomenological
antikaon-nucleon potentials are close to those for the $K^{-}pp$ system,
calculated with the same $V_{\bar{K}N}$ and $V_{NN}$ potentials. The widths of
the four-body states $\Gamma_{K^{-}ppn}\sim 38.2-50.9$ MeV are smaller than
the three-body widths of $K^{-}pp$. Therefore, the neutron added to the
$K^{-}pp$ system slightly influences the binding, but tightens the system.

The four-body binding energy and width of the $\bar{K}NNN$ system strongly
depend on $\bar{K}N$ potential and noticeably - on $NN$ potential, especially
together with the chirally motivated antikaon-nucleon interaction model.

## Acknowledgments

The work was supported by GACR grant 19-19640S.

*

##  Appendix A Elements of $\bar{Z}^{\sigma\rho}_{\alpha}$ and
$\bar{\tau}^{\rho}_{\alpha\beta}$ matrices

After antisymmetrization we come to the system Eq.(24) of $18$ coupled
equations. Elements of the matrices $\bar{Z}^{\sigma\rho}_{\alpha}$ in Eq.
(25) and $\bar{\tau}^{\rho}_{\alpha\beta}$ in Eq.(26) are matrices themselves
with elements
$\bar{Z}^{\sigma\rho(m_{3},n_{3})}_{{\alpha(m_{2},n_{2});}\atop{(i,ss^{\prime})}}$
and
$\bar{\tau}^{\sigma(m_{3})}_{{\alpha\beta(m_{2},m_{2}^{\prime});}\atop{(ii^{\prime},ss^{\prime})}}$.
Indices $m_{3},n_{3}$ denote number of a separable term of the three-body or
$2+2$ amplitudes ($m_{3}=n_{3}=1$), while $m_{2},n_{2}$ are separable indices
of the potentials: $m_{2}=1$ for $V_{\bar{K}N}$ and $m_{2}=1,2$ for $V_{NN}$.
Two-body isospins and spins are denoted $i,i^{\prime}$ and $s,s^{\prime}$,
correspondingly.

The kernel matrices $\bar{Z}^{\sigma\rho}_{\alpha}$ consists of six elements
with $\alpha=NN$:

$\bar{Z}^{12}_{NN}=\left(\begin{tabular}[]{cccc}$\bar{Z}^{12(1,1)}_{{NN(1,1);}\atop{(0,11)}}$&0&$\bar{Z}^{12(1,1)}_{{NN(1,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{12(1,1)}_{{NN(1,1);}\atop{(1,00)}}$&0&$\bar{Z}^{12(1,1)}_{{NN(1,2);}\atop{(1,00)}}$\\\
$\bar{Z}^{12(1,1)}_{{NN(2,1);}\atop{(0,11)}}$&0&$\bar{Z}^{12(1,1)}_{{NN(2,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{12(1,1)}_{{NN(2,1);}\atop{(1,00)}}$&0&$\bar{Z}^{12(1,1)}_{{NN(2,2);}\atop{(1,00)}}$\end{tabular}\right)$
(30)
$\bar{Z}^{13}_{NN}=\left(\begin{tabular}[]{cccc}$\bar{Z}^{13(1,1)}_{{NN(1,1);}\atop{(0,11)}}$&0&$\bar{Z}^{13(1,1)}_{{NN(1,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{13(1,1)}_{{NN(1,1);}\atop{(1,00)}}$&0&$\bar{Z}^{13(1,1)}_{{NN(1,2);}\atop{(1,00)}}$\\\
$\bar{Z}^{13(1,1)}_{{NN(2,1);}\atop{(0,11)}}$&0&$\bar{Z}^{13(1,1)}_{{NN(2,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{13(1,1)}_{{NN(2,1);}\atop{(1,00)}}$&0&$\bar{Z}^{13(1,1)}_{{NN(2,2);}\atop{(1,00)}}$\end{tabular}\right)$
(31)
$\bar{Z}^{21}_{NN}=\left(\begin{tabular}[]{cccc}$\bar{Z}^{21(1,1)}_{{NN(1,1);}\atop{(0,11)}}$&0&$\bar{Z}^{21(1,1)}_{{NN(1,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{21(1,1)}_{{NN(1,1);}\atop{(1,00)}}$&0&$\bar{Z}^{21(1,1)}_{{NN(1,2);}\atop{(1,00)}}$\\\
$\bar{Z}^{21(1,1)}_{{NN(2,1);}\atop{(0,11)}}$&0&$\bar{Z}^{21(1,1)}_{{NN(2,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{21(1,1)}_{{NN(2,1);}\atop{(1,00)}}$&0&$\bar{Z}^{21(1,1)}_{{NN(2,2);}\atop{(1,00)}}$\end{tabular}\right)$
(32)
$\bar{Z}^{23}_{NN}=\left(\begin{tabular}[]{cccc}$\bar{Z}^{23(1,1)}_{{NN(1,1);}\atop{(0,11)}}$&0&$\bar{Z}^{23(1,1)}_{{NN(1,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{23(1,1)}_{{NN(1,1);}\atop{(1,00)}}$&0&$\bar{Z}^{23(1,1)}_{{NN(1,2);}\atop{(1,00)}}$\\\
$\bar{Z}^{23(1,1)}_{{NN(2,1);}\atop{(0,11)}}$&0&$\bar{Z}^{23(1,1)}_{{NN(2,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{23(1,1)}_{{NN(2,1);}\atop{(1,00)}}$&0&$\bar{Z}^{23(1,1)}_{{NN(2,2);}\atop{(1,00)}}$\\\
\end{tabular}\right)$ (33)
$\bar{Z}^{31}_{NN}=\left(\begin{tabular}[]{cccc}$\bar{Z}^{31(1,1)}_{{NN(1,1);}\atop{(0,11)}}$&0&$\bar{Z}^{31(1,1)}_{{NN(1,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{31(1,1)}_{{NN(1,1);}\atop{(1,00)}}$&0&$\bar{Z}^{31(1,1)}_{{NN(1,2);}\atop{(1,00)}}$\\\
$\bar{Z}^{31(1,1)}_{{NN(2,1);}\atop{(0,11)}}$&0&$\bar{Z}^{31(1,1)}_{{NN(2,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{31(1,1)}_{{NN(2,1);}\atop{(1,00)}}$&0&$\bar{Z}^{31(1,1)}_{{NN(2,2);}\atop{(1,00)}}$\end{tabular}\right)$
(34)
$\bar{Z}^{32}_{NN}=\left(\begin{tabular}[]{cccc}$\bar{Z}^{32(1,1)}_{{NN(1,1);}\atop{(0,11)}}$&0&$\bar{Z}^{32(1,1)}_{{NN(1,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{32(1,1)}_{{NN(1,1);}\atop{(1,00)}}$&0&$\bar{Z}^{32(1,1)}_{{NN(1,2);}\atop{(1,00)}}$\\\
$\bar{Z}^{32(1,1)}_{{NN(2,1);}\atop{(0,11)}}$&0&$\bar{Z}^{32(1,1)}_{{NN(2,2);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{32(1,1)}_{{NN(2,1);}\atop{(1,00)}}$&0&$\bar{Z}^{32(1,1)}_{{NN(2,2);}\atop{(1,00)}}$\end{tabular}\right)$
(35)

and three elements with $\alpha=\bar{K}N$:

$\bar{Z}^{22}_{\bar{K}N}=\left(\begin{tabular}[]{cccc}$\bar{Z}^{22(1,1)}_{{\bar{K}N(1,1);}\atop{(0,00)}}$&0&$\bar{Z}^{22(1,1)}_{{\bar{K}N(1,1);}\atop{(0,01)}}$&0\\\
0&$\bar{Z}^{22(1,1)}_{{\bar{K}N(1,1);}\atop{(1,00)}}$&0&$\bar{Z}^{22(1,1)}_{{\bar{K}N(1,1);}\atop{(1,01)}}$\\\
$\bar{Z}^{22(1,1)}_{{\bar{K}N(1,1);}\atop{(0,10)}}$&0&$\bar{Z}^{22(1,1)}_{{\bar{K}N(1,1);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{22(1,1)}_{{\bar{K}N(1,1);}\atop{(1,10)}}$&0&$\bar{Z}^{22(1,1)}_{{\bar{K}N(1,1);}\atop{(1,11)}}$\end{tabular}\right)$
(36)
$\bar{Z}^{23}_{\bar{K}N}=\left(\begin{tabular}[]{cc}$\bar{Z}^{23(1,1)}_{{\bar{K}N(1,1);}\atop{(0,01)}}$&0\\\
0&$\bar{Z}^{23(1,1)}_{{\bar{K}N(1,1);}\atop{(1,00)}}$\\\
$\bar{Z}^{23(1,1)}_{{\bar{K}N(1,1);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{23(1,1)}_{{\bar{K}N(1,1);}\atop{(1,10)}}$\end{tabular}\right)$
(37)
$\bar{Z}^{32}_{\bar{K}N}=\left(\begin{tabular}[]{cccc}$\bar{Z}^{32(1,1)}_{{\bar{K}N(1,1);}\atop{(0,10)}}$&0&$\bar{Z}^{32(1,1)}_{{\bar{K}N(1,1);}\atop{(0,11)}}$&0\\\
0&$\bar{Z}^{32(1,1)}_{{\bar{K}N(1,1);}\atop{(1,00)}}$&0&$\bar{Z}^{32(1,1)}_{{\bar{K}N(1,1);}\atop{(1,01)}}$\end{tabular}\right)$
(38)

Elements of $\bar{\tau}^{\rho}_{\alpha\beta}$ matrix Eq.(26) are parts of the
$NNN$ subsystem ($\rho=1$):

$\bar{\tau}^{1}_{NN,NN}=\left(\begin{tabular}[]{cccc}$\bar{\tau}^{1(1)}_{{NN,NN(1,1);}\atop{(00,11)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(1,1);}\atop{(01,10)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(1,2);}\atop{(00,11)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(1,2);}\atop{(01,10)}}$\\\
$\bar{\tau}^{1(1)}_{{NN,NN(1,1);}\atop{(10,01)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(1,1);}\atop{(11,00)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(1,2);}\atop{(10,01)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(1,2);}\atop{(11,00)}}$\\\
$\bar{\tau}^{1(1)}_{{NN,NN(2,1);}\atop{(00,11)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(2,1);}\atop{(01,10)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(2,2);}\atop{(00,11)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(2,2);}\atop{(01,10)}}$\\\
$\bar{\tau}^{1(1)}_{{NN,NN(2,1);}\atop{(10,01)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(2,1);}\atop{(11,00)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(2,2);}\atop{(10,01)}}$&$\bar{\tau}^{1(1)}_{{NN,NN(2,2);}\atop{(11,00)}}$\end{tabular}\right)$
(39)

of the $\bar{K}NN$ subsystem ($\rho=2$):

$\bar{\tau}^{2}_{NN,NN}=\left(\begin{tabular}[]{cccc}$\bar{\tau}^{2(1)}_{{NN,NN(1,1);}\atop{(00,11)}}$&0&$\bar{\tau}^{2(1)}_{{NN,NN(1,2);}\atop{(00,11)}}$&0\\\
0&$\bar{\tau}^{2(1)}_{{NN,NN(1,1);}\atop{(11,00)}}$&0&$\bar{\tau}^{2(1)}_{{NN,NN(1,2);}\atop{(11,00)}}$\\\
$\bar{\tau}^{2(1)}_{{NN,NN(2,1);}\atop{(00,11)}}$&0&$\bar{\tau}^{2(1)}_{{NN,NN(2,2);}\atop{(00,11)}}$&0\\\
0&$\bar{\tau}^{2(1)}_{{NN,NN(2,1);}\atop{(11,00)}}$&0&$\bar{\tau}^{2(1)}_{{NN,NN(2,2);}\atop{(11,00)}}$\end{tabular}\right)$
(40)
$\bar{\tau}^{2}_{NN,\bar{K}N}=\left(\begin{tabular}[]{cccc}0&0&$\bar{\tau}^{2(1)}_{{NN,\bar{K}N(1,1);}\atop{(00,11)}}$&$\bar{\tau}^{2(1)}_{{NN,\bar{K}N(1,1);}\atop{(01,11)}}$\\\
$\bar{\tau}^{2(1)}_{{NN,\bar{K}N(1,1);}\atop{(10,00)}}$&$\bar{\tau}^{2(1)}_{{NN,\bar{K}N(1,1);}\atop{(11,00)}}$&0&0\\\
0&0&$\bar{\tau}^{2(1)}_{{NN,\bar{K}N(2,1);}\atop{(00,11)}}$&$\bar{\tau}^{2(1)}_{{NN,\bar{K}N(2,1);}\atop{(01,11)}}$\\\
$\bar{\tau}^{2(1)}_{{NN,\bar{K}N(2,1);}\atop{(10,00)}}$&$\bar{\tau}^{2(1)}_{{NN,\bar{K}N(2,1);}\atop{(11,00)}}$&0&0\end{tabular}\right)$
(41)
$\bar{\tau}^{2}_{\bar{K}N,NN}=\left(\begin{tabular}[]{cccc}0&$\bar{\tau}^{2(1)}_{{\bar{K}N,NN(1,1);}\atop{(01,00)}}$&0&$\bar{\tau}^{2(1)}_{{\bar{K}N,NN(1,2);}\atop{(01,00)}}$\\\
0&$\bar{\tau}^{2(1)}_{{\bar{K}N,NN(1,1);}\atop{(11,00)}}$&0&$\bar{\tau}^{2(1)}_{{\bar{K}N,NN(1,2);}\atop{(11,00)}}$\\\
$\bar{\tau}^{2(1)}_{{\bar{K}N,NN(1,1);}\atop{(00,11)}}$&0&$\bar{\tau}^{2(1)}_{{\bar{K}N,NN(1,2);}\atop{(00,11)}}$&0\\\
$\bar{\tau}^{2(1)}_{{\bar{K}N,NN(1,1);}\atop{(10,11)}}$&0&$\bar{\tau}^{2(1)}_{{\bar{K}N,NN(1,2);}\atop{(10,11)}}$&0\end{tabular}\right)$
(42)
$\bar{\tau}^{2}_{\bar{K}N,\bar{K}N}=\left(\begin{tabular}[]{cccc}$\bar{\tau}^{2(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(00,00)}}$&$\bar{\tau}^{2(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(01,00)}}$&0&0\\\
$\bar{\tau}^{2(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(10,00)}}$&$\bar{\tau}^{2(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(11,00)}}$&0&0\\\
0&0&$\bar{\tau}^{2(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(00,11)}}$&$\bar{\tau}^{2(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(01,11)}}$\\\
0&0&$\bar{\tau}^{2(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(10,11)}}$&$\bar{\tau}^{2(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(11,11)}}$\end{tabular}\right)$
(43)

and of the $\bar{K}N+NN$ partition ($\rho=3$):

$\bar{\tau}^{3}_{NN,NN}=\left(\begin{tabular}[]{cccc}$\bar{\tau}^{3(1)}_{{NN,NN(1,1);}\atop{(00,11)}}$&0&$\bar{\tau}^{3(1)}_{{NN,NN(1,2);}\atop{(00,11)}}$&0\\\
0&$\bar{\tau}^{3(1)}_{{NN,NN(1,1);}\atop{(11,00)}}$&0&$\bar{\tau}^{3(1)}_{{NN,NN(1,2);}\atop{(11,00)}}$\\\
$\bar{\tau}^{3(1)}_{{NN,NN(2,1);}\atop{(00,11)}}$&0&$\bar{\tau}^{3(1)}_{{NN,NN(2,2);}\atop{(00,11)}}$&0\\\
0&$\bar{\tau}^{3(1)}_{{NN,NN(2,1);}\atop{(11,00)}}$&0&$\bar{\tau}^{3(1)}_{{NN,NN(2,2);}\atop{(11,00)}}$\end{tabular}\right)$
(44)
$\bar{\tau}^{3}_{NN,\bar{K}N}=\left(\begin{tabular}[]{cc}$\bar{\tau}^{3(1)}_{{NN,\bar{K}N(1,1);}\atop{(00,11)}}$&0\\\
0&$\bar{\tau}^{3(1)}_{{NN,\bar{K}N(1,1);}\atop{(11,00)}}$\\\
$\bar{\tau}^{3(1)}_{{NN,\bar{K}N(2,1);}\atop{(00,11)}}$&0\\\
0&$\bar{\tau}^{3(1)}_{{NN,\bar{K}N(2,1);}\atop{(11,00)}}$\end{tabular}\right)$
(45)
$\bar{\tau}^{3}_{\bar{K}N,NN}=\left(\begin{tabular}[]{cccc}$\bar{\tau}^{3(1)}_{{\bar{K}N,NN(1,1);}\atop{(00,11)}}$&0&$\bar{\tau}^{3(1)}_{{\bar{K}N,NN(1,2);}\atop{(00,11)}}$&0\\\
0&$\bar{\tau}^{3(1)}_{{\bar{K}N,NN(1,1);}\atop{(11,00)}}$&0&$\bar{\tau}^{3(1)}_{{\bar{K}N,NN(1,2);}\atop{(11,00)}}$\end{tabular}\right)$
(46)
$\bar{\tau}^{3}_{\bar{K}N,\bar{K}N}=\left(\begin{tabular}[]{cc}$\bar{\tau}^{3(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(00,11)}}$&0\\\
0&$\bar{\tau}^{3(1)}_{{\bar{K}N,\bar{K}N(1,1);}\atop{(11,00)}}$\end{tabular}\right)$
(47)

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# Hidden Interplay of Current-Induced Spin and Orbital Torques in Bulk
Fe3GeTe2

Tom G. Saunderson<EMAIL_ADDRESS>Institute of Physics, Johannes
Gutenberg-University, 55099 Mainz, Germany  Peter Grünberg Institut and
Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425
Jülich, Germany  Dongwook Go  Institute of Physics, Johannes Gutenberg-
University, 55099 Mainz, Germany  Peter Grünberg Institut and Institute for
Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany
Stefan Blügel  Peter Grünberg Institut and Institute for Advanced Simulation,
Forschungszentrum Jülich and JARA, 52425 Jülich, Germany  Mathias Kläui
Institute of Physics, Johannes Gutenberg-University, 55099 Mainz, Germany
Centre for Quantum Spintronics, Department of Physics, Norwegian University of
Science and Technology, 7491 Trondheim, Norway  Yuriy Mokrousov  Institute of
Physics, Johannes Gutenberg-University, 55099 Mainz, Germany  Peter Grünberg
Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and
JARA, 52425 Jülich, Germany

###### Abstract

Low crystal symmetry of magnetic van der Waals materials naturally promotes
spin-orbital complexity unachievable in common magnetic materials used for
spin-orbit torque switching. Here, using first-principles methods, we
demonstrate that an interplay of spin and orbital degrees of freedom has a
profound impact on spin-orbit torques in the prototypical van der Waals
ferromagnet: Fe3GeTe2. While we show that bulk Fe3GeTe2 hosts strong “hidden”
current-induced torques harvested by each of its layers, we uncover that their
origin alternates between the conventional spin flux torque and the so-called
orbital torque as the magnetization direction is varied. A drastic difference
in the behavior of the two types of torques results in a non-trivial evolution
of switching properties with doping. Our findings promote the design of non-
equilibrium orbital properties as the guiding mechanism for crafting the
properties of spin-orbit torques in layered van der Waals materials.

The discovery of two-dimensional (2D) van der Waals (vdW) ferromagnets
Cr2Ge2Te6 Gong  _et al._ (2017) and CrI3 Huang  _et al._ (2017) has been long-
awaited since the works on how spatial dimensionality affects criticality
Griffiths (1964); Mermin and Wagner (1966). Now, the potential applications of
these discoveries seem vast within the field of spintronics Hu and Xiang
(2020). A 2D layered material could enable the more efficient design of novel
spintronic devices than their metallic bilayer counterparts with similar
symmetries Liu  _et al._ (2021). The recent discovery of Fe3GeTe2 (FGT)
Deiseroth  _et al._ (2006) stands as a significant milestone since its Curie
temperature could be raised to room temperature with ionic liquid gating Deng
_et al._ (2018). FGT has since been researched extensively, showing exciting
characteristics such as nodal line semimetallicity Kim  _et al._ (2018),
skyrmionic spin textures Ding  _et al._ (2020a); Park  _et al._ (2021) and
controllable spin currents Zhou and Charlier (2021). In particular,
experiments have demonstrated magnetization switching in Pt/FGT
heterostructures via current-induced spin-orbit torque Alghamdi  _et al._
(2019); Zhang  _et al._ (2021a).

We note that the magnetization switching demonstrated in Refs. Alghamdi  _et
al._ (2019); Zhang  _et al._ (2021a) utilized the spin-Hall effect in Pt,
where the physical principle is analogous to conventional SOT devices with
transition-metal bilayers. A unique role played by the low crystal symmetry of
FGT was pointed out by Johansen et al. Johansen  _et al._ (2019), which allows
for the generation of a current-induced torque without the need for a heavy
metal interface. Experiments have now established the existence of such an
unconventional torque Zhang  _et al._ (2021b); Martin  _et al._ (2021).
However, these experiments were performed on bulk-like thick samples $\gtrsim
20\ \mathrm{nm}$ of FGT. A key aspect of bulk FGT is the global inversion
symmetry that prevents net torque response. However two vdW layers in the unit
cell of bulk FGT, referred to as A and B layers, are inversion partners with
respect to each other, see Fig. 1. Consequently despite total torque
vanishing, individual layers host local “staggered” torques. We note that a
similar situation is encountered in other centrosymmetric systems, for which
the concept of the “hidden” Rashba effect has been developed Yuan  _et al._
(2019); Atkinson (2020); Lee and Kwon (2020).

Although the symmetry dictates that current-induced torque in bulk FGT is
similar to multiple copies of the result for monolayer FGT Zhang  _et al._
(2021b), the microscopic mechanism in bulk FGT is expected to be different
from that of the monolayer. For instance, while current-induced torque in
monolayer FGT will consist of both spin and orbital accumulations, in bulk
FGT, spin and orbital currents exchanged between layers are also expected to
contribute. In particular, recent theories predict that the orbital current
can be generated much more efficiently than the spin current, e.g. via orbital
Hall effect, not only in transition metals Kontani  _et al._ (2008, 2009); Go
_et al._ (2018); Jo  _et al._ (2018); Salemi and Oppeneer (2022a, b) but also
in 2D materials Canonico  _et al._ (2020a, b); Bhowal and Satpathy (2020,
2020); Cysne  _et al._ (2021); Bhowal and Vignale (2021). Moreover, the
orbital current can exert torque on magnetic moments by transferring orbital
angular momentum Go and Lee (2020); Go  _et al._ (2020). Known as the orbital
torque, it provides a promising alternative route to control the magnetization
through the orbital degree of freedom, and is considered a fresh perspective
Go  _et al._ (2021). So far, the orbital torque has been experimentally
observed mostly in transition-metal-based magnetic heterostructures Ding  _et
al._ (2020b); Kim  _et al._ (2021); Lee  _et al._ (2021a, b); Ding  _et al._
(2022); Liao  _et al._ (2022); Hu  _et al._ (2022); Hayashi  _et al._ (2022);
Santos  _et al._ (2022). However, owing to a highly anisotropic crystal field
potential, strong spin-orbit coupling (SOC), and orbital complexity of the
electronic structure, bulk FGT is expected to exhibit rich entangled dynamics
between spin and orbital degrees of freedom. This motivates our search for the
role of orbital excitations and its interaction with magnetic moments in bulk
FGT, which will enhance the currently underdeveloped understanding of orbital
physics in vdW materials.

In this work, by employing first principles methods, we unveil interplay of
the spin and orbital angular momentum in bulk FGT, which is “hidden” by the
global inversion symmetry. When the magnetization is pointing out of the vdW
plane, we find spin flux exchanged between vdW layers is the major
contribution to the torque on the magnetization, Fig. 1(a). In contrast, as
the magnetization direction gets closer to the vdW plane, current-induced
orbital accumulation becomes strongly pronounced, which leads to a large
orbital torque response, Fig. 1(b). This is the first example of an intrinsic
crossover between spin and orbital torques, making bulk FGT an ideal candidate
for studying non-equilibrium orbital excitations.

Figure 1: A schematic illustration for current-induced torques in Fe3GeTe2
(FGT) with (a) an out-of-plane and (b) in-plane magnetization. In both
instances the current direction is along the $x$-axis. The crystal structure
of Fe3GeTe2, consisting of two individual layers, A and B, is displayed on the
left. The direction of the magnetization at angle $\theta$ with the $z$-axis
is marked with $\mathbf{M}_{\mathrm{A}}$ and $\mathbf{M}_{\mathrm{B}}$ for
each layer. The current-induced torques on each layer are opposite, so that
the overall torque is vanishing. While for out-of-plane FGT, the magnetic
torque is mainly driven by the flow of the spin angular momentum between
layers [marked with arrows in (a)], the magnetic torque is dominated by
prominant orbital accumulation for in-plane FGT [marked with arrows in (b)].
Figure 2: Angular dependence of current-induced torque and the spin and
orbital contributions in FGT. (a) The $\theta$-component of the anti-damping
current-induced torque on the magnetization ($T^{\theta}_{\mathbf{M}}$, black
triangles), the spin contribution ($\varPhi^{\theta}$, red dashed square), and
the orbital contribution ($T^{\theta}_{SO}$, blue dashed circle) for single-
layer FGT as a function of $\theta$. (b) Same as in (a) for the torkance
projected onto the A-layer in bulk FGT. (c) The $y$-component of the current-
induced orbital ($L_{y}$, blue dashed circle) and spin ($S_{y}$, red dashed
square) moment summed over Fe atoms in A-layer of bulk FGT as a function of
the angle $\theta$.  Figure 3: Properties of the current-induced torque and
orbital angular momentum of the A-layer of Fe3GeTe2 for the magnetization
angle of $\theta=30^{\circ}$ [panels (a, c, e, g)] and $\theta=90^{\circ}$
[panels (b,d,f,h)]. In panels (a,b) the plotted band structure is superimposed
with the coloured circles whose color and size represents the expectation
value of the total torkance $T^{\theta}_{\mathbf{M},n\mathbf{k}}$. Similarly,
in (c,d) it is the $y$-component of the current-induced orbital angular
momentum $L_{y,n\mathbf{k}}$ which is represented with the circles.
Additionally, the $\mathbf{k}$-space distribution of $L_{y,\mathbf{k}}$ for
the two angles is shown in (g,h). Panels (e,f) display the total integrated
($T^{\theta}$, solid line), flux ($\varPhi^{\theta}$, red dashed line) and
orbital ($T_{\rm SO}^{\theta}$, blue dashed line) torkance as a function of
band filling for $\theta=30^{\circ}$ (e) and $\theta=90^{\circ}$ (f), where
$E=0$ represents the true Fermi energy of FGT. The prominent current-induced
orbital angular momentum and orbital torque for the in-plane magnetization is
evident.

To describe the current-induced exchange of spin and orbital angular momentum
between A and B layers, which results in torques on the magnetization, we
adopt the spin continuity equation Go  _et al._ (2020):

$\displaystyle\frac{d\mathbf{S}_{\mathrm{A}}}{dt}=\bm{\varPhi}_{\mathrm{A}}+\bm{\mathcal{T}}_{\mathrm{SO,A}}-\bm{\mathcal{T}}_{\mathbf{M},\mathrm{A}},$
(1)

where the subscript stands for the vdW layer A. Here,
$\bm{\varPhi}_{\mathrm{A}}$ is the spin flux from layer B into layer A,
$\bm{\mathcal{T}}_{\mathrm{SO,A}}\sim\mathbf{L}_{\mathrm{A}}\times\mathbf{S}_{\mathrm{A}}$
describes mutual precession between the spin ($\mathbf{S}_{\mathrm{A}}$) and
orbital ($\mathbf{L}_{\mathrm{A}}$) angular momenta at layer A via SOC, and
$\bm{\mathcal{T}}_{\mathbf{M},\mathrm{A}}\sim\hat{\mathbf{M}}_{\rm
A}\times\mathbf{S}_{\mathrm{A}}$ is the torque that the spin
$\mathbf{S}_{\mathrm{A}}$ exerts on A-layer magnetization $\mathbf{M}_{\rm A}$
through the exchange interaction. We remark that the spin flux and local
torques on layers A and B have the same magnitude but are opposite in sign by
inversion symmetry. Thus, we analyze the dynamics of the A-layer only and
remove the subscript A in the following. In a steady state, when
${d\mathbf{S}_{\mathrm{A}}}/{dt}=0$, current-induced torque on the
magnetization is given by two contributions: the spin flux $\bm{\varPhi}$, and
the angular momentum transfer from orbital to spin
$\bm{\mathcal{T}}_{\mathrm{SO}}$. The latter originates from the orbital
accumulation, hence we denote these as spin and orbital contributions to the
magnetic torque respectively. The magnetic torque in 2D systems is often
described by non-equilibrium spin accumulation, but we emphasize that its
origin is the non-equilibrium orbital angular momentum generation.

We assess the two contributions to the torque using a first-principles
description of the system’s electronic structure, and Kubo formalism for
electic field response $\mathbf{E}$, providing the details of our calculations
in Supplemental Material Sup . We use the code FLEUR fle , which implements
full-potential linearized augmented plane wave method Wimmer  _et al._ (1981),
in combination with Wannier interpolation Freimuth  _et al._ (2008); Pizzi
_et al._ (2020) to efficiently compute the anti-damping part of the torkance
tensor $\hat{T}$ defined as $\bm{\mathcal{T}}=\hat{T}\,\mathbf{E}$. When
represented in spherical coordinates we are interested only in the polar
$\theta$-component of the torkance tensor, where $\theta$ is the angle that
the magnetization makes with the $z$-axis, see Fig. 1. Correspondingly, we
assume simplified notations of $T^{\theta}_{\mathbf{M}}$, $\varPhi^{\theta}$
and $T_{\rm SO}^{\theta}$ for the $\theta$-component of the torkance tensor,
and its spin-flux and orbital parts. When studying the angular dependence of
the torque, we consider only the case when the magnetization is tilted away
from the $z$-axis into the $x$-axis while being kept in the $xz$-plane. We
apply the external electric field along $x$.

We first compute the magnitude of the total torkance and its decomposition in
a single A-layer of FGT as a function of angle $\theta$, shown in Fig. 2(a).
As predicted by symmetry Johansen  _et al._ (2019), single-layer FGT hosts a
non-vanishing SOT for $\theta\neq 0$. Interestingly, we find a sign change at
$\theta\approx 60^{\circ}$ with the largest torkance when $\mathbf{M}$ is in-
plane, Fig. 2(a). Markedly, the single-layer SOT is entirely orbital in origin
as we find a complete lack of flux torkance for all magnetization angles. For
the A-layer in bulk, Fig. 2(b), the angular dependence has a similar
functional form to single layer FGT, with an overall suppression in magnitude.
What stands in stark contrast to single-layer FGT is that the spin
contribution, $\varPhi^{\theta}$, constitutes a major component to the torque.
This suggests that the bulk spin contribution may originate from the adjacent
single-layer orbital contribution. However, as $\theta$ gets closer to
$90^{\circ}$, the spin contribution $\varPhi^{\theta}$ is strongly suppressed
and the orbital contribution $T_{\mathrm{SO}}^{\theta}$ becomes dominant. This
results in a peculiar angular dependence of the magnetic torque, which is
significantly different from the prediction based on the leading order
symmetry expansion Johansen  _et al._ (2019); Kurebayashi  _et al._ (2022). We
also remark that bulk FGT is the first example of a material exhibiting
orbital-spin crossover of the magnetic torque as a function of magnetization
angle. This can be attributed to the 2D nature of the crystal structure, which
results in highly anisotropic electronic structure.

In order to understand the origins of spin and orbital contributions to the
magnetic torque, we calculate non-equilibrium spin and orbital accumulation
induced by an external electric field, shown in Fig. 2(c) for the
$y$-component which is relevant for $T_{\mathbf{M}}^{\theta}$. For small
values of $\theta$, both $S_{y}$ and $L_{y}$ are proportional to $M_{x}$,
which is consistent with symmetry analysis in the lowest order of $\mathbf{M}$
Johansen  _et al._ (2019). However, as $\theta$ increases, higher order
contributions become more pronounced. Interestingly, $S_{y}$ and $L_{y}$
exhibit a qualitatively different angular dependence: $S_{y}$ changes sign
when $\theta$ is between $30^{\circ}$ and $45^{\circ}$, while $L_{y}$
monotonically increases as $\theta$ increases. Moreover, $L_{y}$ is an order
of magnitude larger than $S_{y}$ over a wide range of $\theta$. This strongly
supports the idea that orbital accumulation significantly contributes to the
magnetic torque, especially for large values of $\theta$. To understand why
for smaller angles the orbital accumulation does not translate into a large
orbital torque, we proceed to investigate the state-resolved properties to
shed light on the microscopics of the crossover behaviour.

We analyze band-resolved contributions to $T^{\theta}_{\mathbf{M}}$ for
$\theta=30^{\circ}$ [Fig. 3(a)] and $90^{\circ}$ [Fig. 3(b)]. These two angles
are chosen such that they exhibit the strongest $\varPhi^{\theta}$ and
$T_{\mathrm{SO}}^{{\theta}}$, respectively. As expected, the distributions of
$T^{\theta}_{\mathbf{M}}$ for $\theta=30^{\circ}$ and $\theta=90^{\circ}$ are
qualitatively different. In particular, we find that the torque is spread over
wider regions of phase-space for $\theta=30^{\circ}$, while for
$\theta=90^{\circ}$ the torque originates from isolated contributions,
especially for energies above the Fermi level. To characterize the role of
orbital accumulation we plot the state-resolved $L_{y,n\mathbf{k}}$ at
$\theta=30^{\circ}$ [Fig. 3(c)] and $\theta=90^{\circ}$ [Fig. 3(d)]. By
comparing Figs. 3(a) and 3(c), we immediately notice that the isolated
hotspot-like regions for $T^{\theta}_{\mathbf{M}}$ and $L_{y}$ at
$\theta=30^{\circ}$ are either not in the same position or have a different
sign, which is especially visible around the Fermi energy. This is expected
from Fig. 2(b) because the magnetic torque is dominated by the spin flux at
$\theta=30^{\circ}$. On the other hand, at $\theta=90^{\circ}$ the correlation
between $T^{\theta}_{\mathbf{M}}$ and $L_{y}$ is much more pronounced [compare
Figs. 3(b) and 3(d)].

To understand the suppression of orbital torque despite prominent current-
induced orbital magnetization of states at small angles, we compare the
$\mathbf{k}$-resolved contribution for $L_{y}$ summed over all occupied states
at $\theta=30^{\circ}$ [Fig. 3(g)] and $\theta=90^{\circ}$ [Fig. 3(h)]. While
the summation over $\mathbf{k}$ leads to an overall orbital moment at
$\theta=30^{\circ}$, the orbital moment at $\theta=90^{\circ}$ is evidently
greater, fitting with Fig. 2(c). Furthermore, there is asymmetry present at
$\theta=30^{\circ}$ that is not at $\theta=90^{\circ}$, implying a crossover
of orbital characters that could drive the orbital torque. The corresponding
asymmetry of the orbital distribution for $\theta=30^{\circ}$ can be also
directly seen in the band structure plot for energies above the Fermi level,
Fig. 3(c). Conversely, the corresponding current induced spin distribution,
presented in Supplemental Material Sup , displays the opposite behavior. These
findings help us understand the intricate energy-dependent interplay between
the spin and orbital components to the magnetic torque as a function of the
tilting angle.

In previous works, the effect of strong correlations using dynamical mean
field theory are investigated Kim  _et al._ (2018). Within the energy range
chosen in our investigation, the only key difference between both methods is
the position of the Fermi level. Hence, we show the Fermi energy dependence of
$T^{\theta}_{\mathbf{M}}$, $\varPhi^{\theta}$, and
$T_{\mathrm{SO}}^{{\theta}}$ at $\theta=30^{\circ}$ and $90^{\circ}$ in Figs.
3(e) and 3(f), respectively. At $\theta=30^{\circ}$, $\varPhi^{\theta}$ is the
main contribution to $T^{\theta}_{\mathbf{M}}$ although
$T_{\mathrm{SO}}^{{\theta}}$ is also non-vanishing [Fig. 3(e)]. On the other
hand, at $\theta=90^{\circ}$, $T^{\theta}_{\mathbf{M}}$ is mainly governed by
$T_{\mathrm{SO}}^{{\theta}}$ and $\varPhi^{\theta}$ is suppressed over a wide
energy range [Fig. 3(f)]. In Figs. 3(a-d), we notice that the hotspots are
mainly concentrated around the nodal lines near $\Gamma$ at $+0.2\
\mathrm{eV}$ Kim  _et al._ (2018). This implies that the torkance can be
efficiently tuned with band filling. Indeed, we find a substantial increase of
$T^{\theta}_{\mathbf{M}}$ as the Fermi energy is raised by $\sim 0.2\
\mathrm{eV}$ at both $\theta=30^{\circ}$ and $90^{\circ}$. Interestingly, at
$\theta=30^{\circ}$, $T^{\theta}_{\mathbf{M}}$ changes sign from negative to
positive as the Fermi energy is increased by $\approx 0.2\ \mathrm{eV}$. On
the other hand, at $\theta=90^{\circ}$, the sign of $T^{\theta}_{\mathbf{M}}$
remains positive. This suggests a possibility of tuning the sign and magnitude
of the torkance by doping. Such complex anisotropic torkances can be harnessed
to drive the non-trivial excitations of spin textures that FGT hosts Hanke
_et al._ (2020). According to our estimate, raising the energy by $0.2\
\mathrm{eV}$ requires $\approx 1.5$ additional electrons per layer. This may
be achieved e.g. by substitutional doping of Fe by Co.

Let us comment lastly on the relevance of our predictions for the spin-orbit
torque measurements on realistic FGT samples. While the single-layer SOT in
bulk samples can be promoted if global symmetry is broken e.g. by surfaces and
interfaces Martin  _et al._ (2021); Zhang  _et al._ (2021b), inversion
symmetry present in ideal bulk FGT ultimately suppresses the overall torque on
the magnetization. However, the reality is that there are a variety of
“intrinsic” mechanisms which can drive the inversion symmetry breaking in this
bulk system Narita and Fox (2018); Martin  _et al._ (2021); Chakraborty  _et
al._ (2022). One of these effects is that of magnetostriction Narita and Fox
(2018). Here, due to the coupling of the lattice to the magnetic moment, the
rotation of the moment can initiate a crystal structure change, breaking the
inversion symmetry of the perfect crystal. Similarly, the effects of the
electric field on the system could give rise to high frequency magnon modes
capable of canting the spins and driving a torque even in the pristine limit.
It is also plausible to assume that due to the 2D vdW nature of the material
it is possible for the layers to slip over one another. In some instances, if
multiple pairs of layers have displaced in different ways, the inversion
symmetry could be broken in this fashion. Moreover, recent experimental work
reports a discrepancy in the occupancy of the central Fe atom in the A layer
versus the B layer Chakraborty  _et al._ (2022). This implies that the unit
cell is no longer inversion symmetric providing an avenue for bulk torque
generation. Our results can be in principle readily applied to this situation:
Due to the inter-layer occupancy difference being small, from the theoretical
perspective the corresponding torque should be simply given by the difference
in the torques of layers A and B from Fig. 3(e,f) for corresponding
occupancies, and the corresponding effective spin-orbit field can be found in
a similar fashion Johansen  _et al._ (2019).

Finally, the determination of the components of the spin and orbital torque
remain a significant challenge. For example, standard techniques for measuring
the current-induced torques only measure the total torkance, but this can be
overcome by attempting to find an ‘even’-‘odd’ effect such as in MoS2 Cysne
_et al._ (2021) by consecutively removing vdW layers. To probe the orbital
accumulation, a key signature of orbital-driven torkances, one can use optical
probes such as magneto-optical Kerr effect Choi  _et al._ (2021) or x-ray
magnetic circular dichroism Stamm  _et al._ (2019).

To summarize, in our work, we uncovered that in magnetic vdW materials the
microscopics of spin-orbit torques can be extremely rich owing to the
pronounced 2D nature of constituting layers and their orbital complexity. As
we show for FGT, this ultimately results in profoundly anisotropic torque
properties, which may be mediated by an exotic orbital-spin crossover. Given
the fundamentally different properties of spin and orbital angular momentum
out of equilibrium, this suggests exciting possibilities in exploring spin-
orbital dynamics of 2D magnets and corresponding transport manifestations. We
further argue that the detailed knowledge of “hidden” spin-orbit torques in
bulk vdW materials can provide a key to designing their current response via
educated symmetry breaking, which may prove pivotal for the integration of 2D
magnets into the spintronic device setting.

###### Acknowledgements.

The authors appreciate fruitful discussions with Prof. Arne Brataas, Dr.
Martin Gradhand, Maximilian Merte, Fabian Lux and Dr. Sumit Ghosh. We
gratefully acknowledge the Jülich Supercomputing Centre for providing
computational resources under project jiff40. This work was funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) $-$ Grants
No. TRR 173/2 $-$ 268565370 Spin+X (projects A01, BO2 and A11), No. TRR 288
$-$ 422213477 (project B06) and CRC 1238 – 277146847 (project C01).

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