# Contributions of $2\pi$-exchange, $1\pi$-exchange, and contact three-body
forces in NNLO ChEFT to ${}_{\Lambda}^{3}$H

M. Kohno  Research Center for Nuclear Physics, Osaka University, Ibaraki
567-0047, Japan  H. Kamada  Department of Physics, Faculty of Engineering,
Kyushu Institute of Technology, Kitakyushu 804-8550, Japan  K. Miyagawa
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0047, Japan

###### Abstract

Faddeev calculations of hypertriton (${}_{\Lambda}^{3}$H) separation energy
are performed, incorporating all next-to-next-to-leading-order $\Lambda$NN
three-body forces (3BFs) in chiral effective field theory: $2\pi$-exchange,
$1\pi$-exchange, and contact interactions. The $1\pi$-exchange and contact
interactions are rewritten in a form suitable for evaluating partial-wave
matrix elements. The $\Lambda$-deuteron folding potentials constructed from
these 3BFs are evaluated to demonstrate their contributions to
${}_{\Lambda}^{3}$H. The $1\pi$-exchange interaction provides an attractive
effect in which the $d$-state component of the deuteron wave function plays an
important role. The attractive contribution tends to cancel the repulsive ones
from the $2\pi$-exchange and contact 3BFs. Faddeev calculations show that the
net effect of the 3BFs to the ${}_{\Lambda}^{3}$H separation energy is small
in a range between $-5$ to $+20$ keV, depending on the NN interaction used.
Although these results are based on speculative low-energy constants, they can
serve as a reference for further investigations.

##  I Introduction

The hypertriton (${}_{\Lambda}^{3}$H) is an isospin $T=0$ $\Lambda$NN bound
state with small separation energy to $\Lambda$ \+ deuteron. Because a
$\Lambda$N two-body bound system is unlikely, ${}_{\Lambda}^{3}$H is valuable
to investigate $\Lambda$-nucleon interaction. For example, its binding energy
helps resolve the spin singlet and triplet strengths of the $\Lambda$N
interaction to compensate for the lack of scattering data. Although the
separation energy has not been accurately established, the situation will
change shortly due to ongoing new experiments. On the theory side, the studies
of the $\Lambda$NN system bear some ambiguities from the effects of possible
three-body forces (3BFs). In ordinary light nuclei, an attractive contribution
of three-nucleon forces is necessary to explain their binding energies
accurately. Likely, we may expect a non-negligible effect of the 3BFs in
${}_{\Lambda}^{3}$H. It is needed to study whether this contribution is
attractive or repulsive and the order of magnitude.

We have studied the effect of the 3BF in chiral effective field theory (ChEFT)
for the hypertriton in Ref. KKM23 . In that article, the $2\pi$-exchange
$\Lambda$NN interaction derived by Petschauer et al. PET16  at the next-to-
next-to-leading- order (NNLO) was addressed as an initial attempt. Faddeev
calculations were carried out for the first time, incorporating the 3BF
matrix-elements, to show that the contribution is repulsive of the order of 20
keV. This size is small but not negligible compared with the small separation
energy of the hypertriton, the present world average of which is $148\pm 40$
keV MA22 .

The remaining NNLO $\Lambda$NN 3BFs, a $1\pi$-exchange 3BF and a contact 3BF,
are considered in this article. Because the expressions of these 3BFs in
momentum space presented in Ref. PET16  are not readily applicable in
evaluating partial-wave matrix elements, we rewrite them in a form appropriate
for use. As for low-energy coupling constants (LECs), we rely on the
estimation by Petschauer et al. PET17  in a decouplet-dominant model. Although
there are uncertainties, it is worth to evaluate the effect of these 3BFs by
using those LECs as reference values for further studies.

Expressions of the $1\pi$-exchange $\Lambda$NN 3BF and the contact $\Lambda$NN
term are rewritten in Sec. 2 and Sec. 3, respectively, to be suitable for
evaluating partial-wave matrix elements. Before explicit Faddeev calculations
are carried out for ${}_{\Lambda}^{3}$H, it is worth obtaining a preliminary
idea about the contributions of the 3BFs by evaluating a $\Lambda$-deuteron
folding potential as was done for the $2\pi$-exchange 3BF in Ref. KKM23 .
Those $\Lambda$-deuteron folding potentials in momentum space are discussed in
Sec. 4. Results of Faddeev calculations of L3H, which include these 3BFs along
with the 2pi-exchange 3BF, are presented in Sec 5. Summary follows in Sec 6.
Summary follows in Sec. 6.

##  II one-pion exchange $\Lambda$NN

The $1\pi$-exchange $\Lambda$NN interaction in momentum space in the NNLO was
shown by Petschauer et al. in Ref. PET16  as follows:

$\displaystyle V_{1\pi}^{\Lambda
NN}=-\frac{g_{A}}{2f_{0}^{2}}(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
$\displaystyle\times\left\\{\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2^{\prime}2}}{\mbox{\boldmath$q$}_{2^{\prime}2}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{3})\cdot\mbox{\boldmath$q$}_{2^{\prime}2}\right.$
$\displaystyle\hskip
8.53581pt+\frac{\mbox{\boldmath$\sigma$}_{3}\cdot\mbox{\boldmath$q$}_{3^{\prime}3}}{\mbox{\boldmath$q$}_{3^{\prime}3}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{3^{\prime}3}$
$\displaystyle+P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{13}^{(\sigma)}\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{3^{\prime}2}}{\mbox{\boldmath$q$}_{3^{\prime}2}^{2}+m_{\pi}^{2}}\left[-\frac{D_{1}^{\prime}+D_{2}^{\prime}}{2}(\mbox{\boldmath$\sigma$}_{1}+\mbox{\boldmath$\sigma$}_{3})\cdot\mbox{\boldmath$q$}_{3^{\prime}2}\right.$
$\displaystyle\left.\hskip
56.9055pt+\frac{D_{1}^{\prime}-D_{2}^{\prime}}{2}i(\mbox{\boldmath$\sigma$}_{3}\times\mbox{\boldmath$\sigma$}_{1})\cdot\mbox{\boldmath$q$}_{3^{\prime}2}\right]$
$\displaystyle+P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{12}^{(\sigma)}\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2^{\prime}3}}{\mbox{\boldmath$q$}_{2^{\prime}3}^{2}+m_{\pi}^{2}}\left[-\frac{D_{1}^{\prime}+D_{2}^{\prime}}{2}(\mbox{\boldmath$\sigma$}_{1}+\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{2^{\prime}3}\right.$
$\displaystyle\left.\left.\hskip
56.9055pt-\frac{D_{1}^{\prime}-D_{2}^{\prime}}{2}i(\mbox{\boldmath$\sigma$}_{1}\times\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{2^{\prime}3}\right]\right\\},$
(1)

in which a label of 1 assigned to the $\Lambda$ hyperon. $g_{A}$ is the axial-
vector coupling constant, $f_{0}$ is the pion decay constant,
$\mbox{\boldmath$q$}_{i^{\prime}j}$ is a momentum transfer at each $\pi$NN
vertex, and $P_{ij}^{(\sigma)}$ ($P_{ij}^{(\tau)}$) is an exchange operator in
spin (isospin) space. Explicit numbers of the low-energy constants (LECs),
$D_{1}^{\prime}$ and $D_{2}^{\prime}$, are set in Se. IV. Noting
$P_{ij}^{\sigma}=\frac{1}{2}(1+\mbox{\boldmath$\sigma$}_{i}\cdot\mbox{\boldmath$\sigma$}_{j})$
and $\sigma_{k}\sigma_{\ell}=\delta_{k\ell}+i\epsilon_{k\ell m}\sigma_{m}$,
the above expression is rewritten as

$\displaystyle V_{1\pi}^{\Lambda NN}=$
$\displaystyle-\frac{g_{A}}{2f_{0}^{2}}\left(1-P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}\right)(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
$\displaystyle\times\left\\{\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2^{\prime}2}}{\mbox{\boldmath$q$}_{2^{\prime}2}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{3})\cdot\mbox{\boldmath$q$}_{2^{\prime}2}\right.$
$\displaystyle\left.\hskip
10.00002pt+\frac{\mbox{\boldmath$\sigma$}_{3}\cdot\mbox{\boldmath$q$}_{3^{\prime}3}}{\mbox{\boldmath$q$}_{3^{\prime}3}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{3^{\prime}3}\right\\},$
(2)

where $P_{23}^{(p)}$ stands for the exchange operator for the momenta of the
nucleon pair, and $P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}$ means an
exchange operator for the two nucleons. The factor
$1-P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}$ can be taken care of by the
anti-symmetrization of the pair of two-nucleon wave functions. In that case,
$P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}$ is unnecessary, and the
$1\pi$-exchange $\Lambda$NN interaction should be considered as follows.

$\displaystyle V_{1\pi}^{\Lambda NN}=$
$\displaystyle-\frac{g_{A}}{2f_{0}^{2}}(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
$\displaystyle\times\left\\{\frac{\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$q$}_{2^{\prime}2}}{\mbox{\boldmath$q$}_{2^{\prime}2}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{3})\cdot\mbox{\boldmath$q$}_{2^{\prime}2}\right.$
$\displaystyle\left.\hskip
10.00002pt+\frac{\mbox{\boldmath$\sigma$}_{3}\cdot\mbox{\boldmath$q$}_{3^{\prime}3}}{\mbox{\boldmath$q$}_{3^{\prime}3}^{2}+m_{\pi}^{2}}(D_{1}^{\prime}\mbox{\boldmath$\sigma$}_{1}+D_{2}^{\prime}\mbox{\boldmath$\sigma$}_{2})\cdot\mbox{\boldmath$q$}_{3^{\prime}3}\right\\}.$
(3)

The first (second) term in the curly brackets of the above equation
corresponds to the left (right) diagram in Fig. 1.

Figure 1: $1\pi$-exchange $\Lambda NN$ 3BF corresponding to Eq. 3. The double
line means a $\Lambda$N contact part smeared by a cutoff factor.

Now, this expression is transformed to the partial-wave expansion form
applicable in evaluating matrix elements using the expression given in Ref.
KKM22 .

$\displaystyle
V_{1\pi}^{\sigma_{2}}=4\pi(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})\sum_{K=0,1,2}\sum_{\ell_{a}\ell_{b}}$
$\displaystyle\left\\{V_{1\pi,(2,3)}^{K,\ell_{a},\ell_{b}}(p,q)[[\mbox{\boldmath$\sigma$}_{2}\times\mbox{\boldmath$\sigma$}_{3}]^{K}\times[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}]^{0}\right.$
$\displaystyle+V_{1\pi,(1,2)}^{K,\ell_{a},\ell_{b}}(p,q)[[\mbox{\boldmath$\sigma$}_{1}\times\mbox{\boldmath$\sigma$}_{2}]^{K}\times[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}]^{0}$
$\displaystyle+\left.V_{1\pi,(3,1)}^{K,\ell_{a},\ell_{b}}(p,q)[[\mbox{\boldmath$\sigma$}_{3}\times\mbox{\boldmath$\sigma$}_{1}]^{K}\times[Y_{\ell_{a}}(\hat{\mbox{\boldmath$p$}})\times
Y_{\ell_{b}}(\hat{\mbox{\boldmath$q$}})]^{K}]^{0}\right\\},$ (4)

where an abbreviated notation for a tensor product with Clebsch-Gordan
coefficients is used:

$\displaystyle[T_{j_{1}}\times
T_{j_{2}}]_{M}^{J}=\sum_{m_{1}m_{2}}(j_{1}m_{1}j_{2}m_{2}|JM)T_{j_{1}m_{1}}T_{j_{2}m_{2}}.$
(5)

$p$ and $q$ are differences of the final and initial Jacobi momenta. That is,
denoting each momentum of the $i$-th initial particle by
$\mbox{\boldmath$k$}_{i}$, Jacobi momenta are defined as
$\mbox{\boldmath$p$}_{1}=\mbox{\boldmath$k$}_{2}-\mbox{\boldmath$k$}_{3}$ and
$\mbox{\boldmath$q$}_{1}=\mbox{\boldmath$k$}_{1}$ in the center-of-mass frame.
Jacobi momenta for the final configuration are represented with a prime. Then,
$\mbox{\boldmath$p$}\equiv\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1}$
and
$\mbox{\boldmath$q$}^{\prime}\equiv\mbox{\boldmath$q$}_{1}^{\prime}-\mbox{\boldmath$q$}_{1}$.
The explicit expression of $V_{1\pi,(i,j)}^{K,\ell_{a},\ell_{b}}(p,q)$ is
given in Appendix A.

##  III contact term

The contact $\Lambda$NN term in the NNLO was shown by Petschauer et al. in
Ref. PET16  as follows:

$\displaystyle V_{ct}^{\Lambda NN}=$ $\displaystyle
C_{1}^{\prime}(1-\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$\sigma$}_{3})(3+\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
$\displaystyle+C_{2}^{\prime}(\mbox{\boldmath$\sigma$}_{\Lambda}\cdot(\mbox{\boldmath$\sigma$}_{2}+\mbox{\boldmath$\sigma$}_{3}))(1-\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})$
$\displaystyle+C_{3}^{\prime}(3+\mbox{\boldmath$\sigma$}_{2}\cdot\mbox{\boldmath$\sigma$}_{3})(1-\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3}),$
(6)

in which a label of 1 is assigned to the $\Lambda$ hyperon. The exchange of
the nucleon pair is explicitly taken care of, and the expression is
antisymmetric under the exchange of two nucleons; that is,
$P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)}V_{ct}^{\Lambda
NN}=-V_{ct}^{\Lambda NN}$. Therefore, $V_{ct}^{\Lambda NN}$ is written as
$\frac{1}{2}(1-P_{23}^{(\sigma)}P_{23}^{(\tau)}P_{23}^{(p)})V_{ct}^{\Lambda
NN}$. It means that when the interaction is applied to the $\Lambda$NN wave
function in which two nucleons are antisymmetrized, factor $\frac{1}{2}$ is
necessary.

The above expression is rewritten in the following partial-wave expansion
form.

$\displaystyle V_{ct}^{\Lambda
NN}=4\pi\sum_{\tau=0,1}\sum_{k,m,n}V_{ct}^{(k,m,n,\tau)}(\mbox{\boldmath$\tau$}_{2}\cdot\mbox{\boldmath$\tau$}_{3})^{\tau}$
$\displaystyle\times[[\mbox{\boldmath$\sigma$}_{\Lambda}^{(k)}\times[\mbox{\boldmath$\sigma$}_{2}^{(m)}\times\mbox{\boldmath$\sigma$}_{3}^{(n)}]^{k}]^{0}\times[Y_{0}(\hat{\mbox{\boldmath$p$}})\times
Y_{0}(\hat{\mbox{\boldmath$q$}})]^{0}]^{0}.$ (7)

The coefficients $V_{ct}^{(k,m,n,\tau)}$ are given in Table I.

$V_{ct}^{(0,0,0,0)}=\frac{1}{4\pi}\frac{3}{2}(C_{1}^{\prime}+C_{3}^{\prime})$ | $V_{ct}^{(0,0,0,1)}=\frac{1}{4\pi}\frac{1}{2}(C_{1}^{\prime}-3C_{3}^{\prime})$
---|---
$V_{ct}^{(0,1,1,0)}=\frac{1}{4\pi}\frac{\sqrt{3}}{2}(3C_{1}^{\prime}-C_{3}^{\prime})$ | $V_{ct}^{(0,1,1,1)}=\frac{1}{4\pi}\frac{\sqrt{3}}{2}(C_{1}^{\prime}+C_{3}^{\prime})$
$V_{ct}^{(1,1,0,0)}=-\frac{1}{4\pi}\frac{\sqrt{3}}{2}C_{2}^{\prime}$ | $V_{ct}^{(1,1,0,1)}=\frac{1}{4\pi}\frac{\sqrt{3}}{2}C_{2}^{\prime}$
$V_{ct}^{(1,0,1,0)}=-\frac{1}{4\pi}\frac{\sqrt{3}}{2}C_{2}^{\prime}$ | $V_{ct}^{(1,0,1,1)}=\frac{1}{4\pi}\frac{\sqrt{3}}{2}C_{2}^{\prime}$
Table 1: Coefficients $V_{ct}^{(k,m,n,\tau)}$ of the NLO contact $\Lambda$NN
interaction written in the form of Eq. (7). The factor of $\frac{1}{2}$
explained in the text is included. Other coefficients not shown are zero.

##  IV $\Lambda$-deuteron folding potential

It is instructive to evaluate the $\Lambda$-deuteron folding potential
provided by the $1\pi$-exchange and the contact 3BFs to demonstrate the
contribution of these 3BFs in the hypertriton. The folding potential is
calculated by the following integration:

$\displaystyle U_{\Lambda-d}^{J_{t}}(q_{1}^{\prime},q_{1})=$
$\displaystyle\iint p_{1}^{\prime
2}dp_{1}^{\prime}p_{1}^{2}dp_{1}\langle[\Psi_{d}(\mbox{\boldmath$p$}_{1}^{\prime}),(\ell_{\Lambda}^{\prime}1/2)j_{\Lambda}]J_{t}|$
$\displaystyle\times V_{1\pi(ct)}^{\Lambda
NN}|[\Psi_{d}(\mbox{\boldmath$p$}_{1}),(\ell_{\Lambda}1/2)j_{\Lambda}]J_{t}\rangle,$
(8) $\displaystyle\Psi_{d}(\mbox{\boldmath$p$}_{1})=$
$\displaystyle\sum_{\ell_{d}=0,2}\frac{1}{p_{1}}\phi_{\ell_{d}}(p_{1})[Y_{\ell_{d}}(\hat{\mbox{\boldmath$p$}_{1}})\times\chi_{d}^{1}]_{m}^{1},$
(9)

where $\Psi_{d}(\mbox{\boldmath$p$})$ represents a deuteron wave function. A
detailed calculational procedure for the 3BF in the form of Eq. 3 or 7 is
given in Appendix B in Ref. KKM22 .

Figure 2 shows the result of the $1\pi$-exchange $\Lambda$NN 3BF given in Eq.
(3) with the total angular momentum $J_{t}=1/2$. The LECs are taken from the
estimation by Petschauer et al. PET17 : $D_{1}^{\prime}=0$ and
$D_{2}^{\prime}=\frac{2CH^{\prime}}{9\Delta}$ with $C\approx\frac{3}{4}g_{A}$,
$H^{\prime}\approx 1/f_{0}^{2}$, and the decuplet-octet baryon mass splitting
$\Delta$. The numerical value of $D_{2}^{\prime}$ is set as
$D_{2}^{\prime}=3.268\times 10^{-3}$ fm2MeV-1 using $g_{A}=1.29$, $f_{0}=92.4$
MeV, and $\Delta=300$ MeV. As discussed in Ref. PET17 , the sign of
$D_{2}^{\prime}$ could be the opposite. The upper panel of Fig. 2 depicts the
contribution of the $s$-wave pair of the bra and ket deuteron wave functions.
The repulsive magnitude is smaller than the corresponding strength of the
$2\pi$-exchange $\Lambda$NN reported in Ref. KKM23 . The result of the $s$-$d$
pair of the bra and ket deuteron wave functions is shown in the lower panel of
Fig. 2, which is attractive, and the magnitude is unexpectedly large despite
the small $d$-wave component of the deuteron wave function. The $d$-$s$ pair
provided the same contribution. The contribution of the $d$-wave pair of the
bra and ket deuteron wave functions is negligible. The net $\Lambda$-deuteron
folding potential, including the contribution of the $2\pi$-exchange reported
in Ref. KKM23 , is found to be attractive and to have a value of about $-0.35$
MeV at the origin.

Figure 2: $\Lambda$-deuteron folding potential provided by the $1\pi$-exchange
$\Lambda$NN 3BF given in Eq. (3). The coupling constants employed are
$D_{1}^{\prime}=0$ and $D_{2}^{\prime}=3.268\times 10^{-3}$ fm2MeV-1 based on
the estimation by Petschauer et al. in Ref. PET17  using a decouplet
saturation model. The deuteron wave functions are calculated using the N4LO+
NN interactions RKE18  in chiral effective field theory.

Figure 3 shows the result of the contact term of the $\Lambda$NN 3BF given in
Eq. (7). The low-energy constants are set as
$C_{1}^{\prime}=C_{3}^{\prime}=\frac{1}{f_{0}^{4}\Delta}$ and
$C_{2}^{\prime}=0$ with $f_{0}=92.4$ MeV and $\Delta=300$ MeV, following the
estimation by Petschauer et al. PET17 . Other contributions from the $s$-$d$,
$d$-$s$, and $d$-$d$ pairs of the bra and ket deuteron wave functions are
negligibly small.

Figure 3: $\Lambda$-deuteron folding potential provided by the contact
$\Lambda$NN 3BF given in Eq. (7). The coupling constants employed are
$C_{1}^{\prime}=C_{3}^{\prime}=\frac{1}{f_{0}^{4}\Delta}$ and
$C_{2}^{\prime}=0$ with $f_{0}=92.4$ MeV and $\Delta=300$ MeV, which are those
estimated by Petschauer et al. in Ref. PET17  using a decouplet saturation
model.

Therefore, the sum of the contributions of the $1\pi$-exchange and contact
$\Lambda$NN 3BFs is attractive. Even if the repulsive contribution of the
$2\pi$-exchange reported in Ref. KKM23  is included, the net contribution of
the three types of the NLO $\Lambda$NN 3BFs with the LECs estimated by
Petschauer et al. PET17  is expected to be attractive at around $-300$ keV.

##  V Faddeev calculations of ${}_{\Lambda}^{3}$H

In this section, we present the results of the explicit Faddeev calculations
of the separation energy of ${}_{\Lambda}^{3}$H that include all NNLO
$\Lambda$NN interactions in chiral effective field theory. The method of
calculation is explained in Ref. KKM23 . Although the LECs employed are
speculative, the calculated results serve as a reference number for possible
3BF contributions in ${}_{\Lambda}^{3}$H.

Uncertainties in the prediction of the ${}_{\Lambda}^{3}$H separation energy
due to the NN interaction employed were discussed in Ref. KKM23 , which is
qualitatively similar to the finding by Gazda et al. GHF22 . In this article,
we show only the results using N4LO+(550) and N4LO+(400) for the NN
interaction, where the number in parentheses is a cutoff scale in MeV. As for
the YN interactions, two versions of the chiral NLO interactions, NLO13 NLO13
and NLO19 NLO19  with a cutoff scale of 550 MeV parametrized by the Jülich-
Bonn group, are employed. The Nijmegen NSC89 potential NSC89  is also applied,
though it may not be appropriate to use with the regularized chiral
interactions at around 500 MeV.

Calculated results for each YN interaction are depicted in Fig. 4. The
leftmost entry is the separation energy without 3BFs. The second entry from
the left is the result including the $2\pi$-exchange $\Lambda$NN, which was
reported already in Ref. KKM23 . The second entry from the right is the result
with the $1\pi$-exchange 3BF and the $2\pi$-exchange $\Lambda$NN. The
$1\pi$-exchange 3BF acts attractively due to the matrix element of the $s$-$d$
pair of the deuteron wave function, as shown by the folding potential in Sec.
4. The rightmost entry shows the result in which all the 3BFs are included.
The net effect after including NNLO 3BFs turns out to be sensitive to the NN
interaction. In the case of N4LO+(550) in which the separation is narrower,
the 3BFs bring about the attraction in the order of 20 keV for
${}_{\Lambda}^{3}$H. On the other hand, the 3BFs tend to work repulsively in
the case of N4LO+(400). It seems that the 3BFs work attractively in the case
that the ${}_{\Lambda}^{3}$H wave function is wide-spreading. With the wave
function shrinking, the net attraction begins to diminish. The trend of the
3BFs contribution is opposite in the NSC89 YN potential, for which the
interpretation is difficult because the NN interactions are defined in a low
momentum with the cutoff scale of around 500 MeV and the NSC potential in an
entire space.

Figure 4: Results of the ${}_{\Lambda}^{3}$H separation energy obtained by
Faddeev calculations. The NN interactions are the semilocal N4LO+
interactionRKE18  with its cutoff scale of 550 and 400 MeV. The lower, middle,
and upper panels show the results for the chiral NLO19, chiral NLO13, and
Nijmegen NSC89 YN interactions, respectively. The cutoff scale of the chiral
YN interactions is 550 MeV. In each figure, the leftmost entry is the
calculation with the NN interactions only. Other entries are results in which
the $2\pi$-exchange, $1\pi$-exchange, and contact 3BFs are added separately.

##  VI Summary

In Ref. KKM23 , we investigated the effect of the $2\pi$-exchange $\Lambda$NN
3BF in ChEFT in the ${}_{\Lambda}^{3}$H hypernuclei by carrying out Faddeev
calculations that include the 3BF for the first time. In the present article,
the remaining $1\pi$-exchange and contact term $\Lambda$NN 3BFs derived by
Petschauer et al. PET17  at the NNLO level are addressed. Because the
expressions of these 3BFs in Ref. PET16  are not readily applicable in
calculating matrix elements of $\Lambda$ hypernuclei, we rewrite them to be
applied in the formula given in Ref. KKM22 . Then, Faddeev calculations for
${}_{\Lambda}^{3}$H are carried out, including NNLO 3BFs.

The net effect of these 3BFs is found to be small because of the cancellation
between the attractive contribution of the $1\pi$-exchange 3BF and the
repulsive ones of the other 3BFs under the tentative assignment for the sign
of the coupling constant $D_{2}^{\prime}$. It is interesting to see that the
difference in the separation energies with N4LO+(550) and N4LO+(400) becomes
smaller when the 3BFs are included for the chiral YN interactions. The
attraction from the $1\pi$-exchange 3BF is responsible for the matrix element
between the $s$\- and $d$-wave components of the NN wave function due to the
tensor part of the $1\pi$-exchange 3BF. The coupling effect is sizable despite
the small $d$-wave component, as is demonstrated by the folding potential
discussed in Sec. 3. Naturally, the quantitative results serve only as a
reference because of the speculative nature of the LECs employed. However, the
result provides basic information about the contribution of the 3BFs because
it enables us to infer what changes are induced by modifying each coupling
constant. It is necessary to do similar ab-initio calculations in heavier
$\Lambda$-hypernulcei, although the task is computationally demanding.

Acknowledgements. This work is supported by JSPS KAKENHI Grants No. JP19K03849
and No. JP22K03597.

##  Appendix A Tensor-product decomposition of $\Lambda NN$ three-body
interactions

$V_{1\pi,(i,j)}^{(K,\ell_{a},\ell_{b})}(p,q)$ in Eq. (4) for $K=0$, 1 and 2 of
the NNLO $1\pi$-exchange 3BF are given in the following.
($x\equiv\frac{p^{2}+r_{NN}^{2}q^{2}+m_{\pi}^{2}}{2r_{NN}pq}$ with
$r_{NN}=\frac{1}{2}$ and
$\delta_{\ell_{a},\mbox{even}}=\frac{1+(-1)^{\ell_{a}}}{2}$). Note that
$\Lambda$ hyperon receives a label of 1.

$\displaystyle V_{1\pi,(1,2)}^{K=0,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle\delta_{\ell_{a}\ell_{b}}(-1)^{\ell_{a}}\frac{g_{A}D_{1}^{\prime}}{2f_{0}^{2}}\sqrt{\frac{\hat{\ell_{a}}}{3}}\left[\delta_{\ell_{a}0}-\frac{m_{\pi}^{2}}{2r_{NN}pq}Q_{\ell_{a}}(x)\right],$
(10) $\displaystyle V_{1\pi,(1,2)}^{K=1,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle 0,$ (11) $\displaystyle
V_{1\pi,(1,2)}^{K=2,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle-\frac{g_{A}D_{1}^{\prime}}{2f_{0}^{2}}\left[\sqrt{\frac{2\hat{\ell_{a}}\hat{\ell_{b}}}{15}}(\ell_{a}0\ell_{b}0|20)\left(\frac{p^{2}}{2r_{NN}pq}Q_{\ell_{b}}(x)+\frac{r_{NN}^{2}q^{2}}{2r_{NN}pq}Q_{\ell_{a}}(x)\right)\right.$
$\displaystyle\left.\hskip
40.00006pt-\sum_{\ell}(-1)^{\ell}\sqrt{5}\hat{\ell}(\ell 010|\ell_{a}0)(\ell
010|\ell_{b}0)\begin{Bmatrix}\ell_{b}&1&\ell\\\
1&\ell_{a}&2\end{Bmatrix}Q_{\ell}(x)\right],$ (12) $\displaystyle
V_{1\pi,(2,3)}^{K=0,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle\delta_{\ell_{a}\ell_{b}}\delta_{\ell_{a},\mbox{even}}\frac{2g_{A}D_{2}^{\prime}}{2f_{0}^{2}}\sqrt{\frac{\hat{\ell_{a}}}{3}}\left[\delta_{\ell_{a}0}-\frac{m_{\pi}^{2}}{2r_{NN}pq}Q_{\ell_{a}}(x)\right],$
(13) $\displaystyle V_{1\pi,(2,3)}^{K=1,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle 0,$ (14) $\displaystyle
V_{1\pi,(2,3)}^{K=2,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle-\delta_{\ell_{a},\mbox{even}}\frac{2g_{A}D_{2}^{\prime}}{2f_{0}^{2}}\left[\sqrt{\frac{2\hat{\ell_{a}}\hat{\ell_{b}}}{15}}(\ell_{a}0\ell_{b}0|20)\left(\frac{p^{2}}{2r_{NN}pq}Q_{\ell_{b}}(x)+\frac{r_{NN}^{2}q^{2}}{2r_{NN}pq}Q_{\ell_{a}}(x)\right)\right.$
$\displaystyle\left.\hskip 40.00006pt+\sum_{\ell}\sqrt{5}\hat{\ell}(\ell
010|\ell_{a}0)(\ell 010|\ell_{b}0)\begin{Bmatrix}\ell_{b}&1&\ell\\\
1&\ell_{a}&2\end{Bmatrix}Q_{\ell}(x)\right],$ (15) $\displaystyle
V_{1\pi,(3,1)}^{K=0,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle\delta_{\ell_{a}\ell_{b}}\frac{g_{A}D_{1}^{\prime}}{2f_{0}^{2}}\sqrt{\frac{\hat{\ell_{a}}}{3}}\left[\delta_{\ell_{a}0}-\frac{m_{\pi}^{2}}{2r_{NN}pq}Q_{\ell_{a}}(x)\right],$
(16) $\displaystyle V_{1\pi,(3,1)}^{K=1,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle 0,$ (17) $\displaystyle
V_{1\pi,(3,1)}^{K=2,\ell_{a},\ell_{b}}(p,q)=$
$\displaystyle-\frac{g_{A}D_{1}^{\prime}}{2f_{0}^{2}}\left[(-1)^{\ell_{a}}\sqrt{\frac{2\hat{\ell_{a}}\hat{\ell_{b}}}{15}}(\ell_{a}0\ell_{b}0|20)\left(\frac{p^{2}}{2r_{NN}pq}Q_{\ell_{b}}(x)+\frac{r_{NN}^{2}q^{2}}{2r_{NN}pq}Q_{\ell_{a}}(x)\right)\right.$
$\displaystyle\left.\hskip 40.00006pt+\sum_{\ell}\sqrt{5}\hat{\ell}(\ell
010|\ell_{a}0)(\ell 010|\ell_{b}0)\begin{Bmatrix}\ell_{b}&1&\ell\\\
1&\ell_{a}&2\end{Bmatrix}Q_{\ell}(x)\right].$ (18)

## References

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  * (2) S. Petschauer, N. Kaiser, J. Haidenbauer, Ulf-G. Meißner, and W. Weise, ”Leading three-baryon forces from SU(3) chiral effective field theory”, Phys. Rev. C 93, 014001 (2016).
  * (3) P. Eckert, et al., ”Commissioning of the hypertriton binding energy measurement at MAMI”, EPJ Web Conf. 271, 01006 (2022).
  * (4) S. Petschauer, J. Haidenbauer, N. Kaiser, Ulf-G. Meißner, and W. Weise, ”Density-dependent effective baryon-baryon interaction from chiral three-baryon forces”, Nucl. Phys. A 957, 347 (2017).
  * (5) M. Kohno, H. Kamada, and K. Miyagawa, ”Partial-wave expansion of $\Lambda$NN three-baryon interactions in chiral effective field theory”, Phys. Rev. C 106, 054004 (2022).
  * (6) P. Reinert, and H. Krebs, and E. Epelbaum, ”Semilocal momentum-space regularized chiral two-nucleon potentials up to fifth order”, Eur. Phys. J. A 54, 86 (2018).
  * (7) E. Epelbaum, H.-W. Hammer, Ulf-G. Meißner, ”Modern theory of nuclear force,” Rev. Mod. Phys. 81, 1773 (2009).
  * (8) D. Gazda, T.Y. Htun, and C. Forssén, “Nuclear physics uncertainties in light hypernuclei”, Phys. Rev. C 106, 054001 (2022).
  * (9) J. Haidenbauer, S. Petschauer, N. Kaiser, U.-G. Meißner, A. Nogga, and W. Weise, ”Hyperon-nucleon interaction at next-to-leading order in chiral effective field theory,” Nucl. Phys. A 915, 24 (2013).
  * (10) J. Haidenbauer, U.-G. Meißner, and A. Nogga, ”Hyperon-nucleon interaction within chiral effective field theory revisited”, Eur. Phys. J. A 56, 91 (2020).
  * (11) P.M.M. Maessen, Th.A. Rijken, and Y. J.J. de Swart, “Soft-core baryon-baryon one-boson-exchange models. II. Hyperon-nucleon potential”, Phys. Rev. C 40, 2226 (1989).


# Vacancy-Induced Topological Fano Resonance in Kane-Mele Nanoribbons: Design,
Control, and Sensing Applications

S. Jalilvand1 M. Soltani1<EMAIL_ADDRESS>Z. Noorinejad2 M. Amini1 E.
Ghanbari-Adivi1<EMAIL_ADDRESS>$1$ Faculty of Physics, University of
Isfahan, Isfahan 81746-73441, Iran  $2$ Department of Physics, Islamic Azad
University - Shahreza Branch, Shahreza, Iran

###### Abstract

The concept of topological Fano resonance, characterized by an ultrasharp
asymmetric line shape, is a promising candidate for robust sensing
applications due to its sensitivity to external parameters and immunity to
structural disorder. In this study, the vacancy-induced topological Fano
resonance in a nanoribbon made up of a hexagonal lattice with armchair sides
is examined by introducing several on-site vacancies, which are deliberately
created at regular distances, along a zigzag chain that stretches across the
width of the ribbon. The presence of the on-site vacancies can create
localized energy states within the electronic band structure, leading to the
formation of an impurity band, which can result in Fano resonance phenomena by
forming a conductivity channel between the edge modes on both armchair sides
of the ribbon. Consequently, an ultracompact tunable on-chip integrated
topological Fano resonance derived from the graphene-based nanomechanical
phononic crystals is proposed. The Fano resonance arises from the interference
between topologically protected even and odd edge modes at the interface
between trivial and nontrivial insulators in a nanoribbon structure governed
by the Kane-Mele model describing the quantum spin Hall effect in hexagonal
lattices. The simulation of the topological Fano resonance is performed
analytically using the Lippmann-Schwinger scattering formulation. One of the
advantages of the present study is that the related calculations are carried
out analytically, and in addition to the simplicity and directness, it
reproduces the results obtained from the Landauer-Büttiker formulation very
well both quantitatively and qualitatively. The findings open up possibilities
for the design of highly sensitive and accurate robust sensors for detecting
extremely tiny forces, masses, and spatial positions.

##  I Introduction

Fano resonance Fano01  is a quantum phenomenon that occurs when a discrete
energy level interacts with a continuum of energy states, resulting in the
formation of a characteristic asymmetric line shape known as the Fano profile
in the transmission or absorption spectrum. The distinctive shape arises from
the constructive and destructive interference between the discrete state and
the states in the continuum. Fano resonances have been observed in various
physical systems, including atomic and solid-state physics, electromagnetism,
electronics, Aharonov-Bohm interferometers, and quantum dots Miroshnichenko01
; Ujsaghy01 ; Johnson01 ; Kobayashi01 ; Kamenetskii01 ; Attaran01 ; Lv01 ;
Gores01 . The characteristic shape of Fano resonances makes them useful in a
range of applications, including sensors, optical devices, and the study of
quantum interference phenomena. Recently, researchers have introduced the
concept of topological Fano resonance, whose ultra-sharp asymmetric line shape
is guaranteed by design and protected against geometrical disorder, while
remaining sensitive to external parameters. Topologically protected Fano
resonances have been observed in various systems, including acoustic,
microwave, optical, and plasmonic systems, and they open up exciting frontiers
for the generation of various reliable wave-based devices, including low-loss
perfect absorbers, ultrafast switches or modulators, and highly accurate
interferometers, by circumventing the performance degradations caused by
inadvertent fabrication flaws Bandopadhyay01 ; Stassi01 ; Overviga01 ;
Lukyanchuk01 .

In a lattice or crystal, electrons exhibit wave-like behavior. When a defect
is introduced, such as an impurity or a vacancy, it creates a localized state
within the crystal’s band structure. This localized state has the potential to
interact with the continuum of extended states in the crystal, leading to the
formation of Fano interference. In this regard, studies utilizing discrete
lattice models have demonstrated the presence of Fano resonance in the
transmission spectrum of discrete lattice models with impurities attached to
the side Deo01 ; Miroshnichenko02 ; Chakrabarti01 . For instance, the
electronic transmission through a one-dimensional tight-binding lattice with a
quantum dot coupled to the side to explore the Fano anti-resonances in the
transport through quantum dots is studied in Refs. Torio01 ; Rodriguez01 ;
Guevara01 . Recently, the theory for Fano anti-resonances induced by coupling
between vacancy states and edge states of the zigzag phosphorene nanoribbons
is investigated analytically Amini01 ; Amini02 .

In contrast to conventional Fano resonances, which often suffer from
sensitivity to disorder and imperfections, the advent of topological systems
Hasan01  featuring protected states presents a promising alternative which is
called topological Fano resonance ZangenehNejad01 . The topological protection
ensures that the unique features of Fano resonance, such as asymmetric line
shapes and sensitivity to environmental changes, remain intact even in the
presence of fabrication imperfections. This robustness can lead to the
development of more reliable and efficient devices, including sensors,
modulators, and interferometers, where the extreme sensitivity of Fano
resonances can be harnessed without being compromised by fabrication
challenges Ji01 ; Wang01 ; Sun01 . The concept of topological Fano resonance
and its potential applications in the development of robust and reliable
devices is an active area of research, and the search results provide valuable
insights into the control and influence of impurities on Fano resonances in
discrete lattice models.

In this paper, the possibility of creating and controlling Fano resonances in
a system that exhibits the quantum spin Hall (QSH) effect is explored. The
considered system is a two-dimensional topological phase of matter
characterized by the presence of robust and exotic edge states that are immune
to local perturbations and disorder. In regard of the task, a nanoribbon of
honeycomb structure is considered in which the spin-orbit coupling is
described by the Kane-Mele model. This model is the first theoretical model to
predict the QSH effect in graphene. It is shown that introducing a line of
vacancy defect across the width of the ribbon induces Fano anti-resonances in
the transmission of the edge states, which are topologically protected. In
fact, the presence of the on-site vacancies leads to emergence of some zero-
energy localized states surrounding the vacant sites. Overlapping these
created zero-energy localized states and their coupling with the edge modes of
the armchair sides generate an impurity band in the ribbon’s band structure.
As a result, a conduction channel is formed between the topologically
protected edge modes localized at the edges of these armchair sides. The
presence of this channel causes the edge modes on both sides of the ribbon to
be connected. Subsequently, this connection creates the substrate for
occurring the backscattering processes which causes the observation of the
topological Fano resonance in the scattering region created across the width
of the ribbon. Since, the Fano resonance is a widespread wave scattering
phenomenon, the Lippmann–Schwinger(LS) equation in the formal scattering
theory along with the concepts such as the Green function and the transition
operator, can be employed to calculate the transmission and backscattering
probabilities of the electrons during their passing through the channel
between the edge wires. Analytically, the topological Fano resonance
simulation utilizes the LS scattering formulation. This study offers an
advantage by conducting related calculations analytically, resulting in
outcomes that closely align with those achieved through the Landauer-Büttiker
(LB) formulation, demonstrating both quantitative and qualitative accuracy.

Consequently, this research aims to investigate the design, control and
sensing applications of topological Fano resonances in Kane-Mele nano ribbons,
thereby contributing to the advancement of nano scale sensing technologies.
Our results reveal a novel way to manipulate and control the edge state
transport in topological insulators, and to create reliable and tunable Fano-
based devices.

The article is organized as follows. In section 2, the geometry of the
specified system, the Kane-Mele Hamiltonian describing the electronic
properties of the topological insulators, and the topological Fano resonance
profile are explained to give a vision of the vacancy-induced topological Fano
resonance. In section 3, first the discrete energy levels associated to a line
of the on-site vacancy defects are explained and discussed. Then, employing
the formal scattering theory and using the LS equation as well as the Green,
and transition operators, the electron transmission and backscattering
probabilities in the scattering region created across the width of the ribbon
are calculated. The obtained results are provided and discussed. Finally, in
the last section, a summary along with the conclusion remarks are presented.

##  II Vacancy-induced topological Fano resonances

In this section, we delve into the intricacies of vacancy-induced topological
Fano resonances, exploring the nuanced interplay between the topological
characteristics of the Kane-Mele model Kane01 ; Kane02  and the localized
states induced by these engineered defects. Through systematic investigation,
we unravel the distinct features and behavior exhibited by the Fano resonances
in the presence of such vacancies, shedding light on their potential
implications for quantum transport and edge state engineering in topological
materials. The section unfolds with an examination of the theoretical
underpinnings, followed by a detailed presentation of numerical results and
insightful discussions on the observed phenomena.

Figure 1: Schematic representation of the Kane-Mele nanoribbon with the
honeycomb lattice structure. The ribbon is infinite in the $x$-direction and
finite in the $y$-direction with armchair edges. The system is connected to
left and right leads. A scattering region at the center hosts vacancy defects
arranged on a zigzag chain highlighted by a red rectangle box on the diagram.
Armchair chain numbers are labeled by $N$. The vacant sites are denoted by
open red circles which are introduced on armchair chains of $N=1,\ 4,\ 7,\
10$.

###  II.1 Model Hamiltonian and system geometry

The Kane-Mele model Hamiltonian captures the topological properties of the
system, considering the interplay between hopping terms, spin-orbit coupling,
and staggered on-site potentials. For a honeycomb lattice ribbon with armchair
edges, infinite in the $x$-direction and finite in the $y$-direction, shown in
Fig. 1, the tight-binding Hamiltonian is given by Kane01 ; Kane02 :

$\begin{split}{\cal H}_{KM}&=-t\sum_{\langle
m,n\rangle,\sigma}c_{m,\sigma}^{\dagger}c_{n,\sigma}\\\
&-i\lambda_{SO}\sum_{\langle\langle
m,n\rangle\rangle,\sigma,\sigma^{\prime}}e^{i\nu_{mn}\phi}c_{m,\sigma}^{\dagger}(\bm{\sigma}_{\sigma,\sigma^{\prime}})_{z}c_{n,\sigma^{\prime}}\\\
&+M\sum_{m,\sigma}(-1)^{m}c_{m,\sigma}^{\dagger}c_{m,\sigma}\end{split}$ (1)

Here, $c_{m,\sigma}^{\dagger}~{}(c_{m,\sigma})$ creates (annihilates) an
electron at site $i$ with spin $\sigma$, $t$ is the nearest-neighbor hopping
amplitude, $\lambda_{SO}$ is the spin-orbit coupling strength, $M$ represents
the staggered on-site potential, and $e^{i\nu_{mn}\phi}$ is a phase factor in
which $\nu_{mn}=\pm 1$ indicates the direction of the next-nearest-neighbor
hopping. The sums are taken over nearest neighbors $\langle m,n\rangle$ and
next-nearest neighbors $\langle\langle m,n\rangle\rangle$.
${\bf\sigma}=\sigma_{x}{\bf e}_{x}+\sigma_{y}{\bf e}_{y}$ is the well-known
Pauli spin operator. The system is configured as a ribbon attached to left and
right leads, forming a scattering region in the center. Figure 1 illustrates
the geometry with vacancy defects arranged in a line which is shown by a red
rectangle box. Importantly, placing a vacancy defect on a given site is
equivalent to turning off the hopping terms that connect this site to its
neighbors. These vacancies play a pivotal role in inducing topological Fano
resonances, as will be explored in subsequent sections.

###  II.2 The topological Fano profile

In the Kane-Mele model, the topological nature of the system manifests in the
existence of topologically protected edge states within the energy gap. These
edge states, forming helical edge bands, give rise to an edge current that is
impervious to backscattering, a hallmark of the topological phase. In the
absence of mechanisms facilitating backscattering, the edge current remains
unaffected even in the presence of nonmagnetic defects like vacancies.
However, the introduction of vacancies in a specific arrangement, forming a
conduction channel that connects the upper and lower edges of the ribbon, can
lead to the emergence of topological Fano resonance. This channel is
effectively created by engineering the defects in a line across the width of
the ribbon. It is noteworthy that the presence of vacancy defects, in contrast
to on-site impurities with finite potential Rahmati01 , induces discrete
energy-bound states around zero energy. Due to the particle-hole symmetry of
the system the energy of these bound states will be symmetrically distributed
around $E=0$.

Our focus now shifts to the quantum transport through the ribbon in the
presence of vacancy defects. In current investigation, the armchair chains are
labeled by integer numbers $N=1~{},2,~{}3,~{}\cdots$, and four vacancy defects
are introduced on a zigzag chain, situated on armchair chains $N=1,\ 4,\ 7,\
10$ as illustrated in Fig. 1. We specifically consider a ribbon with a width
of $N=10$, indicating the number of armchair chains across the width. The
vacancies are regularly arranged so that the distance between two consecutive
vacant sites is equal to three zigzag chains. In order to check the accuracy
of our analytical simulation of the vacancy-induced topological Fano resenance
in the specified structure, which will be detailed in next section, we obtain
the transmission spectrum of the system by employing the LB formalism for
conduction in confined structures Gamayun01 . For our analysis, we set the
hopping amplitude $t=1$ as the unit of energy, and to maintain the system in
the topological phase, we choose a fixed spin-orbit coupling of
$\lambda_{\text{SO}}=0.2$, while keeping the staggered potential $M=0$ for
simplicity. The line of vacancy defects is highlighted by a red rectangle box.
Our findings reveal distinctive dips in the transmission spectrum through the
ribbon which is shown in Fig. 2. Intriguingly, these dips appear in pairs and
exhibit symmetrical distribution in the energy space around $E=0$. It is
important to note that upon closer examination of the profile, it is evident
that the dips do not exhibit symmetrical behavior with respect to the energies
at which they occur. Fano resonance phenomenon often reveals itself through
the asymmetrical dips in occurrence energies, a unique fingerprint of this
effect. This phenomenon signifies the occurrence of topological Fano resonance
in the Kane-Mele model, forming the crux of our study, which we delve into
further details in the subsequent sections.

Figure 2: Transmission spectrum through a Kane-Mele nanoribbon of width
$N=10$, calculated using the LB formalism for conduction in confined
structures. The ribbon is in the topological phase with a fixed hopping
amplitude $t=1$ as the energy unit, spin-orbit coupling
$\lambda_{\text{SO}}=0.2$, and staggered potential $M=0$. Four vacancy defects
are strategically placed in a line on armchair chains $m=1,4,7,10$, forming a
conduction channel connecting the upper and lower edges. The transmission
spectrum exhibits pairs of dips symmetrically distributed in the energy space
around $E=0$, indicative of the topological Fano resonance.

##  III Transmission analysis

In the presence of multiple vacancy defects arranged in a line, each defect
introduces a discrete energy level within the gap region of the Kane-Mele
nanoribbon. These localized states, associated with the vacancies, exhibit
localization in real space, forming a spatial pattern that aligns with the
arrangement of the defects. Crucially, when defects are strategically placed,
the associated states overlap, creating a conduction channel that connects the
upper and lower edges of the ribbon. To understand the coupling of these
localized states with the edge states and the emergence of the topological
Fano resonance, we employ the LS approach. This theoretical framework allows
us to obtain an explicit analytical expression for the scattering amplitude of
the edge electrons through the presence of multiple vacancy defects. The
resulting scattering amplitude reveals a Fano resonance profile in the
transmission spectrum, characterized by tunable dips symmetrically distributed
around $E=0$. The position and width of these resonances are influenced by the
arrangement of the vacancy defects, offering a means to control and engineer
the topological Fano resonance in the Kane-Mele nanoribbon.

The combination of the discrete energy levels induced by vacancy defects,
their coupling to the edge bands, and the resulting Fano resonance in the
transmission spectrum forms the basis for our exploration of vacancy-induced
topological Fano resonances in the Kane-Mele nanoribbon. In the subsequent
sections, we will provide a detailed analysis of these phenomena, shedding
light on the rich interplay between topology, defects, and quantum transport
in two-dimensional materials.

###  III.1 Discrete energy levels associated with a line of vacancy defects

While the theorem governing the creation of zero energy modes in bipartite
lattices due to a single vacancy site is well-established for honeycomb
lattices in the absence of spin-orbit coupling $(\lambda_{SO}=0)$ Brouwer01 ;
Pereira01 , its applicability to more complex lattice structures, such as the
Kane-Mele model, requires further examination.

For graphene, where only nearest-neighbor hopping is considered, the
Hamiltonian can be represented in the matrix block form as

$\Big{(}\begin{array}[]{cc}{\bf 1}&H_{AB}\\\ H_{AB}^{\dagger}&{\bf
1}\end{array}\Big{)},$ (2)

satisfying the conditions for the creation of a localized zero energy mode
Pereira01 . Here $A$ and $B$ refers to two sublattices of the bipartite
lattice, and $H_{AB}$ and ${\bf 1}$ stand the hopping and identity matrices,
respectively. The extension of this result to materials like phosphorene, with
next-nearest hopping amplitudes, yields localized states with non-zero energy
Amini01 . In our specific case of the Kane-Mele Hamiltonian, ${\cal H}_{KM}$,
the direct applicability of Eq. (2) is not immediately evident. However, our
numerical simulations demonstrate that even in the presence of spin-orbit
coupling ($\lambda_{SO}\neq 0$), a single vacancy defect gives rise to a zero-
energy localized state Rahmati01 ; Sadeghizadeh01 ; AlShuwaili01 .

Figure 3: The spatial profile of the local density of states (LDOS)
corresponds to the configuration shown in Fig. 1, as it is clearly visible
from the figure, the presence of the vacancies creates a conduction channel
between the edge states at the upper and lower armchair edges of the ribbon.
This channel provides the necessary platform for the occurrence of the
backscattering events and the observation of topological Fano resonance in the
transmission spectrum of the system. The vacancies are marked with cross
symbols.

To validate the mentioned behavior, the Kane-Mele model for the honeycomb
lattice was considered and the local density of states around a single vacancy
defect was examined. The obtained results verify the emergence of a localized
state with zero energy, affirming the presence of vacancy-induced discrete
energy levels Rahmati01 ; Sadeghizadeh01 ; AlShuwaili01 .

It is worth noting that the expressions for the edge states in the presence of
spin-orbit interaction Rahmati01 ; Sadeghizadeh01  become notably more
intricate compared to the case without it ($\lambda_{\text{SO}}=0$)
AlShuwaili01 . Consequently, deriving an exact analytic expression for the
localized wave function associated with a single vacancy defect in the
presence of spin-orbit coupling ($\lambda_{\text{SO}}\neq 0$) would entail
increased complexity, surpassing the scope of this current study. The detailed
analysis of such wave functions remains an avenue for future investigations.

Let us now proceed to the scenario of an infinite number of vacancy defects
periodically placed in the bulk honeycomb lattice. The resultant band
structure of the Kane-Mele model in the presence of spin-orbit interaction has
been examined in details by using both the numerical and semi-analytical
methods in Refs. Rahmati01 ; Sadeghizadeh01 ; AlShuwaili01 . Using the
information reported in these research works, it is possible to examine the
periodic arrangement involves placing the vacancy defects such that there are
three armchair chains between adjacent defects, essentially mirroring the
configuration illustrated in Fig. 1 but extending infinitely in the
$y$-direction. For this examination, a non-zero spin-orbit coupling strength
of ($\lambda_{SO}\neq 0$) has been considered. The resulting band structure
reveals an additional energy band within the original energy gap, attributed
to the formation of an impurity band. Each vacancy defect induces a localized
wave function akin to a single atomic orbital. These localized wave functions,
resembling atomic orbitals, overlap with neighboring wave functions from
adjacent vacancy defects, giving rise to the impurity band. Remarkably, the
obtained impurity band can be well-described by a simple tight-binding band
dispersion, represented as $E(k)=2\Delta\sin(ka)$, where $\Delta$ and $a$ are
the band width parameter and the lattice constant, respectively.

The presence of a vacancy defect introduces a discrete energy level at zero
with a localized wave function reminiscent of a single-level atom. In the case
of a finite number of closely situated vacancy defects, effective overlap
occurs between these wave functions, facilitating effective tunneling,
$\Delta$, and resulting in finite discrete energy levels symmetrically
distributed around zero. These energy levels exhibit extended wave functions
across the width of the ribbon, connecting the upper and lower edges and
allowing for backscattering. This phenomenon manifests as Fano anti-resoances
in the transmission spectrum of the system which will be described in the
following.

For a closer examination of the conductive channel created between the edge
modes at the upper and lower weirs around the ribbon depicted in Fig. 1, the
spatial profile of the local density of states is displayed in Fig. 3. The
illustrated profile corresponds to that arrangement of the vacant sites shown
in Fig. 1. As can be seen from the figure evidently, the overlap of the zero-
energy localized modes around the vacant sites forms a conduction channel
between the upper and lower edges of the ribbon. Consequently, the
backscattering of the electrons during their passing through this conductive
channel, causes the observation of the topological Fano resonance in the
considered system.

###  III.2 Calculation of the Fano anti-resonances using the
Lippmann–Schwinger approach

The purpose of this section is to simulate the topological Fano resonance
observed in the transmission spectrum shown in figure 2. To model the quantum
transport through the specified system and calculate Fano anti-resonances, the
Hamiltonian of

${\cal H}={\cal H}_{E}+{\cal H}_{V}+{\cal H}_{EV},$ (3)

is employed in which Hamiltonian ${\cal H}_{E}$ corresponds to the helical
edge states of the Kane-Mele model and is expressed as:

$\begin{split}{\cal H}_{E}&=\int_{-\pi}^{+\pi}\hbar{\rm
v}_{F}k_{U}|k_{U}\rangle\langle k_{U}|dk_{U}\\\
&\qquad\qquad+\int_{-\pi}^{+\pi}\hbar{\rm v}_{F}k_{L}|k_{L}\rangle\langle
k_{L}|dk_{L}.\end{split}$ (4)

in which it is assumed that at the upper side, the continuum edge state
travels to the right with the Fermi speed of ${\rm v}_{F}$, while at the lower
side, it is traveling to the left at the same speed. Additionally, the lattice
constant, $a$, is assumed to be one. Here, $k_{U}$ and $k_{L}$ represent,
respectively, the wave vectors for edge modes at the upper and lower edges
associated to the ribbon displayed in Fig. 1. Hereafter, the $U$ and $L$
indices refer to the upper and lower edges.

The Hamiltonian ${\cal H}_{V}$ describes the line of vacancy defects:

${\cal H}_{V}=E_{0}|v\rangle\langle v|,$ (5)

describing a quantum dot with energy of $E_{0}$. Here, $|v\rangle$ is the ket-
state corresponding to one of the discrete energy states which are induced by
the defects. It is assumed that this vacancy state has been coupled with the
continuum edge states.

The coupling interaction between the edge states and the defect state is given
by ${\cal H}_{EV}$:

$\begin{split}{\cal
H}_{EV}&=\int_{-\pi}^{+\pi}u_{U}(k_{U})|k_{U}\rangle\langle v|dk_{U}\\\
&+\int_{-\pi}^{+\pi}u_{L}(k_{L})|k_{L}\rangle\langle v|dk_{L}+H.C.\end{split}$
(6)

in which $H.C.$ stands for the Hermitian conjugate and the related term
ensures the hermiticity of the Hamiltonian. Also, $u_{U}(k_{U})$ and
$u_{L}(k_{L})$ represent the coupling strengthes between the discrete vacancy
state of $|v\rangle$, and the continuum edge modes of $|k_{U}\rangle$ and
$|k_{L}\rangle$ at the upper and lower edges of the considered ribbon,
respectively.

The LS formulation of the formal scattering theory can be used to calculate
the transmission coefficient. This equation, which is a key concept in quantum
mechanics, particularly in the context of scattering, reads

$|\psi_{out}\rangle=|\psi_{in}\rangle+G_{0}(E)V|\psi_{out}\rangle,$ (7)

where $|\psi_{in}\rangle$ is the incoming or unperturbed wave state,
$|\psi_{out}\rangle$ is the full output or scattered wave function, $E$ is the
energy of the specified scattering system, $G_{0}(E)$ is the free-particle
Green’s function, and $V$ is the interaction potential.

By defining the transition operator, ${\cal T}(E)$, as

$V|\psi_{out}\rangle={\cal T}(E)|\psi_{in}\rangle,$ (8)

the LS equation can be rewritten as follows:

$|\psi_{out}\rangle=|\psi_{in}\rangle+G_{0}(E){\cal T}(E)|\psi_{in}\rangle.$
(9)

By multiplying both sides of the above equation by $V$, it can be converted
into an equation to find the transition operator, ${\cal T}(E)$, as:

${\cal T}(E)=V+VG_{0}(E){\cal T}(E).$ (10)

Usually, using the iteration method, the solution of the transition operator
can be written as an infinite expansion of the Green’s operator $G_{0}(E)$ and
the interaction potential $V$ as

${\cal T}(E)=V+VG_{0}(E)V+VG_{0}(E)VG_{0}(E)V+\cdots.$ (11)

In the considered problem, the interaction potential $V$ is equivalent to the
term of ${\cal H}_{EV}$ in the entire Hamiltonian and the Green operator,
$G_{0}(E)$, is the associated to the rest of the Hamiltonian, ${\cal
H}_{E}+{\cal H}_{V}$. Consequently, it is possible to write:

$G_{0}(E)=G_{E}(E)+G_{V}(E),$ (12)

in which

$\begin{split}G_{E}(E)=&\int_{-\pi}^{+\pi}{|k_{U}\rangle\langle k_{U}|\over
E-\hbar{\rm v}_{F}k_{U}+i0^{+}}dk_{U}\\\
&\qquad\qquad+\int_{-\pi}^{+\pi}{|k_{L}\rangle\langle k_{L}|\over E+\hbar{\rm
v}_{F}k_{L}+i0^{+}}dk_{L},\end{split}$ (13)

and

$G_{V}={|v\rangle\langle v|\over E-E_{0}+i0^{+}}.$ (14)

After some algebraic operation, it can be shown that

$VG_{0}(E)VG_{0}(E)V=\alpha V,$ (15)

where $\alpha$ is given by the expression of

$\begin{split}\alpha&=g_{0}(E)\Big{(}\int_{-\pi}^{+\pi}{\big{|}u_{U}(k_{U})\big{|}^{2}\over
E-\hbar{\rm v}_{F}k_{U}+i0^{+}}dk_{U}\\\
&\qquad\qquad+\int_{-\pi}^{+\pi}{\big{|}u_{L}(k_{L})\big{|}^{2}\over
E+\hbar{\rm v}_{F}k_{L}+i0^{+}}dk_{L}\Big{)},\end{split}$ (16)

in which

$g_{0}(E)={1\over E-E_{0}+i0^{+}}.$ (17)

The combination of Eqs. (10) and (15) leads to

${\cal T}(E)={V+VG_{0}(E)V\over 1-\alpha}.$ (18)

In the following calculation, it is assumed that the interaction occurring
during the scattering process is both short-range and local. Also, for
simplicity, it is assumed that $u_{U}(k_{U})$ and $u_{L}(k_{L})$ are some
constant functions denoted by $u_{U}$ and $u_{L}$, respectively. If the
interaction that leads to scattering is completely local, this is an accurate
assumption. But, since in the specified system this interaction is not
completely local, this assumption is an approximation in our study. Of course,
in the following by comparing the simulation results with those obtained from
LB formulation, we will see that this approximation is very good and
efficient. Anyway, this assumption helps us to easily calculate the integrals
appearing in Eqs. (16), so that we have:

$\begin{split}\int_{-\pi}^{+\pi}{1\over E-\hbar{\rm
v}_{F}k_{U}+i0^{+}}dk_{U}={i\pi\over\hbar{\rm v}_{F}},\\\
\int_{-\pi}^{+\pi}{1\over E+\hbar{\rm
v}_{F}k_{L}+i0^{+}}dk_{L}={i\pi\over\hbar{\rm v}_{F}}.\end{split}$ (19)

With the above results, $\alpha$ is reduced to:

$\alpha=\alpha_{U}+\alpha_{L},$ (20)

where

$\begin{split}\alpha_{U}&=g_{0}(E)\Big{(}{\pi i\over\hbar{\rm
v}_{F}}\Big{)}\big{|}u_{U}\big{|}^{2},\\\ \alpha_{L}&=g_{0}(E)\Big{(}{\pi
i\over\hbar{\rm v}_{F}}\Big{)}\big{|}u_{L}\big{|}^{2}.\end{split}$ (21)

To obtain the transmission coefficient, an initial edge state is considered
which travels along the upper edge of the ribbon with an initial momentum of
$k_{0}$. Consequently, at a far distance before the defect, the initial state
can be represented as;

$\langle\ell|\psi_{in}\rangle=\langle\ell|k_{0}\rangle=e^{ik_{0}\ell}.$ (22)

Here, the lattice constant is assumed as the unit of length and
$\ell\to-\infty$. To know the transmission probability, we should calculate
the probability of detecting the particle at a far distance after the central
scattering region and along the upper side of the ribbon. If the particle is
detected in this distance, it means that the output wave function can be
displayed as

$\langle\ell|\psi_{out}\rangle=t\langle\ell|k_{0}\rangle=te^{ik_{0}\ell},$
(23)

in which $t$ is the transmission amplitude and $\ell\to+\infty$. Multiplying
both sides of Eq.(9) by bra-basis of $\langle\ell|$ reads:

$\langle\ell|\psi_{out}\rangle=\langle\ell|\psi_{in}\rangle+\langle\ell|G_{0}(E){\cal
T}(E)|\psi_{in}\rangle.$ (24)

Using the fact that for the considered case

$V=\int_{-\pi}^{+\pi}u_{U}(k_{U})|k_{U}\rangle\langle v|dk_{U}+H.C.,$ (25)

and

$G_{0}(E)=\int_{-\pi}^{+\pi}{|k_{U}\rangle\langle k_{U}|\over
E-E_{0}+i0^{+}}dk_{U},$ (26)

it can be shown that

$\begin{split}\langle\ell|G_{0}(E){\cal T}(E)|k_{0}\rangle=&{g_{0}(E)\over
1-\alpha}\\\
\times\int_{-\pi}^{+\pi}&{u_{U}^{*}(k_{U})u_{U}(k_{0})e^{ik_{0}\ell}\over
E-i{\rm v}_{F}k_{U}+i0^{+}}dk_{U}.\end{split}$ (27)

Considering that $u_{U}(k_{U})$ and $u_{U}(k_{0})$ can be assumed equal to a
constant value of $u_{U}$, the following closed form expression can be derived

$\langle\ell|G_{0}(E){\cal T}(E)|k_{0}\rangle={2\alpha_{U}\over
1-\alpha}e^{ik_{0}\ell},$ (28)

Finally, by inserting Eqs. (22), (23), and (28) in Eq. (24), and performing
some algebraic calculation, the transmission amplitude is drivable as

$t=1+{2\alpha_{U}\over 1-\alpha}={1+\alpha_{U}-\alpha_{L}\over
1-\alpha_{U}-\alpha_{L}}.$ (29)

If the coupling between the discrete vacancy and the continuous upper-edge
states is absent, we have $u_{U}=0$ and consequently $\alpha_{U}=0$. Then, as
expected and is clear from the above equation, $t=1$ which result in a perfect
transmission. On the other hand, if there is no coupling between the discrete
vacancy and the continuous lower-edge states, then $u_{L}=0$ and $t$ is
reduces to

$t={1+\alpha_{U}\over 1-\alpha_{U}}.$ (30)

Since $\alpha_{U}$ is purely imaginary, $T(E)=|t|^{2}=1$ and the transmission
is complete, as is expected. But in the interaction with the lower edge, there
is a possibility of reflection and scattering to the lower wire. This
discussion confirms that for observing the topological Fano resonance in a
Kane-Mele nanoribbon, it is essential that the vacancy discrete states becomes
coupled with the continuous edge modes on both upper and lower sides. Also,
the above fact indicates that the strength of coupling between discrete and
continuous modes affects the quality of observing Fano resonance.

Figure 4:  Comparison of the transmission spectrum of the specified system
obtained from the current simulation based on the LS scattering formalism and
the results obtained from LB formulation. This comparison verifies the
efficiency and accuracy of the simulation method presented in this study
confirming the occurence of the topological Fano resonance in the considered
structur.

In Fig. 4, the analytically simulated transmission probability, $T(E)$, is
plotted as a function of energy, $E$, and is compared with the calculations
obtained from LB formalism. As expected, the two graphs are both qualitatively
and quantitatively very similar and this figure correctly shows the behavior
exhibited in Fig. 2. The very good match of the two graphs shows that the
assumption that $u_{U}(k_{U})$ and $u_{L}(k_{L})$ are constant is an
approximation that is very close to being accurate. This figure shows that
despite the simplicity of the carried simulation based on LS scattering
formulation, it is reasonably efficient and accurate in quantitative and
qualitative prediction of the topological Fano resonance in Kane-Mele
nanoribbons.

##  IV Summary and Conclusion

A nanoribbon of a two-dimensional hexagonal lattice with finite width and
armchair edges, which governed by the Kane-Mele Hamiltonian, was considered.
This Hamiltonian describes the spin-orbit interaction which leads to the
quantum spin Hall effect. It was assumed that the length of the ribbon at the
far distances is limited to two left and right leads. As a result, at the
armchair sides, edge continuum states, which are topologically protected,
flowed to the left and right with Fermi velocities. It was assumed that the
length of the ribbon, at long distances, was limited to two left and right
leads. As a result, at the armchair sides, edge continuum states, which are
topologically protected, traveled to the left and right with Fermi velocities.
This situation was similar to considering the ribbon’s sides as two current-
carrying wires. By creating a regular array of on-site vacancies on a zigzag
chain across the width of the ribbon, a conduction channel between the edge
wires of the ribbon was created. This cannel caused the creation of a central
scattering region around the zigzag chain that contained the vacancy defects.
Also, this conduction channel, which is due to the overlap of zero-energy
localized states surrounding the vacant sites, allows the creation of discrete
vacancy states, the coupling of these discrete states with continuous edge
states, and the occurrence of backscattering events. It saw shown that the
coupling of topologically protected continuous edge states and the discrete
vacancy states and their interference with each other made it possible to
observe topological Fano resonance. First, the Fano topological resonance
profile was produced using the LB formulation. Then, employing the LS
scattering formalism and using the Green’s operator as well as the transition
matrix this profile was simulated, analytically. In the calculations, it was
shown that the transmission spectrum of the desired system is a function of
the energy of the system and the degree of coupling between the discrete
vacant states with the continuous edge states on both sides of the ribbon. By
adjusting the width of the ribbon, choosing the number of vacancies, and
checking how they are arranged on the zigzag chain across the ribbon, it is
possible to adjust and control the vacancy-induced topological Fano resonance
in the Kane-Mele nanoribbons. From an analytical perspective, the simulation
of topological Fano resonance involves using the LS scattering framework. This
research provides a benefit by performing associated calculations
analytically, producing outcomes that closely match those obtained using the
LB approach, showcasing both quantitative and qualitative precision. These
findings contribute to the understanding of the impact of vacancies on the
electronic and transport properties of nanoribbons, which is essential for the
design, control, and sensing applications of such systems.

###### Acknowledgements.

The first author, SJ, would like to acknowledge the office of graduate studies
at the University of Isfahan for their support and research facilities.
Additionally, the fourth author, MA, would like to acknowledge the support
received from the Abdus Salam (ICTP) associateship program.

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